LNTwww - User contributions [en]

Theory of Stochastic Signals/Probability Density Function (PDF)

2023-04-12T09:32:13Z

Hwang:

{{Header
|Untermenü=Continuous Random Variables
|Vorherige Seite=Generation of Discrete Random Variables
|Nächste Seite=Cumulative Distribution Function
}}

== # OVERVIEW OF THE THIRD MAIN CHAPTER # ==
 
We consider here  »'''continuous random variables'''«,  i.e.,  random variables which can assume infinitely many different values,  at least in certain ranges of real numbers. 
*Their applications in information and communication technology are manifold.
*They are used,  among other things,  for the simulation of noise signals and for the description of fading effects.

We restrict ourselves at first to the statistical description of the  »'''amplitude distribution'''«.  In detail,  the following are treated:

*The relationship between  »probability density function«  $\rm (PDF)$  and  »cumulative distribution function«  $\rm (CDF)$;
*the calculation of  »expected values  and  moments«;
*some  »special cases«  of continuous-value distributions:
#uniform distributed random variables, 
#Gaussian distributed random variables, 
#exponential distributed random variables, 
#Laplace distributed random variables, 
#Rayleigh distributed random variables, 
#Rice distributed random variables, 
#Cauchy distributed random variables;
*the  »generation of continuous random variables«  on a computer.

»'''Inner statistical dependencies'''«  of the underlying processes  '''are not considered here'''.  For this,  we refer to the following main chapters 4 and 5.

==Properties of continuous random variables==
In the second chapter it was shown that the amplitude distribution of a discrete random variable is completely determined by its  $M$  occurrence probabilities,  where the level number  $M$  usually has a finite value.

{{BlaueBox|TEXT=
$\text{Definition:}$  By a  »'''value-continuous random variable'''«  is meant a random variable whose possible numerical values are uncountable   ⇒   $M \to \infty$.     In the following,  we will often use the short form "continuous random variable".}}

Further it shall hold:
*In the following we denote continuous random variables  (mostly)  with  $x$  in contrast to the discrete random variables,  which are denoted with  $z$  as before.
*No statement is made here about a possible time discretization,  i.e.,  continuous random variables can be discrete in time.
*Further,  we assume for this chapter that there are no statistical bindings between the individual samples  $x_ν$,  or at least leave them out of consideration.

[[File: P_ID41__Sto_T_3_1_S1_neu.png |right|frame| Signal and PDF of a Gaussian noise signal]]
{{GraueBox|TEXT=
$\text{Example 1:}$
The graphic shows a section of a stochastic noise signal  $x(t)$  whose instantaneous value can be taken as a continuous random variable  $x$.

*From the  »probability density function»   $\rm (PDF)$  shown on the right,  it can be seen that instantaneous values around the mean  $m_1$  occur most frequently for this example signal.

*Since there are no statistical dependencies between the samples $x_ν$,  such a signal is also referred to as  »white noise«.}}

==Definition of the probability density function==
For a continuous random variable,  the probabilities that it takes on quite specific values are zero.  Therefore,  to describe a continuous random variable,  we must always refer to the  "probability density function"  $\rm (PDF)$.

{{BlaueBox|TEXT=
$\text{Definition:}$  
The value of the  »'''probability density function'''«  $f_{x}(x)$  at location  $x_\mu$  is equal to the probability that the instantaneous value of the random variable  $x$  lies in an  (infinitesimally small)  interval of width  $Δx$  around  $x_\mu$,  divided by  $Δx$:
:$$f_x(x=x_\mu) = \lim_{\rm \Delta \it x \hspace{0.05cm}\to \hspace{0.05cm}\rm 0}\frac{\rm Pr \{\it x_\mu-\rm \Delta \it x/\rm 2 \le \it x \le x_\mu \rm +\rm \Delta \it x/\rm 2\} }{\rm \Delta \it x}.$$}}

This extremely important descriptive variable has the following properties:

*Although from the time course in  [[Theory_of_Stochastic_Signals/Probability_Density_Function_(PDF)#Properties_of_continuous_random_variables|$\text{Example 1}$]]  it can be seen  that the most frequent signal components lie at  $x = m_1$  and the PDF has its largest value here,  for a continuous random variable the probability  ${\rm Pr}(x = m_1)$,  that the instantaneous value is exactly equal to the mean  $m_1$,  is identically zero.

*For the probability that the random variable lies in the range between  $x_{\rm u}$  and  $x_{\rm o}$:
:$${\rm Pr}(x_{\rm u} \le x \le x_{\rm o})= \int_{x_{\rm u} }^{x_{\rm o} }f_{x}(x) \,{\rm d}x.$$

*As an important normalization property,  this yields for the area under the PDF with the boundary transitions  $x_{\rm u} → \hspace{0.05cm} - \hspace{0.05cm} ∞$  and  $x_{\rm o} → +∞:$
:$$\int_{-\infty}^{+\infty} f_{x}(x) \,{\rm d}x = \rm 1.$$

*The corresponding equation for discrete-value,  $M$-level random variables states that the sum over the  $M$  occurrence probabilities gives the value  $1$.

{{BlaueBox|TEXT=
$\text{Note on nomenclature:}$ 
In the literature,  a distinction is often made between the random variable  $X$  and its realizations  $x ∈ X$.

Thus, the above definition equation is
:$$f_{X}(X=x) = \lim_{ {\rm \Delta} x \hspace{0.05cm}\to \hspace{0.05cm} 0}\frac{ {\rm Pr} \{ x - {\rm \Delta} x/2 \le X \le x +{\rm \Delta} x/ 2\} }{ {\rm \Delta} x}.$$

We have largely dispensed with this more precise nomenclature in our learning tutorial  $\rm LNTwww$  so as not to use up two letters for one quantity.
*Lowercase letters  $($as  $x)$  often denote signals and uppercase letters  $($as  $X)$  the associated spectra in our case.
*Nevertheless,  today (2017)  we have to honestly admit that the 2001 decision was not entirely happy.}}

==PDF definition for discrete random variables==
For reasons of a uniform representation of all random variables  (both discrete-value and continuous-value),  it is convenient to define the probability density function also for discrete random variables.

{{BlaueBox|TEXT=
$\text{Definition:}$  
Applying the definition equation of the last section to discrete random variables,  the PDF takes infinitely large values at some points  $x_\mu$  due to the nonvanishingly small probability value and the limit transition  $Δx → 0$.

Thus,  the PDF results in a sum of  [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|$\text{Dirac delta functions}$]]   ⇒   "distributions":
:$$f_{x}(x)=\sum_{\mu=1}^{M}p_\mu\cdot {\rm \delta}( x-x_\mu).$$

The weights of these Dirac delta functions are equal to the probabilities  $p_\mu = {\rm Pr}(x = x_\mu$).}}

Here is another note to help classify the different descriptive quantities for discrete and continuous random variables:   Probability and probability density function are related in a similar way as in the book  [[Signal_Representation|"Signal Representation"]]
*a discrete spectral component of a harmonic oscillation ⇒ line spectrum,  and
*a continuous spectrum of an energy-limited  (pulse-shaped)  signal.

[[File:P_ID40__Sto_T_3_1_S3_NEU.png|right|frame|Signal and PDF of a ternary signal]]
{{GraueBox|TEXT=
$\text{Example 2:}$  Below is a section
*of a rectangular signal with three possible values,
*where the signal value  $0 \ \rm V$  occurs twice as often as the outer signal values  $(\pm 1 \ \rm V)$.

Thus,  the corresponding  PDF  (values from top to bottom)  is:
:$$f_{x}(x) = 0.25 \cdot \delta(x - {\rm 1 V})+ 0.5\cdot \delta(x) + 0.25\cdot \delta (x + 1\rm V).$$}}

For a more in-depth look at the topic covered here,  we recommend the following  (German language)  learning video:

:[[Wahrscheinlichkeit_und_WDF_(Lernvideo)|"Wahrscheinlichkeit und WDF"]]   ⇒   "Probability and probability density function"

==Numerical determination of the PDF==
You can see here a scheme for the numerical determination of the probability density function: 

Assuming that the random variable  $x$  at hand has negligible values outside the range from  $x_{\rm min} = -4.02$  to  $x_{\rm max} = +4.02$,  proceed as follows:
[[File:EN_Sto_T_3_1_S4_neu.png |right|frame| For numerical determination of the PDF]]
#Divide the range of  $x$-values into  $I$  intervals of equal width  $Δx$  and define a field  $\text{PDF}[0 : I-1]$.  In the sketch  $I = 201$  and accordingly  $Δx = 0.04$  is chosen.
#The random variable  $x$  is now called  $N$  times in succession,  each time checking to which interval  $i_{\rm act}$  the current random variable  $x_{\rm act}$ belongs:     $i_{\rm act} = ({\rm int})((x + x_{\rm max})/Δx).$
#The corresponding field element PDF( $i_{\rm act}$) is then incremented by  $1$. 
#After $N$ iterations, $\text{PDF}[i_{\rm act}]$ then contains the number of random numbers belonging to the interval $i_{\rm act}$.
#The actual PDF values are obtained if,  at the end,  all field elements  $\text{PDF}[i]$  with  $0 ≤ i ≤ I-1$  are still divided by  $N \cdot Δx$.
 
{{GraueBox|TEXT=
$\text{Example 3:}$ 
From the drawn green arrows in the graph above,  one can see:
*The value  $x_{\rm act} = 0.07$  leads to the result  $i_{\rm act} =$ (int) ((0.07 + 4.02)/0.04) = (int) $102.25$.
* Here  "(int)"  means an integer conversion after float division   ⇒   $i_{\rm act} = 102$.
*The same interval  $i_{\rm act} = 102$  results for  $0.06 < x_{\rm act} < 0.10$,  so for example also for  $x_{\rm act} = 0.09$. }}

==Exercises for the chapter==

[[Aufgaben:Exercise_3.1:_cos²-PDF_and_PDF_with_Dirac_Functions|Exercise 3.1: Cosine-square PDF and PDF with Dirac Functions]]

[[Aufgaben:Exercise_3.1Z:_Triangular_PDF|Exercise 3.1Z: Triangular PDF]]

{{Display}}

Digital Signal Transmission/System Components of a Baseband Transmission System GS

2023-04-12T09:30:47Z

Hwang:

{{FirstPage}}
{{Header|
Untermenü=Digital Signal Transmission under Idealized Conditions
|Nächste Seite=Fehlerwahrscheinlichkeit bei Basisbandübertragung
}}

== # OVERVIEW OF THE FIRST MAIN CHAPTER # ==
 
The first main chapter introduces the broad field of digital signal transmission,  with some simplifying assumptions:  a redundancy-free binary transmitted signal,  no intersymbol interference.  Although the description is mainly in baseband, most of the results can be applied to the digital carrier frequency systems as well.

In particular,  the following are dealt with:
#  The  »basic structure and components«  of a baseband transmission system,
#  the definitions of  »bit error probability«  and  »bit error rate«,
#  the characteristics of  »Nyquist systems«  that allow intersymbol interference-free transmission,
#  the  »optimization of the binary baseband systems«  under power and peak constraints,
#  the generalization of the results to  »carrier frequency systems«,  and
#  the largely common description of  »ASK, BPSK, and 4-QAM«.

== Simplified system model ==
 
Throughout the first chapter,  the following block diagram is assumed for the digital system as described in  [TS87]<ref>Tröndle, K.; Söder, G.:  Optimization of Digital Transmission Systems.  Boston – London: Artech House, 1987.</ref>:

[[File:EN_Dig_T_1_1_S1xxx.png|right|frame|Simplified system model of a digital transmission system|class=fit]]

The block diagram is constructed in exactly the same way as an  [[Modulation_Methods/Objectives_of_Modulation_and_Demodulation#The_communication_system_under_consideration|"analog transmission system"]]  according to the description in the book  "Modulation Methods",  consisting of

#source   ⇒   German:  "Quelle",  marking:  "Q",
#transmitter   ⇒   German:  "Sender",  marking:  "S",
#channel   ⇒   German:  "Kanal",  marking:  "K",
#interference/noise   ⇒   German:  "Störung",  marking:  "N",
#receiver   ⇒   German:  "Empfänger",  marking:  "E",
#sink  ⇒   German:  "Sinke",  marking:  "V".

The corresponding signals are adapted to these labels,  but use lower case letters,  e.g. source signal  $q(t)$, ... ,  sink signal  $v(t)$.

In comparison to an analog transmission system,  the following similarities and differences can be recognized in this simplified system model:

*Also in the digital transmission system,  the received signal  $r(t)$  is continuous in time and value due to stochastic effects,  e.g. noise.  The transmitted signal  $s(t)$  can be discrete in time and value,  but does not have to be.
*In contrast to the book  "Modulation Methods",  however,  the source signal  $q(t)$  and the sink signal  $v(t)$  are always digital signals.  Accordingly,  they are both discrete-time and discrete-value.
*All information about  $q(t)$  and  $v(t)$  can thus also be expressed by the  "source symbol sequence"  $〈q_ν〉$  and the  "sink symbol sequence"  $〈v_ν〉$  together with the symbol duration  $T$. 
*A digital receiver differs fundamentally from the receiver of an analog system in that it must also include a  '''decision component'''  for obtaining the digital sink signal  $v(t)$  from the analog received signal  $r(t)$. 
*In the first three chapters of this book,  we consider  '''digital baseband transmission''',  which means that the signal  $q(t)$  is transmitted without prior frequency conversion  (modulation with a carrier wave).
*Therefore,  $s(t)$  and  $r(t)$  are low-pass signals here,  and the channel  (including interferences)  must always be assumed to have low-pass characteristics as well. 
 
In the following,  the characteristics of the individual system components are described in detail,  suitably considering the idealizing assumptions for this chapter.

== Descriptive variables of the digital source ==
 
The  '''digital source'''  generates the source symbol sequence  $〈q_ν〉$,  which is to be transmitted to the sink as error-free as possible.  In general,  each symbol of the temporal sequence  $〈q_ν〉$  with  $\nu = 1, 2,$ ...  from a symbol set  $\{q_\mu\}$  with  $\mu = 1$, ... , $M$,  where  $M$  is called the  "source symbol set size"  or the  "level number". 

For the present first main chapter of this book,  the following assumptions are made:
*The source is  "binary"  $(\hspace{-0.05cm}M= 2)$  and the two possible symbols are  $\rm L$  ("Low")  and  $\rm H$  ("High").  We have chosen this somewhat unusual nomenclature in order to be able to describe both unipolar and bipolar signaling in the same way.  Please see the note before  $\text{Example 1}$.
*The source symbols are  "statistically independent",  that is,  the probability  ${\rm Pr}(q_\nu = q_\mu)$,  that the  $\nu$–th symbol of the sequence  $〈q_ν〉$  is equal to the  $\mu$–th symbol of the symbol set  $\{q_\mu\}$  does not depend on  $\nu$. 
*Given these two assumptions,  the digital source is completely described by the  "'symbol probabilities"  $p_{\rm L} = {\rm Pr}(q_\nu = {\rm L}) $  and  $p_{\rm H} = {\rm Pr}(q_\nu = {\rm H}) = 1- p_{\rm L}$.  If  $p_{\rm L} =p_{\rm H}= 0.5$  is still valid,  the source is  "redundancy-free".  Mostly – but not always – such a redundancy-free binary source is assumed in the present first chapter.
*Let the time interval between two symbols be  $T$.  This quantity is called the  "symbol duration"  and the reciprocal value is the  "symbol rate"  $R = 1/T$.  For binary sources  $(\hspace{-0.05cm}M= 2)$  these quantities are also called  "bit duration"  and  "bit rate",  resp.
*With a system-theoretical view to digital baseband transmission,  the source signal is best described by a sequence of weighted and shifted Dirac delta impulses:
::<math>q(t) = \sum_{(\nu)} a_\nu \cdot {\rm \delta} ( t - \nu \cdot T)\hspace{0.05cm}. </math>
*Here,  we refer to  $a_\nu$  as the  '''amplitude coefficients'''.  In the case of  "binary unipolar"  digital signal transmission:
::<math>a_\nu = \left\{ \begin{array}{c} 1 \\
0 \\ \end{array} \right.\quad
\begin{array}{*{1}c} {\rm{for}}
\\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c}
q_\nu = \mathbf{H} \hspace{0.05cm}, \\
q_\nu = \mathbf{L} \hspace{0.05cm}. \\
\end{array}</math>
*Correspondingly,  in the case of a  "binary bipolar"  system:
::<math>a_\nu = \left\{ \begin{array}{c} +1 \\
-1 \\ \end{array} \right.\quad
\begin{array}{*{1}c} {\rm{for}}
\\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c}
q_\nu = \mathbf{H} \hspace{0.05cm}, \\
q_\nu = \mathbf{L} \hspace{0.05cm}. \\
\end{array}</math>
:The following description is mostly for this second case. 

{{BlaueBox|TEXT=
$\text{Note on nomenclature:}$ 
# In the literature,  our symbol  $\rm H$  is often denoted by  $\mathbf{0}$.
#In unipolar signaling,  the symbol  $\mathbf{0}$  is then represented by the amplitude coefficient  $a_\nu =1$  and the symbol  $\rm L$  by the numerical value  $a_\nu =0$. 
#To avoid this unattractive situation in our  "LNTwww",  the symbol  $\mathbf{1}$  is denoted by  $\rm H$,  where  "High"  expresses the situation correctly.}}

{{GraueBox|TEXT=
$\text{Example 1:}$  The graphic shows four binary Dirac-shaped source signals in the range from  $-4 \ \rm µ s$  to  $+4 \ \rm µ s$,  each based on the source symbol sequence
[[File:P_ID127_8.png|right|frame|Description of  "unipolar"  and  "bipolar"  digital source signals]]
:$$\langle q_\nu \rangle = \langle \text{...}\hspace{0.05cm}, \mathbf{L}, \mathbf{H}, \mathbf{H}, \mathbf{L},\hspace{0.15cm}\mathbf{H}, \hspace{0.15cm} \mathbf{L},\mathbf{L},
\mathbf{H},\mathbf{L},\hspace{0.05cm} \text{...} \rangle \hspace{0.05cm} $$

The middle symbol  $($marked in the equation by larger character spacing$)$  refers in each case to the time  $t = 0$.

*The two upper signals are suitable for describing unipolar systems,  the lower ones for bipolar (antipodal) digital signal transmission.
*For the diagrams on the left,  $T = 1\ \rm µ s$  is assumed.  For the two right ones,  however,   $T = 2\ \rm µ s$  and thus half the symbol rate applies.}}

== Characteristics of the digital transmitter==
 
The  '''transmitter'''  of a digital transmission system has the task of generating a suitable transmitted signal  $s(t)$  from the  (Dirac-shaped)  source signal,  which contains the message of the source completely and is adapted to the characteristics of the transmission channel,  the interferences as well as all technical receiving equipment.  In addition,  the transmitter ensures the provision of a sufficiently large transmission power.

As a descriptive quantity for the transmitter,  we use the '''basic transmitter pulse'''  $g_s(t)$.  Due to the definition of the source signal  $q(t)$  as a sum of weighted and shifted Dirac delta functions,  the transmitted signal can be represented with the amplitude coefficients  $a_\nu$  in the following way:

::<math>s(t) = q(t) \star g_s(t) = \sum_{(\nu)} a_\nu \cdot g_s ( t - \nu \cdot T)\hspace{0.05cm}.</math>

Often the basic transmitter pulse  $g_s(t)$  is assumed to be rectangular with
*the pulse height  $s_0 = g_s(t = 0)$  and
*the  (absolute)  pulse duration  $T_{\rm S}$.

{{BlaueBox|TEXT=
$\text{Definition:}$  If  $T_{\rm S} < T$  applies,  this is referred to as an  '''RZ pulse'''  ("return–to–zero"),  and if  $T_{\rm S} = T$,  this is referred to as an
 '''NRZ pulse'''  ("non–return–to–zero").}}

With a different basic transmitter pulse,  for example
*a  [[Signal_Representation/Special_Cases_of_Pulses#Gaussian_pulse|"Gaussian pulse"]],
*a  [[Aufgaben:Exercise_3.4Z:_Trapezoid,_Rectangle_and_Triangle|"trapezoidal pulse"]],
*a  [[Aufgaben:Exercise_1.1:_Basic_Transmission_Pulses|"cosine–square pulse"]] or
*a  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems|"root Nyquist pulse"]],

the  [[Signal_Representation/Special_Cases_of_Pulses#Gaussian_pulse|"equivalent pulse duration"]]  defined by the equal-area rectangle is usually used as description parameter instead of the absolute pulse duration  $T_{\rm S}$: 
:$$\Delta t_{\rm S} = \frac {\int ^{+\infty} _{-\infty} \hspace{0.15cm} g_s(t)\,{\rm d}t}{{\rm Max} \hspace{0.05cm}[g_s(t)]} \le T_{\rm S} \hspace{0.05cm}.$$

Only in case of the rectangular basic transmitter pulse  $\Delta t_{\rm S} = T_{\rm S}$  is valid.

If the height of the basic transmitter pulse  $g_s(t)$  differs from the maximum value  $s_0$  of the transmitted signal  $s(t)$,  we denote the pulse amplitude by  $A_{\rm S}$.  This is true for the Gaussian pulse,  for example.

The interaction module  [[Applets:Pulses_and_Spectra|"Pulses and Spectra"]]  shows some common basic transmitter pulses   $g_s(t)$  and the corresponding spectra  $G_s(f)$.

{{GraueBox|TEXT=
$\text{Example 2:}$  The following graphic is always based on the source symbol sequence
$\langle q_\nu \rangle = \langle \text{...}\hspace{0.05cm}, \mathbf{L}, \mathbf{H}, \mathbf{H}, \mathbf{L},\hspace{0.15cm}\mathbf{H}, \hspace{0.15cm}\mathbf{L},\mathbf{L}, \mathbf{H},\mathbf{L},\hspace{0.05cm} \text{...} \rangle $.  It shows three transmitted signals,

[[File:P_ID1251__Dig_T_1_1_S3_v3_80.png|right|frame|Binary transmitted signals with different pulse shapes]]

*a bipolar transmitted signal  $s_{\rm A}(t)$  with NRZ rectangular pulses,
*a bipolar transmitted signal  $s_{\rm B}(t)$  with RZ rectangular pulses, and
*a unipolar transmitted signal  $s_{\rm C}(t)$  with Gaussian pulses.

In the following descriptions,  the bipolar NRZ rectangular  ("square-wave")  signal  $s_{\rm A}(t)$  is usually assumed. The duration  $T_{\rm S}$  of the basic transmitter pulse  $g_s(t)$  shown in red in the diagram is equal to the distance  $T$  between two successive pulses.

From the additional diagrams one recognizes:
*For the RZ transmitted signal  $s_{\rm B}(t)$,  the pulse duration  $T_{\rm S}$  differs from the pulse spacing  $T$.  The diagram applies to the duty cycle  $T_{\rm S}/T = 0.5$.  Although  $s_{\rm B}(t)$  is also a binary signal,  there are three possible signal values here,  namely  $+s_0$,  $-s_0$  and  $0$.
*An advantage is that even with a long  $\rm H$  or  $\rm L$  sequence there is no DC signal,  which makes clock synchronization easier.  A disadvantage of RZ signaling is the wider spectrum as well as the lower energy per symbol,  which leads to a higher bit error rate.
*The signal  $s_{\rm C}(t)$  is unipolar and uses a Gaussian basic pulse  $g_s(t)$.  Such a signal is found,  for example,  in optical systems with intensity modulation,  since a laser or an LED  ("Light Emitting Diode")  cannot generate negative pulses in principle and a rectangular pulse is technologically more difficult to achieve than the Gaussian form.
*In case of a  "real Gaussian pulse"  the absolute pulse duration is always  $T_{\rm S} \to \infty$.  The  (normalized)  equivalent pulse duration is chosen here with  $\Delta t_{\rm S} /T = 0.3$  relatively small,  so that the maximum value  $s_0$  of the transmitted signal is approximately equal to the pulse amplitude  $A_{\rm S}$. 
*For wider Gaussian pulses these overlap;  the approximation  $s_0 \approx A_{\rm S}$  no longer applies in this case.}}

== Transmission channel and interference==
 
The  '''transmission channel'''  includes all the equipment located between the transmitter and the receiver.  The main component of the channel is the transmission medium,  which can be,  for example, 
*a symmetrical double line,
*a coaxial cable,
*an optical fiber  ("glass fiber"),  or
*a radio field.

In addition,  the transmission channel includes various equipment necessary for operational reasons,  such as power supply,  lightning protection and fault location. 

In the most general case,  the following physical effects must be taken into account:
*The transmission characteristics may be  "time-dependent",  especially in the case of a moving transmitter and/or receiver,  as described in detail in the first main chapter  "Time-Variant Transmission Channels"  of the book  [[Mobile_Communications/Distance_Dependent_Attenuation_and_Shading#Physical_description_of_the_mobile_communication_channel|"Mobile Communications"]].  In this book,  the channel is always assumed to be  '''linear and time-invariant'''   $\rm (LTI)$.
*The characteristics of the LTI channel can be frequency dependent,  characterized by the frequency response  $H_{\rm K}(f)$.  In conducted transmission,   $H_{\rm K}(f) \ne \rm const.$  always holds and distortion occurs,  as discussed in the section [[Digital_Signal_Transmission/Causes_and_Effects_of_Intersymbol_Interference#Definition_of_the_term_.22Intersymbol_Interference.22|"Definition of the term Intersymbol Interference"]].
*Stochastic interference  $n(t)$ is superimposed on the deterministic signal,  for example the unavoidable thermal noise,  pulse interference,  and crosstalk interference from other subscribers. 

For this first main chapter,  $H_{\rm K}(f) =1$  is always assumed,  which means that the first two points mentioned are excluded for the time being.
[[File:P_ID3131__Dig_T_1_1_S4_v1.png|right|frame|Center:  AWGN channel model,     on the left:  Power-spectral density  $\rm (PSD)$,    on the right:  Probability density function  $\rm (PDF)$]]
Thus,  in the following, for the signal at the channel output always holds:
:$$r(t) = s(t) + n(t).$$

The simplest realistic assumption for the transmission channel of a communication system is  '''Additive White Gaussian Noise'''  $\rm (AWGN)$,  as already stated in other LNTwww books,
*in the book  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#General_description|"Theory of Stochastic Signals"]],
*in the section  [[Modulation_Methods/Quality_Criteria|"Quality Criteria"]]  in  "Modulation Methods".
 
{{BlaueBox|TEXT=
$\text{The AWGN channel model  can be summarized as follows:}$ 
*The letter  $\rm N$  indicates that only noise is considered by the AWGN model.  Distortions of the useful signal  $s(t)$  are not accounted for by this simple model.
*Although noise is generally caused by a variety of noise sources along the entire transmission path,  for linear systems it can be accounted for by a single additive noise term at the channel output  $($letter  $\rm A)$.
*The noise includes all frequencies equally.  It has a constant white  $\rm (W)$  [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Wiener-Khintchine_Theorem|"power-spectral density"]]  $\rm (PSD)$  and a Dirac-shaped  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function|"auto-correlation function"]]  $\rm (ACF)$:
:$${\it \Phi}_n(f) = {N_0}/{2}\hspace{0.15cm}
\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.15cm}
\varphi_n(\tau) = {N_0}/{2} \cdot \delta
(\tau)\hspace{0.05cm}.$$
:The factor  $1/2$  on both sides of this Fourier transform equation accounts for the two-sided spectral representation.
*For example,  in the case of thermal noise,  for the physical noise power density  $($that is:   one-sided view)  with noise figure  $F \ge 1$  and absolute temperature  $\theta$:
::<math>{N_0}= F \cdot k_{\rm B} \cdot \theta , \hspace{0.3cm}k_{\rm B} =
1.38 \cdot 10^{-23} \hspace{0.2cm}{ \rm Ws}/{\rm K}\hspace{0.2cm}{\text{(Boltzmann constant)} }\hspace{0.05cm}.</math>
*True white noise would result in infinitely large power.  Therefore,  a band limit on  $B$  must always be considered.  The following applies to the effective noise power:
::<math>N = \sigma_n^2 = {N_0} \cdot B \hspace{0.05cm}.</math>
*The noise signal  $n(t)$  has a  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Probability_density_function_.E2.80.93_Cumulative_density_function|"Gaussian probability density function"]]  $\rm (PDF)$,  which is expressed by the letter  $\rm G$:
::<math>f_n(n) = \frac{1}{\sqrt{2\pi}\cdot\sigma_n}\cdot {\rm e}^{ - {\it
n^{\rm 2} }/{(2\sigma_n^2)} }.</math>
We would like to refer you here to the  (German language)  three-part learning video  [[Der_AWGN-Kanal_(Lernvideo)|"AWGN Channel"]],  in which the AWGN properties are clarified again.}}

== Receiver filter and threshold decision==
 
The simplest receiver for binary transmission via the AWGN channel consists of
[[File:EN_Dig_T_1_1_S5a_v2.png|right|frame|Receiver of a binary baseband transmission system]]
*a  '''receiver filter'''  (German:  "Empfangsfilter"   ⇒   subscript:  "E")  with frequency response  $H_{\rm E}(f)$,
*and a  '''threshold decision'''  for obtaining the binary signal.

This receiver structure can be justified as follows:  The signal  $d(t)$  after the receiver filter   ⇒   "detection signal"  can be divided at least mentally in two parts:  
:$$d(t) = d_{\rm S}(t)+d_{\rm N}(t).$$

*The portion  $d_{\rm S}(t)$  is due solely to the receiver input signal  $r(t)=s(t)$    ⇒   $n(t)= 0$.  The subscript  "S"  stands for "signal".  In the following,  we also refer to  $d_{\rm S}(t)$  as the  "signal component"  of  $d(t)$. 
*With the impulse response  $h_{\rm E}(t)$  as the Fourier retransform of the frequency response  $H_{\rm E}(f)$  holds:
:$$d_{\rm S}(t) = s(t) \star h_{\rm E} (t)\hspace{0.05cm}.$$
*For the second part  $d_{\rm N}(t)$,  on the other hand,  one assumes the receiver input signal  $r(t)=n(t)$    ⇒   $s(t)= 0$. We also refer to this portion  $d_{\rm N}(t)$  as the  "noise component"  of  $d(t)$.  The following convolution operation applies to it:
:$$d_{\rm N}(t) = n(t) \star h_{\rm E} (t)\hspace{0.05cm}.$$

*The white noise  $n(t)$  at the receiver input has theoretically an infinitely large power  $($practically:  an unnecessarily large power$)$.  The low-pass filter with frequency response  $H_{\rm E}(f)$  limits this to the squared expected value of the "noise component"  $d_{\rm N}(t)$   ⇒   "variance"   ⇒   "noise power":
::<math>\sigma_d^2 = {\rm E}\big[d_{\rm N}(t)^2\big] \hspace{0.05cm}.</math>
*However,  it should be noted that the low-pass  $H_{\rm E}(f)$  alters not only the noise  $n(t)$,  but also the transmitted signal  $s(t)$.  As a result,  the individual transmission pulses are broadened and reduced in amplitude.  According to the prerequisites for this chapter,  it must be ensured that  [[Digital_Signal_Transmission/Ursachen_und_Auswirkungen_von_Impulsinterferenzen|"intersymbol interference"]]  does not occur.
*The decider's task is to generate the discrete–value and discrete–time sink signal  $v(t)$  from the continuous–value and continuous–time detection signal  $d(t)$,  which should reproduce the message of the transmitted signal  $s(t)$  as well as possible.  The operation of the  (binary)  threshold decision is described in $\text{Example 3}$.

{{GraueBox|TEXT=
$\text{Example 3:}$  The upper graphic
[[File:EN_Dig_T_1_1_S5b_neu_sat2.png|right|frame|Signals in an optimal binary system|class=fit]]
*shows in red the rectangular transmitted signal  $s(t)$  normalized to  $\pm 1$, 
*which is superimposed by additive noise  $n(t)$. 
*Shown in blue is the received signal  $r(t) = s(t) + n(t)$.

To this graphic it is to be noted further:
#After the receiver filter with a rectangular impulse response of duration  $T$,  the signal  $d(t)$  shown in the middle figure  (green curve)  is obtained.
#In this special case  ("matched filter"),  the part  $d_{\rm S}(t)$,  which is exclusively due to the transmitted signal  $s(t)$,  has the section–wise linear course shown in red dots. 
#The difference  $d(t) - d_{\rm S}(t)$  is the noise component  $d_{\rm N}(t)$,  which originates from the AWGN term  $n(t)$. 
#The subsequent threshold decision evaluates the detection signal  $d(t)$.  For this purpose, it compares its samples at the equidistant detection times – marked by yellow arrows in the graphic – with the threshold value  $E = 0$. 
#Accordingly,  the decider sets the sink signal  $v(t)$  in the range  $\nu \cdot T$ ... $(\nu + 1) \cdot T$  to  $+1$  or  $-1$,  depending on whether the detection sample  $d(t)$  is larger or smaller than the decision threshold  $E$.
#If the decision unit makes always the correct decision,  as in the example shown,  its output signal is  $v(t) = s(t-T/2)$.
#The delay time of half a symbol duration  $(T/2)$  is  due to the fact that the detection signal  $d(t)$  is sensibly decided in the middle of the symbol,  but the provision of the sink signal  $v(t)$  can only take place afterwards for reasons of causality.}}

== Block diagram and prerequisites for the first main chapter==
 
The following block diagram is used as a basis for the further sections of this first main chapter.  Unless explicitly stated otherwise  the following prerequisites apply:
[[File:EN_Dig_T_1_1_S6_v1.png|right|frame|Equivalent block diagram for the investigation of binary baseband transmission systems   ⇒  
$q(t)$:  source signal   ⇒   binary, bipolar and redundancy-free,  bit rate  $R = 1/T$,
   ⇒  
$s(t)$:  transmitted signal   ⇒   NRZ–rectangular,  amplitude  $s_0$,  pulse duration  $T$,
   ⇒  
$g_s(t)$:  basic transmitter pulse   ⇒   spectrum  $G_s(f)= s_0 \cdot T \cdot {\rm sinc}(f \hspace{0.05cm}T)$,
   ⇒  
$r(t)= s(t) + n(t)$:  received signal   ⇒   channel frequency response  $H_{\rm K}(f) =1$,
   ⇒  
$n(t)$:  noise signal   ⇒   "AWGN":  constant one-sided noise power density  $N_0$,
   ⇒  
$d(t) = r(t) \star h_{\rm E} (t)$:  detection signal   ⇒   after low-pass filtering,
   ⇒  
$h_{\rm E}(t) = {\rm F}^{-1}\big[H_{\rm E}(f)\big]$:  impulse response of the receiver filter  $H_{\rm E}(f)$,
   ⇒  
$v(t)$:  sink signal   ⇒   after threshold decision, parameter: $E = 0$,  at times  $\nu \cdot T$.]]

*The transmission is binary,  bipolar and redundancy-free with bit rate  $R = 1/T$.  Coded and/or multilevel transmission is dealt with in the  [[Digital_Signal_Transmission|"second main chapter"]]. 
*The transmitted signal  $s(t) = q(t) \star g_s(t)$  is equal to  $ \pm s_0$ at all times $t$,  i.e.:   The basic transmitter pulse  $g_s(t)$  is NRZ–rectangular with amplitude  $s_0$  and pulse duration  $T$.
*Let for the received signal  $r(t) = s(t) + n(t)$.  Thus,  the channel frequency response is always  $H_{\rm K}(f) =1$.  $n(t)$  is characterized by the constant one-sided  (physical)  noise power density  $N_0$.
*For the detection signal generally applies  $d(t) = r(t) \star h_{\rm E} (t)$,  where  $h_{\rm E}(t) = {\rm F}^{-1}\big[H_{\rm E}(f)\big]$  is the impulse response of the receiver filter with low-pass frequency response  $H_{\rm E}(f)$. 
*$h_{\rm E}(t)$  is optimally matched to the basic transmitter pulse  $g_s(t)$,  so that  [[Digital_Signal_Transmission/Causes_and_Effects_of_Intersymbol_Interference|"intersymbol interference"]]  does not play a role.  Equalization methods are discussed in the [[Digital_Signal_Transmission|"third main chapter"]]  of this book. 
*The parameters of the  (binary)  threshold decision are optimally chosen:  Threshold:  $E = 0$,  detection times:  $\nu \cdot T$.

== Exercises for the chapter==
 
[[Aufgaben:Exercise_1.1:_Basic_Transmission_Pulses|Exercise 1.1: Basic Transmitter Pulses]]

[[Aufgaben:Exercise_1.1Z:_Nonredundant_Binary_Source|Exercise 1.1Z: Non-redundant Binary Source]]

==References==
{{Display}}

Examples of Communication Systems

2023-03-31T10:06:09Z

Hwang:

===Brief summary===

{{BlueBox|TEXT=In the last book of our  »e-learning series  $\rm LNTwww$«  it is shown by  »'''realized and in their time commercially successful messaging systems'''«  that the theoretical foundations of the previous eight books are quite relevant for practice.

Here are some keywords from the book contents:

# $\rm ISDN$  $($"$\rm I$ntegrated $\rm S$ervices $\rm D$igital $\rm N$etwork"$)$:  Objectives & characteristics,  Services & features,  Basic access & primary multiplex access,  Broadband ISDN.
# $\rm xDSL$  $($"$\rm D$igital $\rm S$ubscriber $\rm L$ine"$)$:  Historical development and standardizations,  Reference model,  ADSL, VDSL and HDSL,  WAM and DMT realization.
# $\rm GSM$  $($"Global System for Mobile Communications"$)$:  Second generation mobile communications system,  based on FDMA,  TDMA and GMSK.  Voice '''KORREKTUR: Speech''' coding.
# $\rm UMTS$  $($"$\rm U$niversal  $\rm M$obile  $\rm T$elecommunications  $\rm S$ystem"$)$:  Third-generation mobile communications system, based on CDMA.  Further development HSPA.

Notes:

*The basics are no longer treated in detail here,  rather, reference is made to the relevant reference books.

*GSM and UMTS also show the transience of technological developments.  In some countries, GSM and UMTS are already switched off after a good 20 years.

⇒   Here first a  »'''contents overview'''«  based on the  »'''four main chapters'''«  with a total of  »'''17 individual chapters'''«  and  »'''164 sections'''«.}}

===Contents===

{{Collapsible-Kopf}}
{{Collapse1| header=ISDN - Integrated Services Digital Network
| submenu=
*[[/General Description of ISDN/]]
*[[/ISDN Basic Access/]]
*[[/ISDN Primary Multiplex Connection/]]
*[[/Further Developments of ISDN/]]
}}
{{Collapse2 | header=DSL – Digital Subscriber Line
|submenu=
*[[/General Description of DSL/]]
*[[/xDSL Systems/]]
*[[/xDSL as Transmission Technology/]]
*[[/Methods to Reduce the Bit Error Rate in DSL/]]
}}
{{Collapse3 | header=GSM – Global System for Mobile Communications
|submenu=
*[[/General Description of GSM/]]
*[[/Radio Interface/]]
*[[/Speech Coding/]]
*[[/Entire GSM Transmission System/]]
*[[/Further Developments of the GSM/]]
}}
{{Collapse4 | header=UMTS – Universal Mobile Telecommunications System
|submenu=
*[[/General Description of UMTS/]]
*[[/UMTS Network Architecture/]]
*[[/Telecommunications Aspects of UMTS/]]
*[[/Further Developments of UMTS/]]
}}
{{Collapsible-Fuß}}

===Exercises and multimedia modules===

{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Examples_of_Communication_Systems:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_Videos_Related_to_"Examples_of_Communication_Systems"|$\text{Learning videos}$]]

$(3)$    [[LNTwww:Applets_Related_to_"Examples_of_Communication_Systems"|$\text{Applets}$]] }}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Examples_of_Communication_Systems"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:General_notes_about_"Examples_of_Communication_Systems"|$\text{Impressum}$]] }}
 
[[LNTwww:Authors#Beispiele_von_Nachrichtensystemen|$\text{Hinweise zu den Autoren und den Materialien, von denen bei der Erstellung des Buches ausgegangen wurde.}$]]

{{Display}}

Mobile Communications

2023-03-31T09:44:05Z

Hwang:

===Brief summary===

{{BlueBox|TEXT=The book  »Mobile Communication«  deals with the essential differences compared to systems for the fixed network,  when transmitters and/or receivers are moving.  Thus, impulse '''KORREKTUR: intersymbol''' interference is not caused by frequency dependence of the medium »electrical line« or »optical fiber»,  but by multipath propagation due to reflections,  resulting in constructive or destructive superpositions of the electromagnetic wave with its echoes.  Here are some of the topics covered:

# Time-variant transmission channels:  Distance-dependent attenuation,  shadowing,  Rayleigh fading  $($at line-of-sight$)$,  Rice fading  $($without line-of-sight$)$.
# Frequency-selective transmission channels:  Two-dimensional impulse response and transfer function,  multipath reception in mobile radio,  GWSSUS model.
# Second and third generation mobile radio systems:  Characteristics of GSM  $\rm (2G)$  and UMTS  $\rm (3G)$,  Similarities between the two, further developments.
# The 4G–LTE mobile standard  $($"Long Term Evolution"$)$,  similarities and differences between  "OFDMA"  $($e.g.  at DSL$)$  and  "OFDMA"  $($e.g. at LTE$)$.

Notes:
*Many of the fundamentals covered in the book  »[[Signal Representation]]«  are still valid.  Differences arise from the radio channel,  which is mostly time-varying.

*More detailed descriptions of second and third generation mobile radio systems  $($GSM,  UMTS$)$  can be found in the book  [[Examples_of_Communication_Systems|»Examples of Communication Systems»]].

*It should be noted,  however,  that their importance today  $($2023$)$  is no longer very great;  in some countries GSM and UMTS have already been switched off.

⇒   Here first a  »'''contents overview'''«  based on the  »'''four main chapters'''«  with a total of  »'''sixteen individual chapters'''«  and  »'''121 sections'''«.}}

===Content===
{{Collapsible-Kopf}}
{{Collapse1| header=Time-Variant Transmission Channels
| submenu=
*[[/Distance Dependent Attenuation and Shading/]]
*[[/Probability Density of Rayleigh Fading/]]
*[[/Statistical Bindings within the Rayleigh Process/]]
*[[/Non-Frequency-Selective Fading With Direct Component/]]
}}
{{Collapse2 | header=Frequency-Selective Transmission Channels
|submenu=
*[[/General Description of Time Variant Systems/]]
*[[/Multi-Path Reception in Mobile Communications/]]
*[[/The GWSSUS Channel Model/]]
}}
{{Collapse3 | header=Mobile Radio Systems of the 2nd and 3rd Generation - an Overview
|submenu=
*[[/History and Development of Mobile Communication Systems/]]
*[[/Similarities between GSM and UMTS/]]
*[[/Characteristics of GSM/]]
*[[/Characteristics of UMTS/]]
}}
{{Collapse4 | header=LTE – Long Term Evolution
|submenu=
*[[/General Information on the LTE Mobile Communications Standard/]]
*[[/Technical Innovations of LTE/]]
*[[/The Application of OFDMA and SC-FDMA in LTE/]]
*[[/Physical Layer for LTE/]]
*[[/LTE-Advanced - a Further Development of LTE/]]
}}
{{Collapsible-Fuß}}

===Exercises and multimedia===
{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Mobile_Communications:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_Videos_to_"Mobile_Communications"|$\text{Learning videos}$]]

$(3)$    [[LNTwww:Applets_to_"Mobile_Communications"|$\text{Applets}$]] }}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Mobile_Communications"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:Imprint_for_the_book_"Mobile_Communications"|$\text{Impressum}$]]}}
 

{{Display}}

Mobile Communications

2023-03-31T09:43:41Z

Hwang:

===Brief summary===

{{BlueBox|TEXT=The book  »Mobile Communication«  deals with the essential differences compared to systems for the fixed network,  when transmitters and/or receivers are moving.  Thus, impulse interference is not caused by frequency dependence of the medium »electrical line« or »optical fiber»,  but by multipath propagation due to reflections,  resulting in constructive or destructive superpositions of the electromagnetic wave with its echoes.  Here are some of the topics covered:

# Time-variant transmission channels:  Distance-dependent attenuation,  shadowing,  Rayleigh fading  $($at line-of-sight$)$,  Rice fading  $($without line-of-sight$)$.
# Frequency-selective transmission channels:  Two-dimensional impulse response and transfer function,  multipath reception in mobile radio,  GWSSUS model.
# Second and third generation mobile radio systems:  Characteristics of GSM  $\rm (2G)$  and UMTS  $\rm (3G)$,  Similarities between the two, further developments.
# The 4G–LTE mobile standard  $($"Long Term Evolution"$)$,  similarities and differences between  "OFDMA"  $($e.g.  at DSL$)$  and  "OFDMA"  $($e.g. at LTE$)$.

Notes:
*Many of the fundamentals covered in the book  »[[Signal Representation]]«  are still valid.  Differences arise from the radio channel,  which is mostly time-varying.

*More detailed descriptions of second and third generation mobile radio systems  $($GSM,  UMTS$)$  can be found in the book  [[Examples_of_Communication_Systems|»Examples of Communication Systems»]].

*It should be noted,  however,  that their importance today  $($2023$)$  is no longer very great;  in some countries GSM and UMTS have already been switched off.

⇒   Here first a  »'''contents overview'''«  based on the  »'''four main chapters'''«  with a total of  »'''sixteen individual chapters'''«  and  »'''121 sections'''«.}}

===Content===
{{Collapsible-Kopf}}
{{Collapse1| header=Time-Variant Transmission Channels
| submenu=
*[[/Distance Dependent Attenuation and Shading/]]
*[[/Probability Density of Rayleigh Fading/]]
*[[/Statistical Bindings within the Rayleigh Process/]]
*[[/Non-Frequency-Selective Fading With Direct Component/]]
}}
{{Collapse2 | header=Frequency-Selective Transmission Channels
|submenu=
*[[/General Description of Time Variant Systems/]]
*[[/Multi-Path Reception in Mobile Communications/]]
*[[/The GWSSUS Channel Model/]]
}}
{{Collapse3 | header=Mobile Radio Systems of the 2nd and 3rd Generation - an Overview
|submenu=
*[[/History and Development of Mobile Communication Systems/]]
*[[/Similarities between GSM and UMTS/]]
*[[/Characteristics of GSM/]]
*[[/Characteristics of UMTS/]]
}}
{{Collapse4 | header=LTE – Long Term Evolution
|submenu=
*[[/General Information on the LTE Mobile Communications Standard/]]
*[[/Technical Innovations of LTE/]]
*[[/The Application of OFDMA and SC-FDMA in LTE/]]
*[[/Physical Layer for LTE/]]
*[[/LTE-Advanced - a Further Development of LTE/]]
}}
{{Collapsible-Fuß}}

===Exercises and multimedia===
{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Mobile_Communications:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_Videos_to_"Mobile_Communications"|$\text{Learning videos}$]]

$(3)$    [[LNTwww:Applets_to_"Mobile_Communications"|$\text{Applets}$]] }}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Mobile_Communications"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:Imprint_for_the_book_"Mobile_Communications"|$\text{Impressum}$]]}}
 

{{Display}}

Digital Signal Transmission

2023-03-31T09:31:56Z

Hwang:

===Brief summary===

{{BlueBox|TEXT=The focus of this book is the  »'''calculation of the error probability'''«,  which is the decisive quality feature for digital systems.  The description is mainly in baseband,  but most of the results can also be applied to the digital carrier frequency systems.
# Error probability and optimization of baseband transmission systems.  Properties of Nyquist systems.  First and second Nyquist criteria.
# Fundamentals of line coding:  Redundancy-free codes,  blockwise  $($4B3T$)$  and  symbolwise encoding  $($pseudoternary codes$)$.  Power spectral density.
# Optimization considering intersymbol interference:  Eye diagram,  linear Nyquist equalization,  decision feedback equalization,  Viterbi receiver.
# Generalized description of digital modulation schemes:  Basis functions and vector spaces,  coherent and incoherent demodulation.
# Digital channel models:  Error distance distribution and error correlation function.  BSC model.  Gilbert/Elliott and McCullough bundle error models.

Notes:
*A basic knowledge of  "[[Signal Representation]]"  and  [[Theory_of_Stochastic_Signals|"Stochastic Signal Theory"]]  $($Books 1 and 3$)$  is assumed for understanding this book.

*There are certain,  quite intentional overlaps with the book  [[Modulation Methods|"Modulation Methods"]].

*Mostly a time-variant channel is assumed.  Time invariance is dealt with in the books  [[Mobile_Communications|"Mobile Communications"]]  and  [[Examples_of_Communication_Systems|"Examples of Communication Systems"]] .

⇒   Here first an  »'''overview of contents'''«  on the basis of the  »'''five main chapters'''«  with a total of  »'''26 individual chapters'''«  and  »'''201 sections''''«.}}

===Contents===
{{Collapsible-Kopf}}
{{Collapse1| header=Digital Signal Transmission under Idealized Conditions
| submenu=
*[[/System Components of a Baseband Transmission System/]]
*[[/Error Probability for Baseband Transmission/]]
*[[/Properties of Nyquist Systems/]]
*[[/Optimization of Baseband Transmission Systems/]]
*[[/Linear Digital Modulation - Coherent Demodulation/]]
}}
{{Collapse2 | header=Coded and Multilevel Transmission
|submenu=
*[[/Basics of Coded Transmission/]]
*[[/Redundancy-Free Coding/]]
*[[/Block Coding with 4B3T Codes/]]
*[[/Symbolwise Coding with Pseudo-Ternary Codes/]]
}}
{{Collapse3 | header=Intersymbol Interfering and Equalization Methods
|submenu=
*[[/Causes and Effects of Intersymbol Interference/]]
*[[/Error Probability with Intersymbol Interference/]]
*[[/Consideration of Channel Distortion and Equalization/]]
*[[/Intersymbol Interference for Multi-Level Transmission/]]
*[[/Linear Nyquist Equalization/]]
*[[/Decision Feedback/]]
*[[/Optimal Receiver Strategies/]]
*[[/Viterbi Receiver/]]
}}
{{Collapse4 | header=Generalized Description of Digital Modulation Methods
|submenu=
*[[/Signals, Basis Functions and Vector Spaces/]]
*[[/Structure of the Optimal Receiver/]]
*[[/Approximation of the Error Probability/]]
*[[/Carrier Frequency Systems with Coherent Demodulation/]]
*[[/Carrier Frequency Systems with Non-Coherent Demodulation/]]
}}
{{Collapse5 | header=Digital Channel Models
|submenu=
*[[/Parameters of Digital Channel Models/]]
*[[/Binary Symmetric Channel/]]
*[[/Burst Error Channels/]]
*[[/Applications for Multimedia Files/]]
}}

{{Collapsible-Fuß}}
===Exercises and multimedia===

{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Digital_Signal_Transmission:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_Videos_to_"Digital_Signal_Transmission"|$\text{Learning videos}$]]

$(3)$    [[LNTwww:Applets_to_"Digital_Signal_Transmission"|$\text{Applets}$]] }}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Digital_Signal_Transmission"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:Imprint_for_the_book_"Digital_Signal_Transmission"|$\text{Impressum}$]]}}
 

{{Display}}

Modulation Methods

2023-03-30T19:26:23Z

Hwang:

===Brief summary===

{{BlueBox|TEXT=This book describes essential fundamentals of both analogue and digital transmission systems,  in particular:
# Objective of modulation and demodulation:  Frequency and time division multiplexing.  Ideal,  distortion-free and distorting system.  Quality criteria.
# Amplitude modulation and associated demodulators:  Double sideband– and single sideband modulation.  Envelope and synchronous demodulation.
# Angle modulation and associated demodulators:  Phase modulation and frequency modulation.  Signal-to-noise ratio with angle modulation.
# Digital modulation techniques:  Pulse code modulation,  Linear modulation techniques  $($ASK,  BPSK,  BPSK$)$,  Nonlinear '''KORREKTUR: Non-linear''' modulation  $($FSK,  MSK,  GMSK$)$.
# Multiple access methods:  FDMA, TDMA, CDMA.  Multiplexers and demultiplexers.  CDMA spread sequences.  PN modulation:  Principle and realization.

Notes:
*The description builds on the books  »[[Signal Representation]]«  and  »[[Linear_and_Time_Invariant_Systems]]«. 

*Knowledge of  »[[Theory_of_Stochastic_Signals]]«  is helpful,  but not essential.

*Further details on digital signal transmission  $($BER computation,  system optimisation, ...$)$  are covered in the book  »[[Digital Signal Transmission]]«.

⇒   Here first an  »'''content overview'''«  by means of the  »'''five main chapters'''«  with a total of  »'''23 individual chapters'''«  and  »'''192 sections'''«.}}

===Content===
{{Collapsible-Kopf}}
{{Collapse1| header=General Description
| submenu=
*[[/Objectives of Modulation and Demodulation/]]
*[[/Quality Criteria/]]
*[[/General Model of Modulation/]]
}}
{{Collapse2 | header=Amplitude Modulation and Demodulation
|submenu=
*[[/Double-Sideband Amplitude Modulation/]]
*[[/Synchronous Demodulation/]]
*[[/Envelope Demodulation/]]
*[[/Single-Sideband Modulation/]]
*[[/Further AM Variants/]]
}}
{{Collapse3 | header=Angle Modulation and Demodulation
|submenu=
*[[/Phase Modulation (PM)/]]
*[[/Frequency Modulation (FM)/]]
*[[/Influence of Noise on Systems with Angle Modulation/]]
}}
{{Collapse4 | header=Digital Modulation Methods
|submenu=
*[[/Pulse Code Modulation/]]
*[[/Linear Digital Modulation/]]
*[[/Quadrature Amplitude Modulation/]]
*[[/Non-Linear Digital Modulation/]]
}}
{{Collapse5 | header=Multiple Access Methods
|submenu=
*[[/Tasks and Classification/]]
*[[/Direct-Sequence Spread Spectrum Modulation/]]
*[[/Spreading Sequences for CDMA/]]
*[[/Error Probability of Direct-Sequence Spread Spectrum Modulation/]]
*[[/General Description of OFDM/]]
*[[/Implementation of OFDM Systems/]]
*[[/OFDM for 4G Networks/]]
*[[/Further OFDM Applications/]]
}}
{{Collapsible-Fuß}}

===Exercises and multimedia===

{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Modulation_Methods:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_Videos_to_"Modulation_Methods"|$\text{Learning videos}$]]

$(3)$    [[LNTwww:LNTwww:Applets_to_"Modulation_Methods"|$\text{Applets}$]] }}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Modulation_Methods"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:Imprint_for_the_book_"Modulation_Methods"|$\text{Impressum}$]]}}
 

{{Display}}

Modulation Methods

2023-03-30T19:22:58Z

Hwang:

===Brief summary===

{{BlueBox|TEXT=This book describes essential fundamentals of both analogue and digital transmission systems,  in particular:
# Objective of modulation and demodulation:  Frequency and time division multiplexing.  Ideal,  distortion-free and distorting system.  Quality criteria.
# Amplitude modulation and associated demodulators:  Double sideband– and single sideband modulation.  Envelope and synchronous demodulation.
# Angle modulation and associated demodulators:  Phase modulation and frequency modulation.  Signal-to-noise ratio with angle modulation.
# Digital modulation techniques:  Pulse code modulation,  Linear modulation techniques  $($ASK,  BPSK,  BPSK$)$,  Nonlinear modulation  $($FSK,  MSK,  GMSK$)$.
# Multiple access methods:  FDMA, TDMA, CDMA.  Multiplexers and demultiplexers.  CDMA spread sequences.  PN modulation:  Principle and realization.

Notes:
*The description builds on the books  »[[Signal Representation]]«  and  »[[Linear_and_Time_Invariant_Systems]]«. 

*Knowledge of  »[[Theory_of_Stochastic_Signals]]«  is helpful,  but not essential.

*Further details on digital signal transmission  $($BER computation,  system optimisation, ...$)$  are covered in the book  »[[Digital Signal Transmission]]«.

⇒   Here first an  »'''content overview'''«  by means of the  »'''five main chapters'''«  with a total of  »'''23 individual chapters'''«  and  »'''192 sections'''«.}}

===Content===
{{Collapsible-Kopf}}
{{Collapse1| header=General Description
| submenu=
*[[/Objectives of Modulation and Demodulation/]]
*[[/Quality Criteria/]]
*[[/General Model of Modulation/]]
}}
{{Collapse2 | header=Amplitude Modulation and Demodulation
|submenu=
*[[/Double-Sideband Amplitude Modulation/]]
*[[/Synchronous Demodulation/]]
*[[/Envelope Demodulation/]]
*[[/Single-Sideband Modulation/]]
*[[/Further AM Variants/]]
}}
{{Collapse3 | header=Angle Modulation and Demodulation
|submenu=
*[[/Phase Modulation (PM)/]]
*[[/Frequency Modulation (FM)/]]
*[[/Influence of Noise on Systems with Angle Modulation/]]
}}
{{Collapse4 | header=Digital Modulation Methods
|submenu=
*[[/Pulse Code Modulation/]]
*[[/Linear Digital Modulation/]]
*[[/Quadrature Amplitude Modulation/]]
*[[/Non-Linear Digital Modulation/]]
}}
{{Collapse5 | header=Multiple Access Methods
|submenu=
*[[/Tasks and Classification/]]
*[[/Direct-Sequence Spread Spectrum Modulation/]]
*[[/Spreading Sequences for CDMA/]]
*[[/Error Probability of Direct-Sequence Spread Spectrum Modulation/]]
*[[/General Description of OFDM/]]
*[[/Implementation of OFDM Systems/]]
*[[/OFDM for 4G Networks/]]
*[[/Further OFDM Applications/]]
}}
{{Collapsible-Fuß}}

===Exercises and multimedia===

{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Modulation_Methods:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_Videos_to_"Modulation_Methods"|$\text{Learning videos}$]]

$(3)$    [[LNTwww:LNTwww:Applets_to_"Modulation_Methods"|$\text{Applets}$]] }}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Modulation_Methods"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:Imprint_for_the_book_"Modulation_Methods"|$\text{Impressum}$]]}}
 

{{Display}}

Information Theory

2023-03-30T19:10:33Z

Hwang:

===Brief summary===

{{BlueBox|TEXT=From the earliest beginnings of message transmission as an engineering discipline,  it has been the endeavour of many engineers and mathematicians  to find a quantitative measure for the
*contained  $\rm information$  $($quite generally:  "the knowledge about something"$)$

*in a  $\rm message$  $($here we mean  "a collection of symbols and/or states"$)$.

The  $($abstract$)$  information is communicated by the  $($concrete$)$  message and can be conceived as the interpretation of a message.

[https://en.wikipedia.org/wiki/Claude_Shannon '''Claude Elwood Shannon''']  succeeded in 1948,  in establishing a consistent theory about the information content of messages,  which was revolutionary in its time and created a new,  still highly topical field of science:  »'''Shannon's information theory«'''  named after him.» '''KORREKTUR: »?'''

This is what the fourth book in the  $\rm LNTww$ series deals with,  in particular:
# Entropy of discrete-value sources with and without memory,  as well as natural message sources:  Definition,  meaning and computational possibilities.
# Source coding and data compression,  especially the   "Lempel–Ziv–Welch method"   and   "Huffman's entropy encoding".
# Various entropies of two-dimensional discrete-value random quantities.  Mutual information and channel capacity.  Application to digital signal transmission.
# Discrete-value information theory.  Differential entropy.  AWGN channel capacity with continuous-valued as well as discrete-valued input.

⇒   First a  »'''content overview'''«  on the basis of the  »'''four main chapters'''«  with a total of  »'''13 individual chapters'''«  and  »'''106 sections'''«:}}

===Content===

{{Collapsible-Kopf}}
{{Collapse1| header=Entropy of Discrete Sources
| submenu=
*[[/Discrete Memoryless Sources/]]
*[[/Discrete Sources with Memory/]]
*[[/Natural Discrete Sources/]]
}}
{{Collapse2 | header=Source Coding - Data Compression
|submenu=
*[[/General Description/]]
*[[/Compression According to Lempel, Ziv and Welch/]]
*[[/Entropy Coding According to Huffman/]]
*[[/Further Source Coding Methods/]]
}}
{{Collapse3 | header=Mutual Information Between Two Discrete Random Variables
|submenu=
*[[/Some Preliminary Remarks on Two-Dimensional Random Variables/]]
*[[/Different Entropy Measures of Two-Dimensional Random Variables/]]
*[[/Application to Digital Signal Transmission/]]
}}
{{Collapse4 | header=Information Theory for Continuous Random Variables
|submenu=
*[[/Differential Entropy/]]
*[[/AWGN Channel Capacity for Continuous-Valued Input/]]
*[[/AWGN Channel Capacity for Discrete-Valued Input/]]
}}
{{Collapsible-Fuß}}

===Exercises and multimedia===

{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Information_Theory:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_videos_to_"Information_Theory"|$\text{Learning videos}$]]

$(3)$    [[LNTwww:Applets_to_"Information_Theory"|$\text{Applets}$]] }}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Information_Theory"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:Imprint_for_the_book_"Information_Theory"|$\text{Impressum}$]]}}
 

__NOTOC__
__NOEDITSECTION__

Information Theory

2023-03-30T19:10:09Z

Hwang:

===Brief summary===

{{BlueBox|TEXT=From the earliest beginnings of message transmission as an engineering discipline,  it has been the endeavour of many engineers and mathematicians  to find a quantitative measure for the
*contained  $\rm information$  $($quite generally:  "the knowledge about something"$)$

*in a  $\rm message$  $($here we mean  "a collection of symbols and/or states"$)$.

The  $($abstract$)$  information is communicated by the  $($concrete$)$  message and can be conceived as the interpretation of a message.

[https://en.wikipedia.org/wiki/Claude_Shannon '''Claude Elwood Shannon''']  succeeded in 1948,  in establishing a consistent theory about the information content of messages,  which was revolutionary in its time and created a new,  still highly topical field of science:  »'''Shannon's information theory«'''  named after him.» '''KORREKTUR: »?'''

This is what the fourth book in the  $\rm LNTww$ series deals with,  in particular:
# Entropy of discrete-value sources with and withott memory,  as well as natural message sources:  Definition,  meaning and computational possibilities.
# Source coding and data compression,  especially the   "Lempel–Ziv–Welch method"   and   "Huffman's entropy encoding".
# Various entropies of two-dimensional discrete-value random quantities.  Mutual information and channel capacity.  Application to digital signal transmission.
# Discrete-value information theory.  Differential entropy.  AWGN channel capacity with continuous-valued as well as discrete-valued input.

⇒   First a  »'''content overview'''«  on the basis of the  »'''four main chapters'''«  with a total of  »'''13 individual chapters'''«  and  »'''106 sections'''«:}}

===Content===

{{Collapsible-Kopf}}
{{Collapse1| header=Entropy of Discrete Sources
| submenu=
*[[/Discrete Memoryless Sources/]]
*[[/Discrete Sources with Memory/]]
*[[/Natural Discrete Sources/]]
}}
{{Collapse2 | header=Source Coding - Data Compression
|submenu=
*[[/General Description/]]
*[[/Compression According to Lempel, Ziv and Welch/]]
*[[/Entropy Coding According to Huffman/]]
*[[/Further Source Coding Methods/]]
}}
{{Collapse3 | header=Mutual Information Between Two Discrete Random Variables
|submenu=
*[[/Some Preliminary Remarks on Two-Dimensional Random Variables/]]
*[[/Different Entropy Measures of Two-Dimensional Random Variables/]]
*[[/Application to Digital Signal Transmission/]]
}}
{{Collapse4 | header=Information Theory for Continuous Random Variables
|submenu=
*[[/Differential Entropy/]]
*[[/AWGN Channel Capacity for Continuous-Valued Input/]]
*[[/AWGN Channel Capacity for Discrete-Valued Input/]]
}}
{{Collapsible-Fuß}}

===Exercises and multimedia===

{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Information_Theory:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_videos_to_"Information_Theory"|$\text{Learning videos}$]]

$(3)$    [[LNTwww:Applets_to_"Information_Theory"|$\text{Applets}$]] }}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Information_Theory"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:Imprint_for_the_book_"Information_Theory"|$\text{Impressum}$]]}}
 

__NOTOC__
__NOEDITSECTION__

Information Theory

2023-03-30T19:09:35Z

Hwang:

===Brief summary===

{{BlueBox|TEXT=From the earliest beginnings of message transmission as an engineering discipline,  it has been the endeavour of many engineers and mathematicians  to find a quantitative measure for the
*contained  $\rm information$  $($quite generally:  "the knowledge about something"$)$

*in a  $\rm message$  $($here we mean  "a collection of symbols and/or states"$)$.

The  $($abstract$)$  information is communicated by the  $($concrete$)$  message and can be conceived as the interpretation of a message.

[https://en.wikipedia.org/wiki/Claude_Shannon '''Claude Elwood Shannon''']  succeeded in 1948,  in establishing a consistent theory about the information content of messages,  which was revolutionary in its time and created a new,  still highly topical field of science:  »'''Shannon's information theory«'''  named after him.»

This is what the fourth book in the  $\rm LNTww$ series deals with,  in particular:
# Entropy of discrete-value sources with and withott memory,  as well as natural message sources:  Definition,  meaning and computational possibilities.
# Source coding and data compression,  especially the   "Lempel–Ziv–Welch method"   and   "Huffman's entropy encoding".
# Various entropies of two-dimensional discrete-value random quantities.  Mutual information and channel capacity.  Application to digital signal transmission.
# Discrete-value information theory.  Differential entropy.  AWGN channel capacity with continuous-valued as well as discrete-valued input.

⇒   First a  »'''content overview'''«  on the basis of the  »'''four main chapters'''«  with a total of  »'''13 individual chapters'''«  and  »'''106 sections'''«:}}

===Content===

{{Collapsible-Kopf}}
{{Collapse1| header=Entropy of Discrete Sources
| submenu=
*[[/Discrete Memoryless Sources/]]
*[[/Discrete Sources with Memory/]]
*[[/Natural Discrete Sources/]]
}}
{{Collapse2 | header=Source Coding - Data Compression
|submenu=
*[[/General Description/]]
*[[/Compression According to Lempel, Ziv and Welch/]]
*[[/Entropy Coding According to Huffman/]]
*[[/Further Source Coding Methods/]]
}}
{{Collapse3 | header=Mutual Information Between Two Discrete Random Variables
|submenu=
*[[/Some Preliminary Remarks on Two-Dimensional Random Variables/]]
*[[/Different Entropy Measures of Two-Dimensional Random Variables/]]
*[[/Application to Digital Signal Transmission/]]
}}
{{Collapse4 | header=Information Theory for Continuous Random Variables
|submenu=
*[[/Differential Entropy/]]
*[[/AWGN Channel Capacity for Continuous-Valued Input/]]
*[[/AWGN Channel Capacity for Discrete-Valued Input/]]
}}
{{Collapsible-Fuß}}

===Exercises and multimedia===

{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Information_Theory:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_videos_to_"Information_Theory"|$\text{Learning videos}$]]

$(3)$    [[LNTwww:Applets_to_"Information_Theory"|$\text{Applets}$]] }}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Information_Theory"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:Imprint_for_the_book_"Information_Theory"|$\text{Impressum}$]]}}
 

__NOTOC__
__NOEDITSECTION__

Theory of Stochastic Signals

2023-03-30T19:07:38Z

Hwang:

===Brief summary===

{{BlaueBox|TEXT=This third book of our learning tutorial deals in detail with stochastic signals and their modelling. Knowledge of stochastic signal theory is an important prerequisite for understanding the following books, which focus on transmission aspects.
# Fundamentals and definitions of probability theory;  set-theoretic description;  Statistical dependence;  Markov chains.
# Properties of discrete-valued random variables and their computational generation.  Examples:  Binomial and Poisson distribution.  Moments calculation.
# Description of continuous-valued random variables:  Probability density function,  distribution function,  moment calculation.  special distributions.
# Two- and multi-dimensional random variables:  Autocorrelation function,  power density spectrum, '''KORREKTUR: power-spectral density'''  correlation coefficient,  cross-correlation function.
# Filtering of stochastic signals   ⇒   »Stochastic System Theory«;  digital filters;  properties of matched filter and Wiener–Kolmogorov filter.

Knowledge of the first two  $\text{LNTwww}$-books,  which describe the  [[Signal Representation|»representation of deterministic signals«]]  as well as the  [[Linear_and_Time_Invariant_Systems|"description of linear and time-invariant systems»]],  are helpful for the understanding of the present book,  but not required.

⇒   First a  »'''content overview'''«  on the basis of the  »'''five main chapters'''«  with a total of  »'''28 individual chapters'''«  and  »'''166 sections'''«:}}

===Content===
{{Collapsible-Kopf}}
{{Collapse1| header=Probability Calculation
| submenu=
*[[/Some Basic Definitions/]]
*[[/Set Theory Basics/]]
*[[/Statistical Dependence and Independence/]]
*[[/Markov Chains/]]
}}
{{Collapse2 | header=Discrete Random Variables
|submenu=
*[[/From Random Experiment to Random Variable/]]
*[[/Moments of a Discrete Random Variable/]]
*[[/Binomial Distribution/]]
*[[/Poisson Distribution/]]
*[[/Generation of Discrete Random Variables/]]
}}
{{Collapse3 | header=Continuous Random Variables
|submenu=
*[[/Probability Density Function/]]
*[[/Cumulative Distribution Function/]]
*[[/Expected Values and Moments/]]
*[[/Uniformly Distributed Random Variables/]]
*[[/Gaussian Distributed Random Variables/]]
*[[/Exponentially Distributed Random Variables/]]
*[[/Further Distributions/]]
}}
{{Collapse4 | header=Random Variables with Statistical Dependence
|submenu=
*[[/Two-Dimensional Random Variables/]]
*[[/Two-Dimensional Gaussian Random Variables/]]
*[[/Linear Combinations of Random Variables/]]
*[[/Auto-Correlation Function/]]
*[[/Power-Spectral Density/]]
*[[/Cross-Correlation Function and Cross Power-Spectral Density/]]
*[[/Generalization to N-Dimensional Random Variables/]]
}}
{{Collapse5 | header=Filtering of Stochastic Signals
|submenu=
*[[/Stochastic System Theory/]]
*[[/Digital Filters/]]
*[[/Creation of Predefined ACF Properties/]]
*[[/Matched Filter/]]
*[[/Wiener–Kolmogorow Filter/]]
}}
{{Collapsible-Fuß}}

===Exercises and multimedia===

{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Theory_of_Stochastic_Signals:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_videos_to_"Theory_of_Stochastic_Signals"|$\text{Learning videos}$]]

$(3)$    [[LNTwww:Applets_to_"Theory_of_Stochastic_Signals"|$\text{Applets}$]] }}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Theory_of_Stochastic_Signals"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:Imprint_for_the_book_"Stochastic_Signal_Theory"|$\text{Impressum}$]]}}
 

{{Display}}

Theory of Stochastic Signals

2023-03-30T18:59:43Z

Hwang:

===Brief summary===

{{BlaueBox|TEXT=This third book of our learning tutorial deals in detail with stochastic signals and their modelling. Knowledge of stochastic signal theory is an important prerequisite for understanding the following books, which focus on transmission aspects.
# Fundamentals and definitions of probability theory;  set-theoretic description;  Statistical dependence;  Markov chains.
# Properties of discrete-valued random variables and their computational generation.  Examples:  Binomial and Poisson distribution.  Moments calculation.
# Description of continuous-valued random variables:  Probability density function,  distribution function,  moment calculation.  special distributions.
# Two- and multi-dimensional random variables:  Autocorrelation function,  power density spectrum, '''KORREKTUR: power-spectral density'''  correlation coefficient,  cross-correlation function.
# Filtering of stochastic signals   ⇒   »Stochastic System Theory«;  digital filters;  properties of matched filter and Wiener–Kolmogorov–filter.

Knowledge of the first two  $\text{LNTwww}$-books,  which describe the  [[Signal Representation|»representation of deterministic signals«]]  as well as the  [[Linear_and_Time_Invariant_Systems|"description of linear and time-invariant systems»]],  are helpful for the understanding of the present book,  but not required.

⇒   First a  »'''content overview'''«  on the basis of the  »'''five main chapters'''«  with a total of  »'''28 individual chapters'''«  and  »'''166 sections'''«:}}

===Content===
{{Collapsible-Kopf}}
{{Collapse1| header=Probability Calculation
| submenu=
*[[/Some Basic Definitions/]]
*[[/Set Theory Basics/]]
*[[/Statistical Dependence and Independence/]]
*[[/Markov Chains/]]
}}
{{Collapse2 | header=Discrete Random Variables
|submenu=
*[[/From Random Experiment to Random Variable/]]
*[[/Moments of a Discrete Random Variable/]]
*[[/Binomial Distribution/]]
*[[/Poisson Distribution/]]
*[[/Generation of Discrete Random Variables/]]
}}
{{Collapse3 | header=Continuous Random Variables
|submenu=
*[[/Probability Density Function/]]
*[[/Cumulative Distribution Function/]]
*[[/Expected Values and Moments/]]
*[[/Uniformly Distributed Random Variables/]]
*[[/Gaussian Distributed Random Variables/]]
*[[/Exponentially Distributed Random Variables/]]
*[[/Further Distributions/]]
}}
{{Collapse4 | header=Random Variables with Statistical Dependence
|submenu=
*[[/Two-Dimensional Random Variables/]]
*[[/Two-Dimensional Gaussian Random Variables/]]
*[[/Linear Combinations of Random Variables/]]
*[[/Auto-Correlation Function/]]
*[[/Power-Spectral Density/]]
*[[/Cross-Correlation Function and Cross Power-Spectral Density/]]
*[[/Generalization to N-Dimensional Random Variables/]]
}}
{{Collapse5 | header=Filtering of Stochastic Signals
|submenu=
*[[/Stochastic System Theory/]]
*[[/Digital Filters/]]
*[[/Creation of Predefined ACF Properties/]]
*[[/Matched Filter/]]
*[[/Wiener–Kolmogorow Filter/]]
}}
{{Collapsible-Fuß}}

===Exercises and multimedia===

{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Theory_of_Stochastic_Signals:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_videos_to_"Theory_of_Stochastic_Signals"|$\text{Learning videos}$]]

$(3)$    [[LNTwww:Applets_to_"Theory_of_Stochastic_Signals"|$\text{Applets}$]] }}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Theory_of_Stochastic_Signals"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:Imprint_for_the_book_"Stochastic_Signal_Theory"|$\text{Impressum}$]]}}
 

{{Display}}

Theory of Stochastic Signals

2023-03-30T18:59:18Z

Hwang:

===Brief summary===

{{BlaueBox|TEXT=This third book of our learning tutorial deals in detail with stochastic signals and their modelling. Knowledge of stochastic signal theory is an important prerequisite for understanding the following books, which focus on transmission aspects.
# Fundamentals and definitions of probability theory;  set-theoretic description;  Statistical dependence;  Markov chains.
# Properties of discrete-valued random variables and their computational generation.  Examples:  Binomial and Poisson distribution.  Moments calculation.
# Description of continuous-valued random variables:  Probability density function,  distribution function,  moment calculation.  special distributions.
# Two- and multi-dimensional random variables:  Autocorrelation function,  power density spectrum, '''KORREKTUR: power-spectral density'''  correlation coefficient,  cross-correlation function.
# Filtering of stochastic signals   ⇒   »Stochastic System Theory";  digital filters;  properties of matched filter and Wiener–Kolmogorov–filter.

Knowledge of the first two  $\text{LNTwww}$-books,  which describe the  [[Signal Representation|»representation of deterministic signals«]]  as well as the  [[Linear_and_Time_Invariant_Systems|"description of linear and time-invariant systems»]],  are helpful for the understanding of the present book,  but not required.

⇒   First a  »'''content overview'''«  on the basis of the  »'''five main chapters'''«  with a total of  »'''28 individual chapters'''«  and  »'''166 sections'''«:}}

===Content===
{{Collapsible-Kopf}}
{{Collapse1| header=Probability Calculation
| submenu=
*[[/Some Basic Definitions/]]
*[[/Set Theory Basics/]]
*[[/Statistical Dependence and Independence/]]
*[[/Markov Chains/]]
}}
{{Collapse2 | header=Discrete Random Variables
|submenu=
*[[/From Random Experiment to Random Variable/]]
*[[/Moments of a Discrete Random Variable/]]
*[[/Binomial Distribution/]]
*[[/Poisson Distribution/]]
*[[/Generation of Discrete Random Variables/]]
}}
{{Collapse3 | header=Continuous Random Variables
|submenu=
*[[/Probability Density Function/]]
*[[/Cumulative Distribution Function/]]
*[[/Expected Values and Moments/]]
*[[/Uniformly Distributed Random Variables/]]
*[[/Gaussian Distributed Random Variables/]]
*[[/Exponentially Distributed Random Variables/]]
*[[/Further Distributions/]]
}}
{{Collapse4 | header=Random Variables with Statistical Dependence
|submenu=
*[[/Two-Dimensional Random Variables/]]
*[[/Two-Dimensional Gaussian Random Variables/]]
*[[/Linear Combinations of Random Variables/]]
*[[/Auto-Correlation Function/]]
*[[/Power-Spectral Density/]]
*[[/Cross-Correlation Function and Cross Power-Spectral Density/]]
*[[/Generalization to N-Dimensional Random Variables/]]
}}
{{Collapse5 | header=Filtering of Stochastic Signals
|submenu=
*[[/Stochastic System Theory/]]
*[[/Digital Filters/]]
*[[/Creation of Predefined ACF Properties/]]
*[[/Matched Filter/]]
*[[/Wiener–Kolmogorow Filter/]]
}}
{{Collapsible-Fuß}}

===Exercises and multimedia===

{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Theory_of_Stochastic_Signals:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_videos_to_"Theory_of_Stochastic_Signals"|$\text{Learning videos}$]]

$(3)$    [[LNTwww:Applets_to_"Theory_of_Stochastic_Signals"|$\text{Applets}$]] }}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Theory_of_Stochastic_Signals"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:Imprint_for_the_book_"Stochastic_Signal_Theory"|$\text{Impressum}$]]}}
 

{{Display}}

Linear and Time Invariant Systems

2023-03-30T18:54:53Z

Hwang:

===Brief summary===

{{BlaueBox|TEXT=Based on the book  [[Signaldarstellung|»Signal Representation«]],  here it is described how to mathematically capture the influence of a filter on deterministic signals.
# System theory analyses a quadripole  $($»system«$)$  using  "cause"   ⇒   $[$input   $ X(f)\bullet\! \!-\!-\!\!\circ\, x( t )]$  and  »effect»   ⇒   $[$output  $ Y(f)\bullet\!-\!-\!\circ\, y( t )]$.
# Indicator in frequency domain is the  »frequency response«  $ H(f)=Y(f)/X(f)$,  in time domain the  »impulse response»  $ h(t)$,  where  $ y(t)=x(t)\star h(t)$.
# System distortions   ⇒   $ y(t)\ne K \cdot x(t - \tau)$;  distortion-free system:  output and input differ by attenuation/gain and delay time.
# Linear distortions   ⇒   $ Y(f)=X(f)\cdot H(f)$  $($possibly reversible$)$;  non-linear distortions   ⇒   emergence of new frequencies  $($irreversible distortuions '''KORREKTUR: distortions'''$)$.
# Peculiarities of causal systems   ⇒   $ h(t<0)\equiv 0$;  Hilbert transform,  Laplace transform; inverse Laplace transform   ⇒   residue theorem.
#Some results of line theory '''KORREKTUR: line transmission theory''';  coaxial cable systems   ⇒   "white noise";  copper twisted pairs   ⇒   dominant is  "near-end crosstalk".

The filter influence on a random signal is only dealt with in the last chapter of the book  [[Theory_of_Stochastic_Signals|»Theory of Stochastic Signals»]].

⇒   First a  »'''content overview'''«  on the basis of the  »'''four main chapters'''«  with a total of  »'''twelve individual chapters'''«  and  »'''93 sections'''«.}}

===Contents===
{{Collapsible-Kopf}}
{{Collapse1| header= Basics of System Theory
| submenu=
*[[/System Description in Frequency Domain/]]
*[[/System Description in Time Domain/]]
*[[/Some Low-Pass Functions in Systems Theory/]]
}}
{{Collapse2 | header=Signal Distortion and Equalization
|submenu=
*[[/Classification of the Distortions/]]
*[[/Nonlinear Distortions/]]
*[[/Linear Distortions/]]
}}
{{Collapse3 | header=Description of Causal Realizable Systems
|submenu=
*[[/Conclusions from the Allocation Theorem/]]
*[[/Laplace Transform and p-Transfer Function/]]
*[[/Inverse Laplace Transform/]]
}}
{{Collapse4 | header=Properties of Electrical Cables
|submenu=
*[[/Some Results from Line Transmission Theory/]]
*[[/Properties of Coaxial Cables/]]
*[[/Properties of Balanced Copper Pairs/]]
}}
{{Collapsible-Fuß}}

===Exercises and multimedia===

{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Linear_and_Time-Invariant_Systems:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_Videos_to_Linear_and_Time_Invariant_Systems|$\text{Learning videos}$]]

$(3)$    [[LNTwww:LNTwww:Applets_to_"Linear_and_Time_Invariant_Systems"|$\text{Applets}$]]}}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Linear_and_Time_Invariant_Systems"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:Imprint_for_the_book_"Linear_and_Time_Invariant_Systems"|$\text{Impressum}$]] }}
 

{{Display}}

Linear and Time Invariant Systems

2023-03-30T18:52:01Z

Hwang:

===Brief summary===

{{BlaueBox|TEXT=Based on the book  [[Signaldarstellung|»Signal Representation«]],  here it is described how to mathematically capture the influence of a filter on deterministic signals.
# System theory analyses a quadripole  $($»system«$)$  using  "cause"   ⇒   $[$input   $ X(f)\bullet\! \!-\!-\!\!\circ\, x( t )]$  and  »effect»   ⇒   $[$output  $ Y(f)\bullet\!-\!-\!\circ\, y( t )]$.
# Indicator in frequency domain is the  »frequency response«  $ H(f)=Y(f)/X(f)$,  in time domain the  »impulse response»  $ h(t)$,  where  $ y(t)=x(t)\star h(t)$.
# System distortions   ⇒   $ y(t)\ne K \cdot x(t - \tau)$;  distortion-free system:  output and input differ by attenuation/gain and delay time.
# Linear distortions   ⇒   $ Y(f)=X(f)\cdot H(f)$  $($possibly reversible$)$;  non-linear distortions   ⇒   emergence of new frequencies  $($irreversible distortuions '''KORREKTUR: distortions'''$)$.
# Peculiarities of causal systems   ⇒   $ h(t<0)\equiv 0$;  Hilbert transform,  Laplace transform; inverse Laplace transform   ⇒   residue theorem.
#Some results of line theory;  coaxial cable systems   ⇒   "white noise";  copper twisted pairs   ⇒   dominant is  "near-end crosstalk".

The filter influence on a random signal is only dealt with in the last chapter of the book  [[Theory_of_Stochastic_Signals|»Theory of Stochastic Signals»]].

⇒   First a  »'''content overview'''«  on the basis of the  »'''four main chapters'''«  with a total of  »'''twelve individual chapters'''«  and  »'''93 sections'''«.}}

===Contents===
{{Collapsible-Kopf}}
{{Collapse1| header= Basics of System Theory
| submenu=
*[[/System Description in Frequency Domain/]]
*[[/System Description in Time Domain/]]
*[[/Some Low-Pass Functions in Systems Theory/]]
}}
{{Collapse2 | header=Signal Distortion and Equalization
|submenu=
*[[/Classification of the Distortions/]]
*[[/Nonlinear Distortions/]]
*[[/Linear Distortions/]]
}}
{{Collapse3 | header=Description of Causal Realizable Systems
|submenu=
*[[/Conclusions from the Allocation Theorem/]]
*[[/Laplace Transform and p-Transfer Function/]]
*[[/Inverse Laplace Transform/]]
}}
{{Collapse4 | header=Properties of Electrical Cables
|submenu=
*[[/Some Results from Line Transmission Theory/]]
*[[/Properties of Coaxial Cables/]]
*[[/Properties of Balanced Copper Pairs/]]
}}
{{Collapsible-Fuß}}

===Exercises and multimedia===

{{BlaueBox|TEXT=
In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    [https://en.lntwww.de/Category:Linear_and_Time-Invariant_Systems:_Exercises $\text{Exercises}$]

$(2)$    [[LNTwww:Learning_Videos_to_Linear_and_Time_Invariant_Systems|$\text{Learning videos}$]]

$(3)$    [[LNTwww:LNTwww:Applets_to_"Linear_and_Time_Invariant_Systems"|$\text{Applets}$]]}}

===Further links===

{{BlaueBox|TEXT=
$(4)$    [[LNTwww:Bibliography_to_"Linear_and_Time_Invariant_Systems"|$\text{Bibliography}$]]

$(5)$    [[LNTwww:Imprint_for_the_book_"Linear_and_Time_Invariant_Systems"|$\text{Impressum}$]] }}
 

{{Display}}

Examples of Communication Systems/Methods to Reduce the Bit Error Rate in DSL

2023-03-21T20:17:39Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=xDSL as Transmission Technology
|Nächste Seite=General Description of GSM
}}

==Transmission properties of copper cables ==
 
As already mentioned in the chapter  [[Examples_of_Communication_Systems/General_Description_of_DSL|"General Description of DSL"]] , the telephone line network of Deutsche Telekom mainly uses balanced copper pairs with a diameter of  $\text{0.4 mm}$.  The  "last mile"  is divided into three segments:
*the main cable,

*the branch cable,

*the house connection cable.

On average,  the line length is less than four kilometers.  In cities,  the copper line is shorter than  $\text{2.8 km}$  in  $90\%$  of all cases.

[[File:EN_LZI_T_4_3_S2_neu.png| right|frame|Structure of the local loop area]]

The $\rm xDSL$ variants discussed here were developed specifically for use on such symmetrical balanced copper pairs in the cable network. In order to better understand the technical requirements for the xDSL systems, a closer look must be taken at the transmission characteristics and interference on the conductor pairs.

This topic has already been dealt with in detail in the fourth main chapter  "Properties of Electrical Lines"  of the book  "[[Linear_and_Time_Invariant_Systems]]"  and is therefore only briefly summarized here using the  [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory#Wave_impedance_and_reflections|"equivalent circuit diagram"]] :
*Line transmission properties are fully characterized by the generally complex
:*  "characteristic impedance"  $Z_{\rm W}(f)$  and
:*  "complex propagation function per unit length"   ⇒   $γ(f)$.

*The even  "attenuation function $($per unit length$)$"  $α(f)$  is the real part of  $γ(f)$  and describes the attenuation of the wave propagating along the line:
:$$α(-f)=α(f) .$$

*The odd imaginary part  $β(f)$  of  $γ(f)$  is called  "phase function  $($per unit length$)$"  and gives the phase rotation of the signal wave along the line:
:$$β(-f)=-β(f) .$$

{{GraueBox|TEXT=
$\text{Example 1:}$  As an example,  we consider the function  $\alpha(f)$  shown on the right,  which is based on empirical investigations by  "Deutsche Telekom".
[[File:P_ID1955__Bei_T_2_4_S1b_v1.png|right|frame|Attenuation function per unit length of balanced copper pairs]]

The curves were obtained by averaging over a large number of measured lines of one kilometer length in the frequency range up to  $\text{30 MHz}$.  One can see:
# The attenuation function  $($per unit length$)$   $α(f)$  increases approximately proportionally with the square root of the frequency and decreases with increasing conductor diameter  $d$.
# The attenuation function  $a(f)$  increases linearly with cable length  $l$:
::$$a(f) = α(f) · l.$$

Note the difference between
*$a(f)$   speak "a"   $($for the attenuation function$)$,

*$\alpha(f)$   speak "a" $($for the attenuation function per unit length$)$.}}

For the line diameter  $\text{0.4 mm}$  was given in  [PW95]<ref name ='PW95'>Pollakowski, M.; Wellhausen, H.W.:  Properties of symmetrical local access cables in the frequency range up to 30 MHz.  Communication from the Research and Technology Center of Deutsche Telekom AG, Darmstadt, Verlag für Wissenschaft und Leben Georg Heidecker, 1995.</ref>  an empirical approximation formula for the attenuation function per unit length:

:$$\alpha(f) = \left [ 5.1 + 14.3 \cdot \left (\frac{f}{\rm 1\,MHz}\right )^{0.59} \right ] \frac{\rm dB}{\rm km}
\hspace{0.05cm}.$$

Evaluating this equation,  the following exemplary values hold:
*The attenuation function  $a(f)$  of a balanced copper wire of length  $l = 1 \ \rm km$  with diameter  $0.4 \ \rm mm$  is slightly more than  $60\ \rm dB$  for the signal frequency  $10\ \rm MHz$.

*At twice the frequency  $(20 \ \rm MHz)$  the attenuation value increases to over  $90 \ \rm dB$.  It can be seen that the attenuation does not increase exactly with the root of the frequency,  as would be the case if the skin effect were considered alone,  since several other effects also contribute to the attenuation.

*If the cable length is doubled to   $l = 2 \ \rm km$   the attenuation reaches a value of more than  $120 \ \rm dB$  $($at  $10 \ \rm MHz)$,  which corresponds to an amplitude attenuation factor smaller than  $10^{-6}$.

*Due to the frequency dependence of   $α(f)$  and  $β(f)$:   »'''intersymbol interference'''«  $\rm (ISI)$  as well as  »'''intercarrier interference'''  $\rm (ICI)$  occur. 

*Suitable equalization must therefore be provided for xDSL.

Note:
#In the  [[Linear_and_Time_Invariant_Systems/Properties_of_Balanced_Copper_Pairs|"Properties of balanced copper pairs"]]  chapter of the book  "Linear Time-Invariant Systems"  this topic is treated in detail.
#We refer here to the interactive applet  [[Applets:Dämpfung_von_Kupferkabeln|"Attenuation of copper cables"]].

==Disturbances during transmission==
 
Every transmission system is affected by disturbances,  which usually results primarily from thermal resistance noise.  In addition,  for a two-wire line there are:
*»'''Reflections'''«:   The counter-propagating wave increases the attenuation of copper pairs,  which is taken into account in the  [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory#Influence_of_reflections_-_operational_attenuation|"operational attenuation"]]  of the line.  To prevent such reflection,  the terminating resistor  $Z_{\rm E}(f)$  would have to be chosen identical to the  $($complex and frequency-dependent$)$  characteristic impedance  $Z_{\rm W}(f)$.  This is difficult in practice.  Therefore,  the terminating resistors are chosen to be real and constant,  and the resulting reflections are combated by technical means  – if possible.

[[File:EN_LZI_T_4_3_S6_v2.png|right|frame|On the emergence of crosstalk]]

*»'''Crosstalk'''«:   This is dominant interference in conducted transmission.  Crosstalk occurs when inductive and capacitive couplings between adjacent cores of a cable bundle cause mutual interference during signal transmission.

:Crosstalk is divided into two types (see graphic):

:*'''Near-end Crosstalk'''  $\rm (NEXT)$:  The interfering transmitter and the interfered receiver are on the same side of the cable.

:*'''Far-end Crosstalk'''  $\rm (FEXT)$:  The interfering transmitter and the interfered receiver are on opposite sides of the cable.

:Far-end crosstalk decreases sharply with increasing cable length due to attenuation,  so that near-end crosstalk is dominant even with DSL.
 
{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
To summarize:
#As frequency increases and spacing between line pairs decreases  – as within a star quad –  near-end crosstalk increases.  It is less critical if the conductors are in different basic bundles. 
#Depending on the stranding technique used,  the shielding and the manufacturing accuracy of the cable,  this effect occurs to varying degrees.  The cable length,  on the other hand,  does not play a role in near-end crosstalk:   The own transmitter is not attenuated by the cable. 
#Crosstalk can be significantly reduced by clever assignment,  for example by assigning different services to adjacent pairs,  using different frequency bands with as little overlap as possible.}}

==Signal–to–noise ratio, range and transmission rate ==
 
To evaluate the quality of a transmission system,  the signal-to-noise ratio  $\rm (SNR)$  is usually used.  This is also a measure of the expected bit error rate  $\rm (BER)$.
*Signal and noise in the same frequency band reduce the SNR and lead to a higher bit error rate or  – for a given bit error rate –  to a lower transmission bit rate.

*The relationships between transmit power,  channel quality  $($cable attenuation and noise power$)$  and achievable transmission rate can be illustrated very well by Shannon's channel capacity formula:

:$$C \left [ \frac{\rm bit}{\rm symbol} \right ] = \frac {1}{2} \cdot \log_2 \left ( 1 + \frac{P_{\rm E}}{P_{\rm N}} \right )=
\frac {1}{2} \cdot \log_2 \left ( 1 + \frac{\alpha_{\rm K}^2 \cdot P_{\rm S}}{P_{\rm N}} \right ) \hspace{0.05cm}.$$

The  »'''channel capacity'''«  $C$  denotes the maximum transmission bit rate at which transmission is possible under ideal conditions  $($among others,  the best possible coding with infinite block length$)$   ⇒   »'''channel coding theorem'''». For more details,  see the fourth main chapter  [[Information_Theory/AWGN_Channel_Capacity_for_Continuous-Valued_Input|"AWGN Channel Capacity for Continuous-Valued Input"]].

We assume that the bandwidth is fixed by the xDSL variant and that near-end crosstalk is the dominant interference.  Then the transmission rate can be improved by the following measures:
#For a given transmitted power  $P_{\rm S}$  and a given medium  $($e.g. balanced copper pairs with 0.4 mm diameter$)$,  the received power  $P_{\rm E}$  $($that can be used for demodulation$)$  is increased only by a shorter line length.
#One reduces the interference power  $P_{\rm N}$,  which for a given bandwidth  $B$  would be achieved by increased crosstalk attenuation,  which in turn also depends on the transmission method on the adjacent line pairs.
#Increasing the transmitted power  $P_{\rm S}$  would not be effective here,  since a larger transmitted power would at the same time have an unfavorable effect on the crosstalk.  This measure would only be successful for an AWGN channel  $($example:  coaxial cable$)$.

This listing shows that with xDSL there is a direct correlation between
*line length,

*transmission rate,  and

*the transmission method used.

{{GraueBox|TEXT=
$\text{Example 2:}$  From this graph, which refers to measurements with  "$\rm 1-DA xDSL$"  methods and  $\text{0.4 mm}$  copper cables in test systems with realistic interference conditions,  one can clearly see these dependencies.
[[File:EN_Bei_T_2_4_S3b.png|right|frame|Range and total bit rate for ADSL and VDSL]]

The graph shows for some ADSL and VDSL variants
*the range  $($maximum cable length$)$  $l_{\rm max}$  and

*the total transmission rate  $R_{\rm total}$ 
#of upstream $($first indication$)$
# and downstream $($second indication$)$.

The total transmission rate for the systems considered is between  $2.2 \ \rm Mbit/s$  and  $53\ \rm Mbit/s$.

*The trend of the measured values is shown in this graph as a solid  $($blue$)$  curve and can be formulated as a rough approximation as follows:

:$$l_{\rm max}\,{\rm \big [in}\,\,{\rm km \big ] } = \frac {20}{4 + R_{\rm total}\,{\rm \big [in}\,\,{\rm Mbit/s \big ] } } \hspace{0.05cm}.$$

*It can be seen that the range of all current systems  $($approximately between half a kilometer and three and a half kilometers of line length$)$  differs from this rule of thumb by a maximum of  $±25\%$  $($dashed curves$)$.}}

{{GraueBox|TEXT=
$\text{Example 3:}$ 
The diagram below shows the total data bit rates of  "ADSL2+"  and  "VDSL(2)"  as a function of line length,
[[File:EN_Bei_T_2_4_S3b_neu.png|right|frame|Data bit rates vs. cable lengths for xDSL systems]]
*with the  $($different$)$  red curves referring to the downstream

*and the two blue curves to the upstream.

This is based on a worst-case interference scenario with the following boundary conditions:
*Cable bundle with  $50$  copper pairs  $(0.4$  mm diameter$)$,  PE insulated,

*target symbol error rate  $p_{\rm S}=10^{-7},\ 6 \ \text{dB}$  margin $($reserve SNR to reach target data rate$)$,

*simultaneous operation of the following transmission methods:
#     $25$  times  "ADSL2+ over ISDN",
#    $14$  times ISDN,  four times  "SHDSL"  $(R= \text{1 Mbit/s)}$,
#    five times each  "SHDS"L  $(R= \text{2 Mbit/s)}$  and  "$(\text{VDSL2 band plan 998}$",
#    twice  "HDSL".

You can see from this diagram:
*For short line lengths,  the achievable data rates for VDSL(2) are significantly higher than for ADSL2+.

*From a line length of  $\approx 1800$  meters,  ADSL2+ is significantly better than VDSL(2).

*This is due to the fact that VDSL(2) operates in the lower frequency bands with significantly lower power in order to interfere less with neighboring systems.

*As the line length increases,  the higher frequency subchannels become unusable due to increasing attenuation,  which explains the crash in data rate.}}

==Overview of DSL error correction measures==
 
In order to reduce the BER of xDSL systems,  a number of techniques have been cleverly combined in the specifications to counteract the two most common causes of errors:
*Transmission errors due to pulse and crosstalk interference on the line:   Especially at high data rates,  adjacent symbols in the QAM signal space are close together,  which significantly increases the bit error probability.

*Cutting off of signal peaks due to lack of dynamic range of the transmitter amplifiers:  This  "clipping"  also corresponds to pulse noise and acts as an additional colored noise that noticeably degrades the SNR.

With the DMT method,  two paths are implemented for error correction in the signal processors.  The bit assignment to these paths is done by a multiplexer with sync control.
[[File:EN_Bei_T_2_4_S4_v9.png|right|frame|Complete DSL/DMT system]]

*In the case of  »'''fast path'''«,  low waiting times  $($"latency"$)$  are used.

*With  »'''interleaved path'''«,  low bit error rates are in the foreground.  Here the latency is higher due to the use of an interleaver.

*"Dual latency"  means the simultaneous use of both paths.  The  "ADSL Transceiver Units"  must support dual latency at least in the downstream.

The remaining chapter sections discuss error protection procedures for both paths. 

For other modulation methods,  the error protection measures described here are the same in principle,  but different in detail.

#The transmission chain starts with the  "cyclic redundancy check"  $\rm (CRC)$,  which forms a checksum over an overframe that is evaluated at the receiver.
#Task of the scrambler is to convert long sequences of  "ones"  and  "zeros"  to produce more frequent signal changes.
#This is followed by the "forward error correction"  $\rm (FEC)$  to detect/correct byte errors at the receiving end.  Often used for xDSL:  Reed-Solomon and Trellis coding.
#Task of the  "interleaver"  is to distribute the received code words over a larger time range in order to distribute transmission errors over several code words.
#After passing through the individual bit protection procedures,  the data streams from fast and interleaved paths are combined and processed in  "tone ordering".
#In addition,  a guard interval and cyclic prefix are inserted in the DMT transmitter after the IDFT,  which is removed again in the DMT receiver.
#This represents a very simple realization of signal equalization in the frequency domain when the channel is distorted.

==Cyclic redundancy check==
 
The  "cyclic redundancy check"  $\rm (CRC)$  is a simple bit-level procedure to check the integrity of data during transmission or duplication.  The CRC principle has already been described in detail in the  [[Examples_of_Communication_Systems/ISDN_Primary_Multiplex_Connection#Frame_synchronization|"ISDN chapter"]].  Here follows a brief summary,  using the nomenclature used in the xDSL specifications:
*For each data block  $D(x)$  with  $k$  bit  $(d_0$, ... , $d_{k-1})$,  a parity-check value  $C(x)$  with eight bits is formed prior to data transmission and appended to the original data sequence  $($the variable  $x$  denotes here the delay operator$)$.

*$C(x)$  is obtained as the division remainder of the modulo-2 polynomial division of  $D(x)$  by the given parity-check polynomial  $G(x)$:
:$$D(x) = d_0 \cdot x^{k-1} + d_1 \cdot x^{k-2} + ... + d_{k-2} \cdot x + d_{k-1}\hspace{0.05cm},$$
:$$G(x) = x^8 + x^4 + x^3 + x^2 + 1 \hspace{0.05cm},$$
:$$C(x) = D(x) \cdot x^8 \,\,{\rm mod }\,\, G(x) = c_0 \cdot x^7 + c_1 \cdot x^6 + \text{...} + c_6 \cdot x + c_7
\hspace{0.05cm}.$$

*Another CRC value is formed at the receiver using the same procedure and compared with the transmitted CRC value.  If both are different,  at least one bit error happened during transmission.

*By this way,  bit errors can be detected if they are not too much clustered.  In ADSL practice,  the CRC procedure is sufficient for bit error detection.

The graph shows an exemplary circuit for the CRC value generation with the generator polynomial  $G(x)$  specified for ADSL – realizable in hardware or software:
[[File:EN_Bei_T_2_4_S5neu.png|right|frame|Cyclic Redundancy Check for ADSL]]

#The data block  $D(x)$  to be tested is introduced into the circuit from the left,  the output is fed back and exclusively-or-linked to the digits of the generator polynomial  $G(x)$.
#After passing through the entire data block,  the memory elements contain the CRC parity-check value  $C(x)$.
#It should be noted that with ADSL the data is split into so-called  "superframes"  of 68 frames each.
#Each frame contains data from the  "fast path"  and the  "interleaved path".  In addition,  management and synchronization bits are transmitted in specific frames.
#Eight CRC bits are formed per ADSL superframe and per path and are transmitted as  "fast byte"  resp.  "sync byte" as the first byte of frame  "$0$"  of the next superframe.

==Scrambler and de–scrambler==
 
Task of the scrambler is to convert long sequences of  "ones"  and  "zeros"  in such a way that frequent symbol changes occur.
*A possible realization is a shift register circuit with feedback exclusive-or-linked branches.

*In order to produce the original binary sequence at the receiver,  a mirror-image self-synchronizing  "de-scrambler"  must be used there.

The graph shows on the left an example of a scrambler actually used at DSL with  $23$  memory elements.  The corresponding de-scrambler is shown on the right.

[[File:EN_Bei_T_2_4_S6.png|right|frame|Scrambler and de-scrambler in a DSL/DMT system     $(1)$   scrambler's input        $(3)$   de-scrambler's input     $(2)$   scrambler's output     $(4)$   de-scrambler's output]]

The transmitter-side shift register is loaded with an arbitrary initial value that has no further effect on the operation of the circuit. Here:
:$$11001'10011'00110'01100'110.$$

If we denote
* by  $e_n$  the bits of the binary input sequence,  and

*by  $a_n$  the bits at the output, 

the following relation holds:

:$$a_n = e_n \oplus a_{n- 18}\oplus a_{n- 23}\hspace{0.05cm}.$$

In the example,  the scrambler input sequence consists of  80  consecutive  "ones"  $($upper left gray background$)$,  which are shifted bit by bit into the scrambler.  The output bit sequence then has frequent  "one-zero"  changes,  as desired.

The de-scrambler  $($shown on the right$)$  can be started at any time with any starting value,  which means that no synchronization is required between the two circuits.  Here:
:$$10111'011110'11101'11011'101.$$
The de-scrambler output data stream shows,
*that the de-scrambler initially outputs some  $($up to a maximum of $23)$  erroneous bits,  but then

*synchronizes automatically,  and then

*recovers the original bit sequence  $($only  "ones"$)$  without errors.

==Forward error correction==
 
For  "forward error correction"  $\rm (FEC)$,  all xDSL variants use a  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes|"Reed-Solomon-Code"]].  In some systems  "trellis code modulation"  $\rm (TCM)$  has been made mandatory as an additional error protection measure,  even though it has only been specified as  "optional"  by the international bodies.

Both methods are discussed in detail in the book  [[Channel_Coding|"Channel Coding"]].  Here follows a brief summary of Reed-Solomon coding with respect to its application to DSL:
*With Reed-Solomon encoding,  redundancy bytes are generated for fixed agreed interpolation points of the payload polynomial.  With systematic Reed-Solomon encoding,  a parity-check value is calculated similar to the CRC procedure and appended to the data block to be protected.

*However,  the data is no longer processed  "bit by bit",  but  "byte by byte".  Consequently,  arithmetic operations are no longer performed in the Galois field  $\rm GF( 2 )$  but in  $\rm GF(2^8)$.

The Reed-Solomon parity-check byte can also be determined as the division remainder of a polynomial division,  for xDSL with the following parameters:
#Number  $S$  of DMT symbols to be monitored per Reed-Solomon code word  $(S \ge 1$  for the fast buffer,  $S =2^0$, ... , $2^4$  for the interleaved buffer$)$,
#number  $K$  of user data bytes in the  $S$  DMT symbols,  defined as a polynomial  $B(x)$  of degree  $K$,  where the  "B"  indicates  "bytes",
#Number  $R$  of Reed-Solomon parity-check bytes  $($even number between  $2$  to  $16)$  per parity-check value  $($"fast"  or  "interleaved"$)$,
#sum  $N = K + R$  of the user data bytes and check bytes of the Reed-Solomon code word.

The specifics of Reed-Solomon encoding for xDSL are given here without further comment:
*For xDSL,  the number  $R$  of check bytes must be an integer multiple of the number  $S$  of symbols so that they can be evenly distributed in the payload polynomial.

*The so-called  [https://en.wikipedia.org/wiki/Singleton_bound#MDS_codes "Maximum Distance Separable  $\rm (MDS)$  codes"]  – a subclass of Reed-Solomon codes –  allow the correction of  $R/2$  falsified user data bytes.

*From the selected Reed-Solomon code for the DMT systems,  the constraint is a maximum code word length of  $2^8-1 = 255$  bytes corresponding to  $2040$  bits.

*The redundancy of Reed-Solomon codes can generate a considerable amount of data if the parameters are unfavorable,  thus considerably reducing the net data rate.

*It is recommended that the data amount  $($"gross data rate"$)$  be divided judiciously into useful data  $($"net data rate,  payload"$)$  and error protection data  $($"overhead"$)$.

*Reed-Solomon coding achieves a  "high coding gain".  A system without coding would have to have a signal-to-noise ratio  $\rm (SNR)$  larger by  $3 \ \rm dB$  for same bit error rate.

*By  "trellis-encoded modulation"  $\rm (TCM)$  in combination with other error protection measures,  the coding gain is highly variable; it ranges between  $0 \ \rm dB$  and  $6 \ \rm dB$.

==Interleaving and de–interleaving==
 
The common task of  "interleaver"  $($at the transmitter$)$  and  "de-interleaver"  $($at the receiver$)$  is
*to spread the received Reed-Solomon code words over a larger time range

*in order to distribute any transmission errors over several code words

*and thus increase the chance of correct decoding.

xDSL interleaving is characterized by the parameter  $D$  $($"depth"$)$,  which can take values between  $2^0$  and  $2^9$ .

{{GraueBox|TEXT=
$\text{Example 4:}$  The graph illustrates the principle using the Reed-Solomon code words  $A$,  $B$,  $C$  with five bytes each and the interleaver depth  $D = 2$.
[[File:EN_Bei_T_2_4_S8a.png|right|frame|For DSL interleaving with  $D = 2$]]

Each byte  $B_i$  of the middle Reed-Solomon code word  $B$  is delayed by  $V_i = (D - 1) \cdot i$  bytes. 

Two interleaver blocks are formed:
*The first block contains the bytes  $B_0$,  $B_1$  and  $B_2$  together with the bytes  $A_3$  and  $A_4$  of the previous code word.

*The second block contains the bytes  $B_3$  and  $B_4$  along with the bytes  $C_0$,  $C_1$  and  $C_2$  of the following code word.}}

This  "scrambling"  has the following advantages  $($provided,  $D$  is sufficiently large$)$:
#The error correction capabilities of the Reed-Solomon code are improved.
#The user data rate remains the same,  i.e. is not reduced $($redundancy-free$)$.
#In the event of errors,  entire packets do not have to be repeated at the protocol level.

A disadvantage is that with increasing interleaver depth  $D$  there can be noticeable delay times  $($on the order of milliseconds$)$,  which causes major problems for real-time applications.  However,  interleaving with low depth is only useful if the signal-to-noise ratio is sufficiently high.

{{GraueBox|TEXT=
$\text{Example 5:}$ 
An example of the advantages of interleaver/de-interleaver in the presence of bundle errors is shown in the following graph:
[[File:EN_Bei_T_2_4_S8b.png|right|frame|DSL interleaving and de–interleaving with  $D = 3$]]

*In the first row,  the transmitted byte sequence is shown according to Reed-Solomon encoding,  with each code word consisting of seven bytes as an example.

*In the middle row,  the data bytes are shifted by interleaving with  $D = 3$  so that between  $C_i$  and  $C_{i+1}$  there are two foreign bytes and the green code word is distributed over three blocks.

*Now suppose that during transmission a pulse glitch corrupted '''KORREKTUR: falsified''' three consecutive bytes in a single data block.

*After the de-interleaver,  the original byte sequence of the Reed-Solomon code words is restored,  with the three corrupted '''KORREKTUR: falsified''' bytes distributed among three independent code words.

*If two redundancy bytes were inserted in each case during the Reed-Solomon encoding,  the now separated byte corruptions '''KORREKTUR: falsifications''' can be completely corrected.}}

==Gain scaling and tone ordering==
 
A particularly advantageous feature of DMT is the possibility
#to adjust the bins individually to the existing channel characteristics and
#possibly to switch off "bins" with unfavorable SNR completely.

[[File:EN_Bei_T_2_4_S9.png|right|frame|Bit-bin assignment based on SNR]]
The procedure is as follows:

*Before starting the transmission  – and possibly also dynamically during operation –  the DMT modem measures the channel characteristics for each  "bin"  and sets the maximum transmission rate individually according to the SNR  $($see graphic$)$.

*During initialization,  the  "ADSL Transceiver Units"  exchange bin information,  for example the respective  "bits/bin"  and the required transmission power  $($'"gain"$)$.

*Thereby the  $\text{ATU-C}$  sends information about the upstream and the  $\text{ATU-R}$  sends information about the downstream.

*This message is of the format  $\{b_i, g_i\}$  where  $b_i$  $($four bits$)$  indicates the constellation size.  For the upstream,  the index  $i = 1$, ... , $31$  and for the downstream  $i = 1$, ... , $255$.

*The gain  $g_i$  is a fixed-point number with twelve bits.  For example  $g_i = 001.010000000$  represents the decimal value  $1 + 1/4 =1.25$.

*This indicates that the signal power of channel  $i$  must be higher by  $1.94 \ \rm dB$  than the power of the test signal transmitted during the channel analysis.
 
When operating the fast path and the interleaved path simultaneously $($see  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Overview_of_DSL_error_correction_measures|"graphic"]]  in the section  "DSL error correction measures"$)$,  the bit error rate can be further reduced by optimized carrier frequency allocation  $($"Tone Ordering"$)$.  The background of this measure is again  "clipping"  $($truncation of voltage peaks$)$,  which worsens the overall SNR. This procedure is based on the following rules:
*Bins with dense constellation  $($many bits/bin   ⇒   larger clipping probability$)$  are assigned to the interleaved branch,  since this is per se more reliable due to the additional interleaver.  Accordingly,  the subchannels with low order allocation  $($few bits/bin$)$  are reserved for the fast data buffer.

*New tables are then sent for upstream and downstream,  in which the bins are no longer ordered by index,  but according to the bits/bin ratios.  Based on this new table,  it is possible for the  $\text{ATU-C}$  or  $\text{ATU-R}$  to perform bit extraction successfully.

==Inserting guard interval and cyclic prefix ==
 
In the chapter  [[Modulation_Methods/Implementation_of_OFDM_Systems#Guard_interval_to_reduce_intersymbol_interference| "Realization of OFDM systems"]]  of the book "Modulation Methods" it has already been shown that by inserting a guard interval. The bit error rate can be decisively improved in the presence of linear channel distortion.

We assume that the cable impulse response  $h_{\rm K}(t)$  extends over the time duration  $T_{\rm K}$  . Ideally  $h_{\rm K}(t) = δ(t)$  and accordingly an infinitely short extension:   $T_{\rm K} = 0$. For distorting channel  $(T_{\rm K} > 0 )$  holds:
*By inserting a ''guard interval''  of duration  $T_{\rm G}$  ''intersymbol interference''  between each DSL frame can be avoided as long as  $T_{\rm G}$ ≥ $T_{\rm K}$  holds. However, this measure leads to a rate loss by a factor  $T/(T + T_{\rm G})$  with symbol duration  $T = {1}/{f_0}$.
*But with this, there is still ''inter-carrier interference''  between each subcarrier within the same frame, that is, the  [[Modulation_Methods/General_Description_of_OFDM#System_consideration_in_the_frequency_domain_with_causal_basic_pulse|"DMT individual spectra"]]  are no longer  $\rm si$-shaped '''KORREKTUR: sinc?''' and de-orthogonalization occurs.
*By a  [[Modulation_Methods/Implementation_of_OFDM_Systems#Cyclic_Prefix|"cyclic prefix"]]  also this disturbing effect can be avoided. Here one extends the transmission vector  $\mathbf{s}$  forward by the last  $L$  samples of the IDFT output, where the minimum value for $L$ is given by the duration  $T_{\rm K}$  of the cable impulse response.

{{GraueBox|TEXT=
$\text{Example 6:}$ 
The graphic shows this measure with the DSL/DMT method, for which the parameter  $L = 32$  has been set.

[[File:P_ID1964__Bei_T_2_4_S10a_v1.png|right|frame|DMT transmission signal with cyclic prefix]]
*The samples  $s_{480}$ , ... , $s_{511}$  are added as prefix  $(s_{-32}$ , ... , $s_{-1})$  to the IDFT output vector  $(s_0$ , ... , $s_{511})$  .
*The transmitted signal  $s(t)$  now has the resulting duration  $T ≈ 232 \ {\rm µ s}$  instead of the symbol duration  $T + T_{\rm G} = 1.0625 \cdot T ≈ 246 \ {\rm µ s}$. This reduces the rate by a factor of  $0.94$ .
*In the receive-side evaluation, one is restricted to the time range from  $0$  to  $T$. In this time interval the disturbing influence of the impulse response has already decayed and the subchannels are orthogonal to each other - just as with ideal channel.
*The sample values  $s_{-32}$ , ... , $s_{-1}$  are discarded at the receiver - a rather simple realization of signal equalization.}}

The last diagram in this chapter shows the entire DMT transmission system, but without the  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Overview_of_DSL_error_correction_measures| "error protection measures"]] described earlier. You can see:

[[File:P_ID1967__Bei_T_2_4_S10_v1.png|right|frame|DMT–System with cyclic prefix '''Korrektur''' das Bild muss ich noch machen]]
*In the "Add cyclic prefix" block, the samples  $s_{480}$, ... , $s_{511}$  as  $s_{-32}$, ... , $s_{-1}$  added. The transmit '''KORREKTUR: transmitted''' signal  $s(t)$  thus has the course shown in  $\text{Example 6}$  .
*The received signal  $r(t)$  results from the convolution of  $s(t)$  with  $h_{\rm K}(t)$. After A/D conversion and removal of the cyclic prefix, the input values  $r_0$, ... , $ r_{511}$  for the DFT.
*The (complex) output values  $D_k\hspace{0.01cm}'$  of the DFT depend only on the particular (complex) data value  $D_k$  . Independently of other data  $D_κ (κ ≠ k)$  holds with the noise value  $n_k\hspace{0.01cm}'$:

:$${D}_k\hspace{0.01cm}' = \alpha_k \cdot {D}_k + {n}_k\hspace{0.01cm}', \hspace{0.2cm}\alpha_k = H_{\rm K}( f = f_k)
\hspace{0.05cm}. $$

*Each carrier  $D_k$  is modified in its amplitude and phase by its own (complex) factor  $α_k$, which depends only on the channel. The frequency domain equalizer has only the task of multiplying the coefficient  $D_k\hspace{0.01cm}'$  by the inverse value  ${1}/{α_k}$  . Finally, one obtains:

:$$ \hat{D}_k = {D}_k + {n}_k \hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*This simple realization possibility of the complete equalization of the strongly distorting cable frequency response was one of the decisive criteria that for  $\rm xDSL$  the  $\rm DMT$ method has prevailed over  $\rm QAM$  and  $\rm CAP$ .
*Mostly an additional pre-equalization in the time domain takes place directly after the A/D conversion to avoid also the intersymbol interference between adjacent frames.}}

==Exercises for the chapter==
 
[[Exercise_2.5:_DSL_Error_Protection|Exercise 2.5: DSL Error Protection]]

[[Exercise_2.5Z:_Reach_and_Bit_Rate_with_ADSL|Exercise 2.5Z: Reach and Bit Rate with ADSL]]

[[Exercise_2.6:_Cyclic_Prefix|Exercise 2.6: Cyclic Prefix]]
==References==
<references />

{{Display}}

Examples of Communication Systems/Methods to Reduce the Bit Error Rate in DSL

2023-03-21T20:16:47Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=xDSL as Transmission Technology
|Nächste Seite=General Description of GSM
}}

==Transmission properties of copper cables ==
 
As already mentioned in the chapter  [[Examples_of_Communication_Systems/General_Description_of_DSL|"General Description of DSL"]] , the telephone line network of Deutsche Telekom mainly uses balanced copper pairs with a diameter of  $\text{0.4 mm}$.  The  "last mile"  is divided into three segments:
*the main cable,

*the branch cable,

*the house connection cable.

On average,  the line length is less than four kilometers.  In cities,  the copper line is shorter than  $\text{2.8 km}$  in  $90\%$  of all cases.

[[File:EN_LZI_T_4_3_S2_neu.png| right|frame|Structure of the local loop area]]

The $\rm xDSL$ variants discussed here were developed specifically for use on such symmetrical balanced copper pairs in the cable network. In order to better understand the technical requirements for the xDSL systems, a closer look must be taken at the transmission characteristics and interference on the conductor pairs.

This topic has already been dealt with in detail in the fourth main chapter  "Properties of Electrical Lines"  of the book  "[[Linear_and_Time_Invariant_Systems]]"  and is therefore only briefly summarized here using the  [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory#Wave_impedance_and_reflections|"equivalent circuit diagram"]] :
*Line transmission properties are fully characterized by the generally complex
:*  "characteristic impedance"  $Z_{\rm W}(f)$  and
:*  "complex propagation function per unit length"   ⇒   $γ(f)$.

*The even  "attenuation function $($per unit length$)$"  $α(f)$  is the real part of  $γ(f)$  and describes the attenuation of the wave propagating along the line:
:$$α(-f)=α(f) .$$

*The odd imaginary part  $β(f)$  of  $γ(f)$  is called  "phase function  $($per unit length$)$"  and gives the phase rotation of the signal wave along the line:
:$$β(-f)=-β(f) .$$

{{GraueBox|TEXT=
$\text{Example 1:}$  As an example,  we consider the function  $\alpha(f)$  shown on the right,  which is based on empirical investigations by  "Deutsche Telekom".
[[File:P_ID1955__Bei_T_2_4_S1b_v1.png|right|frame|Attenuation function per unit length of balanced copper pairs]]

The curves were obtained by averaging over a large number of measured lines of one kilometer length in the frequency range up to  $\text{30 MHz}$.  One can see:
# The attenuation function  $($per unit length$)$   $α(f)$  increases approximately proportionally with the square root of the frequency and decreases with increasing conductor diameter  $d$.
# The attenuation function  $a(f)$  increases linearly with cable length  $l$:
::$$a(f) = α(f) · l.$$

Note the difference between
*$a(f)$   speak "a"   $($for the attenuation function$)$,

*$\alpha(f)$   speak "a" $($for the attenuation function per unit length$)$.}}

For the line diameter  $\text{0.4 mm}$  was given in  [PW95]<ref name ='PW95'>Pollakowski, M.; Wellhausen, H.W.:  Properties of symmetrical local access cables in the frequency range up to 30 MHz.  Communication from the Research and Technology Center of Deutsche Telekom AG, Darmstadt, Verlag für Wissenschaft und Leben Georg Heidecker, 1995.</ref>  an empirical approximation formula for the attenuation function per unit length:

:$$\alpha(f) = \left [ 5.1 + 14.3 \cdot \left (\frac{f}{\rm 1\,MHz}\right )^{0.59} \right ] \frac{\rm dB}{\rm km}
\hspace{0.05cm}.$$

Evaluating this equation,  the following exemplary values hold:
*The attenuation function  $a(f)$  of a balanced copper wire of length  $l = 1 \ \rm km$  with diameter  $0.4 \ \rm mm$  is slightly more than  $60\ \rm dB$  for the signal frequency  $10\ \rm MHz$.

*At twice the frequency  $(20 \ \rm MHz)$  the attenuation value increases to over  $90 \ \rm dB$.  It can be seen that the attenuation does not increase exactly with the root of the frequency,  as would be the case if the skin effect were considered alone,  since several other effects also contribute to the attenuation.

*If the cable length is doubled to   $l = 2 \ \rm km$   the attenuation reaches a value of more than  $120 \ \rm dB$  $($at  $10 \ \rm MHz)$,  which corresponds to an amplitude attenuation factor smaller than  $10^{-6}$.

*Due to the frequency dependence of   $α(f)$  and  $β(f)$:   »'''intersymbol interference'''«  $\rm (ISI)$  as well as  »'''intercarrier interference'''  $\rm (ICI)$  occur. 

*Suitable equalization must therefore be provided for xDSL.

Note:
#In the  [[Linear_and_Time_Invariant_Systems/Properties_of_Balanced_Copper_Pairs|"Properties of balanced copper pairs"]]  chapter of the book  "Linear Time-Invariant Systems"  this topic is treated in detail.
#We refer here to the interactive applet  [[Applets:Dämpfung_von_Kupferkabeln|"Attenuation of copper cables"]].

==Disturbances during transmission==
 
Every transmission system is affected by disturbances,  which usually results primarily from thermal resistance noise.  In addition,  for a two-wire line there are:
*»'''Reflections'''«:   The counter-propagating wave increases the attenuation of copper pairs,  which is taken into account in the  [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory#Influence_of_reflections_-_operational_attenuation|"operational attenuation"]]  of the line.  To prevent such reflection,  the terminating resistor  $Z_{\rm E}(f)$  would have to be chosen identical to the  $($complex and frequency-dependent$)$  characteristic impedance  $Z_{\rm W}(f)$.  This is difficult in practice.  Therefore,  the terminating resistors are chosen to be real and constant,  and the resulting reflections are combated by technical means  – if possible.

[[File:EN_LZI_T_4_3_S6_v2.png|right|frame|On the emergence of crosstalk]]

*»'''Crosstalk'''«:   This is dominant interference in conducted transmission.  Crosstalk occurs when inductive and capacitive couplings between adjacent cores of a cable bundle cause mutual interference during signal transmission.

:Crosstalk is divided into two types (see graphic):

:*'''Near-end Crosstalk'''  $\rm (NEXT)$:  The interfering transmitter and the interfered receiver are on the same side of the cable.

:*'''Far-end Crosstalk'''  $\rm (FEXT)$:  The interfering transmitter and the interfered receiver are on opposite sides of the cable.

:Far-end crosstalk decreases sharply with increasing cable length due to attenuation,  so that near-end crosstalk is dominant even with DSL.
 
{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
To summarize:
#As frequency increases and spacing between line pairs decreases  – as within a star quad –  near-end crosstalk increases.  It is less critical if the conductors are in different basic bundles. 
#Depending on the stranding technique used,  the shielding and the manufacturing accuracy of the cable,  this effect occurs to varying degrees.  The cable length,  on the other hand,  does not play a role in near-end crosstalk:   The own transmitter is not attenuated by the cable. 
#Crosstalk can be significantly reduced by clever assignment,  for example by assigning different services to adjacent pairs,  using different frequency bands with as little overlap as possible.}}

==Signal–to–noise ratio, range and transmission rate ==
 
To evaluate the quality of a transmission system,  the signal-to-noise ratio  $\rm (SNR)$  is usually used.  This is also a measure of the expected bit error rate  $\rm (BER)$.
*Signal and noise in the same frequency band reduce the SNR and lead to a higher bit error rate or  – for a given bit error rate –  to a lower transmission bit rate.

*The relationships between transmit power,  channel quality  $($cable attenuation and noise power$)$  and achievable transmission rate can be illustrated very well by Shannon's channel capacity formula:

:$$C \left [ \frac{\rm bit}{\rm symbol} \right ] = \frac {1}{2} \cdot \log_2 \left ( 1 + \frac{P_{\rm E}}{P_{\rm N}} \right )=
\frac {1}{2} \cdot \log_2 \left ( 1 + \frac{\alpha_{\rm K}^2 \cdot P_{\rm S}}{P_{\rm N}} \right ) \hspace{0.05cm}.$$

The  »'''channel capacity'''«  $C$  denotes the maximum transmission bit rate at which transmission is possible under ideal conditions  $($among others,  the best possible coding with infinite block length$)$   ⇒   »'''channel coding theorem'''». For more details,  see the fourth main chapter  [[Information_Theory/AWGN_Channel_Capacity_for_Continuous-Valued_Input|"AWGN Channel Capacity for Continuous-Valued Input"]].

We assume that the bandwidth is fixed by the xDSL variant and that near-end crosstalk is the dominant interference.  Then the transmission rate can be improved by the following measures:
#For a given transmitted power  $P_{\rm S}$  and a given medium  $($e.g. balanced copper pairs with 0.4 mm diameter$)$,  the received power  $P_{\rm E}$  $($that can be used for demodulation$)$  is increased only by a shorter line length.
#One reduces the interference power  $P_{\rm N}$,  which for a given bandwidth  $B$  would be achieved by increased crosstalk attenuation,  which in turn also depends on the transmission method on the adjacent line pairs.
#Increasing the transmitted power  $P_{\rm S}$  would not be effective here,  since a larger transmitted power would at the same time have an unfavorable effect on the crosstalk.  This measure would only be successful for an AWGN channel  $($example:  coaxial cable$)$.

This listing shows that with xDSL there is a direct correlation between
*line length,

*transmission rate,  and

*the transmission method used.

{{GraueBox|TEXT=
$\text{Example 2:}$  From this graph, which refers to measurements with  "$\rm 1-DA xDSL$"  methods and  $\text{0.4 mm}$  copper cables in test systems with realistic interference conditions,  one can clearly see these dependencies.
[[File:EN_Bei_T_2_4_S3b.png|right|frame|Range and total bit rate for ADSL and VDSL]]

The graph shows for some ADSL and VDSL variants
*the range  $($maximum cable length$)$  $l_{\rm max}$  and

*the total transmission rate  $R_{\rm total}$ 
#of upstream $($first indication$)$
# and downstream $($second indication$)$.

The total transmission rate for the systems considered is between  $2.2 \ \rm Mbit/s$  and  $53\ \rm Mbit/s$.

*The trend of the measured values is shown in this graph as a solid  $($blue$)$  curve and can be formulated as a rough approximation as follows:

:$$l_{\rm max}\,{\rm \big [in}\,\,{\rm km \big ] } = \frac {20}{4 + R_{\rm total}\,{\rm \big [in}\,\,{\rm Mbit/s \big ] } } \hspace{0.05cm}.$$

*It can be seen that the range of all current systems  $($approximately between half a kilometer and three and a half kilometers of line length$)$  differs from this rule of thumb by a maximum of  $±25\%$  $($dashed curves$)$.}}

{{GraueBox|TEXT=
$\text{Example 3:}$ 
The diagram below shows the total data bit rates of  "ADSL2+"  and  "VDSL(2)"  as a function of line length,
[[File:EN_Bei_T_2_4_S3b_neu.png|right|frame|Data bit rates vs. cable lengths for xDSL systems]]
*with the  $($different$)$  red curves referring to the downstream

*and the two blue curves to the upstream.

This is based on a worst-case interference scenario with the following boundary conditions:
*Cable bundle with  $50$  copper pairs  $(0.4$  mm diameter$)$,  PE insulated,

*target symbol error rate  $p_{\rm S}=10^{-7},\ 6 \ \text{dB}$  margin $($reserve SNR to reach target data rate$)$,

*simultaneous operation of the following transmission methods:
#     $25$  times  "ADSL2+ over ISDN",
#    $14$  times ISDN,  four times  "SHDSL"  $(R= \text{1 Mbit/s)}$,
#    five times each  "SHDS"L  $(R= \text{2 Mbit/s)}$  and  "$(\text{VDSL2 band plan 998}$",
#    twice  "HDSL".

You can see from this diagram:
*For short line lengths,  the achievable data rates for VDSL(2) are significantly higher than for ADSL2+.

*From a line length of  $\approx 1800$  meters,  ADSL2+ is significantly better than VDSL(2).

*This is due to the fact that VDSL(2) operates in the lower frequency bands with significantly lower power in order to interfere less with neighboring systems.

*As the line length increases,  the higher frequency subchannels become unusable due to increasing attenuation,  which explains the crash in data rate.}}

==Overview of DSL error correction measures==
 
In order to reduce the BER of xDSL systems,  a number of techniques have been cleverly combined in the specifications to counteract the two most common causes of errors:
*Transmission errors due to pulse and crosstalk interference on the line:   Especially at high data rates,  adjacent symbols in the QAM signal space are close together,  which significantly increases the bit error probability.

*Cutting off of signal peaks due to lack of dynamic range of the transmitter amplifiers:  This  "clipping"  also corresponds to pulse noise and acts as an additional colored noise that noticeably degrades the SNR.

With the DMT method,  two paths are implemented for error correction in the signal processors.  The bit assignment to these paths is done by a multiplexer with sync control.
[[File:EN_Bei_T_2_4_S4_v9.png|right|frame|Complete DSL/DMT system]]

*In the case of  »'''fast path'''«,  low waiting times  $($"latency"$)$  are used.

*With  »'''interleaved path'''«,  low bit error rates are in the foreground.  Here the latency is higher due to the use of an interleaver.

*"Dual latency"  means the simultaneous use of both paths.  The  "ADSL Transceiver Units"  must support dual latency at least in the downstream.

The remaining chapter sections discuss error protection procedures for both paths. 

For other modulation methods,  the error protection measures described here are the same in principle,  but different in detail.

#The transmission chain starts with the  "cyclic redundancy check"  $\rm (CRC)$,  which forms a checksum over an overframe that is evaluated at the receiver.
#Task of the scrambler is to convert long sequences of  "ones"  and  "zeros"  to produce more frequent signal changes.
#This is followed by the "forward error correction"  $\rm (FEC)$  to detect/correct byte errors at the receiving end.  Often used for xDSL:  Reed-Solomon and Trellis coding.
#Task of the  "interleaver"  is to distribute the received code words over a larger time range in order to distribute transmission errors over several code words.
#After passing through the individual bit protection procedures,  the data streams from fast and interleaved paths are combined and processed in  "tone ordering".
#In addition,  a guard interval and cyclic prefix are inserted in the DMT transmitter after the IDFT,  which is removed again in the DMT receiver.
#This represents a very simple realization of signal equalization in the frequency domain when the channel is distorted.

==Cyclic redundancy check==
 
The  "cyclic redundancy check"  $\rm (CRC)$  is a simple bit-level procedure to check the integrity of data during transmission or duplication.  The CRC principle has already been described in detail in the  [[Examples_of_Communication_Systems/ISDN_Primary_Multiplex_Connection#Frame_synchronization|"ISDN chapter"]].  Here follows a brief summary,  using the nomenclature used in the xDSL specifications:
*For each data block  $D(x)$  with  $k$  bit  $(d_0$, ... , $d_{k-1})$,  a parity-check value  $C(x)$  with eight bits is formed prior to data transmission and appended to the original data sequence  $($the variable  $x$  denotes here the delay operator$)$.

*$C(x)$  is obtained as the division remainder of the modulo-2 polynomial division of  $D(x)$  by the given parity-check polynomial  $G(x)$:
:$$D(x) = d_0 \cdot x^{k-1} + d_1 \cdot x^{k-2} + ... + d_{k-2} \cdot x + d_{k-1}\hspace{0.05cm},$$
:$$G(x) = x^8 + x^4 + x^3 + x^2 + 1 \hspace{0.05cm},$$
:$$C(x) = D(x) \cdot x^8 \,\,{\rm mod }\,\, G(x) = c_0 \cdot x^7 + c_1 \cdot x^6 + \text{...} + c_6 \cdot x + c_7
\hspace{0.05cm}.$$

*Another CRC value is formed at the receiver using the same procedure and compared with the transmitted CRC value.  If both are different,  at least one bit error happened during transmission.

*By this way,  bit errors can be detected if they are not too much clustered.  In ADSL practice,  the CRC procedure is sufficient for bit error detection.

The graph shows an exemplary circuit for the CRC value generation with the generator polynomial  $G(x)$  specified for ADSL – realizable in hardware or software:
[[File:EN_Bei_T_2_4_S5neu.png|right|frame|Cyclic Redundancy Check for ADSL]]

#The data block  $D(x)$  to be tested is introduced into the circuit from the left,  the output is fed back and exclusively-or-linked to the digits of the generator polynomial  $G(x)$.
#After passing through the entire data block,  the memory elements contain the CRC parity-check value  $C(x)$.
#It should be noted that with ADSL the data is split into so-called  "superframes"  of 68 frames each.
#Each frame contains data from the  "fast path"  and the  "interleaved path".  In addition,  management and synchronization bits are transmitted in specific frames.
#Eight CRC bits are formed per ADSL superframe and per path and are transmitted as  "fast byte"  resp.  "sync byte" as the first byte of frame  "$0$"  of the next superframe.

==Scrambler and de–scrambler==
 
Task of the scrambler is to convert long sequences of  "ones"  and  "zeros"  in such a way that frequent symbol changes occur.
*A possible realization is a shift register circuit with feedback exclusive-or-linked branches.

*In order to produce the original binary sequence at the receiver,  a mirror-image self-synchronizing  "de-scrambler"  must be used there.

The graph shows on the left an example of a scrambler actually used at DSL with  $23$  memory elements.  The corresponding de-scrambler is shown on the right.

[[File:EN_Bei_T_2_4_S6.png|right|frame|Scrambler and de-scrambler in a DSL/DMT system     $(1)$   scrambler's input        $(3)$   de-scrambler's input     $(2)$   scrambler's output     $(4)$   de-scrambler's output]]

The transmitter-side shift register is loaded with an arbitrary initial value that has no further effect on the operation of the circuit. Here:
:$$11001'10011'00110'01100'110.$$

If we denote
* by  $e_n$  the bits of the binary input sequence,  and

*by  $a_n$  the bits at the output, 

the following relation holds:

:$$a_n = e_n \oplus a_{n- 18}\oplus a_{n- 23}\hspace{0.05cm}.$$

In the example,  the scrambler input sequence consists of  80  consecutive  "ones"  $($upper left gray background$)$,  which are shifted bit by bit into the scrambler.  The output bit sequence then has frequent  "one-zero"  changes,  as desired.

The de-scrambler  $($shown on the right$)$  can be started at any time with any starting value,  which means that no synchronization is required between the two circuits.  Here:
:$$10111'011110'11101'11011'101.$$
The de-scrambler output data stream shows,
*that the de-scrambler initially outputs some  $($up to a maximum of $23)$  erroneous bits,  but then

*synchronizes automatically,  and then

*recovers the original bit sequence  $($only  "ones"$)$  without errors.

==Forward error correction==
 
For  "forward error correction"  $\rm (FEC)$,  all xDSL variants use a  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes|"Reed-Solomon-Code"]].  In some systems  "trellis code modulation"  $\rm (TCM)$  has been made mandatory as an additional error protection measure,  even though it has only been specified as  "optional"  by the international bodies.

Both methods are discussed in detail in the book  [[Channel_Coding|"Channel Coding"]].  Here follows a brief summary of Reed-Solomon coding with respect to its application to DSL:
*With Reed-Solomon encoding,  redundancy bytes are generated for fixed agreed interpolation points of the payload polynomial.  With systematic Reed-Solomon encoding,  a parity-check value is calculated similar to the CRC procedure and appended to the data block to be protected.

*However,  the data is no longer processed  "bit by bit",  but  "byte by byte".  Consequently,  arithmetic operations are no longer performed in the Galois field  $\rm GF( 2 )$  but in  $\rm GF(2^8)$.

The Reed-Solomon parity-check byte can also be determined as the division remainder of a polynomial division,  for xDSL with the following parameters:
#Number  $S$  of DMT symbols to be monitored per Reed-Solomon code word  $(S \ge 1$  for the fast buffer,  $S =2^0$, ... , $2^4$  for the interleaved buffer$)$,
#number  $K$  of user data bytes in the  $S$  DMT symbols,  defined as a polynomial  $B(x)$  of degree  $K$,  where the  "B"  indicates  "bytes",
#Number  $R$  of Reed-Solomon parity-check bytes  $($even number between  $2$  to  $16)$  per parity-check value  $($"fast"  or  "interleaved"$)$,
#sum  $N = K + R$  of the user data bytes and check bytes of the Reed-Solomon code word.

The specifics of Reed-Solomon encoding for xDSL are given here without further comment:
*For xDSL,  the number  $R$  of check bytes must be an integer multiple of the number  $S$  of symbols so that they can be evenly distributed in the payload polynomial.

*The so-called  [https://en.wikipedia.org/wiki/Singleton_bound#MDS_codes "Maximum Distance Separable  $\rm (MDS)$  codes"]  – a subclass of Reed-Solomon codes –  allow the correction of  $R/2$  falsified user data bytes.

*From the selected Reed-Solomon code for the DMT systems,  the constraint is a maximum code word length of  $2^8-1 = 255$  bytes corresponding to  $2040$  bits.

*The redundancy of Reed-Solomon codes can generate a considerable amount of data if the parameters are unfavorable,  thus considerably reducing the net data rate.

*It is recommended that the data amount  $($"gross data rate"$)$  be divided judiciously into useful data  $($"net data rate,  payload"$)$  and error protection data  $($"overhead"$)$.

*Reed-Solomon coding achieves a  "high coding gain".  A system without coding would have to have a signal-to-noise ratio  $\rm (SNR)$  larger by  $3 \ \rm dB$  for same bit error rate.

*By  "trellis-encoded modulation"  $\rm (TCM)$  in combination with other error protection measures,  the coding gain is highly variable; it ranges between  $0 \ \rm dB$  and  $6 \ \rm dB$.

==Interleaving and de–interleaving==
 
The common task of  "interleaver"  $($at the transmitter$)$  and  "de-interleaver"  $($at the receiver$)$  is
*to spread the received Reed-Solomon code words over a larger time range

*in order to distribute any transmission errors over several code words

*and thus increase the chance of correct decoding.

xDSL interleaving is characterized by the parameter  $D$  $($"depth"$)$,  which can take values between  $2^0$  and  $2^9$ .

{{GraueBox|TEXT=
$\text{Example 4:}$  The graph illustrates the principle using the Reed-Solomon code words  $A$,  $B$,  $C$  with five bytes each and the interleaver depth  $D = 2$.
[[File:EN_Bei_T_2_4_S8a.png|right|frame|For DSL interleaving with  $D = 2$]]

Each byte  $B_i$  of the middle Reed-Solomon code word  $B$  is delayed by  $V_i = (D - 1) \cdot i$  bytes. 

Two interleaver blocks are formed:
*The first block contains the bytes  $B_0$,  $B_1$  and  $B_2$  together with the bytes  $A_3$  and  $A_4$  of the previous code word.

*The second block contains the bytes  $B_3$  and  $B_4$  along with the bytes  $C_0$,  $C_1$  and  $C_2$  of the following code word.}}

This  "scrambling"  has the following advantages  $($provided,  $D$  is sufficiently large$)$:
#The error correction capabilities of the Reed-Solomon code are improved.
#The user data rate remains the same,  i.e. is not reduced $($redundancy-free$)$.
#In the event of errors,  entire packets do not have to be repeated at the protocol level.

A disadvantage is that with increasing interleaver depth  $D$  there can be noticeable delay times  $($on the order of milliseconds$)$,  which causes major problems for real-time applications.  However,  interleaving with low depth is only useful if the signal-to-noise ratio is sufficiently high.

{{GraueBox|TEXT=
$\text{Example 5:}$ 
An example of the advantages of interleaver/de-interleaver in the presence of bundle errors is shown in the following graph:
[[File:EN_Bei_T_2_4_S8b.png|right|frame|DSL interleaving and de–interleaving with  $D = 3$]]

*In the first row,  the transmitted byte sequence is shown according to Reed-Solomon encoding,  with each code word consisting of seven bytes as an example.

*In the middle row,  the data bytes are shifted by interleaving with  $D = 3$  so that between  $C_i$  and  $C_{i+1}$  there are two foreign bytes and the green code word is distributed over three blocks.

*Now suppose that during transmission a pulse glitch corrupted '''KORREKTUR: falsified''' three consecutive bytes in a single data block.

*After the de-interleaver,  the original byte sequence of the Reed-Solomon code words is restored,  with the three corrupted '''KORREKTUR: falsified''' bytes distributed among three independent code words.

*If two redundancy bytes were inserted in each case during the Reed-Solomon encoding,  the now separated byte corruptions '''KORREKTUR: falsifications''' can be completely corrected.}}

==Gain scaling and tone ordering==
 
A particularly advantageous feature of DMT is the possibility
#to adjust the bins individually to the existing channel characteristics and
#possibly to switch off "bins" with unfavorable SNR completely.

[[File:EN_Bei_T_2_4_S9.png|right|frame|Bit-bin assignment based on SNR]]
The procedure is as follows:

*Before starting the transmission  – and possibly also dynamically during operation –  the DMT modem measures the channel characteristics for each  "bin"  and sets the maximum transmission rate individually according to the SNR  $($see graphic$)$.

*During initialization,  the  "ADSL Transceiver Units"  exchange bin information,  for example the respective  "bits/bin"  and the required transmission power  $($'"gain"$)$.

*Thereby the  $\text{ATU-C}$  sends information about the upstream and the  $\text{ATU-R}$  sends information about the downstream.

*This message is of the format  $\{b_i, g_i\}$  where  $b_i$  $($four bits$)$  indicates the constellation size.  For the upstream,  the index  $i = 1$, ... , $31$  and for the downstream  $i = 1$, ... , $255$.

*The gain  $g_i$  is a fixed-point number with twelve bits.  For example  $g_i = 001.010000000$  represents the decimal value  $1 + 1/4 =1.25$.

*This indicates that the signal power of channel  $i$  must be higher by  $1.94 \ \rm dB$  than the power of the test signal transmitted during the channel analysis.
 
When operating the fast path and the interleaved path simultaneously $($see  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Overview_of_DSL_error_correction_measures|"graphic"]]  in the section  "DSL error correction measures"$)$,  the bit error rate can be further reduced by optimized carrier frequency allocation  $($"Tone Ordering"$)$.  The background of this measure is again  "clipping"  $($truncation of voltage peaks$)$,  which worsens the overall SNR. This procedure is based on the following rules:
*Bins with dense constellation  $($many bits/bin   ⇒   larger clipping probability$)$  are assigned to the interleaved branch,  since this is per se more reliable due to the additional interleaver.  Accordingly,  the subchannels with low order allocation  $($few bits/bin$)$  are reserved for the fast data buffer.

*New tables are then sent for upstream and downstream,  in which the bins are no longer ordered by index,  but according to the bits/bin ratios.  Based on this new table,  it is possible for the  $\text{ATU-C}$  or  $\text{ATU-R}$  to perform bit extraction successfully.

==Inserting guard interval and cyclic prefix ==
 
In the chapter  [[Modulation_Methods/Implementation_of_OFDM_Systems#Guard_interval_to_reduce_intersymbol_interference| "Realization of OFDM systems"]]  of the book "Modulation Methods" it has already been shown that by inserting a guard interval. The bit error rate can be decisively improved in the presence of linear channel distortion.

We assume that the cable impulse response  $h_{\rm K}(t)$  extends over the time duration  $T_{\rm K}$  . Ideally  $h_{\rm K}(t) = δ(t)$  and accordingly an infinitely short extension:   $T_{\rm K} = 0$. For distorting channel  $(T_{\rm K} > 0 )$  holds:
*By inserting a ''guard interval''  of duration  $T_{\rm G}$  ''intersymbol interference''  between each DSL frame can be avoided as long as  $T_{\rm G}$ ≥ $T_{\rm K}$  holds. However, this measure leads to a rate loss by a factor  $T/(T + T_{\rm G})$  with symbol duration  $T = {1}/{f_0}$.
*But with this, there is still ''inter-carrier interference''  between each subcarrier within the same frame, that is, the  [[Modulation_Methods/General_Description_of_OFDM#System_consideration_in_the_frequency_domain_with_causal_basic_pulse|"DMT individual spectra"]]  are no longer  $\rm si$-shaped '''KORREKTUR: sinc?''' and de-orthogonalization occurs.
*By a  [[Modulation_Methods/Implementation_of_OFDM_Systems#Cyclic_Prefix|"cyclic prefix"]]  also this disturbing effect can be avoided. Here one extends the transmission vector  $\mathbf{s}$  forward by the last  $L$  samples of the IDFT output, where the minimum value for $L$ is given by the duration  $T_{\rm K}$  of the cable impulse response.

{{GraueBox|TEXT=
$\text{Example 6:}$ 
The graphic shows this measure with the DSL/DMT method, for which the parameter  $L = 32$  has been set.

[[File:P_ID1964__Bei_T_2_4_S10a_v1.png|right|frame|DMT transmission signal with cyclic prefix]]
*The samples  $s_{480}$ , ... , $s_{511}$  are added as prefix  $(s_{-32}$ , ... , $s_{-1})$  to the IDFT output vector  $(s_0$ , ... , $s_{511})$  .
*The transmitted signal  $s(t)$  now has the resulting duration  $T ≈ 232 \ {\rm µ s}$  instead of the symbol duration  $T + T_{\rm G} = 1.0625 \cdot T ≈ 246 \ {\rm µ s}$. This reduces the rate by a factor of  $0.94$ .
*In the receive-side evaluation, one is restricted to the time range from  $0$  to  $T$. In this time interval the disturbing influence of the impulse response has already decayed and the subchannels are orthogonal to each other - just as with ideal channel.
*The sample values  $s_{-32}$ , ... , $s_{-1}$  are discarded at the receiver - a rather simple realization of signal equalization.}}

The last diagram in this chapter shows the entire DMT transmission system, but without the  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Overview_of_DSL_error_correction_measures| "error protection measures"]] described earlier. You can see:

[[File:P_ID1967__Bei_T_2_4_S10_v1.png|right|frame|DMT–System with cyclic prefix '''Korrektur''' das Bild muss ich noch machen]]
*In the "Add cyclic prefix" block, the samples  $s_{480}$, ... , $s_{511}$  as  $s_{-32}$, ... , $s_{-1}$  added. The transmit '''KORREKTUR: transmitted''' signal  $s(t)$  thus has the course shown in  $\text{Example 6}$  .
*The received signal  $r(t)$  results from the convolution of  $s(t)$  with  $h_{\rm K}(t)$. After A/D conversion and removal of the cyclic prefix, the input values  $r_0$, ... , $ r_{511}$  for the DFT.
*The (complex) output values  $D_k\hspace{0.01cm}'$  of the DFT depend only on the particular (complex) data value  $D_k$  . Independently of other data  $D_κ (κ ≠ k)$  holds with the noise value  $n_k\hspace{0.01cm}'$:

:$${D}_k\hspace{0.01cm}' = \alpha_k \cdot {D}_k + {n}_k\hspace{0.01cm}', \hspace{0.2cm}\alpha_k = H_{\rm K}( f = f_k)
\hspace{0.05cm}. $$

*Each carrier  $D_k$  is modified in its amplitude and phase by its own (complex) factor  $α_k$, which depends only on the channel. The frequency domain equalizer has only the task of multiplying the coefficient  $D_k\hspace{0.01cm}'$  by the inverse value  ${1}/{α_k}$  . Finally, one obtains:

:$$ \hat{D}_k = {D}_k + {n}_k \hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*This simple realization possibility of the complete equalization of the strongly distorting cable frequency response was one of the decisive criteria that for  $\rm xDSL$  the  $\rm DMT$ method has prevailed over  $\rm QAM$  and  $\rm CAP$ .
*Mostly an additional pre-equalization in the time domain takes place directly after the A/D conversion to avoid also the intersymbol interference between adjacent frames.}}

==Exercises for the chapter==
 
[[Exercise_2.5:_DSL_Error_Protection|"Exercise 2.5: DSL Error Protection"]]

[[Exercise_2.5Z:_Reach_and_Bit_Rate_with_ADSL|"Exercise 2.5Z: Reach and Bit Rate with ADSL"]]

[[Exercise_2.6:_Cyclic_Prefix|"Exercise 2.6: Cyclic Prefix"]]
==References==
<references />

{{Display}}

Examples of Communication Systems/Methods to Reduce the Bit Error Rate in DSL

2023-03-21T20:15:53Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=xDSL as Transmission Technology
|Nächste Seite=General Description of GSM
}}

==Transmission properties of copper cables ==
 
As already mentioned in the chapter  [[Examples_of_Communication_Systems/General_Description_of_DSL|"General Description of DSL"]] , the telephone line network of Deutsche Telekom mainly uses balanced copper pairs with a diameter of  $\text{0.4 mm}$.  The  "last mile"  is divided into three segments:
*the main cable,

*the branch cable,

*the house connection cable.

On average,  the line length is less than four kilometers.  In cities,  the copper line is shorter than  $\text{2.8 km}$  in  $90\%$  of all cases.

[[File:EN_LZI_T_4_3_S2_neu.png| right|frame|Structure of the local loop area]]

The $\rm xDSL$ variants discussed here were developed specifically for use on such symmetrical balanced copper pairs in the cable network. In order to better understand the technical requirements for the xDSL systems, a closer look must be taken at the transmission characteristics and interference on the conductor pairs.

This topic has already been dealt with in detail in the fourth main chapter  "Properties of Electrical Lines"  of the book  "[[Linear_and_Time_Invariant_Systems]]"  and is therefore only briefly summarized here using the  [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory#Wave_impedance_and_reflections|"equivalent circuit diagram"]] :
*Line transmission properties are fully characterized by the generally complex
:*  "characteristic impedance"  $Z_{\rm W}(f)$  and
:*  "complex propagation function per unit length"   ⇒   $γ(f)$.

*The even  "attenuation function $($per unit length$)$"  $α(f)$  is the real part of  $γ(f)$  and describes the attenuation of the wave propagating along the line:
:$$α(-f)=α(f) .$$

*The odd imaginary part  $β(f)$  of  $γ(f)$  is called  "phase function  $($per unit length$)$"  and gives the phase rotation of the signal wave along the line:
:$$β(-f)=-β(f) .$$

{{GraueBox|TEXT=
$\text{Example 1:}$  As an example,  we consider the function  $\alpha(f)$  shown on the right,  which is based on empirical investigations by  "Deutsche Telekom".
[[File:P_ID1955__Bei_T_2_4_S1b_v1.png|right|frame|Attenuation function per unit length of balanced copper pairs]]

The curves were obtained by averaging over a large number of measured lines of one kilometer length in the frequency range up to  $\text{30 MHz}$.  One can see:
# The attenuation function  $($per unit length$)$   $α(f)$  increases approximately proportionally with the square root of the frequency and decreases with increasing conductor diameter  $d$.
# The attenuation function  $a(f)$  increases linearly with cable length  $l$:
::$$a(f) = α(f) · l.$$

Note the difference between
*$a(f)$   speak "a"   $($for the attenuation function$)$,

*$\alpha(f)$   speak "a" $($for the attenuation function per unit length$)$.}}

For the line diameter  $\text{0.4 mm}$  was given in  [PW95]<ref name ='PW95'>Pollakowski, M.; Wellhausen, H.W.:  Properties of symmetrical local access cables in the frequency range up to 30 MHz.  Communication from the Research and Technology Center of Deutsche Telekom AG, Darmstadt, Verlag für Wissenschaft und Leben Georg Heidecker, 1995.</ref>  an empirical approximation formula for the attenuation function per unit length:

:$$\alpha(f) = \left [ 5.1 + 14.3 \cdot \left (\frac{f}{\rm 1\,MHz}\right )^{0.59} \right ] \frac{\rm dB}{\rm km}
\hspace{0.05cm}.$$

Evaluating this equation,  the following exemplary values hold:
*The attenuation function  $a(f)$  of a balanced copper wire of length  $l = 1 \ \rm km$  with diameter  $0.4 \ \rm mm$  is slightly more than  $60\ \rm dB$  for the signal frequency  $10\ \rm MHz$.

*At twice the frequency  $(20 \ \rm MHz)$  the attenuation value increases to over  $90 \ \rm dB$.  It can be seen that the attenuation does not increase exactly with the root of the frequency,  as would be the case if the skin effect were considered alone,  since several other effects also contribute to the attenuation.

*If the cable length is doubled to   $l = 2 \ \rm km$   the attenuation reaches a value of more than  $120 \ \rm dB$  $($at  $10 \ \rm MHz)$,  which corresponds to an amplitude attenuation factor smaller than  $10^{-6}$.

*Due to the frequency dependence of   $α(f)$  and  $β(f)$:   »'''intersymbol interference'''«  $\rm (ISI)$  as well as  »'''intercarrier interference'''  $\rm (ICI)$  occur. 

*Suitable equalization must therefore be provided for xDSL.

Note:
#In the  [[Linear_and_Time_Invariant_Systems/Properties_of_Balanced_Copper_Pairs|"Properties of balanced copper pairs"]]  chapter of the book  "Linear Time-Invariant Systems"  this topic is treated in detail.
#We refer here to the interactive applet  [[Applets:Dämpfung_von_Kupferkabeln|"Attenuation of copper cables"]].

==Disturbances during transmission==
 
Every transmission system is affected by disturbances,  which usually results primarily from thermal resistance noise.  In addition,  for a two-wire line there are:
*»'''Reflections'''«:   The counter-propagating wave increases the attenuation of copper pairs,  which is taken into account in the  [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory#Influence_of_reflections_-_operational_attenuation|"operational attenuation"]]  of the line.  To prevent such reflection,  the terminating resistor  $Z_{\rm E}(f)$  would have to be chosen identical to the  $($complex and frequency-dependent$)$  characteristic impedance  $Z_{\rm W}(f)$.  This is difficult in practice.  Therefore,  the terminating resistors are chosen to be real and constant,  and the resulting reflections are combated by technical means  – if possible.

[[File:EN_LZI_T_4_3_S6_v2.png|right|frame|On the emergence of crosstalk]]

*»'''Crosstalk'''«:   This is dominant interference in conducted transmission.  Crosstalk occurs when inductive and capacitive couplings between adjacent cores of a cable bundle cause mutual interference during signal transmission.

:Crosstalk is divided into two types (see graphic):

:*'''Near-end Crosstalk'''  $\rm (NEXT)$:  The interfering transmitter and the interfered receiver are on the same side of the cable.

:*'''Far-end Crosstalk'''  $\rm (FEXT)$:  The interfering transmitter and the interfered receiver are on opposite sides of the cable.

:Far-end crosstalk decreases sharply with increasing cable length due to attenuation,  so that near-end crosstalk is dominant even with DSL.
 
{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
To summarize:
#As frequency increases and spacing between line pairs decreases  – as within a star quad –  near-end crosstalk increases.  It is less critical if the conductors are in different basic bundles. 
#Depending on the stranding technique used,  the shielding and the manufacturing accuracy of the cable,  this effect occurs to varying degrees.  The cable length,  on the other hand,  does not play a role in near-end crosstalk:   The own transmitter is not attenuated by the cable. 
#Crosstalk can be significantly reduced by clever assignment,  for example by assigning different services to adjacent pairs,  using different frequency bands with as little overlap as possible.}}

==Signal–to–noise ratio, range and transmission rate ==
 
To evaluate the quality of a transmission system,  the signal-to-noise ratio  $\rm (SNR)$  is usually used.  This is also a measure of the expected bit error rate  $\rm (BER)$.
*Signal and noise in the same frequency band reduce the SNR and lead to a higher bit error rate or  – for a given bit error rate –  to a lower transmission bit rate.

*The relationships between transmit power,  channel quality  $($cable attenuation and noise power$)$  and achievable transmission rate can be illustrated very well by Shannon's channel capacity formula:

:$$C \left [ \frac{\rm bit}{\rm symbol} \right ] = \frac {1}{2} \cdot \log_2 \left ( 1 + \frac{P_{\rm E}}{P_{\rm N}} \right )=
\frac {1}{2} \cdot \log_2 \left ( 1 + \frac{\alpha_{\rm K}^2 \cdot P_{\rm S}}{P_{\rm N}} \right ) \hspace{0.05cm}.$$

The  »'''channel capacity'''«  $C$  denotes the maximum transmission bit rate at which transmission is possible under ideal conditions  $($among others,  the best possible coding with infinite block length$)$   ⇒   »'''channel coding theorem'''». For more details,  see the fourth main chapter  [[Information_Theory/AWGN_Channel_Capacity_for_Continuous-Valued_Input|"AWGN Channel Capacity for Continuous-Valued Input"]].

We assume that the bandwidth is fixed by the xDSL variant and that near-end crosstalk is the dominant interference.  Then the transmission rate can be improved by the following measures:
#For a given transmitted power  $P_{\rm S}$  and a given medium  $($e.g. balanced copper pairs with 0.4 mm diameter$)$,  the received power  $P_{\rm E}$  $($that can be used for demodulation$)$  is increased only by a shorter line length.
#One reduces the interference power  $P_{\rm N}$,  which for a given bandwidth  $B$  would be achieved by increased crosstalk attenuation,  which in turn also depends on the transmission method on the adjacent line pairs.
#Increasing the transmitted power  $P_{\rm S}$  would not be effective here,  since a larger transmitted power would at the same time have an unfavorable effect on the crosstalk.  This measure would only be successful for an AWGN channel  $($example:  coaxial cable$)$.

This listing shows that with xDSL there is a direct correlation between
*line length,

*transmission rate,  and

*the transmission method used.

{{GraueBox|TEXT=
$\text{Example 2:}$  From this graph, which refers to measurements with  "$\rm 1-DA xDSL$"  methods and  $\text{0.4 mm}$  copper cables in test systems with realistic interference conditions,  one can clearly see these dependencies.
[[File:EN_Bei_T_2_4_S3b.png|right|frame|Range and total bit rate for ADSL and VDSL]]

The graph shows for some ADSL and VDSL variants
*the range  $($maximum cable length$)$  $l_{\rm max}$  and

*the total transmission rate  $R_{\rm total}$ 
#of upstream $($first indication$)$
# and downstream $($second indication$)$.

The total transmission rate for the systems considered is between  $2.2 \ \rm Mbit/s$  and  $53\ \rm Mbit/s$.

*The trend of the measured values is shown in this graph as a solid  $($blue$)$  curve and can be formulated as a rough approximation as follows:

:$$l_{\rm max}\,{\rm \big [in}\,\,{\rm km \big ] } = \frac {20}{4 + R_{\rm total}\,{\rm \big [in}\,\,{\rm Mbit/s \big ] } } \hspace{0.05cm}.$$

*It can be seen that the range of all current systems  $($approximately between half a kilometer and three and a half kilometers of line length$)$  differs from this rule of thumb by a maximum of  $±25\%$  $($dashed curves$)$.}}

{{GraueBox|TEXT=
$\text{Example 3:}$ 
The diagram below shows the total data bit rates of  "ADSL2+"  and  "VDSL(2)"  as a function of line length,
[[File:EN_Bei_T_2_4_S3b_neu.png|right|frame|Data bit rates vs. cable lengths for xDSL systems]]
*with the  $($different$)$  red curves referring to the downstream

*and the two blue curves to the upstream.

This is based on a worst-case interference scenario with the following boundary conditions:
*Cable bundle with  $50$  copper pairs  $(0.4$  mm diameter$)$,  PE insulated,

*target symbol error rate  $p_{\rm S}=10^{-7},\ 6 \ \text{dB}$  margin $($reserve SNR to reach target data rate$)$,

*simultaneous operation of the following transmission methods:
#     $25$  times  "ADSL2+ over ISDN",
#    $14$  times ISDN,  four times  "SHDSL"  $(R= \text{1 Mbit/s)}$,
#    five times each  "SHDS"L  $(R= \text{2 Mbit/s)}$  and  "$(\text{VDSL2 band plan 998}$",
#    twice  "HDSL".

You can see from this diagram:
*For short line lengths,  the achievable data rates for VDSL(2) are significantly higher than for ADSL2+.

*From a line length of  $\approx 1800$  meters,  ADSL2+ is significantly better than VDSL(2).

*This is due to the fact that VDSL(2) operates in the lower frequency bands with significantly lower power in order to interfere less with neighboring systems.

*As the line length increases,  the higher frequency subchannels become unusable due to increasing attenuation,  which explains the crash in data rate.}}

==Overview of DSL error correction measures==
 
In order to reduce the BER of xDSL systems,  a number of techniques have been cleverly combined in the specifications to counteract the two most common causes of errors:
*Transmission errors due to pulse and crosstalk interference on the line:   Especially at high data rates,  adjacent symbols in the QAM signal space are close together,  which significantly increases the bit error probability.

*Cutting off of signal peaks due to lack of dynamic range of the transmitter amplifiers:  This  "clipping"  also corresponds to pulse noise and acts as an additional colored noise that noticeably degrades the SNR.

With the DMT method,  two paths are implemented for error correction in the signal processors.  The bit assignment to these paths is done by a multiplexer with sync control.
[[File:EN_Bei_T_2_4_S4_v9.png|right|frame|Complete DSL/DMT system]]

*In the case of  »'''fast path'''«,  low waiting times  $($"latency"$)$  are used.

*With  »'''interleaved path'''«,  low bit error rates are in the foreground.  Here the latency is higher due to the use of an interleaver.

*"Dual latency"  means the simultaneous use of both paths.  The  "ADSL Transceiver Units"  must support dual latency at least in the downstream.

The remaining chapter sections discuss error protection procedures for both paths. 

For other modulation methods,  the error protection measures described here are the same in principle,  but different in detail.

#The transmission chain starts with the  "cyclic redundancy check"  $\rm (CRC)$,  which forms a checksum over an overframe that is evaluated at the receiver.
#Task of the scrambler is to convert long sequences of  "ones"  and  "zeros"  to produce more frequent signal changes.
#This is followed by the "forward error correction"  $\rm (FEC)$  to detect/correct byte errors at the receiving end.  Often used for xDSL:  Reed-Solomon and Trellis coding.
#Task of the  "interleaver"  is to distribute the received code words over a larger time range in order to distribute transmission errors over several code words.
#After passing through the individual bit protection procedures,  the data streams from fast and interleaved paths are combined and processed in  "tone ordering".
#In addition,  a guard interval and cyclic prefix are inserted in the DMT transmitter after the IDFT,  which is removed again in the DMT receiver.
#This represents a very simple realization of signal equalization in the frequency domain when the channel is distorted.

==Cyclic redundancy check==
 
The  "cyclic redundancy check"  $\rm (CRC)$  is a simple bit-level procedure to check the integrity of data during transmission or duplication.  The CRC principle has already been described in detail in the  [[Examples_of_Communication_Systems/ISDN_Primary_Multiplex_Connection#Frame_synchronization|"ISDN chapter"]].  Here follows a brief summary,  using the nomenclature used in the xDSL specifications:
*For each data block  $D(x)$  with  $k$  bit  $(d_0$, ... , $d_{k-1})$,  a parity-check value  $C(x)$  with eight bits is formed prior to data transmission and appended to the original data sequence  $($the variable  $x$  denotes here the delay operator$)$.

*$C(x)$  is obtained as the division remainder of the modulo-2 polynomial division of  $D(x)$  by the given parity-check polynomial  $G(x)$:
:$$D(x) = d_0 \cdot x^{k-1} + d_1 \cdot x^{k-2} + ... + d_{k-2} \cdot x + d_{k-1}\hspace{0.05cm},$$
:$$G(x) = x^8 + x^4 + x^3 + x^2 + 1 \hspace{0.05cm},$$
:$$C(x) = D(x) \cdot x^8 \,\,{\rm mod }\,\, G(x) = c_0 \cdot x^7 + c_1 \cdot x^6 + \text{...} + c_6 \cdot x + c_7
\hspace{0.05cm}.$$

*Another CRC value is formed at the receiver using the same procedure and compared with the transmitted CRC value.  If both are different,  at least one bit error happened during transmission.

*By this way,  bit errors can be detected if they are not too much clustered.  In ADSL practice,  the CRC procedure is sufficient for bit error detection.

The graph shows an exemplary circuit for the CRC value generation with the generator polynomial  $G(x)$  specified for ADSL – realizable in hardware or software:
[[File:EN_Bei_T_2_4_S5neu.png|right|frame|Cyclic Redundancy Check for ADSL]]

#The data block  $D(x)$  to be tested is introduced into the circuit from the left,  the output is fed back and exclusively-or-linked to the digits of the generator polynomial  $G(x)$.
#After passing through the entire data block,  the memory elements contain the CRC parity-check value  $C(x)$.
#It should be noted that with ADSL the data is split into so-called  "superframes"  of 68 frames each.
#Each frame contains data from the  "fast path"  and the  "interleaved path".  In addition,  management and synchronization bits are transmitted in specific frames.
#Eight CRC bits are formed per ADSL superframe and per path and are transmitted as  "fast byte"  resp.  "sync byte" as the first byte of frame  "$0$"  of the next superframe.

==Scrambler and de–scrambler==
 
Task of the scrambler is to convert long sequences of  "ones"  and  "zeros"  in such a way that frequent symbol changes occur.
*A possible realization is a shift register circuit with feedback exclusive-or-linked branches.

*In order to produce the original binary sequence at the receiver,  a mirror-image self-synchronizing  "de-scrambler"  must be used there.

The graph shows on the left an example of a scrambler actually used at DSL with  $23$  memory elements.  The corresponding de-scrambler is shown on the right.

[[File:EN_Bei_T_2_4_S6.png|right|frame|Scrambler and de-scrambler in a DSL/DMT system     $(1)$   scrambler's input        $(3)$   de-scrambler's input     $(2)$   scrambler's output     $(4)$   de-scrambler's output]]

The transmitter-side shift register is loaded with an arbitrary initial value that has no further effect on the operation of the circuit. Here:
:$$11001'10011'00110'01100'110.$$

If we denote
* by  $e_n$  the bits of the binary input sequence,  and

*by  $a_n$  the bits at the output, 

the following relation holds:

:$$a_n = e_n \oplus a_{n- 18}\oplus a_{n- 23}\hspace{0.05cm}.$$

In the example,  the scrambler input sequence consists of  80  consecutive  "ones"  $($upper left gray background$)$,  which are shifted bit by bit into the scrambler.  The output bit sequence then has frequent  "one-zero"  changes,  as desired.

The de-scrambler  $($shown on the right$)$  can be started at any time with any starting value,  which means that no synchronization is required between the two circuits.  Here:
:$$10111'011110'11101'11011'101.$$
The de-scrambler output data stream shows,
*that the de-scrambler initially outputs some  $($up to a maximum of $23)$  erroneous bits,  but then

*synchronizes automatically,  and then

*recovers the original bit sequence  $($only  "ones"$)$  without errors.

==Forward error correction==
 
For  "forward error correction"  $\rm (FEC)$,  all xDSL variants use a  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes|"Reed-Solomon-Code"]].  In some systems  "trellis code modulation"  $\rm (TCM)$  has been made mandatory as an additional error protection measure,  even though it has only been specified as  "optional"  by the international bodies.

Both methods are discussed in detail in the book  [[Channel_Coding|"Channel Coding"]].  Here follows a brief summary of Reed-Solomon coding with respect to its application to DSL:
*With Reed-Solomon encoding,  redundancy bytes are generated for fixed agreed interpolation points of the payload polynomial.  With systematic Reed-Solomon encoding,  a parity-check value is calculated similar to the CRC procedure and appended to the data block to be protected.

*However,  the data is no longer processed  "bit by bit",  but  "byte by byte".  Consequently,  arithmetic operations are no longer performed in the Galois field  $\rm GF( 2 )$  but in  $\rm GF(2^8)$.

The Reed-Solomon parity-check byte can also be determined as the division remainder of a polynomial division,  for xDSL with the following parameters:
#Number  $S$  of DMT symbols to be monitored per Reed-Solomon code word  $(S \ge 1$  for the fast buffer,  $S =2^0$, ... , $2^4$  for the interleaved buffer$)$,
#number  $K$  of user data bytes in the  $S$  DMT symbols,  defined as a polynomial  $B(x)$  of degree  $K$,  where the  "B"  indicates  "bytes",
#Number  $R$  of Reed-Solomon parity-check bytes  $($even number between  $2$  to  $16)$  per parity-check value  $($"fast"  or  "interleaved"$)$,
#sum  $N = K + R$  of the user data bytes and check bytes of the Reed-Solomon code word.

The specifics of Reed-Solomon encoding for xDSL are given here without further comment:
*For xDSL,  the number  $R$  of check bytes must be an integer multiple of the number  $S$  of symbols so that they can be evenly distributed in the payload polynomial.

*The so-called  [https://en.wikipedia.org/wiki/Singleton_bound#MDS_codes "Maximum Distance Separable  $\rm (MDS)$  codes"]  – a subclass of Reed-Solomon codes –  allow the correction of  $R/2$  falsified user data bytes.

*From the selected Reed-Solomon code for the DMT systems,  the constraint is a maximum code word length of  $2^8-1 = 255$  bytes corresponding to  $2040$  bits.

*The redundancy of Reed-Solomon codes can generate a considerable amount of data if the parameters are unfavorable,  thus considerably reducing the net data rate.

*It is recommended that the data amount  $($"gross data rate"$)$  be divided judiciously into useful data  $($"net data rate,  payload"$)$  and error protection data  $($"overhead"$)$.

*Reed-Solomon coding achieves a  "high coding gain".  A system without coding would have to have a signal-to-noise ratio  $\rm (SNR)$  larger by  $3 \ \rm dB$  for same bit error rate.

*By  "trellis-encoded modulation"  $\rm (TCM)$  in combination with other error protection measures,  the coding gain is highly variable; it ranges between  $0 \ \rm dB$  and  $6 \ \rm dB$.

==Interleaving and de–interleaving==
 
The common task of  "interleaver"  $($at the transmitter$)$  and  "de-interleaver"  $($at the receiver$)$  is
*to spread the received Reed-Solomon code words over a larger time range

*in order to distribute any transmission errors over several code words

*and thus increase the chance of correct decoding.

xDSL interleaving is characterized by the parameter  $D$  $($"depth"$)$,  which can take values between  $2^0$  and  $2^9$ .

{{GraueBox|TEXT=
$\text{Example 4:}$  The graph illustrates the principle using the Reed-Solomon code words  $A$,  $B$,  $C$  with five bytes each and the interleaver depth  $D = 2$.
[[File:EN_Bei_T_2_4_S8a.png|right|frame|For DSL interleaving with  $D = 2$]]

Each byte  $B_i$  of the middle Reed-Solomon code word  $B$  is delayed by  $V_i = (D - 1) \cdot i$  bytes. 

Two interleaver blocks are formed:
*The first block contains the bytes  $B_0$,  $B_1$  and  $B_2$  together with the bytes  $A_3$  and  $A_4$  of the previous code word.

*The second block contains the bytes  $B_3$  and  $B_4$  along with the bytes  $C_0$,  $C_1$  and  $C_2$  of the following code word.}}

This  "scrambling"  has the following advantages  $($provided,  $D$  is sufficiently large$)$:
#The error correction capabilities of the Reed-Solomon code are improved.
#The user data rate remains the same,  i.e. is not reduced $($redundancy-free$)$.
#In the event of errors,  entire packets do not have to be repeated at the protocol level.

A disadvantage is that with increasing interleaver depth  $D$  there can be noticeable delay times  $($on the order of milliseconds$)$,  which causes major problems for real-time applications.  However,  interleaving with low depth is only useful if the signal-to-noise ratio is sufficiently high.

{{GraueBox|TEXT=
$\text{Example 5:}$ 
An example of the advantages of interleaver/de-interleaver in the presence of bundle errors is shown in the following graph:
[[File:EN_Bei_T_2_4_S8b.png|right|frame|DSL interleaving and de–interleaving with  $D = 3$]]

*In the first row,  the transmitted byte sequence is shown according to Reed-Solomon encoding,  with each code word consisting of seven bytes as an example.

*In the middle row,  the data bytes are shifted by interleaving with  $D = 3$  so that between  $C_i$  and  $C_{i+1}$  there are two foreign bytes and the green code word is distributed over three blocks.

*Now suppose that during transmission a pulse glitch corrupted '''KORREKTUR: falsified''' three consecutive bytes in a single data block.

*After the de-interleaver,  the original byte sequence of the Reed-Solomon code words is restored,  with the three corrupted '''KORREKTUR: falsified''' bytes distributed among three independent code words.

*If two redundancy bytes were inserted in each case during the Reed-Solomon encoding,  the now separated byte corruptions '''KORREKTUR: falsifications''' can be completely corrected.}}

==Gain scaling and tone ordering==
 
A particularly advantageous feature of DMT is the possibility
#to adjust the bins individually to the existing channel characteristics and
#possibly to switch off "bins" with unfavorable SNR completely.

[[File:EN_Bei_T_2_4_S9.png|right|frame|Bit-bin assignment based on SNR]]
The procedure is as follows:

*Before starting the transmission  – and possibly also dynamically during operation –  the DMT modem measures the channel characteristics for each  "bin"  and sets the maximum transmission rate individually according to the SNR  $($see graphic$)$.

*During initialization,  the  "ADSL Transceiver Units"  exchange bin information,  for example the respective  "bits/bin"  and the required transmission power  $($'"gain"$)$.

*Thereby the  $\text{ATU-C}$  sends information about the upstream and the  $\text{ATU-R}$  sends information about the downstream.

*This message is of the format  $\{b_i, g_i\}$  where  $b_i$  $($four bits$)$  indicates the constellation size.  For the upstream,  the index  $i = 1$, ... , $31$  and for the downstream  $i = 1$, ... , $255$.

*The gain  $g_i$  is a fixed-point number with twelve bits.  For example  $g_i = 001.010000000$  represents the decimal value  $1 + 1/4 =1.25$.

*This indicates that the signal power of channel  $i$  must be higher by  $1.94 \ \rm dB$  than the power of the test signal transmitted during the channel analysis.
 
When operating the fast path and the interleaved path simultaneously $($see  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Overview_of_DSL_error_correction_measures|"graphic"]]  in the section  "DSL error correction measures"$)$,  the bit error rate can be further reduced by optimized carrier frequency allocation  $($"Tone Ordering"$)$.  The background of this measure is again  "clipping"  $($truncation of voltage peaks$)$,  which worsens the overall SNR. This procedure is based on the following rules:
*Bins with dense constellation  $($many bits/bin   ⇒   larger clipping probability$)$  are assigned to the interleaved branch,  since this is per se more reliable due to the additional interleaver.  Accordingly,  the subchannels with low order allocation  $($few bits/bin$)$  are reserved for the fast data buffer.

*New tables are then sent for upstream and downstream,  in which the bins are no longer ordered by index,  but according to the bits/bin ratios.  Based on this new table,  it is possible for the  $\text{ATU-C}$  or  $\text{ATU-R}$  to perform bit extraction successfully.

==Inserting guard interval and cyclic prefix ==
 
In the chapter  [[Modulation_Methods/Implementation_of_OFDM_Systems#Guard_interval_to_reduce_intersymbol_interference| "Realization of OFDM systems"]]  of the book "Modulation Methods" it has already been shown that by inserting a guard interval. The bit error rate can be decisively improved in the presence of linear channel distortion.

We assume that the cable impulse response  $h_{\rm K}(t)$  extends over the time duration  $T_{\rm K}$  . Ideally  $h_{\rm K}(t) = δ(t)$  and accordingly an infinitely short extension:   $T_{\rm K} = 0$. For distorting channel  $(T_{\rm K} > 0 )$  holds:
*By inserting a ''guard interval''  of duration  $T_{\rm G}$  ''intersymbol interference''  between each DSL frame can be avoided as long as  $T_{\rm G}$ ≥ $T_{\rm K}$  holds. However, this measure leads to a rate loss by a factor  $T/(T + T_{\rm G})$  with symbol duration  $T = {1}/{f_0}$.
*But with this, there is still ''inter-carrier interference''  between each subcarrier within the same frame, that is, the  [[Modulation_Methods/General_Description_of_OFDM#System_consideration_in_the_frequency_domain_with_causal_basic_pulse|"DMT individual spectra"]]  are no longer  $\rm si$-shaped '''KORREKTUR: sinc?''' and de-orthogonalization occurs.
*By a  [[Modulation_Methods/Implementation_of_OFDM_Systems#Cyclic_Prefix|"cyclic prefix"]]  also this disturbing effect can be avoided. Here one extends the transmission vector  $\mathbf{s}$  forward by the last  $L$  samples of the IDFT output, where the minimum value for $L$ is given by the duration  $T_{\rm K}$  of the cable impulse response.

{{GraueBox|TEXT=
$\text{Example 6:}$ 
The graphic shows this measure with the DSL/DMT method, for which the parameter  $L = 32$  has been set.

[[File:P_ID1964__Bei_T_2_4_S10a_v1.png|right|frame|DMT transmission signal with cyclic prefix]]
*The samples  $s_{480}$ , ... , $s_{511}$  are added as prefix  $(s_{-32}$ , ... , $s_{-1})$  to the IDFT output vector  $(s_0$ , ... , $s_{511})$  .
*The transmitted signal  $s(t)$  now has the resulting duration  $T ≈ 232 \ {\rm µ s}$  instead of the symbol duration  $T + T_{\rm G} = 1.0625 \cdot T ≈ 246 \ {\rm µ s}$. This reduces the rate by a factor of  $0.94$ .
*In the receive-side evaluation, one is restricted to the time range from  $0$  to  $T$. In this time interval the disturbing influence of the impulse response has already decayed and the subchannels are orthogonal to each other - just as with ideal channel.
*The sample values  $s_{-32}$ , ... , $s_{-1}$  are discarded at the receiver - a rather simple realization of signal equalization.}}

The last diagram in this chapter shows the entire DMT transmission system, but without the  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Overview_of_DSL_error_correction_measures| "error protection measures"]] described earlier. You can see:

[[File:P_ID1967__Bei_T_2_4_S10_v1.png|right|frame|DMT–System with cyclic prefix '''Korrektur''' das Bild muss ich noch machen]]
*In the "Add cyclic prefix" block, the samples  $s_{480}$, ... , $s_{511}$  as  $s_{-32}$, ... , $s_{-1}$  added. The transmit signal  $s(t)$  thus has the course shown in  $\text{Example 6}$  .
*The received signal  $r(t)$  results from the convolution of  $s(t)$  with  $h_{\rm K}(t)$. After A/D conversion and removal of the cyclic prefix, the input values  $r_0$, ... , $ r_{511}$  for the DFT.
*The (complex) output values  $D_k\hspace{0.01cm}'$  of the DFT depend only on the particular (complex) data value  $D_k$  . Independently of other data  $D_κ (κ ≠ k)$  holds with the noise value  $n_k\hspace{0.01cm}'$:

:$${D}_k\hspace{0.01cm}' = \alpha_k \cdot {D}_k + {n}_k\hspace{0.01cm}', \hspace{0.2cm}\alpha_k = H_{\rm K}( f = f_k)
\hspace{0.05cm}. $$

*Each carrier  $D_k$  is modified in its amplitude and phase by its own (complex) factor  $α_k$, which depends only on the channel. The frequency domain equalizer has only the task of multiplying the coefficient  $D_k\hspace{0.01cm}'$  by the inverse value  ${1}/{α_k}$  . Finally, one obtains:

:$$ \hat{D}_k = {D}_k + {n}_k \hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*This simple realization possibility of the complete equalization of the strongly distorting cable frequency response was one of the decisive criteria that for  $\rm xDSL$  the  $\rm DMT$ method has prevailed over  $\rm QAM$  and  $\rm CAP$ .
*Mostly an additional pre-equalization in the time domain takes place directly after the A/D conversion to avoid also the intersymbol interference between adjacent frames.}}

==Exercises for the chapter==
 
[[Exercise_2.5:_DSL_Error_Protection|"Exercise 2.5: DSL Error Protection"]]

[[Exercise_2.5Z:_Reach_and_Bit_Rate_with_ADSL|"Exercise 2.5Z: Reach and Bit Rate with ADSL"]]

[[Exercise_2.6:_Cyclic_Prefix|"Exercise 2.6: Cyclic Prefix"]]
==References==
<references />

{{Display}}

Examples of Communication Systems/Methods to Reduce the Bit Error Rate in DSL

2023-03-21T20:12:53Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=xDSL as Transmission Technology
|Nächste Seite=General Description of GSM
}}

==Transmission properties of copper cables ==
 
As already mentioned in the chapter  [[Examples_of_Communication_Systems/General_Description_of_DSL|"General Description of DSL"]] , the telephone line network of Deutsche Telekom mainly uses balanced copper pairs with a diameter of  $\text{0.4 mm}$.  The  "last mile"  is divided into three segments:
*the main cable,

*the branch cable,

*the house connection cable.

On average,  the line length is less than four kilometers.  In cities,  the copper line is shorter than  $\text{2.8 km}$  in  $90\%$  of all cases.

[[File:EN_LZI_T_4_3_S2_neu.png| right|frame|Structure of the local loop area]]

The $\rm xDSL$ variants discussed here were developed specifically for use on such symmetrical balanced copper pairs in the cable network. In order to better understand the technical requirements for the xDSL systems, a closer look must be taken at the transmission characteristics and interference on the conductor pairs.

This topic has already been dealt with in detail in the fourth main chapter  "Properties of Electrical Lines"  of the book  "[[Linear_and_Time_Invariant_Systems]]"  and is therefore only briefly summarized here using the  [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory#Wave_impedance_and_reflections|"equivalent circuit diagram"]] :
*Line transmission properties are fully characterized by the generally complex
:*  "characteristic impedance"  $Z_{\rm W}(f)$  and
:*  "complex propagation function per unit length"   ⇒   $γ(f)$.

*The even  "attenuation function $($per unit length$)$"  $α(f)$  is the real part of  $γ(f)$  and describes the attenuation of the wave propagating along the line:
:$$α(-f)=α(f) .$$

*The odd imaginary part  $β(f)$  of  $γ(f)$  is called  "phase function  $($per unit length$)$"  and gives the phase rotation of the signal wave along the line:
:$$β(-f)=-β(f) .$$

{{GraueBox|TEXT=
$\text{Example 1:}$  As an example,  we consider the function  $\alpha(f)$  shown on the right,  which is based on empirical investigations by  "Deutsche Telekom".
[[File:P_ID1955__Bei_T_2_4_S1b_v1.png|right|frame|Attenuation function per unit length of balanced copper pairs]]

The curves were obtained by averaging over a large number of measured lines of one kilometer length in the frequency range up to  $\text{30 MHz}$.  One can see:
# The attenuation function  $($per unit length$)$   $α(f)$  increases approximately proportionally with the square root of the frequency and decreases with increasing conductor diameter  $d$.
# The attenuation function  $a(f)$  increases linearly with cable length  $l$:
::$$a(f) = α(f) · l.$$

Note the difference between
*$a(f)$   speak "a"   $($for the attenuation function$)$,

*$\alpha(f)$   speak "a" $($for the attenuation function per unit length$)$.}}

For the line diameter  $\text{0.4 mm}$  was given in  [PW95]<ref name ='PW95'>Pollakowski, M.; Wellhausen, H.W.:  Properties of symmetrical local access cables in the frequency range up to 30 MHz.  Communication from the Research and Technology Center of Deutsche Telekom AG, Darmstadt, Verlag für Wissenschaft und Leben Georg Heidecker, 1995.</ref>  an empirical approximation formula for the attenuation function per unit length:

:$$\alpha(f) = \left [ 5.1 + 14.3 \cdot \left (\frac{f}{\rm 1\,MHz}\right )^{0.59} \right ] \frac{\rm dB}{\rm km}
\hspace{0.05cm}.$$

Evaluating this equation,  the following exemplary values hold:
*The attenuation function  $a(f)$  of a balanced copper wire of length  $l = 1 \ \rm km$  with diameter  $0.4 \ \rm mm$  is slightly more than  $60\ \rm dB$  for the signal frequency  $10\ \rm MHz$.

*At twice the frequency  $(20 \ \rm MHz)$  the attenuation value increases to over  $90 \ \rm dB$.  It can be seen that the attenuation does not increase exactly with the root of the frequency,  as would be the case if the skin effect were considered alone,  since several other effects also contribute to the attenuation.

*If the cable length is doubled to   $l = 2 \ \rm km$   the attenuation reaches a value of more than  $120 \ \rm dB$  $($at  $10 \ \rm MHz)$,  which corresponds to an amplitude attenuation factor smaller than  $10^{-6}$.

*Due to the frequency dependence of   $α(f)$  and  $β(f)$:   »'''intersymbol interference'''«  $\rm (ISI)$  as well as  »'''intercarrier interference'''  $\rm (ICI)$  occur. 

*Suitable equalization must therefore be provided for xDSL.

Note:
#In the  [[Linear_and_Time_Invariant_Systems/Properties_of_Balanced_Copper_Pairs|"Properties of balanced copper pairs"]]  chapter of the book  "Linear Time-Invariant Systems"  this topic is treated in detail.
#We refer here to the interactive applet  [[Applets:Dämpfung_von_Kupferkabeln|"Attenuation of copper cables"]].

==Disturbances during transmission==
 
Every transmission system is affected by disturbances,  which usually results primarily from thermal resistance noise.  In addition,  for a two-wire line there are:
*»'''Reflections'''«:   The counter-propagating wave increases the attenuation of copper pairs,  which is taken into account in the  [[Linear_and_Time_Invariant_Systems/Some_Results_from_Line_Transmission_Theory#Influence_of_reflections_-_operational_attenuation|"operational attenuation"]]  of the line.  To prevent such reflection,  the terminating resistor  $Z_{\rm E}(f)$  would have to be chosen identical to the  $($complex and frequency-dependent$)$  characteristic impedance  $Z_{\rm W}(f)$.  This is difficult in practice.  Therefore,  the terminating resistors are chosen to be real and constant,  and the resulting reflections are combated by technical means  – if possible.

[[File:EN_LZI_T_4_3_S6_v2.png|right|frame|On the emergence of crosstalk]]

*»'''Crosstalk'''«:   This is dominant interference in conducted transmission.  Crosstalk occurs when inductive and capacitive couplings between adjacent cores of a cable bundle cause mutual interference during signal transmission.

:Crosstalk is divided into two types (see graphic):

:*'''Near-end Crosstalk'''  $\rm (NEXT)$:  The interfering transmitter and the interfered receiver are on the same side of the cable.

:*'''Far-end Crosstalk'''  $\rm (FEXT)$:  The interfering transmitter and the interfered receiver are on opposite sides of the cable.

:Far-end crosstalk decreases sharply with increasing cable length due to attenuation,  so that near-end crosstalk is dominant even with DSL.
 
{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
To summarize:
#As frequency increases and spacing between line pairs decreases  – as within a star quad –  near-end crosstalk increases.  It is less critical if the conductors are in different basic bundles. 
#Depending on the stranding technique used,  the shielding and the manufacturing accuracy of the cable,  this effect occurs to varying degrees.  The cable length,  on the other hand,  does not play a role in near-end crosstalk:   The own transmitter is not attenuated by the cable. 
#Crosstalk can be significantly reduced by clever assignment,  for example by assigning different services to adjacent pairs,  using different frequency bands with as little overlap as possible.}}

==Signal–to–noise ratio, range and transmission rate ==
 
To evaluate the quality of a transmission system,  the signal-to-noise ratio  $\rm (SNR)$  is usually used.  This is also a measure of the expected bit error rate  $\rm (BER)$.
*Signal and noise in the same frequency band reduce the SNR and lead to a higher bit error rate or  – for a given bit error rate –  to a lower transmission bit rate.

*The relationships between transmit power,  channel quality  $($cable attenuation and noise power$)$  and achievable transmission rate can be illustrated very well by Shannon's channel capacity formula:

:$$C \left [ \frac{\rm bit}{\rm symbol} \right ] = \frac {1}{2} \cdot \log_2 \left ( 1 + \frac{P_{\rm E}}{P_{\rm N}} \right )=
\frac {1}{2} \cdot \log_2 \left ( 1 + \frac{\alpha_{\rm K}^2 \cdot P_{\rm S}}{P_{\rm N}} \right ) \hspace{0.05cm}.$$

The  »'''channel capacity'''«  $C$  denotes the maximum transmission bit rate at which transmission is possible under ideal conditions  $($among others,  the best possible coding with infinite block length$)$   ⇒   »'''channel coding theorem'''». For more details,  see the fourth main chapter  [[Information_Theory/AWGN_Channel_Capacity_for_Continuous-Valued_Input|"AWGN Channel Capacity for Continuous-Valued Input"]].

We assume that the bandwidth is fixed by the xDSL variant and that near-end crosstalk is the dominant interference.  Then the transmission rate can be improved by the following measures:
#For a given transmitted power  $P_{\rm S}$  and a given medium  $($e.g. balanced copper pairs with 0.4 mm diameter$)$,  the received power  $P_{\rm E}$  $($that can be used for demodulation$)$  is increased only by a shorter line length.
#One reduces the interference power  $P_{\rm N}$,  which for a given bandwidth  $B$  would be achieved by increased crosstalk attenuation,  which in turn also depends on the transmission method on the adjacent line pairs.
#Increasing the transmitted power  $P_{\rm S}$  would not be effective here,  since a larger transmitted power would at the same time have an unfavorable effect on the crosstalk.  This measure would only be successful for an AWGN channel  $($example:  coaxial cable$)$.

This listing shows that with xDSL there is a direct correlation between
*line length,

*transmission rate,  and

*the transmission method used.

{{GraueBox|TEXT=
$\text{Example 2:}$  From this graph, which refers to measurements with  "$\rm 1-DA xDSL$"  methods and  $\text{0.4 mm}$  copper cables in test systems with realistic interference conditions,  one can clearly see these dependencies.
[[File:EN_Bei_T_2_4_S3b.png|right|frame|Range and total bit rate for ADSL and VDSL]]

The graph shows for some ADSL and VDSL variants
*the range  $($maximum cable length$)$  $l_{\rm max}$  and

*the total transmission rate  $R_{\rm total}$ 
#of upstream $($first indication$)$
# and downstream $($second indication$)$.

The total transmission rate for the systems considered is between  $2.2 \ \rm Mbit/s$  and  $53\ \rm Mbit/s$.

*The trend of the measured values is shown in this graph as a solid  $($blue$)$  curve and can be formulated as a rough approximation as follows:

:$$l_{\rm max}\,{\rm \big [in}\,\,{\rm km \big ] } = \frac {20}{4 + R_{\rm total}\,{\rm \big [in}\,\,{\rm Mbit/s \big ] } } \hspace{0.05cm}.$$

*It can be seen that the range of all current systems  $($approximately between half a kilometer and three and a half kilometers of line length$)$  differs from this rule of thumb by a maximum of  $±25\%$  $($dashed curves$)$.}}

{{GraueBox|TEXT=
$\text{Example 3:}$ 
The diagram below shows the total data bit rates of  "ADSL2+"  and  "VDSL(2)"  as a function of line length,
[[File:EN_Bei_T_2_4_S3b_neu.png|right|frame|Data bit rates vs. cable lengths for xDSL systems]]
*with the  $($different$)$  red curves referring to the downstream

*and the two blue curves to the upstream.

This is based on a worst-case interference scenario with the following boundary conditions:
*Cable bundle with  $50$  copper pairs  $(0.4$  mm diameter$)$,  PE insulated,

*target symbol error rate  $p_{\rm S}=10^{-7},\ 6 \ \text{dB}$  margin $($reserve SNR to reach target data rate$)$,

*simultaneous operation of the following transmission methods:
#     $25$  times  "ADSL2+ over ISDN",
#    $14$  times ISDN,  four times  "SHDSL"  $(R= \text{1 Mbit/s)}$,
#    five times each  "SHDS"L  $(R= \text{2 Mbit/s)}$  and  "$(\text{VDSL2 band plan 998}$",
#    twice  "HDSL".

You can see from this diagram:
*For short line lengths,  the achievable data rates for VDSL(2) are significantly higher than for ADSL2+.

*From a line length of  $\approx 1800$  meters,  ADSL2+ is significantly better than VDSL(2).

*This is due to the fact that VDSL(2) operates in the lower frequency bands with significantly lower power in order to interfere less with neighboring systems.

*As the line length increases,  the higher frequency subchannels become unusable due to increasing attenuation,  which explains the crash in data rate.}}

==Overview of DSL error correction measures==
 
In order to reduce the BER of xDSL systems,  a number of techniques have been cleverly combined in the specifications to counteract the two most common causes of errors:
*Transmission errors due to pulse and crosstalk interference on the line:   Especially at high data rates,  adjacent symbols in the QAM signal space are close together,  which significantly increases the bit error probability.

*Cutting off of signal peaks due to lack of dynamic range of the transmitter amplifiers:  This  "clipping"  also corresponds to pulse noise and acts as an additional colored noise that noticeably degrades the SNR.

With the DMT method,  two paths are implemented for error correction in the signal processors.  The bit assignment to these paths is done by a multiplexer with sync control.
[[File:EN_Bei_T_2_4_S4_v9.png|right|frame|Complete DSL/DMT system]]

*In the case of  »'''fast path'''«,  low waiting times  $($"latency"$)$  are used.

*With  »'''interleaved path'''«,  low bit error rates are in the foreground.  Here the latency is higher due to the use of an interleaver.

*"Dual latency"  means the simultaneous use of both paths.  The  "ADSL Transceiver Units"  must support dual latency at least in the downstream.

The remaining chapter sections discuss error protection procedures for both paths. 

For other modulation methods,  the error protection measures described here are the same in principle,  but different in detail.

#The transmission chain starts with the  "cyclic redundancy check"  $\rm (CRC)$,  which forms a checksum over an overframe that is evaluated at the receiver.
#Task of the scrambler is to convert long sequences of  "ones"  and  "zeros"  to produce more frequent signal changes.
#This is followed by the "forward error correction"  $\rm (FEC)$  to detect/correct byte errors at the receiving end.  Often used for xDSL:  Reed-Solomon and Trellis coding.
#Task of the  "interleaver"  is to distribute the received code words over a larger time range in order to distribute transmission errors over several code words.
#After passing through the individual bit protection procedures,  the data streams from fast and interleaved paths are combined and processed in  "tone ordering".
#In addition,  a guard interval and cyclic prefix are inserted in the DMT transmitter after the IDFT,  which is removed again in the DMT receiver.
#This represents a very simple realization of signal equalization in the frequency domain when the channel is distorted.

==Cyclic redundancy check==
 
The  "cyclic redundancy check"  $\rm (CRC)$  is a simple bit-level procedure to check the integrity of data during transmission or duplication.  The CRC principle has already been described in detail in the  [[Examples_of_Communication_Systems/ISDN_Primary_Multiplex_Connection#Frame_synchronization|"ISDN chapter"]].  Here follows a brief summary,  using the nomenclature used in the xDSL specifications:
*For each data block  $D(x)$  with  $k$  bit  $(d_0$, ... , $d_{k-1})$,  a parity-check value  $C(x)$  with eight bits is formed prior to data transmission and appended to the original data sequence  $($the variable  $x$  denotes here the delay operator$)$.

*$C(x)$  is obtained as the division remainder of the modulo-2 polynomial division of  $D(x)$  by the given parity-check polynomial  $G(x)$:
:$$D(x) = d_0 \cdot x^{k-1} + d_1 \cdot x^{k-2} + ... + d_{k-2} \cdot x + d_{k-1}\hspace{0.05cm},$$
:$$G(x) = x^8 + x^4 + x^3 + x^2 + 1 \hspace{0.05cm},$$
:$$C(x) = D(x) \cdot x^8 \,\,{\rm mod }\,\, G(x) = c_0 \cdot x^7 + c_1 \cdot x^6 + \text{...} + c_6 \cdot x + c_7
\hspace{0.05cm}.$$

*Another CRC value is formed at the receiver using the same procedure and compared with the transmitted CRC value.  If both are different,  at least one bit error happened during transmission.

*By this way,  bit errors can be detected if they are not too much clustered.  In ADSL practice,  the CRC procedure is sufficient for bit error detection.

The graph shows an exemplary circuit for the CRC value generation with the generator polynomial  $G(x)$  specified for ADSL – realizable in hardware or software:
[[File:EN_Bei_T_2_4_S5neu.png|right|frame|Cyclic Redundancy Check for ADSL]]

#The data block  $D(x)$  to be tested is introduced into the circuit from the left,  the output is fed back and exclusively-or-linked to the digits of the generator polynomial  $G(x)$.
#After passing through the entire data block,  the memory elements contain the CRC parity-check value  $C(x)$.
#It should be noted that with ADSL the data is split into so-called  "superframes"  of 68 frames each.
#Each frame contains data from the  "fast path"  and the  "interleaved path".  In addition,  management and synchronization bits are transmitted in specific frames.
#Eight CRC bits are formed per ADSL superframe and per path and are transmitted as  "fast byte"  resp.  "sync byte" as the first byte of frame  "$0$"  of the next superframe.

==Scrambler and de–scrambler==
 
Task of the scrambler is to convert long sequences of  "ones"  and  "zeros"  in such a way that frequent symbol changes occur.
*A possible realization is a shift register circuit with feedback exclusive-or-linked branches.

*In order to produce the original binary sequence at the receiver,  a mirror-image self-synchronizing  "de-scrambler"  must be used there.

The graph shows on the left an example of a scrambler actually used at DSL with  $23$  memory elements.  The corresponding de-scrambler is shown on the right.

[[File:EN_Bei_T_2_4_S6.png|right|frame|Scrambler and de-scrambler in a DSL/DMT system     $(1)$   scrambler's input        $(3)$   de-scrambler's input     $(2)$   scrambler's output     $(4)$   de-scrambler's output]]

The transmitter-side shift register is loaded with an arbitrary initial value that has no further effect on the operation of the circuit. Here:
:$$11001'10011'00110'01100'110.$$

If we denote
* by  $e_n$  the bits of the binary input sequence,  and

*by  $a_n$  the bits at the output, 

the following relation holds:

:$$a_n = e_n \oplus a_{n- 18}\oplus a_{n- 23}\hspace{0.05cm}.$$

In the example,  the scrambler input sequence consists of  80  consecutive  "ones"  $($upper left gray background$)$,  which are shifted bit by bit into the scrambler.  The output bit sequence then has frequent  "one-zero"  changes,  as desired.

The de-scrambler  $($shown on the right$)$  can be started at any time with any starting value,  which means that no synchronization is required between the two circuits.  Here:
:$$10111'011110'11101'11011'101.$$
The de-scrambler output data stream shows,
*that the de-scrambler initially outputs some  $($up to a maximum of $23)$  erroneous bits,  but then

*synchronizes automatically,  and then

*recovers the original bit sequence  $($only  "ones"$)$  without errors.

==Forward error correction==
 
For  "forward error correction"  $\rm (FEC)$,  all xDSL variants use a  [[Channel_Coding/Definition_and_Properties_of_Reed-Solomon_Codes|"Reed-Solomon-Code"]].  In some systems  "trellis code modulation"  $\rm (TCM)$  has been made mandatory as an additional error protection measure,  even though it has only been specified as  "optional"  by the international bodies.

Both methods are discussed in detail in the book  [[Channel_Coding|"Channel Coding"]].  Here follows a brief summary of Reed-Solomon coding with respect to its application to DSL:
*With Reed-Solomon encoding,  redundancy bytes are generated for fixed agreed interpolation points of the payload polynomial.  With systematic Reed-Solomon encoding,  a parity-check value is calculated similar to the CRC procedure and appended to the data block to be protected.

*However,  the data is no longer processed  "bit by bit",  but  "byte by byte".  Consequently,  arithmetic operations are no longer performed in the Galois field  $\rm GF( 2 )$  but in  $\rm GF(2^8)$.

The Reed-Solomon parity-check byte can also be determined as the division remainder of a polynomial division,  for xDSL with the following parameters:
#Number  $S$  of DMT symbols to be monitored per Reed-Solomon code word  $(S \ge 1$  for the fast buffer,  $S =2^0$, ... , $2^4$  for the interleaved buffer$)$,
#number  $K$  of user data bytes in the  $S$  DMT symbols,  defined as a polynomial  $B(x)$  of degree  $K$,  where the  "B"  indicates  "bytes",
#Number  $R$  of Reed-Solomon parity-check bytes  $($even number between  $2$  to  $16)$  per parity-check value  $($"fast"  or  "interleaved"$)$,
#sum  $N = K + R$  of the user data bytes and check bytes of the Reed-Solomon code word.

The specifics of Reed-Solomon encoding for xDSL are given here without further comment:
*For xDSL,  the number  $R$  of check bytes must be an integer multiple of the number  $S$  of symbols so that they can be evenly distributed in the payload polynomial.

*The so-called  [https://en.wikipedia.org/wiki/Singleton_bound#MDS_codes "Maximum Distance Separable  $\rm (MDS)$  codes"]  – a subclass of Reed-Solomon codes –  allow the correction of  $R/2$  falsified user data bytes.

*From the selected Reed-Solomon code for the DMT systems,  the constraint is a maximum code word length of  $2^8-1 = 255$  bytes corresponding to  $2040$  bits.

*The redundancy of Reed-Solomon codes can generate a considerable amount of data if the parameters are unfavorable,  thus considerably reducing the net data rate.

*It is recommended that the data amount  $($"gross data rate"$)$  be divided judiciously into useful data  $($"net data rate,  payload"$)$  and error protection data  $($"overhead"$)$.

*Reed-Solomon coding achieves a  "high coding gain".  A system without coding would have to have a signal-to-noise ratio  $\rm (SNR)$  larger by  $3 \ \rm dB$  for same bit error rate.

*By  "trellis-encoded modulation"  $\rm (TCM)$  in combination with other error protection measures,  the coding gain is highly variable; it ranges between  $0 \ \rm dB$  and  $6 \ \rm dB$.

==Interleaving and de–interleaving==
 
The common task of  "interleaver"  $($at the transmitter$)$  and  "de-interleaver"  $($at the receiver$)$  is
*to spread the received Reed-Solomon code words over a larger time range

*in order to distribute any transmission errors over several code words

*and thus increase the chance of correct decoding.

xDSL interleaving is characterized by the parameter  $D$  $($"depth"$)$,  which can take values between  $2^0$  and  $2^9$ .

{{GraueBox|TEXT=
$\text{Example 4:}$  The graph illustrates the principle using the Reed-Solomon code words  $A$,  $B$,  $C$  with five bytes each and the interleaver depth  $D = 2$.
[[File:EN_Bei_T_2_4_S8a.png|right|frame|For DSL interleaving with  $D = 2$]]

Each byte  $B_i$  of the middle Reed-Solomon code word  $B$  is delayed by  $V_i = (D - 1) \cdot i$  bytes. 

Two interleaver blocks are formed:
*The first block contains the bytes  $B_0$,  $B_1$  and  $B_2$  together with the bytes  $A_3$  and  $A_4$  of the previous code word.

*The second block contains the bytes  $B_3$  and  $B_4$  along with the bytes  $C_0$,  $C_1$  and  $C_2$  of the following code word.}}

This  "scrambling"  has the following advantages  $($provided,  $D$  is sufficiently large$)$:
#The error correction capabilities of the Reed-Solomon code are improved.
#The user data rate remains the same,  i.e. is not reduced $($redundancy-free$)$.
#In the event of errors,  entire packets do not have to be repeated at the protocol level.

A disadvantage is that with increasing interleaver depth  $D$  there can be noticeable delay times  $($on the order of milliseconds$)$,  which causes major problems for real-time applications.  However,  interleaving with low depth is only useful if the signal-to-noise ratio is sufficiently high.

{{GraueBox|TEXT=
$\text{Example 5:}$ 
An example of the advantages of interleaver/de-interleaver in the presence of bundle errors is shown in the following graph:
[[File:EN_Bei_T_2_4_S8b.png|right|frame|DSL interleaving and de–interleaving with  $D = 3$]]

*In the first row,  the transmitted byte sequence is shown according to Reed-Solomon encoding,  with each code word consisting of seven bytes as an example.

*In the middle row,  the data bytes are shifted by interleaving with  $D = 3$  so that between  $C_i$  and  $C_{i+1}$  there are two foreign bytes and the green code word is distributed over three blocks.

*Now suppose that during transmission a pulse glitch corrupted '''KORREKTUR: falsified''' three consecutive bytes in a single data block.

*After the de-interleaver,  the original byte sequence of the Reed-Solomon code words is restored,  with the three corrupted '''KORREKTUR: falsified''' bytes distributed among three independent code words.

*If two redundancy bytes were inserted in each case during the Reed-Solomon encoding,  the now separated byte corruptions '''KORREKTUR: falsifications''' can be completely corrected.}}

==Gain scaling and tone ordering==
 
A particularly advantageous feature of DMT is the possibility
#to adjust the bins individually to the existing channel characteristics and
#possibly to switch off "bins" with unfavorable SNR completely.

[[File:EN_Bei_T_2_4_S9.png|right|frame|Bit-bin assignment based on SNR]]
The procedure is as follows:

*Before starting the transmission  – and possibly also dynamically during operation –  the DMT modem measures the channel characteristics for each  "bin"  and sets the maximum transmission rate individually according to the SNR  $($see graphic$)$.

*During initialization,  the  "ADSL Transceiver Units"  exchange bin information,  for example the respective  "bits/bin"  and the required transmission power  $($'"gain"$)$.

*Thereby the  $\text{ATU-C}$  sends information about the upstream and the  $\text{ATU-R}$  sends information about the downstream.

*This message is of the format  $\{b_i, g_i\}$  where  $b_i$  $($four bits$)$  indicates the constellation size.  For the upstream,  the index  $i = 1$, ... , $31$  and for the downstream  $i = 1$, ... , $255$.

*The gain  $g_i$  is a fixed-point number with twelve bits.  For example  $g_i = 001.010000000$  represents the decimal value  $1 + 1/4 =1.25$.

*This indicates that the signal power of channel  $i$  must be higher by  $1.94 \ \rm dB$  than the power of the test signal transmitted during the channel analysis.
 
When operating the fast path and the interleaved path simultaneously $($see  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Overview_of_DSL_error_correction_measures|"graphic"]]  in the section  "DSL error correction measures"$)$,  the bit error rate can be further reduced by optimized carrier frequency allocation  $($"Tone Ordering"$)$.  The background of this measure is again  "clipping"  $($truncation of voltage peaks$)$,  which worsens the overall SNR. This procedure is based on the following rules:
*Bins with dense constellation  $($many bits/bin   ⇒   larger clipping probability$)$  are assigned to the interleaved branch,  since this is per se more reliable due to the additional interleaver.  Accordingly,  the subchannels with low order allocation  $($few bits/bin$)$  are reserved for the fast data buffer.

*New tables are then sent for upstream and downstream,  in which the bins are no longer ordered by index,  but according to the bits/bin ratios.  Based on this new table,  it is possible for the  $\text{ATU-C}$  or  $\text{ATU-R}$  to perform bit extraction successfully.

==Inserting guard interval and cyclic prefix ==
 
In the chapter  [[Modulation_Methods/Implementation_of_OFDM_Systems#Guard_interval_to_reduce_intersymbol_interference| "Realization of OFDM systems"]]  of the book "Modulation Methods" it has already been shown that by inserting a guard interval. The bit error rate can be decisively improved in the presence of linear channel distortion.

We assume that the cable impulse response  $h_{\rm K}(t)$  extends over the time duration  $T_{\rm K}$  . Ideally  $h_{\rm K}(t) = δ(t)$  and accordingly an infinitely short extension:   $T_{\rm K} = 0$. For distorting channel  $(T_{\rm K} > 0 )$  holds:
*By inserting a ''guard interval''  of duration  $T_{\rm G}$  ''intersymbol interference''  between each DSL frame can be avoided as long as  $T_{\rm G}$ ≥ $T_{\rm K}$  holds. However, this measure leads to a rate loss by a factor  $T/(T + T_{\rm G})$  with symbol duration  $T = {1}/{f_0}$.
*But with this, there is still ''inter-carrier interference''  between each subcarrier within the same frame, that is, the  [[Modulation_Methods/General_Description_of_OFDM#System_consideration_in_the_frequency_domain_with_causal_basic_pulse|"DMT individual spectra"]]  are no longer  $\rm si$-shaped and de-orthogonalization occurs.
*By a  [[Modulation_Methods/Implementation_of_OFDM_Systems#Cyclic_Prefix|"cyclic prefix"]]  also this disturbing effect can be avoided. Here one extends the transmission vector  $\mathbf{s}$  forward by the last  $L$  samples of the IDFT output, where the minimum value for $L$ is given by the duration  $T_{\rm K}$  of the cable impulse response.

{{GraueBox|TEXT=
$\text{Example 6:}$ 
The graphic shows this measure with the DSL/DMT method, for which the parameter  $L = 32$  has been set.

[[File:P_ID1964__Bei_T_2_4_S10a_v1.png|right|frame|DMT transmission signal with cyclic prefix]]
*The samples  $s_{480}$ , ... , $s_{511}$  are added as prefix  $(s_{-32}$ , ... , $s_{-1})$  to the IDFT output vector  $(s_0$ , ... , $s_{511})$  .
*The transmitted signal  $s(t)$  now has the resulting duration  $T ≈ 232 \ {\rm µ s}$  instead of the symbol duration  $T + T_{\rm G} = 1.0625 \cdot T ≈ 246 \ {\rm µ s}$. This reduces the rate by a factor of  $0.94$ .
*In the receive-side evaluation, one is restricted to the time range from  $0$  to  $T$. In this time interval the disturbing influence of the impulse response has already decayed and the subchannels are orthogonal to each other - just as with ideal channel.
*The sample values  $s_{-32}$ , ... , $s_{-1}$  are discarded at the receiver - a rather simple realization of signal equalization.}}

The last diagram in this chapter shows the entire DMT transmission system, but without the  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Overview_of_DSL_error_correction_measures| "error protection measures"]] described earlier. You can see:

[[File:P_ID1967__Bei_T_2_4_S10_v1.png|right|frame|DMT–System with cyclic prefix '''Korrektur''' das Bild muss ich noch machen]]
*In the "Add cyclic prefix" block, the samples  $s_{480}$, ... , $s_{511}$  as  $s_{-32}$, ... , $s_{-1}$  added. The transmit signal  $s(t)$  thus has the course shown in  $\text{Example 6}$  .
*The received signal  $r(t)$  results from the convolution of  $s(t)$  with  $h_{\rm K}(t)$. After A/D conversion and removal of the cyclic prefix, the input values  $r_0$, ... , $ r_{511}$  for the DFT.
*The (complex) output values  $D_k\hspace{0.01cm}'$  of the DFT depend only on the particular (complex) data value  $D_k$  . Independently of other data  $D_κ (κ ≠ k)$  holds with the noise value  $n_k\hspace{0.01cm}'$:

:$${D}_k\hspace{0.01cm}' = \alpha_k \cdot {D}_k + {n}_k\hspace{0.01cm}', \hspace{0.2cm}\alpha_k = H_{\rm K}( f = f_k)
\hspace{0.05cm}. $$

*Each carrier  $D_k$  is modified in its amplitude and phase by its own (complex) factor  $α_k$, which depends only on the channel. The frequency domain equalizer has only the task of multiplying the coefficient  $D_k\hspace{0.01cm}'$  by the inverse value  ${1}/{α_k}$  . Finally, one obtains:

:$$ \hat{D}_k = {D}_k + {n}_k \hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*This simple realization possibility of the complete equalization of the strongly distorting cable frequency response was one of the decisive criteria that for  $\rm xDSL$  the  $\rm DMT$ method has prevailed over  $\rm QAM$  and  $\rm CAP$ .
*Mostly an additional pre-equalization in the time domain takes place directly after the A/D conversion to avoid also the intersymbol interference between adjacent frames.}}

==Exercises for the chapter==
 
[[Exercise_2.5:_DSL_Error_Protection|"Exercise 2.5: DSL Error Protection"]]

[[Exercise_2.5Z:_Reach_and_Bit_Rate_with_ADSL|"Exercise 2.5Z: Reach and Bit Rate with ADSL"]]

[[Exercise_2.6:_Cyclic_Prefix|"Exercise 2.6: Cyclic Prefix"]]
==References==
<references />

{{Display}}

Examples of Communication Systems/xDSL as Transmission Technology

2023-03-21T19:53:04Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=xDSL Systems
|Nächste Seite=Methods to Reduce the Bit Error Rate in DSL
}}

==Possible bandwidth allocations for xDSL==
 
The xDSL specifications give operators a great deal of freedom with regard to allocation.

For the necessary directional separation of the xDSL signal transmission according to
*the downstream direction from the provider to the customer  $($with the highest possible data rate$)$,

*the upstream direction from the customer to the provider  $($with mostly lower data rate$)$,

two variants have been standardized for this purpose:

{{BlaueBox|TEXT=
$\text{Definition:}$ 
*With the  »'''frequency separation method'''«  the data streams for the two directions are transmitted in two separate frequency bands with the advantage that a simple filter is sufficient to separate the transmission directions,  which simplifies the technical implementation.

*With the  »'''frequency uniform position method'''«  the spectra of upstream and downstream overlap in a certain part.  The separation is done here with help of an echo cancellation circuit.  Advantages of this method:  The lower bandwidth requirement at higher  $($and thus more attenuated$)$  frequencies and longer range.}}

[[File:EN_Bei_T_2_3_S1a_v4.png|right|frame|"Frequency separation method"  &  "Frequency uniform position method"]]
The graph compares these two options.  Basically,  the specifications leave it up to the developers/operators to decide,
*to operate xDSL alone on the subscriber line,  or

*to allow mixed operation of xDSL with the telephone services  $\rm POTS$  $($"Plain Old Telephone Service"$)$  or  $\rm ISDN$  $($"Integrated Services Digital Network"$)$,  and
*thus to exclude or also occupy the lower frequency range occupied by the two telephone services for xDSL.
 
==ADSL bandwidth allocation==
 
Since it is technically much easier to realise, the decision in Germany for "ADSL" and "ADSL2+" was in favour of

[[File:EN_Bei_T_2_3_S1b_v4.png|right|frame|ADSL bandwidth allocation in Germany and most of other European countries]]
*of the  "frequency separation method",  and
*the general reservation of the lower frequency range for ISDN.

The  "frequency uniform position method"  is still used in some cases,  but rather rarely.

The bandwidth available for DSL is not further decomposed for the transmission methods
*[[Modulation_Methods/Quadrature_Amplitude_Modulation|$\rm QAM$]]  $($"Quadrature Amplitude Modulation"$)$  and

*[[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Carrierless_Amplitude_Phase_Modulation|$\rm CAP$]]  $($"Carrierless Amplitude Phase Modulation$)$.

In contrast,  with the multicarrier method  [[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|$\rm DMT$]]  $($"Discrete Multitone Transmission"$),$  the uplink and the downlink channel are divided into  $N_{\rm Up}$  resp.  $N_{\rm Down}$  bins of  $4.3125\rm \ kHz $  each.

In addition to the above graph, it should be noted:
#Phone services  $($POTS or ISDN$)$  and xDSL are in different frequency bands,  which minimizes mutual interference in the trunk cable.  Thus,  the stronger signal ISDN does not interfere with the parallel running xDSL and vice versa.
#The lower frequency range up to  $\text{120 kHz}$  has been reserved for ISDN  $($optionally POTS$)$.  This value results from the first zero of the  [[Digital_Signal_Transmission/Block_Coding_with_4B3T_Codes#ACF_and_PSD_of_the_4B3T_codes|"ISDN spectrum with 4B3T coding"]].  Above $\text{120 kHz}$  the ISDN spectrum is completely suppressed.
#To separate the telephone and xDSL signals,  a  [[Examples_of_Communication_Systems/xDSL_Systems#Components_of_DSL_Internet_access|"splitter"]]  is used at both ends of the two-wire line,  which includes one low-pass filter and one high-pass filter each and also takes into account the following frequency gap up to  $\text{138 kHz}$.
#After this occupancy gap follows the ADSL upstream band from  $\text{138 kHz}$  to $\text{276 kHz}$.  This allows the transmission of  $N_{\rm Up} = 32$  subcarriers with bandwidth  $\text{4.3125 kHz}$  each.  This value is derived from the frame transmission rate.
#The subsequent downstream range extends to  $\text{1104 kHz}$  for ADSL,  with which  $N_{\rm Down} = 256$  subcarriers can be realized.  The separation of the upstream and the downstream channel in xDSL is done by a band-pass filter in the modem.
#However,  the first  $64$  subcarriers  $($this corresponds to $\text{276 kHz)}$  must not be occupied.  With the  [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Possible_bandwidth_allocations_for_xDSL|"frequency uniform position method"]]  only  $32$  subcarriers would have to be left out,  taking into account that the separation of uplink and downlink requires a more complex implementation.
#For  ADSL2+,  the system bandwidth is  $\text{2208 kHz}$   ⇒   $N_{\rm Down} = 512$  subcarriers.  The number of bins to be spared remains unchanged compared to ADSL.  Taking into account that two bins are occupied by control functions  $($e.g. for synchronization$)$,  $190$  $($ADSL$)$  or  $446$  $($ADSL2+$)$ downstream channels remain for users.
#The ISDN reservation prescribed in Germany,  however,  has the consequence for xDSL that the low frequencies,  which are by far the least attenuated in a copper line and would therefore actually be the most suitable,  cannot be used.
#Further,  from the frequency arrangement,  it can be seen that the downstream bins are more attenuated than the upstream bins  $($higher frequencies$)$  and consequently have a smaller signal-to-noise ratio  $($SNR$)$.
#The decision  "upstream below downstream"  is related to the fact that the loss of downstream channels has only a comparatively small impact on the transmission rate.  In the upstream,  such a failure would be much more noticeable in percentage terms.

==VDSL(2) bandwidth allocation==
 
The ITU has defined several profiles for  "VDSL(2)".  At the time of writing this chapter  (2010),  the frequency band assignment specified in the graphic applies to the systems deployed in Germany in accordance with the ITU's  $\text{ VDSL(2) Plan 998b - Profile 17a (Annex B)}$.  The (slightly) lighter coloring at the higher frequencies is meant to indicate that these channels are more attenuated.

Without claiming to be exhaustive, this allocation plan can be characterized as follows:

[[File:EN_Bei_T_2_3_S5a_ganzneu.png|right|frame|VDSL(2) bandwidth allocation in Germany ]]

#To achieve higher bit rates,  eight times as many bins are used here as in  "ADSL2+".
#Thus,  the bandwidth enables  $8 \cdot \text{2208 MHz = 17664 MHz}$,  transmission rates of up to  $\text{100 Mbit/s}$  $($depending on cable length and conditions$)$.
#The same as in ADSL,  the frequency bands for the upstream subchannels are always arranged at the lower frequencies,  since the greater cable attenuation  has a greater percentage influence on the bit rate for upstream than for downstream.
#In VDSL(2),  the  [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Possible_bandwidth_allocations_for_xDSL|"frequency separation method"]]  is always used.  Overlapping of the upstream and downstream frequency bands is categorically excluded in the ITU specification .
#In the VDSL systems,  the lower frequencies are reserved for ISDN.  This is followed by alternating ranges for upstream and downstream.  From the specified limits,  one can recognize the narrower upstream ranges,  compared to the downstream.
#You can see an alternating arrangement of upstream and downstream areas.  One reason is that with this wide spectrum it should be avoided that one direction  $($e.g. downstream$)$  is assigned only strongly attenuated  $($high$)$  frequencies.

The VDSL(2) specification provides for allocation plans up to system bandwidths of  $\text{30 MHz}$  $($according to profile 30a$)$.
* This should enable transmission rates up to about  $\text{ 200 Mbit/s}$  over short distances.

*For this purpose, the bandwidth of the individual subchannels is also doubled compared to ADSL to  $\text{8.625 kHz}$ .

==Overview of xDSL transmission methods==
 
At the beginning of the various standardization procedures for the individual xDSL variants,  different transmission methods were defined as a basis:
# [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modulation"]]  $\text{QAM}$  for  "QAM-ADSL"  and  "QAM-VDSL", 
# [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Carrierless_Amplitude_Phase_Modulation|"Carrierless Amplitude Phase Modulation"]]  $\text{CAP}$  for  "CAP-HDSL"  and  "CAP-ADSL",
#[[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|"Discrete Multitone Transmission"]]  $\text{DMT}$  for  "ADSL",  "ADSL2",  "ADSL2+",  "VDSL"  and",  "VDSL2".

With increasing market demand for higher transmission rates and the associated requirements, two main  suitable processes crystallized,  namely  "QAM/CAP"  and  "DMT".

Since the manufacturers were unable to agree on a common standard from 1997 to 2003,  also for patent reasons  $($one even speaks of  "line code wars"  in this context$)$,  the two competing methods coexisted for a long time.  At the so-called  "DSL Olympics"  in 2003,  '''the decision was finally made in favor of DMT''',
*on the one hand,  because of the slightly better  "performance"  in general,

*but in particular because of its higher robustness against narrowband interference; especially for the USA  $($many overhead telephone lines and related problems with coupled radio signals$)$  this argument played a major role.

First,  however,  the systems  "QAM-xDSL"  and  "CAP-xDSL"  will be considered very briefly.

==Basics of Quadrature Amplitude Modulation==
 

The principle has already been described in detail in the chapter  [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modation"]]  of the book  "Modulation Methods". 
[[File:EN_Bei_T_2_3_S6a.png|right|frame| Linear modulator with  $\rm I$ and  $\rm Q$–components;  signal space for  $\text{16-QAM}$]]
Here follows short summary on the basis of the block diagram on the right:
*QAM is a  "single carrier modulation method"  around the carrier frequency  $f_{\rm T}$.  First,  a blockwise serial/parallel conversion of the bit stream and signal space assignment is performed.

*From each  $b$  binary symbols,  two multilevel '''KORREKTUR: multi-level''' amplitude coefficients  $a_{{\rm I}n}$  and  $a_{{\rm Q}n}$  are derived  $($"in-phase component"  and  "quadrature component"$)$.  Both coefficients can each take one of  $M = 2^{b/2}$  possible amplitude values.

*The example considered here applies to the  $\text{16-QAM}$  with  $b = M = 4$  and correspondingly sixteen  signal space points. For a   $\text{256-QAM}$  would hold:   $b = 8$  and  $M = 16$  $(2^b = M^2 = 256)$.

*The coefficients  $a_{{\rm I}n}$  and  $a_{{\rm Q}n}$  are each impressed on a Dirac delta impulse as weights.  For pulse shaping one usually uses  $($because of the small bandwidth$)$  a raised-cosine filter.

*With the basic transmission pulse  $g_s(t)$  is then valid in the two branches of the block diagram:
:$$ s_{\rm I}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm
I\hspace{0.03cm}\it n} \cdot g_s (t - n \cdot T)\hspace{0.05cm},\hspace{0.5cm}
s_{\rm Q}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm
Q\hspace{0.03cm}\it n} \cdot g_s (t - n \cdot
T)\hspace{0.05cm}.$$
:Note:  Because of the redundancy-free conversion to a higher-level code,  the symbol duration  $T$  of these signals is greater by a factor  $b$  than the bit duration  $T_{\rm B}$  of the binary input sequence.  In the drawn example  $($16-QAM$)$,  $T = 4 ⋅ T_{\rm B}$.
*The  »'''QAM transmitted signal'''«  $s(t)$  is then the sum of the two partial signals  $ s_{\rm I}(t)$  and  $ s_{\rm Q}(t)$,  multiplied by  "cosine"  resp.  "minus-sine"  $($possibly followed by a band limit to prevent interference to adjacent bands, as indicated in the graph below$)$:
:$$s(t) = s_{\rm I}(t) \cdot \cos (2 \pi f_{\rm T}\,t) - s_{\rm Q}(t) \cdot \sin (2 \pi f_{\rm T}\,t)
\hspace{0.05cm}. $$
*The two branches  $(\rm I$  and  $\rm Q)$  can be considered as two completely separate  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#M.E2.80.93level_amplitude_shift_keying_.28M.E2.80.93ASK.29 |"M–level ASK systems"]]  which do not interfere with each other as long as all components are optimally designed.  This means at the same time:  

*Compared to a  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_phase_shift_keying_.28BPSK.29|"Binary Phase Shift Keying"]]  $\rm (BPSK)$  modulation with cosine or sine only,  the  '''QAM allows a doubling of the data rate at constant quality'''.

[[File:EN_Bei_T_2_3_S6b.png|right|frame|Quadrature Amplitude Modulation as band-pass and equivalent low-pass model]]

'''Equivalent low-pass model'''

The graph on the right shows
*above the band-pass model,  identical with the last graph,

*below the  "equivalent low-pass model".

Here,  the inphase and quadrature coefficients are combined to give the complex amplitude coefficient
:$$a_n = a_{\text{I}n} + {\rm j} ⋅ a_{\text{Q}n}.$$
Additionally, the physical transmitted signal  $s(t)$  is replaced by the equivalent low-pass signal
:$$s_{\rm TP}(t) = s_{\rm I}(t) + {\rm j} ⋅ s_{\rm Q}(t).$$

⇒   The representation of the QAM transmitter and receiver is the content of the SWF flash animation  [[Applets:Principle_of_QAM|"Principle of Quadrature Amplitude Modulation"]].
 
{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*As the bit number  $b$  and thus the number of defined symbols  $(M=b^2)$  increases,  the bandwidth efficiency increases,  but also the signal processing overhead increases.

*In addition,  it must be taken into account that a dense QAM allocation is appropriate only if the channel is sufficiently good.}}

'''Reference Model for ADSL-QAM'''

This diagram shows the reference model for  "ADSL-QAM",  whereby we will only deal with the red function blocks  "QAM-Modulator"  and  "'QAM-Demodulator"  here.
[[File:EN_Bei_T_2_3_S5a.png|right|frame|Reference Model for ADSL-QAM]]

*The carrier frequency $f_{\rm T}$  lies within the specified up and down band of the respective xDSL variant.

*Like signal space size  $($between  $M^2=2^2=4$  and  $M^2=2^8=256$  signal space points$)$  and symbol rate, it is determined by channel measurements during transmission initialisation.

*For ADSL-QAM,  the following symbol rates have been specified  $($all numerical values in  "${\rm kBaud}$" $= 10^3 \rm symbols/s)$  :
:*In upstream:  $20$, $40$, $84$, $100$, $120$, $136$;

:*in downstream: $40$, $126$, $160$, $252$, $336$, $504$, $806.4$, $1008$.

==Possible QAM signal space constellations==
 
We still consider possible arrangements of signal space points in quadrature amplitude modulation using three examples.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
An important QAM parameter is the bit number  $b$  processed to the amplitude coefficient pair  $(a_{\rm I}, a_{\rm Q})$.  Here  $b$  is always an even number.
[[File:EN_Bei_T_2_3_S6a_v2.png|right|frame|Signal space constellations  "$\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($left$)$  and  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($right$)$
'''Korrektur''': Jiwoo, ich habe die gelben Punkte durch braune Punkte ersetzt, weil meine schlechten Auge diese nicht erkennen konnten. Könntest u bitte den Text kontrollieren und mit der deutschen Version vergleichen, ob jetzt wirklich alle gelben Punkte braun sind. Das gleiche gilt für Example 3.]]

*If  $b = 2$,  both  $a_{\rm I}$  and  $a_{\rm Q}$  can only take the values  $±1$  resulting in the  $\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$  corresponding to the left constellation.

*According to an ITU recommendation,  the assignment applies here:
:$$q_1 = 0, \ q_0 = 0 \, \Leftrightarrow \,a_{\rm I} = +1, \ a_{\rm Q} = +1,$$
:$$q_1 = 0, \ q_0 = 1 \, \Leftrightarrow \, a_{\rm I} = +1, \ a_{\rm Q} = -1,$$
:$$q_1 = 1, \ q_0 = 0 \, \Leftrightarrow \,a_{\rm I} = -1, \ a_{\rm Q} = +1,$$
:$$q_1 = 1, \ q_0 = 1 \, \Leftrightarrow \, a_{\rm I} = -1, \ a_{\rm Q} = -1.$$

*The point  '''10'''   marked in brown  $(a_{\rm I} = -1, \ a_{\rm Q} = 1)$  thus stands for  $q_1 = 1$  and  $q_0 = 0$.

With  $b = 4$   ⇒   $M = 2^{b/2} = 4$  one arrives at  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  according to the right diagram with the possible amplitude coefficients 
:$$a_{\rm I} ∈ \{±3, ±1\}, \ \ a_{\rm Q} ∈ \{±3, ±1\}.$$

*The assignment can be determined using the auxiliary graph given on the left below, as illustrated by the following numerical examples.

        $\rm (A)$    $q_3 = 1, \ q_2 = 0, \ q_1 = 1,\ q_0 = 1$  $($brown marker$)$:
:#The two most significant bits  $\rm (MSB)$   ⇒   "'''11'''"  determine,  according to the  $\text{4-QAM}$  diagram,  the  $($second$)$  quadrant in which the symbol is located.
:#The two least significant bits  $\rm (LSB)$   ⇒   "'''10'''"  together with the auxiliary graph determine the point within this quadrant.  Result:  $a_{\rm I} = -1$,  $a_{\rm Q} = +3$.
        $\rm (B)$    $q_3 = 0, \ q_2 = 1, \ q_1 = 1,\ q_0 = 0$ $($green marker$)$:
:#The two most significant bits   ⇒   "'''01'''"   refer here to the fourth quadrant.
:#The two least significant bits   ⇒   "'''10'''"  refer to the green dot in the fourth quadrant:   $a_{\rm I} = -3, \ a_{\rm Q} = -3$.}}

{{GraueBox|TEXT=
$\text{Example 2:}$  We use the same graphic as in $\text{Example 1}$.  The decimal value  $D$  provides another way to label the points.
[[File:EN_Bei_T_2_3_S6a_v2.png|right|frame|Signal space constellations  "$\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($left$)$  and  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($right$)$]]

*The brown marked point in the  $\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$ diagram is binary labeled  "'''10'''"    ⇒   decimal:  $D = 2$.  This point simultaneously marks the second quadrant of  $\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$.

*The further subdivision results from the lower left graphic.  There at the brown dot is written  $4D + 3=11$"  $($decimal$)$.

*Therefore,  the upper right dot  $($highlighted in brown$)$  in the upper left quadrant stands for decimal  "$11$   ⇒   binary '''1011'''.

*For the green dot,  $D = 1$  gives the decimal value  $4D + 2 =6$,$11$ which corresponds to the binary representation  "'''0110'''" .
 
According to this scheme,  the signal space constellations
*for  $\rm 64\hspace{0.05cm}–\hspace{-0.02cm}QAM$   ⇒   $(b = 6, \ M = 8)$ 

*and  $\rm 256-QAM$  ⇒   $(b = 8, \ M = 16)$ 

which will be discussed in detail in the  [[Exercise_2.3:_QAM_Signal_Space_Assignment|$\text{Exercise 2.3}$]].  }}

{{GraueBox|TEXT=
$\text{Example 3:}$ 
We still consider for the described  $\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$  $($left graph,  here referred to as  "ITU proposal"$)$  the resulting symbol and bit error probability in the presence of AWGN noise:
#An error can be assumed to result in a horizontally or vertically adjacent symbol,  as indicated for the upper left green dot.
#The symbol error probability $p$  depends on the Euclidean distance of the two points and the AWGN noise power density  $N_0$.
#A distortion to the more distant blue point instead of one of the two neighboring brown points is rather unlikely with Gaussian noise.
[[File:EN Bei T 2 3 S6b neu.png|right|frame|Signal space constellation:  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$";  left: ITU propsal,  right:  Gray encoding. '''KORREKTUR: Symbolfehler, Bitfehler''']]

An analysis shows:
*All corner points  $($green$)$  can only be distorted in two directions.

*In contrast, the inner QAM points  $($blue$)$  have four direct neighbors.

* The remaining symbols  $($brown$)$  have three neighbors.

For the  »'''symbol error probability'''«   $p_{\rm S}$  then holds:

::$$p_{\rm S} = {1}/{16} \cdot (4 \cdot 2 p + 8 \cdot 3 p + 4 \cdot 4 p) = 3p.$$

To calculate the  »'''bit error probability'''«   $p_{\rm B}$  it must now be taken into account that for the left constellation one symbol error leads
* within one quadrant to only  one bit error  $($e.g: "'''0100'''"   ⇒   "'''0110'''"$)$,

*between adjacent quadrants to two bit errors $($e.g: "'''1111'''"   ⇒   "'''0101'''"$)$.

The computation of  $p_{\rm B} $  here involves some effort.

In contrast,  with Gray encoding  $($right diagram$)$:
# Each symbol differs from its neighbors by exactly one bit.
#Each symbol error thus results in exactly one bit error.
#Since each individual symbol contains four bits,  the  "bit error probability" is in this case: 

::$$p_{\rm B} = p_{\rm S}/4 = 3/4 \cdot p. $$}}

==Carrierless Amplitude Phase Modulation==
 
"Carrierless amplitude phase modulation"  $\rm (CAP)$  is a bandwidth-efficient variant of QAM,  which can be realized very easily with digital signal processors.  The only difference to QAM is that the modulation with a carrier signal can be omitted.

[[File:EN_Bei_T_2_3_S7a.png|right|frame|Block diagram of  "Carrierless Amplitude Phase Modulation"]]
[[File:EN_Bei_T_2_3_S7d_neu.png|right|frame|Reference model for  ADSL–CAP]]

*Digital filtering is used instead of multiplication by cosine and minus sine.  $g_{\rm I}(t)$  and  $g_{\rm Q}(t)$  are impulse responses of two transversal bandpass '''KORREKTUR: band-pass''' filters with same amplitude characteristic,  phase-shifted by  $π/2$.

*Both are orthogonal to each other,  which means that the integral of the product  $g_{\rm I}(t) · g_{\rm Q}(t)$  over a symbol duration gives zero.

*The signals  $s_{\rm I}(t)$  and  $s_{\rm Q}(t)$  generated in this way are combined,  converted to a continuous-time signal by a D/A converter. 

*The unwanted high-frequency components generated during D/A conversion are eliminated by a low-pass filter before transmission.

*At the receiver,  the signal  $r(t)$  is first converted to a discrete-time signal using an A/D converter and then the in-phase and quadrature symbols  $a_{\rm I}$  and  $a_{\rm Q}$  are extracted via two  "finite impulse response  $\rm (FIR)$  filters"  and downstream "threshold decision".

CAP was the de facto standard in initial ADSL specifications until 1996.  The graph shows the reference model.
#The frequencies up to  $\text{4 kHz}$  were reserved for POTS/ISDN.
#The up channel occupied the frequency range of  $\text{15 - 160 kHz}$,
#The down channel occupied the frequencies from  $\text{240 - 1500 kHz}$.

One problem with CAP is that a  "bad channel" has dramatic consequences on the transmission quality.  Therefore,  today (2011) ADSL-CAP is only found in a few HDSL variants.

==Basics of Discrete Multitone Transmission==
 
[[File:EN_Bei_T_2_3_S8a_v8.png|right|frame|Spectra at OFDM and DMT;     Note:   For this graph a symmetrical rectangle is assumed.     A rectangle between  $0$  and  $T$  would still result in a phase term.     However,  nothing would change with respect to  $|S(f)|$.]]
"Discrete Multitone Transmission"  $\rm (DMT)$  refers to a multicarrier modulation method that is almost identical to  [[Modulation_Methods/General_Description_of_OFDM|"Orthogonal Frequency Division Multiplexing"]]  $\rm (OFDM)$.  In the case of wired transmission,  one usually refers to  "DMT",  and in the case of wireless transmission,  one refers to  "OFDM".

In both cases,  one divides the entire bandwidth into many narrowband equidistant subchannels.  The respective subcarrier signals  $s_k(t)$  are individually impinged with complex data symbols  $D_k$  and the sum of the modulated subcarrier signals is transmitted as signal  $s(t)$ .

The diagram illustrates the principle of OFDM and DMT in the frequency domain,  partly using the values specified for ADSL–DMT:

*$255$  Subcarriers with carrier frequencies  $k \cdot f_0$  $(k = 1$, ... , $255)$.

*$4000$  data frames are transmitted per second.

*After  $68$  data frames,  one synchronization frame is inserted each.

*Due to the  "cyclic prefix"  $($see chapter  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Inserting_guard_interval_and_cyclic_prefix|"Insertion of Guard–Interval and Cyclic Prefix"]]$)$  the symbol duration  $T = 1/f_0$  must still be shortened by the factor  $16/17$.

*Thus,  the  »'''basic frequency'''«  of DMT is 
:$$f_0 = 4 \cdot (69/68) \cdot (69/68) = 4.3125 \ \rm kHz.$$

An essential difference between OFDM and DMT is that
*in OFDM,  the above drawn spectrum  $S(f)$  in reality describes an  "equivalent low-pass spectrum"  $S_{\rm TP}(f)$  and still the shift around a carrier frequency  $f_{\rm T}$  has to be considered:

:$$S_{\rm TP}(f ) = \sum_{k = 1}^{255} D_k \cdot \delta (f - k \cdot f_0)\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
S(f) = \frac{1}{2} \big [ S_{\rm TP}(f - f_{\rm T}) + S^*_{\rm TP}(-(f + f_{\rm T}))\big ]
\hspace{0.05cm};$$

*in DMT,  on the other hand,  the components at negative frequencies must still be taken into account,  which are to be weighted with the conjugate-complex spectral coefficients:

:$$S(f ) = \sum_{k = 1}^{255} \big [ D_k \cdot \delta (f - k \cdot f_0) + D^*_k \cdot \delta (f + k \cdot f_0) \big ]
\hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Please note:}$ 
#According to these equations,  the complex OFDM signal  $s_{\rm OFDM}(t)$  consists of  $K = 255$  complex exponential oscillations.
#The DMT signal  $s_{\rm DMT}(t)$  is composed of as many cosine oscillations with frequencies  $k \cdot f_0$  $($full occupancy presupposed$)$.
#Despite complex coefficients  $D_k$ resulting from QAM occupancy of the carriers,  the DMT signal is always real because of the conjugate-complex complements at negative frequencies.}}

However,  in both OFDM and DMT,  the transmitted signal  $s(t)$  is precisely limited in time to the symbol duration  $T = 1/f_0 ≈ 232 \ {\rm µs}$  which is equivalent to multiplying by a rectangle of duration $T$.  In the spectral domain,  this corresponds to convolution with the function  $\text{sinc}(fT)$:
*Each Dirac delta function at  $k · f_0$  thus becomes an sinc– function at the same position,  given the time limit,  as shown in the second diagram of the graph above.

*Adjacent subcarrier spectra overlap on the frequency axis,  but exactly at  $k · f_0$  the coefficients  $D_k$  can be seen again,  since all other spectra have zeros here.

{{GraueBox|TEXT=
$\text{Example 4:}$ 
Assuming the conditions favorable for ADSL downstream,  viz.
#$4000$ frames are transmitted per second.
#All subcarriers are active at all times  $(K = 255)$.
#Each carrier is occupied with a 1024-QAM  $(b = 10)$;  according to ITU:  $8 ≤ b ≤ 15$.
#Ideal conditions prevail,  so that the orthogonality evident in the graph is preserved,

then for the  »'''maximum bit rate'''« 
:$$R_{\rm B,\ max} = 4000 · K · b ≈ 10 \ \rm Mbit/s.$$

However,  specified ADSL downstream is only  $2 \ \rm Mbit/s$  due to
*the omission of the  $64$  lowest carriers because of ISDN and upstream,

*the QAM occupancy of the heavily attenuated carriers with less than  $10$  bits,  and

*the consideration of the cyclic prefix,  and some operational reasons.}}

==DMT realization with IDFT/DFT==
 
The upper graph shows the complete DMT system.  For now we will focus on the two red blocks.  The blue blocks will be covered in the [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL|"next chapter"]].  Simplified,  the transmitter and receiver can be represented as shown in the second graph:

[[File:EN_Bei_T_2_3_S9d_ganz_neuV8.png|right|frame|Discrete multitone transmission system]]
[[File:P_ID1951__Bei_T_2_3_S9b_v3.png|right|frame|DMT transmitter and receiver '''Korrektur''' das Bild muss ich noch machen]]

*To perform a DMT modulation,  the transmitter accumulates a block of input bits in a data buffer to be transmitted as one frame.

*The QAM encoder supplies per frame the complex-valued data symbols  $D_1$,  ... ,  $D_{255}$,  which is expanded to the vector  $\mathbf{D}$  of length  $512$  by
#   $D_0 = D_{256} = 0$,
#   $D_k = D^\star_{512-k} \ (k = 257,$ ... , $511)$,
#   the coefficients   $D_{257}$, ... , $D_{511}$  are identical to  $D_{-255}$, ... , $D_{-1}$ $($as consequence of  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Finite_signal_representation|"finite signals"]]  properties$)$.
*The spectral samples  $\mathbf{D}$  are transformed with the [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Inverse_discrete_Fourier_transform|"Inverse Discrete Fourier Transform"]]  to the vector  $\mathbf{s}$  of time signal samples, also with length $512$. Because of the conjugate-complex assignment in the spectral domain   ⇒   $\text{Im}[\mathbf{s}] = 0$.

*After parallel/serial and digital/analog conversion and low-pass filtering of  $\text{Re}[\mathbf{s}]$  the physical  $($real$)$  as well as continuous-time transmitted signal  $s(t)$  results.

*For this,  in the range  $0 ≤ t ≤ T$  $($factor  $2$,  since two coefficients each contribute to cosine/sine$)$:

:$$s(t) = \sum_{k = 1}^{255} \big [ 2 \cdot{\rm Re}\{D_k\} \cdot \cos(2\pi k f_0 t ) - 2 \cdot{\rm Im}\{D_k\} \cdot \sin(2\pi k f_0 t )\big ] \hspace{0.05cm}. $$

*The received signal at the AWGN output is  $r(t) = s(t) + n(t)$.  After A/D and S/P conversion,  $r(t)$  can be expressed by the (real) vector  $\mathbf{r}$ . The  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#From_the_continuous_to_the_discrete_Fourier_transform|"Discrete Fourier Transform"]]  then provides estimates for the transmitted spectral coefficients.
 

{{GraueBox|TEXT=
$\text{Example 5:}$ 
As an example, let's consider the ADSL/DMT downstream.
[[File:EN_Bei_T_2_3_S10_neu.png|right|frame|Allocation of the DMT frequency band with QAM coefficients]]
*In the upper left graph on the right you can see the amounts  $\vert D_k\vert $  of the occupied subchannels  $64$, ... , $255$.  The carriers  $0$, ... , $63$  for the reserved frequency range of ISDN and upstream are set to zero.

*Next to it on the right are the spectral coefficients   $D_{64}$, ... , $D_{255}$   are shown in the complex plane,&nbsp: where the signal space is chosen very large.

*The graph on the left shows the transmitted signal  $s(t)$  for the frame duration   $T = {1}/{f_0} ≈ 232 \ \rm µs$,  which is obtained by low-pass filtering the IDFT values  $s_0$, ... , $s_{511}$  results. This useful signal looks almost like noise.
 
[[File:P_ID1953__Bei_T_2_3_S10b_v1.png|left|frame|Transmitted signal with above DMT allocation]]
 
It can be seen:

#Main problem of DMT is the unfavorable Crest factor   ⇒   ratio of maximum value  $s_{\rm max}$  and rms value  $s_{\rm eff}$.
#The large dynamic range visible in the exemplary signal curve places high demands on the amplifier's linearity.
#When the dynamic range is limited,  the peaks of  $s(t)$  are cut off.
#This acts as a pulse-like disturbance and an additional noise load to the system.}}

{{BlaueBox|TEXT=
$\text{Summarizing:}$ 
*Discrete Multitone Transmission  $\rm (DMT)$  is basically the parallel implementation of many narrowband QAM modems with different carriers and relatively low data transmission rates. 

*The low bandwidth per subcarrier allows a long symbol duration,  thus reducing the influence of intersymbol interference and reducing the development effort for equalization.

*A major reason for the DMT success is its technical ease of implementation.  IDFT and DFT are formed with digital signal processors in real time.  The vectors have the length  $512$  $($power of two$)$.  Therefore the particularly fast FFT algorithm  $($"Fast Fourier Transformation"$)$  can be applied.}}

==Exercises for the chapter ==

[[Exercise_2.3:_QAM_Signal_Space_Assignment|Exercise 2.3: QAM Signal Space Assignment]]

[[Exercise_2.3Z:_xDSL_Frequency_Band|Exercise 2.3Z: xDSL Frequency Band]]

[[Exercise_2.4:_DSL/DMT_with_IDFT/DFT|Exercise 2.4: DSL/DMT with IDFT/DFT]]

[[Exercise_2.4Z:_Repetition_to_IDFT|Exercise 2.4Z: Repetition to IDFT]]

{{Display}}

Examples of Communication Systems/xDSL as Transmission Technology

2023-03-21T19:44:54Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=xDSL Systems
|Nächste Seite=Methods to Reduce the Bit Error Rate in DSL
}}

==Possible bandwidth allocations for xDSL==
 
The xDSL specifications give operators a great deal of freedom with regard to allocation.

For the necessary directional separation of the xDSL signal transmission according to
*the downstream direction from the provider to the customer  $($with the highest possible data rate$)$,

*the upstream direction from the customer to the provider  $($with mostly lower data rate$)$,

two variants have been standardized for this purpose:

{{BlaueBox|TEXT=
$\text{Definition:}$ 
*With the  »'''frequency separation method'''«  the data streams for the two directions are transmitted in two separate frequency bands with the advantage that a simple filter is sufficient to separate the transmission directions,  which simplifies the technical implementation.

*With the  »'''frequency uniform position method'''«  the spectra of upstream and downstream overlap in a certain part.  The separation is done here with help of an echo cancellation circuit.  Advantages of this method:  The lower bandwidth requirement at higher  $($and thus more attenuated$)$  frequencies and longer range.}}

[[File:EN_Bei_T_2_3_S1a_v4.png|right|frame|"Frequency separation method"  &  "Frequency uniform position method"]]
The graph compares these two options.  Basically,  the specifications leave it up to the developers/operators to decide,
*to operate xDSL alone on the subscriber line,  or

*to allow mixed operation of xDSL with the telephone services  $\rm POTS$  $($"Plain Old Telephone Service"$)$  or  $\rm ISDN$  $($"Integrated Services Digital Network"$)$,  and
*thus to exclude or also occupy the lower frequency range occupied by the two telephone services for xDSL.
 
==ADSL bandwidth allocation==
 
Since it is technically much easier to realise, the decision in Germany for "ADSL" and "ADSL2+" was in favour of

[[File:EN_Bei_T_2_3_S1b_v4.png|right|frame|ADSL bandwidth allocation in Germany and most of other European countries]]
*of the  "frequency separation method",  and
*the general reservation of the lower frequency range for ISDN.

The  "frequency uniform position method"  is still used in some cases,  but rather rarely.

The bandwidth available for DSL is not further decomposed for the transmission methods
*[[Modulation_Methods/Quadrature_Amplitude_Modulation|$\rm QAM$]]  $($"Quadrature Amplitude Modulation"$)$  and

*[[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Carrierless_Amplitude_Phase_Modulation|$\rm CAP$]]  $($"Carrierless Amplitude Phase Modulation$)$.

In contrast,  with the multicarrier method  [[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|$\rm DMT$]]  $($"Discrete Multitone Transmission"$),$  the uplink and the downlink channel are divided into  $N_{\rm Up}$  resp.  $N_{\rm Down}$  bins of  $4.3125\rm \ kHz $  each.

In addition to the above graph, it should be noted:
#Phone services  $($POTS or ISDN$)$  and xDSL are in different frequency bands,  which minimizes mutual interference in the trunk cable.  Thus,  the stronger signal ISDN does not interfere with the parallel running xDSL and vice versa.
#The lower frequency range up to  $\text{120 kHz}$  has been reserved for ISDN  $($optionally POTS$)$.  This value results from the first zero of the  [[Digital_Signal_Transmission/Block_Coding_with_4B3T_Codes#ACF_and_PSD_of_the_4B3T_codes|"ISDN spectrum with 4B3T coding"]].  Above $\text{120 kHz}$  the ISDN spectrum is completely suppressed.
#To separate the telephone and xDSL signals,  a  [[Examples_of_Communication_Systems/xDSL_Systems#Components_of_DSL_Internet_access|"splitter"]]  is used at both ends of the two-wire line,  which includes one low-pass filter and one high-pass filter each and also takes into account the following frequency gap up to  $\text{138 kHz}$.
#After this occupancy gap follows the ADSL upstream band from  $\text{138 kHz}$  to $\text{276 kHz}$.  This allows the transmission of  $N_{\rm Up} = 32$  subcarriers with bandwidth  $\text{4.3125 kHz}$  each.  This value is derived from the frame transmission rate.
#The subsequent downstream range extends to  $\text{1104 kHz}$  for ADSL,  with which  $N_{\rm Down} = 256$  subcarriers can be realized.  The separation of the upstream and the downstream channel in xDSL is done by a band-pass filter in the modem.
#However,  the first  $64$  subcarriers  $($this corresponds to $\text{276 kHz)}$  must not be occupied.  With the  [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Possible_bandwidth_allocations_for_xDSL|"frequency uniform position method"]]  only  $32$  subcarriers would have to be left out,  taking into account that the separation of uplink and downlink requires a more complex implementation.
#For  ADSL2+,  the system bandwidth is  $\text{2208 kHz}$   ⇒   $N_{\rm Down} = 512$  subcarriers.  The number of bins to be spared remains unchanged compared to ADSL.  Taking into account that two bins are occupied by control functions  $($e.g. for synchronization$)$,  $190$  $($ADSL$)$  or  $446$  $($ADSL2+$)$ downstream channels remain for users.
#The ISDN reservation prescribed in Germany,  however,  has the consequence for xDSL that the low frequencies,  which are by far the least attenuated in a copper line and would therefore actually be the most suitable,  cannot be used.
#Further,  from the frequency arrangement,  it can be seen that the downstream bins are more attenuated than the upstream bins  $($higher frequencies$)$  and consequently have a smaller signal-to-noise ratio  $($SNR$)$.
#The decision  "upstream below downstream"  is related to the fact that the loss of downstream channels has only a comparatively small impact on the transmission rate.  In the upstream,  such a failure would be much more noticeable in percentage terms.

==VDSL(2) bandwidth allocation==
 
The ITU has defined several profiles for  "VDSL(2)".  At the time of writing this chapter  (2010),  the frequency band assignment specified in the graphic applies to the systems deployed in Germany in accordance with the ITU's  $\text{ VDSL(2) Plan 998b - Profile 17a (Annex B)}$.  The (slightly) lighter coloring at the higher frequencies is meant to indicate that these channels are more attenuated.

Without claiming to be exhaustive, this allocation plan can be characterized as follows:

[[File:EN_Bei_T_2_3_S5a_ganzneu.png|right|frame|VDSL(2) bandwidth allocation in Germany ]]

#To achieve higher bit rates,  eight times as many bins are used here as in  "ADSL2+".
#Thus,  the bandwidth enables  $8 \cdot \text{2208 MHz = 17664 MHz}$,  transmission rates of up to  $\text{100 Mbit/s}$  $($depending on cable length and conditions$)$.
#The same as in ADSL,  the frequency bands for the upstream subchannels are always arranged at the lower frequencies,  since the greater cable attenuation  has a greater percentage influence on the bit rate for upstream than for downstream.
#In VDSL(2),  the  [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Possible_bandwidth_allocations_for_xDSL|"frequency separation method"]]  is always used.  Overlapping of the upstream and downstream frequency bands is categorically excluded in the ITU specification .
#In the VDSL systems,  the lower frequencies are reserved for ISDN.  This is followed by alternating ranges for upstream and downstream.  From the specified limits,  one can recognize the narrower upstream ranges,  compared to the downstream.
#You can see an alternating arrangement of upstream and downstream areas.  One reason is that with this wide spectrum it should be avoided that one direction  $($e.g. downstream$)$  is assigned only strongly attenuated  $($high$)$  frequencies.

The VDSL(2) specification provides for allocation plans up to system bandwidths of  $\text{30 MHz}$  $($according to profile 30a$)$.
* This should enable transmission rates up to about  $\text{ 200 Mbit/s}$  over short distances.

*For this purpose, the bandwidth of the individual subchannels is also doubled compared to ADSL to  $\text{8.625 kHz}$ .

==Overview of xDSL transmission methods==
 
At the beginning of the various standardization procedures for the individual xDSL variants,  different transmission methods were defined as a basis:
# [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modulation"]]  $\text{QAM}$  for  "QAM-ADSL"  and  "QAM-VDSL", 
# [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Carrierless_Amplitude_Phase_Modulation|"Carrierless Amplitude Phase Modulation"]]  $\text{CAP}$  for  "CAP-HDSL"  and  "CAP-ADSL",
#[[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|"Discrete Multitone Transmission"]]  $\text{DMT}$  for  "ADSL",  "ADSL2",  "ADSL2+",  "VDSL"  and",  "VDSL2".

With increasing market demand for higher transmission rates and the associated requirements, two main  suitable processes crystallized,  namely  "QAM/CAP"  and  "DMT".

Since the manufacturers were unable to agree on a common standard from 1997 to 2003,  also for patent reasons  $($one even speaks of  "line code wars"  in this context$)$,  the two competing methods coexisted for a long time.  At the so-called  "DSL Olympics"  in 2003,  '''the decision was finally made in favor of DMT''',
*on the one hand,  because of the slightly better  "performance"  in general,

*but in particular because of its higher robustness against narrowband interference; especially for the USA  $($many overhead telephone lines and related problems with coupled radio signals$)$  this argument played a major role.

First,  however,  the systems  "QAM-xDSL"  and  "CAP-xDSL"  will be considered very briefly.

==Basics of Quadrature Amplitude Modulation==
 

The principle has already been described in detail in the chapter  [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modation"]]  of the book  "Modulation Methods". 
[[File:EN_Bei_T_2_3_S6a.png|right|frame| Linear modulator with  $\rm I$ and  $\rm Q$–components;  signal space for  $\text{16-QAM}$]]
Here follows short summary on the basis of the block diagram on the right:
*QAM is a  "single carrier modulation method"  around the carrier frequency  $f_{\rm T}$.  First,  a blockwise serial/parallel conversion of the bit stream and signal space assignment is performed.

*From each  $b$  binary symbols,  two multilevel '''KORREKTUR: multi-level''' amplitude coefficients  $a_{{\rm I}n}$  and  $a_{{\rm Q}n}$  are derived  $($"in-phase component"  and  "quadrature component"$)$.  Both coefficients can each take one of  $M = 2^{b/2}$  possible amplitude values.

*The example considered here applies to the  $\text{16-QAM}$  with  $b = M = 4$  and correspondingly sixteen  signal space points. For a   $\text{256-QAM}$  would hold:   $b = 8$  and  $M = 16$  $(2^b = M^2 = 256)$.

*The coefficients  $a_{{\rm I}n}$  and  $a_{{\rm Q}n}$  are each impressed on a Dirac delta impulse as weights.  For pulse shaping one usually uses  $($because of the small bandwidth$)$  a raised-cosine filter.

*With the basic transmission pulse  $g_s(t)$  is then valid in the two branches of the block diagram:
:$$ s_{\rm I}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm
I\hspace{0.03cm}\it n} \cdot g_s (t - n \cdot T)\hspace{0.05cm},\hspace{0.5cm}
s_{\rm Q}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm
Q\hspace{0.03cm}\it n} \cdot g_s (t - n \cdot
T)\hspace{0.05cm}.$$
:Note:  Because of the redundancy-free conversion to a higher-level code,  the symbol duration  $T$  of these signals is greater by a factor  $b$  than the bit duration  $T_{\rm B}$  of the binary input sequence.  In the drawn example  $($16-QAM$)$,  $T = 4 ⋅ T_{\rm B}$.
*The  »'''QAM transmitted signal'''«  $s(t)$  is then the sum of the two partial signals  $ s_{\rm I}(t)$  and  $ s_{\rm Q}(t)$,  multiplied by  "cosine"  resp.  "minus-sine"  $($possibly followed by a band limit to prevent interference to adjacent bands, as indicated in the graph below$)$:
:$$s(t) = s_{\rm I}(t) \cdot \cos (2 \pi f_{\rm T}\,t) - s_{\rm Q}(t) \cdot \sin (2 \pi f_{\rm T}\,t)
\hspace{0.05cm}. $$
*The two branches  $(\rm I$  and  $\rm Q)$  can be considered as two completely separate  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#M.E2.80.93level_amplitude_shift_keying_.28M.E2.80.93ASK.29 |"M–level ASK systems"]]  which do not interfere with each other as long as all components are optimally designed.  This means at the same time:  

*Compared to a  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_phase_shift_keying_.28BPSK.29|"Binary Phase Shift Keying"]]  $\rm (BPSK)$  modulation with cosine or sine only,  the  '''QAM allows a doubling of the data rate at constant quality'''.

[[File:EN_Bei_T_2_3_S6b.png|right|frame|Quadrature Amplitude Modulation as band-pass and equivalent low-pass model]]

'''Equivalent low-pass model'''

The graph on the right shows
*above the band-pass model,  identical with the last graph,

*below the  "equivalent low-pass model".

Here,  the inphase and quadrature coefficients are combined to give the complex amplitude coefficient
:$$a_n = a_{\text{I}n} + {\rm j} ⋅ a_{\text{Q}n}.$$
Additionally, the physical transmitted signal  $s(t)$  is replaced by the equivalent low-pass signal
:$$s_{\rm TP}(t) = s_{\rm I}(t) + {\rm j} ⋅ s_{\rm Q}(t).$$

⇒   The representation of the QAM transmitter and receiver is the content of the SWF flash animation  [[Applets:Principle_of_QAM|"Principle of Quadrature Amplitude Modulation"]].
 
{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*As the bit number  $b$  and thus the number of defined symbols  $(M=b^2)$  increases,  the bandwidth efficiency increases,  but also the signal processing overhead increases.

*In addition,  it must be taken into account that a dense QAM allocation is appropriate only if the channel is sufficiently good.}}

'''Reference Model for ADSL-QAM'''

This diagram shows the reference model for  "ADSL-QAM",  whereby we will only deal with the red function blocks  "QAM-Modulator"  and  "'QAM-Demodulator"  here.
[[File:EN_Bei_T_2_3_S5a.png|right|frame|Reference Model for ADSL-QAM]]

*The carrier frequency $f_{\rm T}$  lies within the specified up and down band of the respective xDSL variant.

*Like signal space size  $($between  $M^2=2^2=4$  and  $M^2=2^8=256$  signal space points$)$  and symbol rate, it is determined by channel measurements during transmission initialisation.

*For ADSL-QAM,  the following symbol rates have been specified  $($all numerical values in  "${\rm kBaud}$" $= 10^3 \rm symbols/s)$  :
:*In upstream:  $20$, $40$, $84$, $100$, $120$, $136$;

:*in downstream: $40$, $126$, $160$, $252$, $336$, $504$, $806.4$, $1008$.

==Possible QAM signal space constellations==
 
We still consider possible arrangements of signal space points in quadrature amplitude modulation using three examples.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
An important QAM parameter is the bit number  $b$  processed to the amplitude coefficient pair  $(a_{\rm I}, a_{\rm Q})$.  Here  $b$  is always an even number.
[[File:EN_Bei_T_2_3_S6a_v2.png|right|frame|Signal space constellations  "$\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($left$)$  and  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($right$)$
'''Korrektur''': Jiwoo, ich habe die gelben Punkte durch braune Punkte ersetzt, weil meine schlechten Auge diese nicht erkennen konnten. Könntest u bitte den Text kontrollieren und mit der deutschen Version vergleichen, ob jetzt wirklich alle gelben Punkte braun sind. Das gleiche gilt für Example 3.]]

*If  $b = 2$,  both  $a_{\rm I}$  and  $a_{\rm Q}$  can only take the values  $±1$  resulting in the  $\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$  corresponding to the left constellation.

*According to an ITU recommendation,  the assignment applies here:
:$$q_1 = 0, \ q_0 = 0 \, \Leftrightarrow \,a_{\rm I} = +1, \ a_{\rm Q} = +1,$$
:$$q_1 = 0, \ q_0 = 1 \, \Leftrightarrow \, a_{\rm I} = +1, \ a_{\rm Q} = -1,$$
:$$q_1 = 1, \ q_0 = 0 \, \Leftrightarrow \,a_{\rm I} = -1, \ a_{\rm Q} = +1,$$
:$$q_1 = 1, \ q_0 = 1 \, \Leftrightarrow \, a_{\rm I} = -1, \ a_{\rm Q} = -1.$$

*The point  '''10'''   marked in brown  $(a_{\rm I} = -1, \ a_{\rm Q} = 1)$  thus stands for  $q_1 = 1$  and  $q_0 = 0$.

With  $b = 4$   ⇒   $M = 2^{b/2} = 4$  one arrives at  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  according to the right diagram with the possible amplitude coefficients 
:$$a_{\rm I} ∈ \{±3, ±1\}, \ \ a_{\rm Q} ∈ \{±3, ±1\}.$$

*The assignment can be determined using the auxiliary graph given on the left below, as illustrated by the following numerical examples.

        $\rm (A)$    $q_3 = 1, \ q_2 = 0, \ q_1 = 1,\ q_0 = 1$  $($brown marker$)$:
:#The two most significant bits  $\rm (MSB)$   ⇒   "'''11'''"  determine,  according to the  $\text{4-QAM}$  diagram,  the  $($second$)$  quadrant in which the symbol is located.
:#The two least significant bits  $\rm (LSB)$   ⇒   "'''10'''"  together with the auxiliary graph determine the point within this quadrant.  Result:  $a_{\rm I} = -1$,  $a_{\rm Q} = +3$.
        $\rm (B)$    $q_3 = 0, \ q_2 = 1, \ q_1 = 1,\ q_0 = 0$ $($green marker$)$:
:#The two most significant bits   ⇒   "'''01'''"   refer here to the fourth quadrant.
:#The two least significant bits   ⇒   "'''10'''"  refer to the green dot in the fourth quadrant:   $a_{\rm I} = -3, \ a_{\rm Q} = -3$.}}

{{GraueBox|TEXT=
$\text{Example 2:}$  We use the same graphic as in $\text{Example 1}$.  The decimal value  $D$  provides another way to label the points.
[[File:EN_Bei_T_2_3_S6a_v2.png|right|frame|Signal space constellations  "$\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($left$)$  and  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($right$)$]]

*The brown marked point in the  $\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$ diagram is binary labeled  "'''10'''"    ⇒   decimal:  $D = 2$.  This point simultaneously marks the second quadrant of  $\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$.

*The further subdivision results from the lower left graphic.  There at the brown dot is written  $4D + 3=11$"  $($decimal$)$.

*Therefore,  the upper right dot  $($highlighted in brown$)$  in the upper left quadrant stands for decimal  "$11$   ⇒   binary '''1011'''.

*For the green dot,  $D = 1$  gives the decimal value  $4D + 2 =6$,$11$ which corresponds to the binary representation  "'''0110'''" .
 
According to this scheme,  the signal space constellations
*for  $\rm 64\hspace{0.05cm}–\hspace{-0.02cm}QAM$   ⇒   $(b = 6, \ M = 8)$ 

*and  $\rm 256-QAM$  ⇒   $(b = 8, \ M = 16)$ 

which will be discussed in detail in the  [[Exercise_2.3:_QAM_Signal_Space_Assignment|$\text{Exercise 2.3}$]].  }}

{{GraueBox|TEXT=
$\text{Example 3:}$ 
We still consider for the described  $\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$  $($left graph,  here referred to as  "ITU proposal"$)$  the resulting symbol and bit error probability in the presence of AWGN noise:
#An error can be assumed to result in a horizontally or vertically adjacent symbol,  as indicated for the upper left green dot.
#The symbol error probability $p$  depends on the Euclidean distance of the two points and the AWGN noise power density  $N_0$.
#A distortion to the more distant blue point instead of one of the two neighboring brown points is rather unlikely with Gaussian noise.
[[File:EN Bei T 2 3 S6b neu.png|right|frame|Signal space constellation:  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$";  left: ITU propsal,  right:  Gray encoding. '''KORREKTUR: Symbolfehler, Bitfehler''']]

An analysis shows:
*All corner points  $($green$)$  can only be distorted in two directions.

*In contrast, the inner QAM points  $($blue$)$  have four direct neighbors.

* The remaining symbols  $($brown$)$  have three neighbors.

For the  »'''symbol error probability'''«   $p_{\rm S}$  then holds:

::$$p_{\rm S} = {1}/{16} \cdot (4 \cdot 2 p + 8 \cdot 3 p + 4 \cdot 4 p) = 3p.$$

To calculate the  »'''bit error probability'''«   $p_{\rm B}$  it must now be taken into account that for the left constellation one symbol error leads
* within one quadrant to only  one bit error  $($e.g: "'''0100'''"   ⇒   "'''0110'''"$)$,

*between adjacent quadrants to two bit errors $($e.g: "'''1111'''"   ⇒   "'''0101'''"$)$.

The computation of  $p_{\rm B} $  here involves some effort.

In contrast,  with Gray encoding  $($right diagram$)$:
# Each symbol differs from its neighbors by exactly one bit.
#Each symbol error thus results in exactly one bit error.
#Since each individual symbol contains four bits,  the  "bit error probability" is in this case: 

::$$p_{\rm B} = p_{\rm S}/4 = 3/4 \cdot p. $$}}

==Carrierless Amplitude Phase Modulation==
 
"Carrierless amplitude phase modulation"  $\rm (CAP)$  is a bandwidth-efficient variant of QAM,  which can be realized very easily with digital signal processors.  The only difference to QAM is that the modulation with a carrier signal can be omitted.

[[File:EN_Bei_T_2_3_S7a.png|right|frame|Block diagram of  "Carrierless Amplitude Phase Modulation"]]
[[File:EN_Bei_T_2_3_S7d_neu.png|right|frame|Reference model for  ADSL–CAP]]

*Digital filtering is used instead of multiplication by cosine and minus sine.  $g_{\rm I}(t)$  and  $g_{\rm Q}(t)$  are impulse responses of two transversal bandpass '''KORREKTUR: band-pass''' filters with same amplitude characteristic,  phase-shifted by  $π/2$.

*Both are orthogonal to each other,  which means that the integral of the product  $g_{\rm I}(t) · g_{\rm Q}(t)$  over a symbol duration gives zero.

*The signals  $s_{\rm I}(t)$  and  $s_{\rm Q}(t)$  generated in this way are combined,  converted to a continuous-time signal by a D/A converter. 

*The unwanted high-frequency components generated during D/A conversion are eliminated by a low-pass filter before transmission.

*At the receiver,  the signal  $r(t)$  is first converted to a discrete-time signal using an A/D converter and then the in-phase and quadrature symbols  $a_{\rm I}$  and  $a_{\rm Q}$  are extracted via two  "finite impulse response  $\rm (FIR)$  filters"  and downstream "threshold decision".

CAP was the de facto standard in initial ADSL specifications until 1996.  The graph shows the reference model.
#The frequencies up to  $\text{4 kHz}$  were reserved for POTS/ISDN.
#The up channel occupied the frequency range of  $\text{15 - 160 kHz}$,
#The down channel occupied the frequencies from  $\text{240 - 1500 kHz}$.

One problem with CAP is that a  "bad channel" has dramatic consequences on the transmission quality.  Therefore,  today (2011) ADSL-CAP is only found in a few HDSL variants.

==Basics of Discrete Multitone Transmission==
 
[[File:EN_Bei_T_2_3_S8a_v8.png|right|frame|Spectra at OFDM and DMT;     Note:   For this graph a symmetrical rectangle is assumed.     A rectangle between  $0$  and  $T$  would still result in a phase term.     However,  nothing would change with respect to  $|S(f)|$.]]
"Discrete Multitone Transmission"  $\rm (DMT)$  refers to a multicarrier modulation method that is almost identical to  [[Modulation_Methods/General_Description_of_OFDM|"Orthogonal Frequency Division Multiplexing"]]  $\rm (OFDM)$.  In the case of wired transmission,  one usually refers to  "DMT",  and in the case of wireless transmission,  one refers to  "OFDM".

In both cases,  one divides the entire bandwidth into many narrowband equidistant subchannels.  The respective subcarrier signals  $s_k(t)$  are individually impinged with complex data symbols  $D_k$  and the sum of the modulated subcarrier signals is transmitted as signal  $s(t)$ .

The diagram illustrates the principle of OFDM and DMT in the frequency domain,  partly using the values specified for ADSL–DMT:

*$255$  Subcarriers with carrier frequencies  $k \cdot f_0$  $(k = 1$, ... , $255)$.

*$4000$  data frames are transmitted per second.

*After  $68$  data frames,  one synchronization frame is inserted each.

*Due to the  "cyclic prefix"  $($see chapter  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Insert.C3.BCgen_of_Guard.E2.80.93Interval_and_cyclic_Pr. C3.A4fix|"Insertion of Guard–Interval and Cyclic Prefix"]]$)$  the symbol duration  $T = 1/f_0$  must still be shortened by the factor  $16/17$.

*Thus,  the  »'''basic frequency'''«  of DMT is 
:$$f_0 = 4 \cdot (69/68) \cdot (69/68) = 4.3125 \ \rm kHz.$$

An essential difference between OFDM and DMT is that
*in OFDM,  the above drawn spectrum  $S(f)$  in reality describes an  "equivalent low-pass spectrum"  $S_{\rm TP}(f)$  and still the shift around a carrier frequency  $f_{\rm T}$  has to be considered:

:$$S_{\rm TP}(f ) = \sum_{k = 1}^{255} D_k \cdot \delta (f - k \cdot f_0)\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
S(f) = \frac{1}{2} \big [ S_{\rm TP}(f - f_{\rm T}) + S^*_{\rm TP}(-(f + f_{\rm T}))\big ]
\hspace{0.05cm};$$

*in DMT,  on the other hand,  the components at negative frequencies must still be taken into account,  which are to be weighted with the conjugate-complex spectral coefficients:

:$$S(f ) = \sum_{k = 1}^{255} \big [ D_k \cdot \delta (f - k \cdot f_0) + D^*_k \cdot \delta (f + k \cdot f_0) \big ]
\hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Please note:}$ 
#According to these equations,  the complex OFDM signal  $s_{\rm OFDM}(t)$  consists of  $K = 255$  complex exponential oscillations.
#The DMT signal  $s_{\rm DMT}(t)$  is composed of as many cosine oscillations with frequencies  $k \cdot f_0$  $($full occupancy presupposed$)$.
#Despite complex coefficients  $D_k$ resulting from QAM occupancy of the carriers,  the DMT signal is always real because of the conjugate-complex complements at negative frequencies.}}

However,  in both OFDM and DMT,  the transmitted signal  $s(t)$  is precisely limited in time to the symbol duration  $T = 1/f_0 ≈ 232 \ {\rm µs}$  which is equivalent to multiplying by a rectangle of duration $T$.  In the spectral domain,  this corresponds to convolution with the function  $\text{sinc}(fT)$:
*Each Dirac delta function at  $k · f_0$  thus becomes an sinc– function at the same position,  given the time limit,  as shown in the second diagram of the graph above.

*Adjacent subcarrier spectra overlap on the frequency axis,  but exactly at  $k · f_0$  the coefficients  $D_k$  can be seen again,  since all other spectra have zeros here.

{{GraueBox|TEXT=
$\text{Example 4:}$ 
Assuming the conditions favorable for ADSL downstream,  viz.
#$4000$ frames are transmitted per second.
#All subcarriers are active at all times  $(K = 255)$.
#Each carrier is occupied with a 1024-QAM  $(b = 10)$;  according to ITU:  $8 ≤ b ≤ 15$.
#Ideal conditions prevail,  so that the orthogonality evident in the graph is preserved,

then for the  »'''maximum bit rate'''« 
:$$R_{\rm B,\ max} = 4000 · K · b ≈ 10 \ \rm Mbit/s.$$

However,  specified ADSL downstream is only  $2 \ \rm Mbit/s$  due to
*the omission of the  $64$  lowest carriers because of ISDN and upstream,

*the QAM occupancy of the heavily attenuated carriers with less than  $10$  bits,  and

*the consideration of the cyclic prefix,  and some operational reasons.}}

==DMT realization with IDFT/DFT==
 
The upper graph shows the complete DMT system.  For now we will focus on the two red blocks.  The blue blocks will be covered in the [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL|"next chapter"]].  Simplified,  the transmitter and receiver can be represented as shown in the second graph:

[[File:EN_Bei_T_2_3_S9d_ganz_neuV8.png|right|frame|Discrete multitone transmission system]]
[[File:P_ID1951__Bei_T_2_3_S9b_v3.png|right|frame|DMT transmitter and receiver '''Korrektur''' das Bild muss ich noch machen]]

*To perform a DMT modulation,  the transmitter accumulates a block of input bits in a data buffer to be transmitted as one frame.

*The QAM encoder supplies per frame the complex-valued data symbols  $D_1$,  ... ,  $D_{255}$,  which is expanded to the vector  $\mathbf{D}$  of length  $512$  by
#   $D_0 = D_{256} = 0$,
#   $D_k = D^\star_{512-k} \ (k = 257,$ ... , $511)$,
#   the coefficients   $D_{257}$, ... , $D_{511}$  are identical to  $D_{-255}$, ... , $D_{-1}$ $($as consequence of  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Finite_signal_representation|"finite signals"]]  properties$)$.
*The spectral samples  $\mathbf{D}$  are transformed with the [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Inverse_discrete_Fourier_transform|"Inverse Discrete Fourier Transform"]]  to the vector  $\mathbf{s}$  of time signal samples, also with length $512$. Because of the conjugate-complex assignment in the spectral domain   ⇒   $\text{Im}[\mathbf{s}] = 0$.

*After parallel/serial and digital/analog conversion and low-pass filtering of  $\text{Re}[\mathbf{s}]$  the physical  $($real$)$  as well as continuous-time transmitted signal  $s(t)$  results.

*For this,  in the range  $0 ≤ t ≤ T$  $($factor  $2$,  since two coefficients each contribute to cosine/sine$)$:

:$$s(t) = \sum_{k = 1}^{255} \big [ 2 \cdot{\rm Re}\{D_k\} \cdot \cos(2\pi k f_0 t ) - 2 \cdot{\rm Im}\{D_k\} \cdot \sin(2\pi k f_0 t )\big ] \hspace{0.05cm}. $$

*The received signal at the AWGN output is  $r(t) = s(t) + n(t)$.  After A/D and S/P conversion,  $r(t)$  can be expressed by the (real) vector  $\mathbf{r}$ . The  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#From_the_continuous_to_the_discrete_Fourier_transform|"Discrete Fourier Transform"]]  then provides estimates for the transmitted spectral coefficients.
 

{{GraueBox|TEXT=
$\text{Example 5:}$ 
As an example, let's consider the ADSL/DMT downstream.
[[File:EN_Bei_T_2_3_S10_neu.png|right|frame|Allocation of the DMT frequency band with QAM coefficients]]
*In the upper left graph on the right you can see the amounts  $\vert D_k\vert $  of the occupied subchannels  $64$, ... , $255$.  The carriers  $0$, ... , $63$  for the reserved frequency range of ISDN and upstream are set to zero.

*Next to it on the right are the spectral coefficients   $D_{64}$, ... , $D_{255}$   are shown in the complex plane,&nbsp: where the signal space is chosen very large.

*The graph on the left shows the transmitted signal  $s(t)$  for the frame duration   $T = {1}/{f_0} ≈ 232 \ \rm µs$,  which is obtained by low-pass filtering the IDFT values  $s_0$, ... , $s_{511}$  results. This useful signal looks almost like noise.
 
[[File:P_ID1953__Bei_T_2_3_S10b_v1.png|left|frame|Transmitted signal with above DMT allocation]]
 
It can be seen:

#Main problem of DMT is the unfavorable Crest factor   ⇒   ratio of maximum value  $s_{\rm max}$  and rms value  $s_{\rm eff}$.
#The large dynamic range visible in the exemplary signal curve places high demands on the amplifier's linearity.
#When the dynamic range is limited,  the peaks of  $s(t)$  are cut off.
#This acts as a pulse-like disturbance and an additional noise load to the system.}}

{{BlaueBox|TEXT=
$\text{Summarizing:}$ 
*Discrete Multitone Transmission  $\rm (DMT)$  is basically the parallel implementation of many narrowband QAM modems with different carriers and relatively low data transmission rates. 

*The low bandwidth per subcarrier allows a long symbol duration,  thus reducing the influence of intersymbol interference and reducing the development effort for equalization.

*A major reason for the DMT success is its technical ease of implementation.  IDFT and DFT are formed with digital signal processors in real time.  The vectors have the length  $512$  $($power of two$)$.  Therefore the particularly fast FFT algorithm  $($"Fast Fourier Transformation"$)$  can be applied.}}

==Exercises for the chapter ==

[[Exercise_2.3:_QAM_Signal_Space_Assignment|Exercise 2.3: QAM Signal Space Assignment]]

[[Exercise_2.3Z:_xDSL_Frequency_Band|Exercise 2.3Z: xDSL Frequency Band]]

[[Exercise_2.4:_DSL/DMT_with_IDFT/DFT|Exercise 2.4: DSL/DMT with IDFT/DFT]]

[[Exercise_2.4Z:_Repetition_to_IDFT|Exercise 2.4Z: Repetition to IDFT]]

{{Display}}

Examples of Communication Systems/xDSL as Transmission Technology

2023-03-21T19:43:56Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=xDSL Systems
|Nächste Seite=Methods to Reduce the Bit Error Rate in DSL
}}

==Possible bandwidth allocations for xDSL==
 
The xDSL specifications give operators a great deal of freedom with regard to allocation.

For the necessary directional separation of the xDSL signal transmission according to
*the downstream direction from the provider to the customer  $($with the highest possible data rate$)$,

*the upstream direction from the customer to the provider  $($with mostly lower data rate$)$,

two variants have been standardized for this purpose:

{{BlaueBox|TEXT=
$\text{Definition:}$ 
*With the  »'''frequency separation method'''«  the data streams for the two directions are transmitted in two separate frequency bands with the advantage that a simple filter is sufficient to separate the transmission directions,  which simplifies the technical implementation.

*With the  »'''frequency uniform position method'''«  the spectra of upstream and downstream overlap in a certain part.  The separation is done here with help of an echo cancellation circuit.  Advantages of this method:  The lower bandwidth requirement at higher  $($and thus more attenuated$)$  frequencies and longer range.}}

[[File:EN_Bei_T_2_3_S1a_v4.png|right|frame|"Frequency separation method"  &  "Frequency uniform position method"]]
The graph compares these two options.  Basically,  the specifications leave it up to the developers/operators to decide,
*to operate xDSL alone on the subscriber line,  or

*to allow mixed operation of xDSL with the telephone services  $\rm POTS$  $($"Plain Old Telephone Service"$)$  or  $\rm ISDN$  $($"Integrated Services Digital Network"$)$,  and
*thus to exclude or also occupy the lower frequency range occupied by the two telephone services for xDSL.
 
==ADSL bandwidth allocation==
 
Since it is technically much easier to realise, the decision in Germany for "ADSL" and "ADSL2+" was in favour of

[[File:EN_Bei_T_2_3_S1b_v4.png|right|frame|ADSL bandwidth allocation in Germany and most of other European countries]]
*of the  "frequency separation method",  and
*the general reservation of the lower frequency range for ISDN.

The  "frequency uniform position method"  is still used in some cases,  but rather rarely.

The bandwidth available for DSL is not further decomposed for the transmission methods
*[[Modulation_Methods/Quadrature_Amplitude_Modulation|$\rm QAM$]]  $($"Quadrature Amplitude Modulation"$)$  and

*[[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Carrierless_Amplitude_Phase_Modulation|$\rm CAP$]]  $($"Carrierless Amplitude Phase Modulation$)$.

In contrast,  with the multicarrier method  [[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|$\rm DMT$]]  $($"Discrete Multitone Transmission"$),$  the uplink and the downlink channel are divided into  $N_{\rm Up}$  resp.  $N_{\rm Down}$  bins of  $4.3125\rm \ kHz $  each.

In addition to the above graph, it should be noted:
#Phone services  $($POTS or ISDN$)$  and xDSL are in different frequency bands,  which minimizes mutual interference in the trunk cable.  Thus,  the stronger signal ISDN does not interfere with the parallel running xDSL and vice versa.
#The lower frequency range up to  $\text{120 kHz}$  has been reserved for ISDN  $($optionally POTS$)$.  This value results from the first zero of the  [[Digital_Signal_Transmission/Block_Coding_with_4B3T_Codes#ACF_and_PSD_of_the_4B3T_codes|"ISDN spectrum with 4B3T coding"]].  Above $\text{120 kHz}$  the ISDN spectrum is completely suppressed.
#To separate the telephone and xDSL signals,  a  [[Examples_of_Communication_Systems/xDSL_Systems#Components_of_DSL_Internet_access|"splitter"]]  is used at both ends of the two-wire line,  which includes one low-pass filter and one high-pass filter each and also takes into account the following frequency gap up to  $\text{138 kHz}$.
#After this occupancy gap follows the ADSL upstream band from  $\text{138 kHz}$  to $\text{276 kHz}$.  This allows the transmission of  $N_{\rm Up} = 32$  subcarriers with bandwidth  $\text{4.3125 kHz}$  each.  This value is derived from the frame transmission rate.
#The subsequent downstream range extends to  $\text{1104 kHz}$  for ADSL,  with which  $N_{\rm Down} = 256$  subcarriers can be realized.  The separation of the upstream and the downstream channel in xDSL is done by a band-pass filter in the modem.
#However,  the first  $64$  subcarriers  $($this corresponds to $\text{276 kHz)}$  must not be occupied.  With the  [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Possible_bandwidth_allocations_for_xDSL|"frequency uniform position method"]]  only  $32$  subcarriers would have to be left out,  taking into account that the separation of uplink and downlink requires a more complex implementation.
#For  ADSL2+,  the system bandwidth is  $\text{2208 kHz}$   ⇒   $N_{\rm Down} = 512$  subcarriers.  The number of bins to be spared remains unchanged compared to ADSL.  Taking into account that two bins are occupied by control functions  $($e.g. for synchronization$)$,  $190$  $($ADSL$)$  or  $446$  $($ADSL2+$)$ downstream channels remain for users.
#The ISDN reservation prescribed in Germany,  however,  has the consequence for xDSL that the low frequencies,  which are by far the least attenuated in a copper line and would therefore actually be the most suitable,  cannot be used.
#Further,  from the frequency arrangement,  it can be seen that the downstream bins are more attenuated than the upstream bins  $($higher frequencies$)$  and consequently have a smaller signal-to-noise ratio  $($SNR$)$.
#The decision  "upstream below downstream"  is related to the fact that the loss of downstream channels has only a comparatively small impact on the transmission rate.  In the upstream,  such a failure would be much more noticeable in percentage terms.

==VDSL(2) bandwidth allocation==
 
The ITU has defined several profiles for  "VDSL(2)".  At the time of writing this chapter  (2010),  the frequency band assignment specified in the graphic applies to the systems deployed in Germany in accordance with the ITU's  $\text{ VDSL(2) Plan 998b - Profile 17a (Annex B)}$.  The (slightly) lighter coloring at the higher frequencies is meant to indicate that these channels are more attenuated.

Without claiming to be exhaustive, this allocation plan can be characterized as follows:

[[File:EN_Bei_T_2_3_S5a_ganzneu.png|right|frame|VDSL(2) bandwidth allocation in Germany ]]

#To achieve higher bit rates,  eight times as many bins are used here as in  "ADSL2+".
#Thus,  the bandwidth enables  $8 \cdot \text{2208 MHz = 17664 MHz}$,  transmission rates of up to  $\text{100 Mbit/s}$  $($depending on cable length and conditions$)$.
#The same as in ADSL,  the frequency bands for the upstream subchannels are always arranged at the lower frequencies,  since the greater cable attenuation  has a greater percentage influence on the bit rate for upstream than for downstream.
#In VDSL(2),  the  [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Possible_bandwidth_allocations_for_xDSL|"frequency separation method"]]  is always used.  Overlapping of the upstream and downstream frequency bands is categorically excluded in the ITU specification .
#In the VDSL systems,  the lower frequencies are reserved for ISDN.  This is followed by alternating ranges for upstream and downstream.  From the specified limits,  one can recognize the narrower upstream ranges,  compared to the downstream.
#You can see an alternating arrangement of upstream and downstream areas.  One reason is that with this wide spectrum it should be avoided that one direction  $($e.g. downstream$)$  is assigned only strongly attenuated  $($high$)$  frequencies.

The VDSL(2) specification provides for allocation plans up to system bandwidths of  $\text{30 MHz}$  $($according to profile 30a$)$.
* This should enable transmission rates up to about  $\text{ 200 Mbit/s}$  over short distances.

*For this purpose, the bandwidth of the individual subchannels is also doubled compared to ADSL to  $\text{8.625 kHz}$ .

==Overview of xDSL transmission methods==
 
At the beginning of the various standardization procedures for the individual xDSL variants,  different transmission methods were defined as a basis:
# [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modulation"]]  $\text{QAM}$  for  "QAM-ADSL"  and  "QAM-VDSL", 
# [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Carrierless_Amplitude_Phase_Modulation|"Carrierless Amplitude Phase Modulation"]]  $\text{CAP}$  for  "CAP-HDSL"  and  "CAP-ADSL",
#[[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|"Discrete Multitone Transmission"]]  $\text{DMT}$  for  "ADSL",  "ADSL2",  "ADSL2+",  "VDSL"  and",  "VDSL2".

With increasing market demand for higher transmission rates and the associated requirements, two main  suitable processes crystallized,  namely  "QAM/CAP"  and  "DMT".

Since the manufacturers were unable to agree on a common standard from 1997 to 2003,  also for patent reasons  $($one even speaks of  "line code wars"  in this context$)$,  the two competing methods coexisted for a long time.  At the so-called  "DSL Olympics"  in 2003,  '''the decision was finally made in favor of DMT''',
*on the one hand,  because of the slightly better  "performance"  in general,

*but in particular because of its higher robustness against narrowband interference; especially for the USA  $($many overhead telephone lines and related problems with coupled radio signals$)$  this argument played a major role.

First,  however,  the systems  "QAM-xDSL"  and  "CAP-xDSL"  will be considered very briefly.

==Basics of Quadrature Amplitude Modulation==
 

The principle has already been described in detail in the chapter  [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modation"]]  of the book  "Modulation Methods". 
[[File:EN_Bei_T_2_3_S6a.png|right|frame| Linear modulator with  $\rm I$ and  $\rm Q$–components;  signal space for  $\text{16-QAM}$]]
Here follows short summary on the basis of the block diagram on the right:
*QAM is a  "single carrier modulation method"  around the carrier frequency  $f_{\rm T}$.  First,  a blockwise serial/parallel conversion of the bit stream and signal space assignment is performed.

*From each  $b$  binary symbols,  two multilevel '''KORREKTUR: multi-level''' amplitude coefficients  $a_{{\rm I}n}$  and  $a_{{\rm Q}n}$  are derived  $($"in-phase component"  and  "quadrature component"$)$.  Both coefficients can each take one of  $M = 2^{b/2}$  possible amplitude values.

*The example considered here applies to the  $\text{16-QAM}$  with  $b = M = 4$  and correspondingly sixteen  signal space points. For a   $\text{256-QAM}$  would hold:   $b = 8$  and  $M = 16$  $(2^b = M^2 = 256)$.

*The coefficients  $a_{{\rm I}n}$  and  $a_{{\rm Q}n}$  are each impressed on a Dirac delta impulse as weights.  For pulse shaping one usually uses  $($because of the small bandwidth$)$  a raised-cosine filter.

*With the basic transmission pulse  $g_s(t)$  is then valid in the two branches of the block diagram:
:$$ s_{\rm I}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm
I\hspace{0.03cm}\it n} \cdot g_s (t - n \cdot T)\hspace{0.05cm},\hspace{0.5cm}
s_{\rm Q}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm
Q\hspace{0.03cm}\it n} \cdot g_s (t - n \cdot
T)\hspace{0.05cm}.$$
:Note:  Because of the redundancy-free conversion to a higher-level code,  the symbol duration  $T$  of these signals is greater by a factor  $b$  than the bit duration  $T_{\rm B}$  of the binary input sequence.  In the drawn example  $($16-QAM$)$,  $T = 4 ⋅ T_{\rm B}$.
*The  »'''QAM transmitted signal'''«  $s(t)$  is then the sum of the two partial signals  $ s_{\rm I}(t)$  and  $ s_{\rm Q}(t)$,  multiplied by  "cosine"  resp.  "minus-sine"  $($possibly followed by a band limit to prevent interference to adjacent bands, as indicated in the graph below$)$:
:$$s(t) = s_{\rm I}(t) \cdot \cos (2 \pi f_{\rm T}\,t) - s_{\rm Q}(t) \cdot \sin (2 \pi f_{\rm T}\,t)
\hspace{0.05cm}. $$
*The two branches  $(\rm I$  and  $\rm Q)$  can be considered as two completely separate  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#M.E2.80.93level_amplitude_shift_keying_.28M.E2.80.93ASK.29 |"M–level ASK systems"]]  which do not interfere with each other as long as all components are optimally designed.  This means at the same time:  

*Compared to a  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_phase_shift_keying_.28BPSK.29|"Binary Phase Shift Keying"]]  $\rm (BPSK)$  modulation with cosine or sine only,  the  '''QAM allows a doubling of the data rate at constant quality'''.

[[File:EN_Bei_T_2_3_S6b.png|right|frame|Quadrature Amplitude Modulation as band-pass and equivalent low-pass model]]

'''Equivalent low-pass model'''

The graph on the right shows
*above the band-pass model,  identical with the last graph,

*below the  "equivalent low-pass model".

Here,  the inphase and quadrature coefficients are combined to give the complex amplitude coefficient
:$$a_n = a_{\text{I}n} + {\rm j} ⋅ a_{\text{Q}n}.$$
Additionally, the physical transmitted signal  $s(t)$  is replaced by the equivalent low-pass signal
:$$s_{\rm TP}(t) = s_{\rm I}(t) + {\rm j} ⋅ s_{\rm Q}(t).$$

⇒   The representation of the QAM transmitter and receiver is the content of the SWF flash animation  [[Applets:Principle_of_QAM|"Principle of Quadrature Amplitude Modulation"]].
 
{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*As the bit number  $b$  and thus the number of defined symbols  $(M=b^2)$  increases,  the bandwidth efficiency increases,  but also the signal processing overhead increases.

*In addition,  it must be taken into account that a dense QAM allocation is appropriate only if the channel is sufficiently good.}}

'''Reference Model for ADSL-QAM'''

This diagram shows the reference model for  "ADSL-QAM",  whereby we will only deal with the red function blocks  "QAM-Modulator"  and  "'QAM-Demodulator"  here.
[[File:EN_Bei_T_2_3_S5a.png|right|frame|Reference Model for ADSL-QAM]]

*The carrier frequency $f_{\rm T}$  lies within the specified up and down band of the respective xDSL variant.

*Like signal space size  $($between  $M^2=2^2=4$  and  $M^2=2^8=256$  signal space points$)$  and symbol rate, it is determined by channel measurements during transmission initialisation.

*For ADSL-QAM,  the following symbol rates have been specified  $($all numerical values in  "${\rm kBaud}$" $= 10^3 \rm symbols/s)$  :
:*In upstream:  $20$, $40$, $84$, $100$, $120$, $136$;

:*in downstream: $40$, $126$, $160$, $252$, $336$, $504$, $806.4$, $1008$.

==Possible QAM signal space constellations==
 
We still consider possible arrangements of signal space points in quadrature amplitude modulation using three examples.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
An important QAM parameter is the bit number  $b$  processed to the amplitude coefficient pair  $(a_{\rm I}, a_{\rm Q})$.  Here  $b$  is always an even number.
[[File:EN_Bei_T_2_3_S6a_v2.png|right|frame|Signal space constellations  "$\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($left$)$  and  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($right$)$
'''Korrektur''': Jiwoo, ich habe die gelben Punkte durch braune Punkte ersetzt, weil meine schlechten Auge diese nicht erkennen konnten. Könntest u bitte den Text kontrollieren und mit der deutschen Version vergleichen, ob jetzt wirklich alle gelben Punkte braun sind. Das gleiche gilt für Example 3.]]

*If  $b = 2$,  both  $a_{\rm I}$  and  $a_{\rm Q}$  can only take the values  $±1$  resulting in the  $\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$  corresponding to the left constellation.

*According to an ITU recommendation,  the assignment applies here:
:$$q_1 = 0, \ q_0 = 0 \, \Leftrightarrow \,a_{\rm I} = +1, \ a_{\rm Q} = +1,$$
:$$q_1 = 0, \ q_0 = 1 \, \Leftrightarrow \, a_{\rm I} = +1, \ a_{\rm Q} = -1,$$
:$$q_1 = 1, \ q_0 = 0 \, \Leftrightarrow \,a_{\rm I} = -1, \ a_{\rm Q} = +1,$$
:$$q_1 = 1, \ q_0 = 1 \, \Leftrightarrow \, a_{\rm I} = -1, \ a_{\rm Q} = -1.$$

*The point  '''10'''   marked in brown  $(a_{\rm I} = -1, \ a_{\rm Q} = 1)$  thus stands for  $q_1 = 1$  and  $q_0 = 0$.

With  $b = 4$   ⇒   $M = 2^{b/2} = 4$  one arrives at  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  according to the right diagram with the possible amplitude coefficients 
:$$a_{\rm I} ∈ \{±3, ±1\}, \ \ a_{\rm Q} ∈ \{±3, ±1\}.$$

*The assignment can be determined using the auxiliary graph given on the left below, as illustrated by the following numerical examples.

        $\rm (A)$    $q_3 = 1, \ q_2 = 0, \ q_1 = 1,\ q_0 = 1$  $($brown marker$)$:
:#The two most significant bits  $\rm (MSB)$   ⇒   "'''11'''"  determine,  according to the  $\text{4-QAM}$  diagram,  the  $($second$)$  quadrant in which the symbol is located.
:#The two least significant bits  $\rm (LSB)$   ⇒   "'''10'''"  together with the auxiliary graph determine the point within this quadrant.  Result:  $a_{\rm I} = -1$,  $a_{\rm Q} = +3$.
        $\rm (B)$    $q_3 = 0, \ q_2 = 1, \ q_1 = 1,\ q_0 = 0$ $($green marker$)$:
:#The two most significant bits   ⇒   "'''01'''"   refer here to the fourth quadrant.
:#The two least significant bits   ⇒   "'''10'''"  refer to the green dot in the fourth quadrant:   $a_{\rm I} = -3, \ a_{\rm Q} = -3$.}}

{{GraueBox|TEXT=
$\text{Example 2:}$  We use the same graphic as in $\text{Example 1}$.  The decimal value  $D$  provides another way to label the points.
[[File:EN_Bei_T_2_3_S6a_v2.png|right|frame|Signal space constellations  "$\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($left$)$  and  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($right$)$]]

*The brown marked point in the  $\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$ diagram is binary labeled  "'''10'''"    ⇒   decimal:  $D = 2$.  This point simultaneously marks the second quadrant of  $\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$.

*The further subdivision results from the lower left graphic.  There at the brown dot is written  $4D + 3=11$"  $($decimal$)$.

*Therefore,  the upper right dot  $($highlighted in brown$)$  in the upper left quadrant stands for decimal  "$11$   ⇒   binary '''1011'''.

*For the green dot,  $D = 1$  gives the decimal value  $4D + 2 =6$,$11$ which corresponds to the binary representation  "'''0110'''" .
 
According to this scheme,  the signal space constellations
*for  $\rm 64\hspace{0.05cm}–\hspace{-0.02cm}QAM$   ⇒   $(b = 6, \ M = 8)$ 

*and  $\rm 256-QAM$  ⇒   $(b = 8, \ M = 16)$ 

which will be discussed in detail in the  [[Exercise_2.3:_QAM_Signal_Space_Assignment|$\text{Exercise 2.3}$]].  }}

{{GraueBox|TEXT=
$\text{Example 3:}$ 
We still consider for the described  $\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$  $($left graph,  here referred to as  "ITU proposal"$)$  the resulting symbol and bit error probability in the presence of AWGN noise:
#An error can be assumed to result in a horizontally or vertically adjacent symbol,  as indicated for the upper left green dot.
#The symbol error probability $p$  depends on the Euclidean distance of the two points and the AWGN noise power density  $N_0$.
#A distortion to the more distant blue point instead of one of the two neighboring brown points is rather unlikely with Gaussian noise.
[[File:EN Bei T 2 3 S6b neu.png|right|frame|Signal space constellation:  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$";  left: ITU propsal,  right:  Gray encoding. '''KORREKTUR: Symbolfehler, Bitfehler''']]

An analysis shows:
*All corner points  $($green$)$  can only be distorted in two directions.

*In contrast, the inner QAM points  $($blue$)$  have four direct neighbors.

* The remaining symbols  $($brown$)$  have three neighbors.

For the  »'''symbol error probability'''«   $p_{\rm S}$  then holds:

::$$p_{\rm S} = {1}/{16} \cdot (4 \cdot 2 p + 8 \cdot 3 p + 4 \cdot 4 p) = 3p.$$

To calculate the  »'''bit error probability'''«   $p_{\rm B}$  it must now be taken into account that for the left constellation one symbol error leads
* within one quadrant to only  one bit error  $($e.g: "'''0100'''"   ⇒   "'''0110'''"$)$,

*between adjacent quadrants to two bit errors $($e.g: "'''1111'''"   ⇒   "'''0101'''"$)$.

The computation of  $p_{\rm B} $  here involves some effort.

In contrast,  with Gray encoding  $($right diagram$)$:
# Each symbol differs from its neighbors by exactly one bit.
#Each symbol error thus results in exactly one bit error.
#Since each individual symbol contains four bits,  the  "bit error probability" is in this case: 

::$$p_{\rm B} = p_{\rm S}/4 = 3/4 \cdot p. $$}}

==Carrierless Amplitude Phase Modulation==
 
"Carrierless amplitude phase modulation"  $\rm (CAP)$  is a bandwidth-efficient variant of QAM,  which can be realized very easily with digital signal processors.  The only difference to QAM is that the modulation with a carrier signal can be omitted.

[[File:EN_Bei_T_2_3_S7a.png|right|frame|Block diagram of  "Carrierless Amplitude Phase Modulation"]]
[[File:EN_Bei_T_2_3_S7d_neu.png|right|frame|Reference model for  ADSL–CAP]]

*Digital filtering is used instead of multiplication by cosine and minus sine.  $g_{\rm I}(t)$  and  $g_{\rm Q}(t)$  are impulse responses of two transversal bandpass filters with same amplitude characteristic,  phase-shifted by  $π/2$.

*Both are orthogonal to each other,  which means that the integral of the product  $g_{\rm I}(t) · g_{\rm Q}(t)$  over a symbol duration gives zero.

*The signals  $s_{\rm I}(t)$  and  $s_{\rm Q}(t)$  generated in this way are combined,  converted to a continuous-time signal by a D/A converter. 

*The unwanted high-frequency components generated during D/A conversion are eliminated by a low-pass filter before transmission.

*At the receiver,  the signal  $r(t)$  is first converted to a discrete-time signal using an A/D converter and then the in-phase and quadrature symbols  $a_{\rm I}$  and  $a_{\rm Q}$  are extracted via two  "finite impulse response  $\rm (FIR)$  filters"  and downstream "threshold decision".

CAP was the de facto standard in initial ADSL specifications until 1996.  The graph shows the reference model.
#The frequencies up to  $\text{4 kHz}$  were reserved for POTS/ISDN.
#The up channel occupied the frequency range of  $\text{15 - 160 kHz}$,
#The down channel occupied the frequencies from  $\text{240 - 1500 kHz}$.

One problem with CAP is that a  "bad channel" has dramatic consequences on the transmission quality.  Therefore,  today (2011) ADSL-CAP is only found in a few HDSL variants.

==Basics of Discrete Multitone Transmission==
 
[[File:EN_Bei_T_2_3_S8a_v8.png|right|frame|Spectra at OFDM and DMT;     Note:   For this graph a symmetrical rectangle is assumed.     A rectangle between  $0$  and  $T$  would still result in a phase term.     However,  nothing would change with respect to  $|S(f)|$.]]
"Discrete Multitone Transmission"  $\rm (DMT)$  refers to a multicarrier modulation method that is almost identical to  [[Modulation_Methods/General_Description_of_OFDM|"Orthogonal Frequency Division Multiplexing"]]  $\rm (OFDM)$.  In the case of wired transmission,  one usually refers to  "DMT",  and in the case of wireless transmission,  one refers to  "OFDM".

In both cases,  one divides the entire bandwidth into many narrowband equidistant subchannels.  The respective subcarrier signals  $s_k(t)$  are individually impinged with complex data symbols  $D_k$  and the sum of the modulated subcarrier signals is transmitted as signal  $s(t)$ .

The diagram illustrates the principle of OFDM and DMT in the frequency domain,  partly using the values specified for ADSL–DMT:

*$255$  Subcarriers with carrier frequencies  $k \cdot f_0$  $(k = 1$, ... , $255)$.

*$4000$  data frames are transmitted per second.

*After  $68$  data frames,  one synchronization frame is inserted each.

*Due to the  "cyclic prefix"  $($see chapter  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Insert.C3.BCgen_of_Guard.E2.80.93Interval_and_cyclic_Pr. C3.A4fix|"Insertion of Guard–Interval and Cyclic Prefix"]]$)$  the symbol duration  $T = 1/f_0$  must still be shortened by the factor  $16/17$.

*Thus,  the  »'''basic frequency'''«  of DMT is 
:$$f_0 = 4 \cdot (69/68) \cdot (69/68) = 4.3125 \ \rm kHz.$$

An essential difference between OFDM and DMT is that
*in OFDM,  the above drawn spectrum  $S(f)$  in reality describes an  "equivalent low-pass spectrum"  $S_{\rm TP}(f)$  and still the shift around a carrier frequency  $f_{\rm T}$  has to be considered:

:$$S_{\rm TP}(f ) = \sum_{k = 1}^{255} D_k \cdot \delta (f - k \cdot f_0)\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
S(f) = \frac{1}{2} \big [ S_{\rm TP}(f - f_{\rm T}) + S^*_{\rm TP}(-(f + f_{\rm T}))\big ]
\hspace{0.05cm};$$

*in DMT,  on the other hand,  the components at negative frequencies must still be taken into account,  which are to be weighted with the conjugate-complex spectral coefficients:

:$$S(f ) = \sum_{k = 1}^{255} \big [ D_k \cdot \delta (f - k \cdot f_0) + D^*_k \cdot \delta (f + k \cdot f_0) \big ]
\hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Please note:}$ 
#According to these equations,  the complex OFDM signal  $s_{\rm OFDM}(t)$  consists of  $K = 255$  complex exponential oscillations.
#The DMT signal  $s_{\rm DMT}(t)$  is composed of as many cosine oscillations with frequencies  $k \cdot f_0$  $($full occupancy presupposed$)$.
#Despite complex coefficients  $D_k$ resulting from QAM occupancy of the carriers,  the DMT signal is always real because of the conjugate-complex complements at negative frequencies.}}

However,  in both OFDM and DMT,  the transmitted signal  $s(t)$  is precisely limited in time to the symbol duration  $T = 1/f_0 ≈ 232 \ {\rm µs}$  which is equivalent to multiplying by a rectangle of duration $T$.  In the spectral domain,  this corresponds to convolution with the function  $\text{sinc}(fT)$:
*Each Dirac delta function at  $k · f_0$  thus becomes an sinc– function at the same position,  given the time limit,  as shown in the second diagram of the graph above.

*Adjacent subcarrier spectra overlap on the frequency axis,  but exactly at  $k · f_0$  the coefficients  $D_k$  can be seen again,  since all other spectra have zeros here.

{{GraueBox|TEXT=
$\text{Example 4:}$ 
Assuming the conditions favorable for ADSL downstream,  viz.
#$4000$ frames are transmitted per second.
#All subcarriers are active at all times  $(K = 255)$.
#Each carrier is occupied with a 1024-QAM  $(b = 10)$;  according to ITU:  $8 ≤ b ≤ 15$.
#Ideal conditions prevail,  so that the orthogonality evident in the graph is preserved,

then for the  »'''maximum bit rate'''« 
:$$R_{\rm B,\ max} = 4000 · K · b ≈ 10 \ \rm Mbit/s.$$

However,  specified ADSL downstream is only  $2 \ \rm Mbit/s$  due to
*the omission of the  $64$  lowest carriers because of ISDN and upstream,

*the QAM occupancy of the heavily attenuated carriers with less than  $10$  bits,  and

*the consideration of the cyclic prefix,  and some operational reasons.}}

==DMT realization with IDFT/DFT==
 
The upper graph shows the complete DMT system.  For now we will focus on the two red blocks.  The blue blocks will be covered in the [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL|"next chapter"]].  Simplified,  the transmitter and receiver can be represented as shown in the second graph:

[[File:EN_Bei_T_2_3_S9d_ganz_neuV8.png|right|frame|Discrete multitone transmission system]]
[[File:P_ID1951__Bei_T_2_3_S9b_v3.png|right|frame|DMT transmitter and receiver '''Korrektur''' das Bild muss ich noch machen]]

*To perform a DMT modulation,  the transmitter accumulates a block of input bits in a data buffer to be transmitted as one frame.

*The QAM encoder supplies per frame the complex-valued data symbols  $D_1$,  ... ,  $D_{255}$,  which is expanded to the vector  $\mathbf{D}$  of length  $512$  by
#   $D_0 = D_{256} = 0$,
#   $D_k = D^\star_{512-k} \ (k = 257,$ ... , $511)$,
#   the coefficients   $D_{257}$, ... , $D_{511}$  are identical to  $D_{-255}$, ... , $D_{-1}$ $($as consequence of  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Finite_signal_representation|"finite signals"]]  properties$)$.
*The spectral samples  $\mathbf{D}$  are transformed with the [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Inverse_discrete_Fourier_transform|"Inverse Discrete Fourier Transform"]]  to the vector  $\mathbf{s}$  of time signal samples, also with length $512$. Because of the conjugate-complex assignment in the spectral domain   ⇒   $\text{Im}[\mathbf{s}] = 0$.

*After parallel/serial and digital/analog conversion and low-pass filtering of  $\text{Re}[\mathbf{s}]$  the physical  $($real$)$  as well as continuous-time transmitted signal  $s(t)$  results.

*For this,  in the range  $0 ≤ t ≤ T$  $($factor  $2$,  since two coefficients each contribute to cosine/sine$)$:

:$$s(t) = \sum_{k = 1}^{255} \big [ 2 \cdot{\rm Re}\{D_k\} \cdot \cos(2\pi k f_0 t ) - 2 \cdot{\rm Im}\{D_k\} \cdot \sin(2\pi k f_0 t )\big ] \hspace{0.05cm}. $$

*The received signal at the AWGN output is  $r(t) = s(t) + n(t)$.  After A/D and S/P conversion,  $r(t)$  can be expressed by the (real) vector  $\mathbf{r}$ . The  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#From_the_continuous_to_the_discrete_Fourier_transform|"Discrete Fourier Transform"]]  then provides estimates for the transmitted spectral coefficients.
 

{{GraueBox|TEXT=
$\text{Example 5:}$ 
As an example, let's consider the ADSL/DMT downstream.
[[File:EN_Bei_T_2_3_S10_neu.png|right|frame|Allocation of the DMT frequency band with QAM coefficients]]
*In the upper left graph on the right you can see the amounts  $\vert D_k\vert $  of the occupied subchannels  $64$, ... , $255$.  The carriers  $0$, ... , $63$  for the reserved frequency range of ISDN and upstream are set to zero.

*Next to it on the right are the spectral coefficients   $D_{64}$, ... , $D_{255}$   are shown in the complex plane,&nbsp: where the signal space is chosen very large.

*The graph on the left shows the transmitted signal  $s(t)$  for the frame duration   $T = {1}/{f_0} ≈ 232 \ \rm µs$,  which is obtained by low-pass filtering the IDFT values  $s_0$, ... , $s_{511}$  results. This useful signal looks almost like noise.
 
[[File:P_ID1953__Bei_T_2_3_S10b_v1.png|left|frame|Transmitted signal with above DMT allocation]]
 
It can be seen:

#Main problem of DMT is the unfavorable Crest factor   ⇒   ratio of maximum value  $s_{\rm max}$  and rms value  $s_{\rm eff}$.
#The large dynamic range visible in the exemplary signal curve places high demands on the amplifier's linearity.
#When the dynamic range is limited,  the peaks of  $s(t)$  are cut off.
#This acts as a pulse-like disturbance and an additional noise load to the system.}}

{{BlaueBox|TEXT=
$\text{Summarizing:}$ 
*Discrete Multitone Transmission  $\rm (DMT)$  is basically the parallel implementation of many narrowband QAM modems with different carriers and relatively low data transmission rates. 

*The low bandwidth per subcarrier allows a long symbol duration,  thus reducing the influence of intersymbol interference and reducing the development effort for equalization.

*A major reason for the DMT success is its technical ease of implementation.  IDFT and DFT are formed with digital signal processors in real time.  The vectors have the length  $512$  $($power of two$)$.  Therefore the particularly fast FFT algorithm  $($"Fast Fourier Transformation"$)$  can be applied.}}

==Exercises for the chapter ==

[[Exercise_2.3:_QAM_Signal_Space_Assignment|Exercise 2.3: QAM Signal Space Assignment]]

[[Exercise_2.3Z:_xDSL_Frequency_Band|Exercise 2.3Z: xDSL Frequency Band]]

[[Exercise_2.4:_DSL/DMT_with_IDFT/DFT|Exercise 2.4: DSL/DMT with IDFT/DFT]]

[[Exercise_2.4Z:_Repetition_to_IDFT|Exercise 2.4Z: Repetition to IDFT]]

{{Display}}

Examples of Communication Systems/xDSL as Transmission Technology

2023-03-21T19:35:13Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=xDSL Systems
|Nächste Seite=Methods to Reduce the Bit Error Rate in DSL
}}

==Possible bandwidth allocations for xDSL==
 
The xDSL specifications give operators a great deal of freedom with regard to allocation.

For the necessary directional separation of the xDSL signal transmission according to
*the downstream direction from the provider to the customer  $($with the highest possible data rate$)$,

*the upstream direction from the customer to the provider  $($with mostly lower data rate$)$,

two variants have been standardized for this purpose:

{{BlaueBox|TEXT=
$\text{Definition:}$ 
*With the  »'''frequency separation method'''«  the data streams for the two directions are transmitted in two separate frequency bands with the advantage that a simple filter is sufficient to separate the transmission directions,  which simplifies the technical implementation.

*With the  »'''frequency uniform position method'''«  the spectra of upstream and downstream overlap in a certain part.  The separation is done here with help of an echo cancellation circuit.  Advantages of this method:  The lower bandwidth requirement at higher  $($and thus more attenuated$)$  frequencies and longer range.}}

[[File:EN_Bei_T_2_3_S1a_v4.png|right|frame|"Frequency separation method"  &  "Frequency uniform position method"]]
The graph compares these two options.  Basically,  the specifications leave it up to the developers/operators to decide,
*to operate xDSL alone on the subscriber line,  or

*to allow mixed operation of xDSL with the telephone services  $\rm POTS$  $($"Plain Old Telephone Service"$)$  or  $\rm ISDN$  $($"Integrated Services Digital Network"$)$,  and
*thus to exclude or also occupy the lower frequency range occupied by the two telephone services for xDSL.
 
==ADSL bandwidth allocation==
 
Since it is technically much easier to realise, the decision in Germany for "ADSL" and "ADSL2+" was in favour of

[[File:EN_Bei_T_2_3_S1b_v4.png|right|frame|ADSL bandwidth allocation in Germany and most of other European countries]]
*of the  "frequency separation method",  and
*the general reservation of the lower frequency range for ISDN.

The  "frequency uniform position method"  is still used in some cases,  but rather rarely.

The bandwidth available for DSL is not further decomposed for the transmission methods
*[[Modulation_Methods/Quadrature_Amplitude_Modulation|$\rm QAM$]]  $($"Quadrature Amplitude Modulation"$)$  and

*[[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Carrierless_Amplitude_Phase_Modulation|$\rm CAP$]]  $($"Carrierless Amplitude Phase Modulation$)$.

In contrast,  with the multicarrier method  [[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|$\rm DMT$]]  $($"Discrete Multitone Transmission"$),$  the uplink and the downlink channel are divided into  $N_{\rm Up}$  resp.  $N_{\rm Down}$  bins of  $4.3125\rm \ kHz $  each.

In addition to the above graph, it should be noted:
#Phone services  $($POTS or ISDN$)$  and xDSL are in different frequency bands,  which minimizes mutual interference in the trunk cable.  Thus,  the stronger signal ISDN does not interfere with the parallel running xDSL and vice versa.
#The lower frequency range up to  $\text{120 kHz}$  has been reserved for ISDN  $($optionally POTS$)$.  This value results from the first zero of the  [[Digital_Signal_Transmission/Block_Coding_with_4B3T_Codes#ACF_and_PSD_of_the_4B3T_codes|"ISDN spectrum with 4B3T coding"]].  Above $\text{120 kHz}$  the ISDN spectrum is completely suppressed.
#To separate the telephone and xDSL signals,  a  [[Examples_of_Communication_Systems/xDSL_Systems#Components_of_DSL_Internet_access|"splitter"]]  is used at both ends of the two-wire line,  which includes one low-pass filter and one high-pass filter each and also takes into account the following frequency gap up to  $\text{138 kHz}$.
#After this occupancy gap follows the ADSL upstream band from  $\text{138 kHz}$  to $\text{276 kHz}$.  This allows the transmission of  $N_{\rm Up} = 32$  subcarriers with bandwidth  $\text{4.3125 kHz}$  each.  This value is derived from the frame transmission rate.
#The subsequent downstream range extends to  $\text{1104 kHz}$  for ADSL,  with which  $N_{\rm Down} = 256$  subcarriers can be realized.  The separation of the upstream and the downstream channel in xDSL is done by a band-pass filter in the modem.
#However,  the first  $64$  subcarriers  $($this corresponds to $\text{276 kHz)}$  must not be occupied.  With the  [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Possible_bandwidth_allocations_for_xDSL|"frequency uniform position method"]]  only  $32$  subcarriers would have to be left out,  taking into account that the separation of uplink and downlink requires a more complex implementation.
#For  ADSL2+,  the system bandwidth is  $\text{2208 kHz}$   ⇒   $N_{\rm Down} = 512$  subcarriers.  The number of bins to be spared remains unchanged compared to ADSL.  Taking into account that two bins are occupied by control functions  $($e.g. for synchronization$)$,  $190$  $($ADSL$)$  or  $446$  $($ADSL2+$)$ downstream channels remain for users.
#The ISDN reservation prescribed in Germany,  however,  has the consequence for xDSL that the low frequencies,  which are by far the least attenuated in a copper line and would therefore actually be the most suitable,  cannot be used.
#Further,  from the frequency arrangement,  it can be seen that the downstream bins are more attenuated than the upstream bins  $($higher frequencies$)$  and consequently have a smaller signal-to-noise ratio  $($SNR$)$.
#The decision  "upstream below downstream"  is related to the fact that the loss of downstream channels has only a comparatively small impact on the transmission rate.  In the upstream,  such a failure would be much more noticeable in percentage terms.

==VDSL(2) bandwidth allocation==
 
The ITU has defined several profiles for  "VDSL(2)".  At the time of writing this chapter  (2010),  the frequency band assignment specified in the graphic applies to the systems deployed in Germany in accordance with the ITU's  $\text{ VDSL(2) Plan 998b - Profile 17a (Annex B)}$.  The (slightly) lighter coloring at the higher frequencies is meant to indicate that these channels are more attenuated.

Without claiming to be exhaustive, this allocation plan can be characterized as follows:

[[File:EN_Bei_T_2_3_S5a_ganzneu.png|right|frame|VDSL(2) bandwidth allocation in Germany ]]

#To achieve higher bit rates,  eight times as many bins are used here as in  "ADSL2+".
#Thus,  the bandwidth enables  $8 \cdot \text{2208 MHz = 17664 MHz}$,  transmission rates of up to  $\text{100 Mbit/s}$  $($depending on cable length and conditions$)$.
#The same as in ADSL,  the frequency bands for the upstream subchannels are always arranged at the lower frequencies,  since the greater cable attenuation  has a greater percentage influence on the bit rate for upstream than for downstream.
#In VDSL(2),  the  [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Possible_bandwidth_allocations_for_xDSL|"frequency separation method"]]  is always used.  Overlapping of the upstream and downstream frequency bands is categorically excluded in the ITU specification .
#In the VDSL systems,  the lower frequencies are reserved for ISDN.  This is followed by alternating ranges for upstream and downstream.  From the specified limits,  one can recognize the narrower upstream ranges,  compared to the downstream.
#You can see an alternating arrangement of upstream and downstream areas.  One reason is that with this wide spectrum it should be avoided that one direction  $($e.g. downstream$)$  is assigned only strongly attenuated  $($high$)$  frequencies.

The VDSL(2) specification provides for allocation plans up to system bandwidths of  $\text{30 MHz}$  $($according to profile 30a$)$.
* This should enable transmission rates up to about  $\text{ 200 Mbit/s}$  over short distances.

*For this purpose, the bandwidth of the individual subchannels is also doubled compared to ADSL to  $\text{8.625 kHz}$ .

==Overview of xDSL transmission methods==
 
At the beginning of the various standardization procedures for the individual xDSL variants,  different transmission methods were defined as a basis:
# [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modulation"]]  $\text{QAM}$  for  "QAM-ADSL"  and  "QAM-VDSL", 
# [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Carrierless_Amplitude_Phase_Modulation|"Carrierless Amplitude Phase Modulation"]]  $\text{CAP}$  for  "CAP-HDSL"  and  "CAP-ADSL",
#[[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|"Discrete Multitone Transmission"]]  $\text{DMT}$  for  "ADSL",  "ADSL2",  "ADSL2+",  "VDSL"  and",  "VDSL2".

With increasing market demand for higher transmission rates and the associated requirements, two main  suitable processes crystallized,  namely  "QAM/CAP"  and  "DMT".

Since the manufacturers were unable to agree on a common standard from 1997 to 2003,  also for patent reasons  $($one even speaks of  "line code wars"  in this context$)$,  the two competing methods coexisted for a long time.  At the so-called  "DSL Olympics"  in 2003,  '''the decision was finally made in favor of DMT''',
*on the one hand,  because of the slightly better  "performance"  in general,

*but in particular because of its higher robustness against narrowband interference; especially for the USA  $($many overhead telephone lines and related problems with coupled radio signals$)$  this argument played a major role.

First,  however,  the systems  "QAM-xDSL"  and  "CAP-xDSL"  will be considered very briefly.

==Basics of Quadrature Amplitude Modulation==
 

The principle has already been described in detail in the chapter  [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modation"]]  of the book  "Modulation Methods". 
[[File:EN_Bei_T_2_3_S6a.png|right|frame| Linear modulator with  $\rm I$ and  $\rm Q$–components;  signal space for  $\text{16-QAM}$]]
Here follows short summary on the basis of the block diagram on the right:
*QAM is a  "single carrier modulation method"  around the carrier frequency  $f_{\rm T}$.  First,  a blockwise serial/parallel conversion of the bit stream and signal space assignment is performed.

*From each  $b$  binary symbols,  two multilevel '''KORREKTUR: multi-level''' amplitude coefficients  $a_{{\rm I}n}$  and  $a_{{\rm Q}n}$  are derived  $($"in-phase component"  and  "quadrature component"$)$.  Both coefficients can each take one of  $M = 2^{b/2}$  possible amplitude values.

*The example considered here applies to the  $\text{16-QAM}$  with  $b = M = 4$  and correspondingly sixteen  signal space points. For a   $\text{256-QAM}$  would hold:   $b = 8$  and  $M = 16$  $(2^b = M^2 = 256)$.

*The coefficients  $a_{{\rm I}n}$  and  $a_{{\rm Q}n}$  are each impressed on a Dirac delta impulse as weights.  For pulse shaping one usually uses  $($because of the small bandwidth$)$  a raised-cosine filter.

*With the basic transmission pulse  $g_s(t)$  is then valid in the two branches of the block diagram:
:$$ s_{\rm I}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm
I\hspace{0.03cm}\it n} \cdot g_s (t - n \cdot T)\hspace{0.05cm},\hspace{0.5cm}
s_{\rm Q}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm
Q\hspace{0.03cm}\it n} \cdot g_s (t - n \cdot
T)\hspace{0.05cm}.$$
:Note:  Because of the redundancy-free conversion to a higher-level code,  the symbol duration  $T$  of these signals is greater by a factor  $b$  than the bit duration  $T_{\rm B}$  of the binary input sequence.  In the drawn example  $($16-QAM$)$,  $T = 4 ⋅ T_{\rm B}$.
*The  »'''QAM transmitted signal'''«  $s(t)$  is then the sum of the two partial signals  $ s_{\rm I}(t)$  and  $ s_{\rm Q}(t)$,  multiplied by  "cosine"  resp.  "minus-sine"  $($possibly followed by a band limit to prevent interference to adjacent bands, as indicated in the graph below$)$:
:$$s(t) = s_{\rm I}(t) \cdot \cos (2 \pi f_{\rm T}\,t) - s_{\rm Q}(t) \cdot \sin (2 \pi f_{\rm T}\,t)
\hspace{0.05cm}. $$
*The two branches  $(\rm I$  and  $\rm Q)$  can be considered as two completely separate  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#M.E2.80.93level_amplitude_shift_keying_.28M.E2.80.93ASK.29 |"M–level ASK systems"]]  which do not interfere with each other as long as all components are optimally designed.  This means at the same time:  

*Compared to a  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_phase_shift_keying_.28BPSK.29|"Binary Phase Shift Keying"]]  $\rm (BPSK)$  modulation with cosine or sine only,  the  '''QAM allows a doubling of the data rate at constant quality'''.

[[File:EN_Bei_T_2_3_S6b.png|right|frame|Quadrature Amplitude Modulation as band-pass and equivalent low-pass model]]

'''Equivalent low-pass model'''

The graph on the right shows
*above the band-pass model,  identical with the last graph,

*below the  "equivalent low-pass model".

Here,  the inphase and quadrature coefficients are combined to give the complex amplitude coefficient
:$$a_n = a_{\text{I}n} + {\rm j} ⋅ a_{\text{Q}n}.$$
Additionally, the physical transmitted signal  $s(t)$  is replaced by the equivalent low-pass signal
:$$s_{\rm TP}(t) = s_{\rm I}(t) + {\rm j} ⋅ s_{\rm Q}(t).$$

⇒   The representation of the QAM transmitter and receiver is the content of the SWF flash animation  [[Applets:Principle_of_QAM|"Principle of Quadrature Amplitude Modulation"]].
 
{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*As the bit number  $b$  and thus the number of defined symbols  $(M=b^2)$  increases,  the bandwidth efficiency increases,  but also the signal processing overhead increases.

*In addition,  it must be taken into account that a dense QAM allocation is appropriate only if the channel is sufficiently good.}}

'''Reference Model for ADSL-QAM'''

This diagram shows the reference model for  "ADSL-QAM",  whereby we will only deal with the red function blocks  "QAM-Modulator"  and  "'QAM-Demodulator"  here.
[[File:EN_Bei_T_2_3_S5a.png|right|frame|Reference Model for ADSL-QAM]]

*The carrier frequency $f_{\rm T}$  lies within the specified up and down band of the respective xDSL variant.

*Like signal space size  $($between  $M^2=2^2=4$  and  $M^2=2^8=256$  signal space points$)$  and symbol rate, it is determined by channel measurements during transmission initialisation.

*For ADSL-QAM,  the following symbol rates have been specified  $($all numerical values in  "${\rm kBaud}$" $= 10^3 \rm symbols/s)$  :
:*In upstream:  $20$, $40$, $84$, $100$, $120$, $136$;

:*in downstream: $40$, $126$, $160$, $252$, $336$, $504$, $806.4$, $1008$.

==Possible QAM signal space constellations==
 
We still consider possible arrangements of signal space points in quadrature amplitude modulation using three examples.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
An important QAM parameter is the bit number  $b$  processed to the amplitude coefficient pair  $(a_{\rm I}, a_{\rm Q})$.  Here  $b$  is always an even number.
[[File:EN_Bei_T_2_3_S6a_v2.png|right|frame|Signal space constellations  "$\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($left$)$  and  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($right$)$
'''Korrektur''': Jiwoo, ich habe die gelben Punkte durch braune Punkte ersetzt, weil meine schlechten Auge diese nicht erkennen konnten. Könntest u bitte den Text kontrollieren und mit der deutschen Version vergleichen, ob jetzt wirklich alle gelben Punkte braun sind. Das gleiche gilt für Example 3.]]

*If  $b = 2$,  both  $a_{\rm I}$  and  $a_{\rm Q}$  can only take the values  $±1$  resulting in the  $\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$  corresponding to the left constellation.

*According to an ITU recommendation,  the assignment applies here:
:$$q_1 = 0, \ q_0 = 0 \, \Leftrightarrow \,a_{\rm I} = +1, \ a_{\rm Q} = +1,$$
:$$q_1 = 0, \ q_0 = 1 \, \Leftrightarrow \, a_{\rm I} = +1, \ a_{\rm Q} = -1,$$
:$$q_1 = 1, \ q_0 = 0 \, \Leftrightarrow \,a_{\rm I} = -1, \ a_{\rm Q} = +1,$$
:$$q_1 = 1, \ q_0 = 1 \, \Leftrightarrow \, a_{\rm I} = -1, \ a_{\rm Q} = -1.$$

*The point  '''10'''   marked in brown  $(a_{\rm I} = -1, \ a_{\rm Q} = 1)$  thus stands for  $q_1 = 1$  and  $q_0 = 0$.

With  $b = 4$   ⇒   $M = 2^{b/2} = 4$  one arrives at  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  according to the right diagram with the possible amplitude coefficients 
:$$a_{\rm I} ∈ \{±3, ±1\}, \ \ a_{\rm Q} ∈ \{±3, ±1\}.$$

*The assignment can be determined using the auxiliary graph given on the left below, as illustrated by the following numerical examples.

        $\rm (A)$    $q_3 = 1, \ q_2 = 0, \ q_1 = 1,\ q_0 = 1$  $($brown marker$)$:
:#The two most significant bits  $\rm (MSB)$   ⇒   "'''11'''"  determine,  according to the  $\text{4-QAM}$  diagram,  the  $($second$)$  quadrant in which the symbol is located.
:#The two least significant bits  $\rm (LSB)$   ⇒   "'''10'''"  together with the auxiliary graph determine the point within this quadrant.  Result:  $a_{\rm I} = -1$,  $a_{\rm Q} = +3$.
        $\rm (B)$    $q_3 = 0, \ q_2 = 1, \ q_1 = 1,\ q_0 = 0$ $($green marker$)$:
:#The two most significant bits   ⇒   "'''01'''"   refer here to the fourth quadrant.
:#The two least significant bits   ⇒   "'''10'''"  refer to the green dot in the fourth quadrant:   $a_{\rm I} = -3, \ a_{\rm Q} = -3$.}}

{{GraueBox|TEXT=
$\text{Example 2:}$  We use the same graphic as in $\text{Example 1}$.  The decimal value  $D$  provides another way to label the points.
[[File:EN_Bei_T_2_3_S6a_v2.png|right|frame|Signal space constellations  "$\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($left$)$  and  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($right$)$]]

*The brown marked point in the  $\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$ diagram is binary labeled  "'''10'''"    ⇒   decimal:  $D = 2$.  This point simultaneously marks the second quadrant of  $\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$.

*The further subdivision results from the lower left graphic.  There at the brown dot is written  $4D + 3=11$"  $($decimal$)$.

*Therefore,  the upper right dot  $($highlighted in brown$)$  in the upper left quadrant stands for decimal  "$11$   ⇒   binary '''1011'''.

*For the green dot,  $D = 1$  gives the decimal value  $4D + 2 =6$,$11$ which corresponds to the binary representation  "'''0110'''" .
 
According to this scheme,  the signal space constellations
*for  $\rm 64\hspace{0.05cm}–\hspace{-0.02cm}QAM$   ⇒   $(b = 6, \ M = 8)$ 

*and  $\rm 256-QAM$  ⇒   $(b = 8, \ M = 16)$ 

which will be discussed in detail in the  [[Exercise_2.3:_QAM_Signal_Space_Assignment|$\text{Exercise 2.3}$]].  }}

{{GraueBox|TEXT=
$\text{Example 3:}$ 
We still consider for the described  $\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$  $($left graph,  here referred to as  "ITU proposal"$)$  the resulting symbol and bit error probability in the presence of AWGN noise:
#An error can be assumed to result in a horizontally or vertically adjacent symbol,  as indicated for the upper left green dot.
#The symbol error probability $p$  depends on the Euclidean distance of the two points and the AWGN noise power density  $N_0$.
#A distortion to the more distant blue point instead of one of the two neighboring brown points is rather unlikely with Gaussian noise.
[[File:EN Bei T 2 3 S6b neu.png|right|frame|Signal space constellation:  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$";  left: ITU propsal,  right:  Gray encoding. ]]

An analysis shows:
*All corner points  $($green$)$  can only be distorted in two directions.

*In contrast, the inner QAM points  $($blue$)$  have four direct neighbors.

* The remaining symbols  $($brown$)$  have three neighbors.

For the  »'''symbol error probability'''«   $p_{\rm S}$  then holds:

::$$p_{\rm S} = {1}/{16} \cdot (4 \cdot 2 p + 8 \cdot 3 p + 4 \cdot 4 p) = 3p.$$

To calculate the  »'''bit error probability'''«   $p_{\rm B}$  it must now be taken into account that for the left constellation one symbol error leads
* within one quadrant to only  one bit error  $($e.g: "'''0100'''"   ⇒   "'''0110'''"$)$,

*between adjacent quadrants to two bit errors $($e.g: "'''1111'''"   ⇒   "'''0101'''"$)$.

The computation of  $p_{\rm B} $  here involves some effort.

In contrast,  with Gray encoding  $($right diagram$)$:
# Each symbol differs from its neighbors by exactly one bit.
#Each symbol error thus results in exactly one bit error.
#Since each individual symbol contains four bits,  the  "bit error probability" is in this case: 

::$$p_{\rm B} = p_{\rm S}/4 = 3/4 \cdot p. $$}}

==Carrierless Amplitude Phase Modulation==
 
"Carrierless amplitude phase modulation"  $\rm (CAP)$  is a bandwidth-efficient variant of QAM,  which can be realized very easily with digital signal processors.  The only difference to QAM is that the modulation with a carrier signal can be omitted.

[[File:EN_Bei_T_2_3_S7a.png|right|frame|Block diagram of  "Carrierless Amplitude Phase Modulation"]]
[[File:EN_Bei_T_2_3_S7d_neu.png|right|frame|Reference model for  ADSL–CAP]]

*Digital filtering is used instead of multiplication by cosine and minus sine.  $g_{\rm I}(t)$  and  $g_{\rm Q}(t)$  are impulse responses of two transversal bandpass filters with same amplitude characteristic,  phase-shifted by  $π/2$.

*Both are orthogonal to each other,  which means that the integral of the product  $g_{\rm I}(t) · g_{\rm Q}(t)$  over a symbol duration gives zero.

*The signals  $s_{\rm I}(t)$  and  $s_{\rm Q}(t)$  generated in this way are combined,  converted to a continuous-time signal by a D/A converter. 

*The unwanted high-frequency components generated during D/A conversion are eliminated by a low-pass filter before transmission.

*At the receiver,  the signal  $r(t)$  is first converted to a discrete-time signal using an A/D converter and then the in-phase and quadrature symbols  $a_{\rm I}$  and  $a_{\rm Q}$  are extracted via two  "finite impulse response  $\rm (FIR)$  filters"  and downstream "threshold decision".

CAP was the de facto standard in initial ADSL specifications until 1996.  The graph shows the reference model.
#The frequencies up to  $\text{4 kHz}$  were reserved for POTS/ISDN.
#The up channel occupied the frequency range of  $\text{15 - 160 kHz}$,
#The down channel occupied the frequencies from  $\text{240 - 1500 kHz}$.

One problem with CAP is that a  "bad channel" has dramatic consequences on the transmission quality.  Therefore,  today (2011) ADSL-CAP is only found in a few HDSL variants.

==Basics of Discrete Multitone Transmission==
 
[[File:EN_Bei_T_2_3_S8a_v8.png|right|frame|Spectra at OFDM and DMT;     Note:   For this graph a symmetrical rectangle is assumed.     A rectangle between  $0$  and  $T$  would still result in a phase term.     However,  nothing would change with respect to  $|S(f)|$.]]
"Discrete Multitone Transmission"  $\rm (DMT)$  refers to a multicarrier modulation method that is almost identical to  [[Modulation_Methods/General_Description_of_OFDM|"Orthogonal Frequency Division Multiplexing"]]  $\rm (OFDM)$.  In the case of wired transmission,  one usually refers to  "DMT",  and in the case of wireless transmission,  one refers to  "OFDM".

In both cases,  one divides the entire bandwidth into many narrowband equidistant subchannels.  The respective subcarrier signals  $s_k(t)$  are individually impinged with complex data symbols  $D_k$  and the sum of the modulated subcarrier signals is transmitted as signal  $s(t)$ .

The diagram illustrates the principle of OFDM and DMT in the frequency domain,  partly using the values specified for ADSL–DMT:

*$255$  Subcarriers with carrier frequencies  $k \cdot f_0$  $(k = 1$, ... , $255)$.

*$4000$  data frames are transmitted per second.

*After  $68$  data frames,  one synchronization frame is inserted each.

*Due to the  "cyclic prefix"  $($see chapter  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Insert.C3.BCgen_of_Guard.E2.80.93Interval_and_cyclic_Pr. C3.A4fix|"Insertion of Guard–Interval and Cyclic Prefix"]]$)$  the symbol duration  $T = 1/f_0$  must still be shortened by the factor  $16/17$.

*Thus,  the  »'''basic frequency'''«  of DMT is 
:$$f_0 = 4 \cdot (69/68) \cdot (69/68) = 4.3125 \ \rm kHz.$$

An essential difference between OFDM and DMT is that
*in OFDM,  the above drawn spectrum  $S(f)$  in reality describes an  "equivalent low-pass spectrum"  $S_{\rm TP}(f)$  and still the shift around a carrier frequency  $f_{\rm T}$  has to be considered:

:$$S_{\rm TP}(f ) = \sum_{k = 1}^{255} D_k \cdot \delta (f - k \cdot f_0)\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
S(f) = \frac{1}{2} \big [ S_{\rm TP}(f - f_{\rm T}) + S^*_{\rm TP}(-(f + f_{\rm T}))\big ]
\hspace{0.05cm};$$

*in DMT,  on the other hand,  the components at negative frequencies must still be taken into account,  which are to be weighted with the conjugate-complex spectral coefficients:

:$$S(f ) = \sum_{k = 1}^{255} \big [ D_k \cdot \delta (f - k \cdot f_0) + D^*_k \cdot \delta (f + k \cdot f_0) \big ]
\hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Please note:}$ 
#According to these equations,  the complex OFDM signal  $s_{\rm OFDM}(t)$  consists of  $K = 255$  complex exponential oscillations.
#The DMT signal  $s_{\rm DMT}(t)$  is composed of as many cosine oscillations with frequencies  $k \cdot f_0$  $($full occupancy presupposed$)$.
#Despite complex coefficients  $D_k$ resulting from QAM occupancy of the carriers,  the DMT signal is always real because of the conjugate-complex complements at negative frequencies.}}

However,  in both OFDM and DMT,  the transmitted signal  $s(t)$  is precisely limited in time to the symbol duration  $T = 1/f_0 ≈ 232 \ {\rm µs}$  which is equivalent to multiplying by a rectangle of duration $T$.  In the spectral domain,  this corresponds to convolution with the function  $\text{sinc}(fT)$:
*Each Dirac delta function at  $k · f_0$  thus becomes an sinc– function at the same position,  given the time limit,  as shown in the second diagram of the graph above.

*Adjacent subcarrier spectra overlap on the frequency axis,  but exactly at  $k · f_0$  the coefficients  $D_k$  can be seen again,  since all other spectra have zeros here.

{{GraueBox|TEXT=
$\text{Example 4:}$ 
Assuming the conditions favorable for ADSL downstream,  viz.
#$4000$ frames are transmitted per second.
#All subcarriers are active at all times  $(K = 255)$.
#Each carrier is occupied with a 1024-QAM  $(b = 10)$;  according to ITU:  $8 ≤ b ≤ 15$.
#Ideal conditions prevail,  so that the orthogonality evident in the graph is preserved,

then for the  »'''maximum bit rate'''« 
:$$R_{\rm B,\ max} = 4000 · K · b ≈ 10 \ \rm Mbit/s.$$

However,  specified ADSL downstream is only  $2 \ \rm Mbit/s$  due to
*the omission of the  $64$  lowest carriers because of ISDN and upstream,

*the QAM occupancy of the heavily attenuated carriers with less than  $10$  bits,  and

*the consideration of the cyclic prefix,  and some operational reasons.}}

==DMT realization with IDFT/DFT==
 
The upper graph shows the complete DMT system.  For now we will focus on the two red blocks.  The blue blocks will be covered in the [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL|"next chapter"]].  Simplified,  the transmitter and receiver can be represented as shown in the second graph:

[[File:EN_Bei_T_2_3_S9d_ganz_neuV8.png|right|frame|Discrete multitone transmission system]]
[[File:P_ID1951__Bei_T_2_3_S9b_v3.png|right|frame|DMT transmitter and receiver '''Korrektur''' das Bild muss ich noch machen]]

*To perform a DMT modulation,  the transmitter accumulates a block of input bits in a data buffer to be transmitted as one frame.

*The QAM encoder supplies per frame the complex-valued data symbols  $D_1$,  ... ,  $D_{255}$,  which is expanded to the vector  $\mathbf{D}$  of length  $512$  by
#   $D_0 = D_{256} = 0$,
#   $D_k = D^\star_{512-k} \ (k = 257,$ ... , $511)$,
#   the coefficients   $D_{257}$, ... , $D_{511}$  are identical to  $D_{-255}$, ... , $D_{-1}$ $($as consequence of  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Finite_signal_representation|"finite signals"]]  properties$)$.
*The spectral samples  $\mathbf{D}$  are transformed with the [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Inverse_discrete_Fourier_transform|"Inverse Discrete Fourier Transform"]]  to the vector  $\mathbf{s}$  of time signal samples, also with length $512$. Because of the conjugate-complex assignment in the spectral domain   ⇒   $\text{Im}[\mathbf{s}] = 0$.

*After parallel/serial and digital/analog conversion and low-pass filtering of  $\text{Re}[\mathbf{s}]$  the physical  $($real$)$  as well as continuous-time transmitted signal  $s(t)$  results.

*For this,  in the range  $0 ≤ t ≤ T$  $($factor  $2$,  since two coefficients each contribute to cosine/sine$)$:

:$$s(t) = \sum_{k = 1}^{255} \big [ 2 \cdot{\rm Re}\{D_k\} \cdot \cos(2\pi k f_0 t ) - 2 \cdot{\rm Im}\{D_k\} \cdot \sin(2\pi k f_0 t )\big ] \hspace{0.05cm}. $$

*The received signal at the AWGN output is  $r(t) = s(t) + n(t)$.  After A/D and S/P conversion,  $r(t)$  can be expressed by the (real) vector  $\mathbf{r}$ . The  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#From_the_continuous_to_the_discrete_Fourier_transform|"Discrete Fourier Transform"]]  then provides estimates for the transmitted spectral coefficients.
 

{{GraueBox|TEXT=
$\text{Example 5:}$ 
As an example, let's consider the ADSL/DMT downstream.
[[File:EN_Bei_T_2_3_S10_neu.png|right|frame|Allocation of the DMT frequency band with QAM coefficients]]
*In the upper left graph on the right you can see the amounts  $\vert D_k\vert $  of the occupied subchannels  $64$, ... , $255$.  The carriers  $0$, ... , $63$  for the reserved frequency range of ISDN and upstream are set to zero.

*Next to it on the right are the spectral coefficients   $D_{64}$, ... , $D_{255}$   are shown in the complex plane,&nbsp: where the signal space is chosen very large.

*The graph on the left shows the transmitted signal  $s(t)$  for the frame duration   $T = {1}/{f_0} ≈ 232 \ \rm µs$,  which is obtained by low-pass filtering the IDFT values  $s_0$, ... , $s_{511}$  results. This useful signal looks almost like noise.
 
[[File:P_ID1953__Bei_T_2_3_S10b_v1.png|left|frame|Transmitted signal with above DMT allocation]]
 
It can be seen:

#Main problem of DMT is the unfavorable Crest factor   ⇒   ratio of maximum value  $s_{\rm max}$  and rms value  $s_{\rm eff}$.
#The large dynamic range visible in the exemplary signal curve places high demands on the amplifier's linearity.
#When the dynamic range is limited,  the peaks of  $s(t)$  are cut off.
#This acts as a pulse-like disturbance and an additional noise load to the system.}}

{{BlaueBox|TEXT=
$\text{Summarizing:}$ 
*Discrete Multitone Transmission  $\rm (DMT)$  is basically the parallel implementation of many narrowband QAM modems with different carriers and relatively low data transmission rates. 

*The low bandwidth per subcarrier allows a long symbol duration,  thus reducing the influence of intersymbol interference and reducing the development effort for equalization.

*A major reason for the DMT success is its technical ease of implementation.  IDFT and DFT are formed with digital signal processors in real time.  The vectors have the length  $512$  $($power of two$)$.  Therefore the particularly fast FFT algorithm  $($"Fast Fourier Transformation"$)$  can be applied.}}

==Exercises for the chapter ==

[[Exercise_2.3:_QAM_Signal_Space_Assignment|Exercise 2.3: QAM Signal Space Assignment]]

[[Exercise_2.3Z:_xDSL_Frequency_Band|Exercise 2.3Z: xDSL Frequency Band]]

[[Exercise_2.4:_DSL/DMT_with_IDFT/DFT|Exercise 2.4: DSL/DMT with IDFT/DFT]]

[[Exercise_2.4Z:_Repetition_to_IDFT|Exercise 2.4Z: Repetition to IDFT]]

{{Display}}

Examples of Communication Systems/xDSL as Transmission Technology

2023-03-21T19:19:49Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=xDSL Systems
|Nächste Seite=Methods to Reduce the Bit Error Rate in DSL
}}

==Possible bandwidth allocations for xDSL==
 
The xDSL specifications give operators a great deal of freedom with regard to allocation.

For the necessary directional separation of the xDSL signal transmission according to
*the downstream direction from the provider to the customer  $($with the highest possible data rate$)$,

*the upstream direction from the customer to the provider  $($with mostly lower data rate$)$,

two variants have been standardized for this purpose:

{{BlaueBox|TEXT=
$\text{Definition:}$ 
*With the  »'''frequency separation method'''«  the data streams for the two directions are transmitted in two separate frequency bands with the advantage that a simple filter is sufficient to separate the transmission directions,  which simplifies the technical implementation.

*With the  »'''frequency uniform position method'''«  the spectra of upstream and downstream overlap in a certain part.  The separation is done here with help of an echo cancellation circuit.  Advantages of this method:  The lower bandwidth requirement at higher  $($and thus more attenuated$)$  frequencies and longer range.}}

[[File:EN_Bei_T_2_3_S1a_v4.png|right|frame|"Frequency separation method"  &  "Frequency uniform position method"]]
The graph compares these two options.  Basically,  the specifications leave it up to the developers/operators to decide,
*to operate xDSL alone on the subscriber line,  or

*to allow mixed operation of xDSL with the telephone services  $\rm POTS$  $($"Plain Old Telephone Service"$)$  or  $\rm ISDN$  $($"Integrated Services Digital Network"$)$,  and
*thus to exclude or also occupy the lower frequency range occupied by the two telephone services for xDSL.
 
==ADSL bandwidth allocation==
 
Since it is technically much easier to realise, the decision in Germany for "ADSL" and "ADSL2+" was in favour of

[[File:EN_Bei_T_2_3_S1b_v4.png|right|frame|ADSL bandwidth allocation in Germany and most of other European countries]]
*of the  "frequency separation method",  and
*the general reservation of the lower frequency range for ISDN.

The  "frequency uniform position method"  is still used in some cases,  but rather rarely.

The bandwidth available for DSL is not further decomposed for the transmission methods
*[[Modulation_Methods/Quadrature_Amplitude_Modulation|$\rm QAM$]]  $($"Quadrature Amplitude Modulation"$)$  and

*[[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Carrierless_Amplitude_Phase_Modulation|$\rm CAP$]]  $($"Carrierless Amplitude Phase Modulation$)$.

In contrast,  with the multicarrier method  [[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|$\rm DMT$]]  $($"Discrete Multitone Transmission"$),$  the uplink and the downlink channel are divided into  $N_{\rm Up}$  resp.  $N_{\rm Down}$  bins of  $4.3125\rm \ kHz $  each.

In addition to the above graph, it should be noted:
#Phone services  $($POTS or ISDN$)$  and xDSL are in different frequency bands,  which minimizes mutual interference in the trunk cable.  Thus,  the stronger signal ISDN does not interfere with the parallel running xDSL and vice versa.
#The lower frequency range up to  $\text{120 kHz}$  has been reserved for ISDN  $($optionally POTS$)$.  This value results from the first zero of the  [[Digital_Signal_Transmission/Block_Coding_with_4B3T_Codes#ACF_and_PSD_of_the_4B3T_codes|"ISDN spectrum with 4B3T coding"]].  Above $\text{120 kHz}$  the ISDN spectrum is completely suppressed.
#To separate the telephone and xDSL signals,  a  [[Examples_of_Communication_Systems/xDSL_Systems#Components_of_DSL_Internet_access|"splitter"]]  is used at both ends of the two-wire line,  which includes one low-pass filter and one high-pass filter each and also takes into account the following frequency gap up to  $\text{138 kHz}$.
#After this occupancy gap follows the ADSL upstream band from  $\text{138 kHz}$  to $\text{276 kHz}$.  This allows the transmission of  $N_{\rm Up} = 32$  subcarriers with bandwidth  $\text{4.3125 kHz}$  each.  This value is derived from the frame transmission rate.
#The subsequent downstream range extends to  $\text{1104 kHz}$  for ADSL,  with which  $N_{\rm Down} = 256$  subcarriers can be realized.  The separation of the upstream and the downstream channel in xDSL is done by a band-pass filter in the modem.
#However,  the first  $64$  subcarriers  $($this corresponds to $\text{276 kHz)}$  must not be occupied.  With the  [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Possible_bandwidth_allocations_for_xDSL|"frequency uniform position method"]]  only  $32$  subcarriers would have to be left out,  taking into account that the separation of uplink and downlink requires a more complex implementation.
#For  ADSL2+,  the system bandwidth is  $\text{2208 kHz}$   ⇒   $N_{\rm Down} = 512$  subcarriers.  The number of bins to be spared remains unchanged compared to ADSL.  Taking into account that two bins are occupied by control functions  $($e.g. for synchronization$)$,  $190$  $($ADSL$)$  or  $446$  $($ADSL2+$)$ downstream channels remain for users.
#The ISDN reservation prescribed in Germany,  however,  has the consequence for xDSL that the low frequencies,  which are by far the least attenuated in a copper line and would therefore actually be the most suitable,  cannot be used.
#Further,  from the frequency arrangement,  it can be seen that the downstream bins are more attenuated than the upstream bins  $($higher frequencies$)$  and consequently have a smaller signal-to-noise ratio  $($SNR$)$.
#The decision  "upstream below downstream"  is related to the fact that the loss of downstream channels has only a comparatively small impact on the transmission rate.  In the upstream,  such a failure would be much more noticeable in percentage terms.

==VDSL(2) bandwidth allocation==
 
The ITU has defined several profiles for  "VDSL(2)".  At the time of writing this chapter  (2010),  the frequency band assignment specified in the graphic applies to the systems deployed in Germany in accordance with the ITU's  $\text{ VDSL(2) Plan 998b - Profile 17a (Annex B)}$.  The (slightly) lighter coloring at the higher frequencies is meant to indicate that these channels are more attenuated.

Without claiming to be exhaustive, this allocation plan can be characterized as follows:

[[File:EN_Bei_T_2_3_S5a_ganzneu.png|right|frame|VDSL(2) bandwidth allocation in Germany ]]

#To achieve higher bit rates,  eight times as many bins are used here as in  "ADSL2+".
#Thus,  the bandwidth enables  $8 \cdot \text{2208 MHz = 17664 MHz}$,  transmission rates of up to  $\text{100 Mbit/s}$  $($depending on cable length and conditions$)$.
#The same as in ADSL,  the frequency bands for the upstream subchannels are always arranged at the lower frequencies,  since the greater cable attenuation  has a greater percentage influence on the bit rate for upstream than for downstream.
#In VDSL(2),  the  [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Possible_bandwidth_allocations_for_xDSL|"frequency separation method"]]  is always used.  Overlapping of the upstream and downstream frequency bands is categorically excluded in the ITU specification .
#In the VDSL systems,  the lower frequencies are reserved for ISDN.  This is followed by alternating ranges for upstream and downstream.  From the specified limits,  one can recognize the narrower upstream ranges,  compared to the downstream.
#You can see an alternating arrangement of upstream and downstream areas.  One reason is that with this wide spectrum it should be avoided that one direction  $($e.g. downstream$)$  is assigned only strongly attenuated  $($high$)$  frequencies.

The VDSL(2) specification provides for allocation plans up to system bandwidths of  $\text{30 MHz}$  $($according to profile 30a$)$.
* This should enable transmission rates up to about  $\text{ 200 Mbit/s}$  over short distances.

*For this purpose, the bandwidth of the individual subchannels is also doubled compared to ADSL to  $\text{8.625 kHz}$ .

==Overview of xDSL transmission methods==
 
At the beginning of the various standardization procedures for the individual xDSL variants,  different transmission methods were defined as a basis:
# [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modulation"]]  $\text{QAM}$  for  "QAM-ADSL"  and  "QAM-VDSL", 
# [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Carrierless_Amplitude_Phase_Modulation|"Carrierless Amplitude Phase Modulation"]]  $\text{CAP}$  for  "CAP-HDSL"  and  "CAP-ADSL",
#[[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|"Discrete Multitone Transmission"]]  $\text{DMT}$  for  "ADSL",  "ADSL2",  "ADSL2+",  "VDSL"  and",  "VDSL2".

With increasing market demand for higher transmission rates and the associated requirements, two main  suitable processes crystallized,  namely  "QAM/CAP"  and  "DMT".

Since the manufacturers were unable to agree on a common standard from 1997 to 2003,  also for patent reasons  $($one even speaks of  "line code wars"  in this context$)$,  the two competing methods coexisted for a long time.  At the so-called  "DSL Olympics"  in 2003,  '''the decision was finally made in favor of DMT''',
*on the one hand,  because of the slightly better  "performance"  in general,

*but in particular because of its higher robustness against narrowband interference; especially for the USA  $($many overhead telephone lines and related problems with coupled radio signals$)$  this argument played a major role.

First,  however,  the systems  "QAM-xDSL"  and  "CAP-xDSL"  will be considered very briefly.

==Basics of Quadrature Amplitude Modulation==
 

The principle has already been described in detail in the chapter  [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modation"]]  of the book  "Modulation Methods". 
[[File:EN_Bei_T_2_3_S6a.png|right|frame| Linear modulator with  $\rm I$ and  $\rm Q$–components;  signal space for  $\text{16-QAM}$]]
Here follows short summary on the basis of the block diagram on the right:
*QAM is a  "single carrier modulation method"  around the carrier frequency  $f_{\rm T}$.  First,  a blockwise serial/parallel conversion of the bit stream and signal space assignment is performed.

*From each  $b$  binary symbols,  two multilevel amplitude coefficients  $a_{{\rm I}n}$  and  $a_{{\rm Q}n}$  are derived  $($"in-phase component"  and  "quadrature component"$)$.  Both coefficients can each take one of  $M = 2^{b/2}$  possible amplitude values.

*The example considered here applies to the  $\text{16-QAM}$  with  $b = M = 4$  and correspondingly sixteen  signal space points. For a   $\text{256-QAM}$  would hold:   $b = 8$  and  $M = 16$  $(2^b = M^2 = 256)$.

*The coefficients  $a_{{\rm I}n}$  and  $a_{{\rm Q}n}$  are each impressed on a Dirac delta impulse as weights.  For pulse shaping one usually uses  $($because of the small bandwidth$)$  a raised-cosine filter.

*With the basic transmission pulse  $g_s(t)$  is then valid in the two branches of the block diagram:
:$$ s_{\rm I}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm
I\hspace{0.03cm}\it n} \cdot g_s (t - n \cdot T)\hspace{0.05cm},\hspace{0.5cm}
s_{\rm Q}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm
Q\hspace{0.03cm}\it n} \cdot g_s (t - n \cdot
T)\hspace{0.05cm}.$$
:Note:  Because of the redundancy-free conversion to a higher-level code,  the symbol duration  $T$  of these signals is greater by a factor  $b$  than the bit duration  $T_{\rm B}$  of the binary input sequence.  In the drawn example  $($16-QAM$)$,  $T = 4 ⋅ T_{\rm B}$.
*The  »'''QAM transmitted signal'''«  $s(t)$  is then the sum of the two partial signals  $ s_{\rm I}(t)$  and  $ s_{\rm Q}(t)$,  multiplied by  "cosine"  resp.  "minus-sine"  $($possibly followed by a band limit to prevent interference to adjacent bands, as indicated in the graph below$)$:
:$$s(t) = s_{\rm I}(t) \cdot \cos (2 \pi f_{\rm T}\,t) - s_{\rm Q}(t) \cdot \sin (2 \pi f_{\rm T}\,t)
\hspace{0.05cm}. $$
*The two branches  $(\rm I$  and  $\rm Q)$  can be considered as two completely separate  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#M.E2.80.93level_amplitude_shift_keying_.28M.E2.80.93ASK.29 |"M–level ASK systems"]]  which do not interfere with each other as long as all components are optimally designed.  This means at the same time:  

*Compared to a  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_phase_shift_keying_.28BPSK.29|"Binary Phase Shift Keying"]]  $\rm (BPSK)$  modulation with cosine or sine only,  the  '''QAM allows a doubling of the data rate at constant quality'''.

[[File:EN_Bei_T_2_3_S6b.png|right|frame|Quadrature Amplitude Modulation as band-pass and equivalent low-pass model]]

'''Equivalent low-pass model'''

The graph on the right shows
*above the band-pass model,  identical with the last graph,

*below the  "equivalent low-pass model".

Here,  the inphase and quadrature coefficients are combined to give the complex amplitude coefficient
:$$a_n = a_{\text{I}n} + {\rm j} ⋅ a_{\text{Q}n}.$$
Additionally, the physical transmitted signal  $s(t)$  is replaced by the equivalent low-pass signal
:$$s_{\rm TP}(t) = s_{\rm I}(t) + {\rm j} ⋅ s_{\rm Q}(t).$$

⇒   The representation of the QAM transmitter and receiver is the content of the SWF flash animation  [[Applets:Principle_of_QAM|"Principle of Quadrature Amplitude Modulation"]].
 
{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*As the bit number  $b$  and thus the number of defined symbols  $(M=b^2)$  increases,  the bandwidth efficiency increases,  but also the signal processing overhead increases.

*In addition,  it must be taken into account that a dense QAM allocation is appropriate only if the channel is sufficiently good.}}

'''Reference Model for ADSL-QAM'''

This diagram shows the reference model for  "ADSL-QAM",  whereby we will only deal with the red function blocks  "QAM-Modulator"  and  "'QAM-Demodulator"  here.
[[File:EN_Bei_T_2_3_S5a.png|right|frame|Reference Model for ADSL-QAM]]

*The carrier frequency $f_{\rm T}$  lies within the specified up and down band of the respective xDSL variant.

*Like signal space size  $($between  $M^2=2^2=4$  and  $M^2=2^8=256$  signal space points$)$  and symbol rate, it is determined by channel measurements during transmission initialisation.

*For ADSL-QAM,  the following symbol rates have been specified  $($all numerical values in  "${\rm kBaud}$" $= 10^3 \rm symbols/s)$  :
:*In upstream:  $20$, $40$, $84$, $100$, $120$, $136$;

:*in downstream: $40$, $126$, $160$, $252$, $336$, $504$, $806.4$, $1008$.

==Possible QAM signal space constellations==
 
We still consider possible arrangements of signal space points in quadrature amplitude modulation using three examples.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
An important QAM parameter is the bit number  $b$  processed to the amplitude coefficient pair  $(a_{\rm I}, a_{\rm Q})$.  Here  $b$  is always an even number.
[[File:EN_Bei_T_2_3_S6a_v2.png|right|frame|Signal space constellations  "$\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($left$)$  and  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($right$)$
'''Korrektur''': Jiwoo, ich habe die gelben Punkte durch braune Punkte ersetzt, weil meine schlechten Auge diese nicht erkennen konnten. Könntest u bitte den Text kontrollieren und mit der deutschen Version vergleichen, ob jetzt wirklich alle gelben Punkte braun sind. Das gleiche gilt für Example 3.]]

*If  $b = 2$,  both  $a_{\rm I}$  and  $a_{\rm Q}$  can only take the values  $±1$  resulting in the  $\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$  corresponding to the left constellation.

*According to an ITU recommendation,  the assignment applies here:
:$$q_1 = 0, \ q_0 = 0 \, \Leftrightarrow \,a_{\rm I} = +1, \ a_{\rm Q} = +1,$$
:$$q_1 = 0, \ q_0 = 1 \, \Leftrightarrow \, a_{\rm I} = +1, \ a_{\rm Q} = -1,$$
:$$q_1 = 1, \ q_0 = 0 \, \Leftrightarrow \,a_{\rm I} = -1, \ a_{\rm Q} = +1,$$
:$$q_1 = 1, \ q_0 = 1 \, \Leftrightarrow \, a_{\rm I} = -1, \ a_{\rm Q} = -1.$$

*The point  '''10'''   marked in brown  $(a_{\rm I} = -1, \ a_{\rm Q} = 1)$  thus stands for  $q_1 = 1$  and  $q_0 = 0$.

With  $b = 4$   ⇒   $M = 2^{b/2} = 4$  one arrives at  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  according to the right diagram with the possible amplitude coefficients 
:$$a_{\rm I} ∈ \{±3, ±1\}, \ \ a_{\rm Q} ∈ \{±3, ±1\}.$$

*The assignment can be determined using the auxiliary graph given on the left below, as illustrated by the following numerical examples.

        $\rm (A)$    $q_3 = 1, \ q_2 = 0, \ q_1 = 1,\ q_0 = 1$  $($brown marker$)$:
:#The two most significant bits  $\rm (MSB)$   ⇒   "'''11'''"  determine,  according to the  $\text{4-QAM}$  diagram,  the  $($second$)$  quadrant in which the symbol is located.
:#The two least significant bits  $\rm (LSB)$   ⇒   "'''10'''"  together with the auxiliary graph determine the point within this quadrant.  Result:  $a_{\rm I} = -1$,  $a_{\rm Q} = +3$.
        $\rm (B)$    $q_3 = 0, \ q_2 = 1, \ q_1 = 1,\ q_0 = 0$ $($green marker$)$:
:#The two most significant bits   ⇒   "'''01'''"   refer here to the fourth quadrant.
:#The two least significant bits   ⇒   "'''10'''"  refer to the green dot in the fourth quadrant:   $a_{\rm I} = -3, \ a_{\rm Q} = -3$.}}

{{GraueBox|TEXT=
$\text{Example 2:}$  We use the same graphic as in $\text{Example 1}$.  The decimal value  $D$  provides another way to label the points.
[[File:EN_Bei_T_2_3_S6a_v2.png|right|frame|Signal space constellations  "$\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($left$)$  and  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($right$)$]]

*The brown marked point in the  $\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$ diagram is binary labeled  "'''10'''"    ⇒   decimal:  $D = 2$.  This point simultaneously marks the second quadrant of  $\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$.

*The further subdivision results from the lower left graphic.  There at the brown dot is written  $4D + 3=11$"  $($decimal$)$.

*Therefore,  the upper right dot  $($highlighted in brown$)$  in the upper left quadrant stands for decimal  "$11$   ⇒   binary '''1011'''.

*For the green dot,  $D = 1$  gives the decimal value  $4D + 2 =6$,$11$ which corresponds to the binary representation  "'''0110'''" .
 
According to this scheme,  the signal space constellations
*for  $\rm 64\hspace{0.05cm}–\hspace{-0.02cm}QAM$   ⇒   $(b = 6, \ M = 8)$ 

*and  $\rm 256-QAM$  ⇒   $(b = 8, \ M = 16)$ 

which will be discussed in detail in the  [[Exercise_2.3:_QAM_Signal_Space_Assignment|$\text{Exercise 2.3}$]].  }}

{{GraueBox|TEXT=
$\text{Example 3:}$ 
We still consider for the described  $\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$  $($left graph,  here referred to as  "ITU proposal"$)$  the resulting symbol and bit error probability in the presence of AWGN noise:
#An error can be assumed to result in a horizontally or vertically adjacent symbol,  as indicated for the upper left green dot.
#The symbol error probability $p$  depends on the Euclidean distance of the two points and the AWGN noise power density  $N_0$.
#A distortion to the more distant blue point instead of one of the two neighboring brown points is rather unlikely with Gaussian noise.
[[File:EN Bei T 2 3 S6b neu.png|right|frame|Signal space constellation:  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$";  left: ITU propsal,  right:  Gray encoding. ]]

An analysis shows:
*All corner points  $($green$)$  can only be distorted in two directions.

*In contrast, the inner QAM points  $($blue$)$  have four direct neighbors.

* The remaining symbols  $($brown$)$  have three neighbors.

For the  »'''symbol error probability'''«   $p_{\rm S}$  then holds:

::$$p_{\rm S} = {1}/{16} \cdot (4 \cdot 2 p + 8 \cdot 3 p + 4 \cdot 4 p) = 3p.$$

To calculate the  »'''bit error probability'''«   $p_{\rm B}$  it must now be taken into account that for the left constellation one symbol error leads
* within one quadrant to only  one bit error  $($e.g: "'''0100'''"   ⇒   "'''0110'''"$)$,

*between adjacent quadrants to two bit errors $($e.g: "'''1111'''"   ⇒   "'''0101'''"$)$.

The computation of  $p_{\rm B} $  here involves some effort.

In contrast,  with Gray encoding  $($right diagram$)$:
# Each symbol differs from its neighbors by exactly one bit.
#Each symbol error thus results in exactly one bit error.
#Since each individual symbol contains four bits,  the  "bit error probability" is in this case: 

::$$p_{\rm B} = p_{\rm S}/4 = 3/4 \cdot p. $$}}

==Carrierless Amplitude Phase Modulation==
 
"Carrierless amplitude phase modulation"  $\rm (CAP)$  is a bandwidth-efficient variant of QAM,  which can be realized very easily with digital signal processors.  The only difference to QAM is that the modulation with a carrier signal can be omitted.

[[File:EN_Bei_T_2_3_S7a.png|right|frame|Block diagram of  "Carrierless Amplitude Phase Modulation"]]
[[File:EN_Bei_T_2_3_S7d_neu.png|right|frame|Reference model for  ADSL–CAP]]

*Digital filtering is used instead of multiplication by cosine and minus sine.  $g_{\rm I}(t)$  and  $g_{\rm Q}(t)$  are impulse responses of two transversal bandpass filters with same amplitude characteristic,  phase-shifted by  $π/2$.

*Both are orthogonal to each other,  which means that the integral of the product  $g_{\rm I}(t) · g_{\rm Q}(t)$  over a symbol duration gives zero.

*The signals  $s_{\rm I}(t)$  and  $s_{\rm Q}(t)$  generated in this way are combined,  converted to a continuous-time signal by a D/A converter. 

*The unwanted high-frequency components generated during D/A conversion are eliminated by a low-pass filter before transmission.

*At the receiver,  the signal  $r(t)$  is first converted to a discrete-time signal using an A/D converter and then the in-phase and quadrature symbols  $a_{\rm I}$  and  $a_{\rm Q}$  are extracted via two  "finite impulse response  $\rm (FIR)$  filters"  and downstream "threshold decision".

CAP was the de facto standard in initial ADSL specifications until 1996.  The graph shows the reference model.
#The frequencies up to  $\text{4 kHz}$  were reserved for POTS/ISDN.
#The up channel occupied the frequency range of  $\text{15 - 160 kHz}$,
#The down channel occupied the frequencies from  $\text{240 - 1500 kHz}$.

One problem with CAP is that a  "bad channel" has dramatic consequences on the transmission quality.  Therefore,  today (2011) ADSL-CAP is only found in a few HDSL variants.

==Basics of Discrete Multitone Transmission==
 
[[File:EN_Bei_T_2_3_S8a_v8.png|right|frame|Spectra at OFDM and DMT;     Note:   For this graph a symmetrical rectangle is assumed.     A rectangle between  $0$  and  $T$  would still result in a phase term.     However,  nothing would change with respect to  $|S(f)|$.]]
"Discrete Multitone Transmission"  $\rm (DMT)$  refers to a multicarrier modulation method that is almost identical to  [[Modulation_Methods/General_Description_of_OFDM|"Orthogonal Frequency Division Multiplexing"]]  $\rm (OFDM)$.  In the case of wired transmission,  one usually refers to  "DMT",  and in the case of wireless transmission,  one refers to  "OFDM".

In both cases,  one divides the entire bandwidth into many narrowband equidistant subchannels.  The respective subcarrier signals  $s_k(t)$  are individually impinged with complex data symbols  $D_k$  and the sum of the modulated subcarrier signals is transmitted as signal  $s(t)$ .

The diagram illustrates the principle of OFDM and DMT in the frequency domain,  partly using the values specified for ADSL–DMT:

*$255$  Subcarriers with carrier frequencies  $k \cdot f_0$  $(k = 1$, ... , $255)$.

*$4000$  data frames are transmitted per second.

*After  $68$  data frames,  one synchronization frame is inserted each.

*Due to the  "cyclic prefix"  $($see chapter  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Insert.C3.BCgen_of_Guard.E2.80.93Interval_and_cyclic_Pr. C3.A4fix|"Insertion of Guard–Interval and Cyclic Prefix"]]$)$  the symbol duration  $T = 1/f_0$  must still be shortened by the factor  $16/17$.

*Thus,  the  »'''basic frequency'''«  of DMT is 
:$$f_0 = 4 \cdot (69/68) \cdot (69/68) = 4.3125 \ \rm kHz.$$

An essential difference between OFDM and DMT is that
*in OFDM,  the above drawn spectrum  $S(f)$  in reality describes an  "equivalent low-pass spectrum"  $S_{\rm TP}(f)$  and still the shift around a carrier frequency  $f_{\rm T}$  has to be considered:

:$$S_{\rm TP}(f ) = \sum_{k = 1}^{255} D_k \cdot \delta (f - k \cdot f_0)\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
S(f) = \frac{1}{2} \big [ S_{\rm TP}(f - f_{\rm T}) + S^*_{\rm TP}(-(f + f_{\rm T}))\big ]
\hspace{0.05cm};$$

*in DMT,  on the other hand,  the components at negative frequencies must still be taken into account,  which are to be weighted with the conjugate-complex spectral coefficients:

:$$S(f ) = \sum_{k = 1}^{255} \big [ D_k \cdot \delta (f - k \cdot f_0) + D^*_k \cdot \delta (f + k \cdot f_0) \big ]
\hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Please note:}$ 
#According to these equations,  the complex OFDM signal  $s_{\rm OFDM}(t)$  consists of  $K = 255$  complex exponential oscillations.
#The DMT signal  $s_{\rm DMT}(t)$  is composed of as many cosine oscillations with frequencies  $k \cdot f_0$  $($full occupancy presupposed$)$.
#Despite complex coefficients  $D_k$ resulting from QAM occupancy of the carriers,  the DMT signal is always real because of the conjugate-complex complements at negative frequencies.}}

However,  in both OFDM and DMT,  the transmitted signal  $s(t)$  is precisely limited in time to the symbol duration  $T = 1/f_0 ≈ 232 \ {\rm µs}$  which is equivalent to multiplying by a rectangle of duration $T$.  In the spectral domain,  this corresponds to convolution with the function  $\text{sinc}(fT)$:
*Each Dirac delta function at  $k · f_0$  thus becomes an sinc– function at the same position,  given the time limit,  as shown in the second diagram of the graph above.

*Adjacent subcarrier spectra overlap on the frequency axis,  but exactly at  $k · f_0$  the coefficients  $D_k$  can be seen again,  since all other spectra have zeros here.

{{GraueBox|TEXT=
$\text{Example 4:}$ 
Assuming the conditions favorable for ADSL downstream,  viz.
#$4000$ frames are transmitted per second.
#All subcarriers are active at all times  $(K = 255)$.
#Each carrier is occupied with a 1024-QAM  $(b = 10)$;  according to ITU:  $8 ≤ b ≤ 15$.
#Ideal conditions prevail,  so that the orthogonality evident in the graph is preserved,

then for the  »'''maximum bit rate'''« 
:$$R_{\rm B,\ max} = 4000 · K · b ≈ 10 \ \rm Mbit/s.$$

However,  specified ADSL downstream is only  $2 \ \rm Mbit/s$  due to
*the omission of the  $64$  lowest carriers because of ISDN and upstream,

*the QAM occupancy of the heavily attenuated carriers with less than  $10$  bits,  and

*the consideration of the cyclic prefix,  and some operational reasons.}}

==DMT realization with IDFT/DFT==
 
The upper graph shows the complete DMT system.  For now we will focus on the two red blocks.  The blue blocks will be covered in the [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL|"next chapter"]].  Simplified,  the transmitter and receiver can be represented as shown in the second graph:

[[File:EN_Bei_T_2_3_S9d_ganz_neuV8.png|right|frame|Discrete multitone transmission system]]
[[File:P_ID1951__Bei_T_2_3_S9b_v3.png|right|frame|DMT transmitter and receiver '''Korrektur''' das Bild muss ich noch machen]]

*To perform a DMT modulation,  the transmitter accumulates a block of input bits in a data buffer to be transmitted as one frame.

*The QAM encoder supplies per frame the complex-valued data symbols  $D_1$,  ... ,  $D_{255}$,  which is expanded to the vector  $\mathbf{D}$  of length  $512$  by
#   $D_0 = D_{256} = 0$,
#   $D_k = D^\star_{512-k} \ (k = 257,$ ... , $511)$,
#   the coefficients   $D_{257}$, ... , $D_{511}$  are identical to  $D_{-255}$, ... , $D_{-1}$ $($as consequence of  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Finite_signal_representation|"finite signals"]]  properties$)$.
*The spectral samples  $\mathbf{D}$  are transformed with the [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Inverse_discrete_Fourier_transform|"Inverse Discrete Fourier Transform"]]  to the vector  $\mathbf{s}$  of time signal samples, also with length $512$. Because of the conjugate-complex assignment in the spectral domain   ⇒   $\text{Im}[\mathbf{s}] = 0$.

*After parallel/serial and digital/analog conversion and low-pass filtering of  $\text{Re}[\mathbf{s}]$  the physical  $($real$)$  as well as continuous-time transmitted signal  $s(t)$  results.

*For this,  in the range  $0 ≤ t ≤ T$  $($factor  $2$,  since two coefficients each contribute to cosine/sine$)$:

:$$s(t) = \sum_{k = 1}^{255} \big [ 2 \cdot{\rm Re}\{D_k\} \cdot \cos(2\pi k f_0 t ) - 2 \cdot{\rm Im}\{D_k\} \cdot \sin(2\pi k f_0 t )\big ] \hspace{0.05cm}. $$

*The received signal at the AWGN output is  $r(t) = s(t) + n(t)$.  After A/D and S/P conversion,  $r(t)$  can be expressed by the (real) vector  $\mathbf{r}$ . The  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#From_the_continuous_to_the_discrete_Fourier_transform|"Discrete Fourier Transform"]]  then provides estimates for the transmitted spectral coefficients.
 

{{GraueBox|TEXT=
$\text{Example 5:}$ 
As an example, let's consider the ADSL/DMT downstream.
[[File:EN_Bei_T_2_3_S10_neu.png|right|frame|Allocation of the DMT frequency band with QAM coefficients]]
*In the upper left graph on the right you can see the amounts  $\vert D_k\vert $  of the occupied subchannels  $64$, ... , $255$.  The carriers  $0$, ... , $63$  for the reserved frequency range of ISDN and upstream are set to zero.

*Next to it on the right are the spectral coefficients   $D_{64}$, ... , $D_{255}$   are shown in the complex plane,&nbsp: where the signal space is chosen very large.

*The graph on the left shows the transmitted signal  $s(t)$  for the frame duration   $T = {1}/{f_0} ≈ 232 \ \rm µs$,  which is obtained by low-pass filtering the IDFT values  $s_0$, ... , $s_{511}$  results. This useful signal looks almost like noise.
 
[[File:P_ID1953__Bei_T_2_3_S10b_v1.png|left|frame|Transmitted signal with above DMT allocation]]
 
It can be seen:

#Main problem of DMT is the unfavorable Crest factor   ⇒   ratio of maximum value  $s_{\rm max}$  and rms value  $s_{\rm eff}$.
#The large dynamic range visible in the exemplary signal curve places high demands on the amplifier's linearity.
#When the dynamic range is limited,  the peaks of  $s(t)$  are cut off.
#This acts as a pulse-like disturbance and an additional noise load to the system.}}

{{BlaueBox|TEXT=
$\text{Summarizing:}$ 
*Discrete Multitone Transmission  $\rm (DMT)$  is basically the parallel implementation of many narrowband QAM modems with different carriers and relatively low data transmission rates. 

*The low bandwidth per subcarrier allows a long symbol duration,  thus reducing the influence of intersymbol interference and reducing the development effort for equalization.

*A major reason for the DMT success is its technical ease of implementation.  IDFT and DFT are formed with digital signal processors in real time.  The vectors have the length  $512$  $($power of two$)$.  Therefore the particularly fast FFT algorithm  $($"Fast Fourier Transformation"$)$  can be applied.}}

==Exercises for the chapter ==

[[Exercise_2.3:_QAM_Signal_Space_Assignment|Exercise 2.3: QAM Signal Space Assignment]]

[[Exercise_2.3Z:_xDSL_Frequency_Band|Exercise 2.3Z: xDSL Frequency Band]]

[[Exercise_2.4:_DSL/DMT_with_IDFT/DFT|Exercise 2.4: DSL/DMT with IDFT/DFT]]

[[Exercise_2.4Z:_Repetition_to_IDFT|Exercise 2.4Z: Repetition to IDFT]]

{{Display}}

Examples of Communication Systems/xDSL Systems

2023-03-21T19:07:55Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=General Description of DSL
|Nächste Seite=xDSL as Transmission Technology
}}

==Reference models==
 
[[File:EN_Bei_T_2_2_S1_v3.png|right|frame|xDSL reference model of the ITU]]

Based on the following general ITU reference model,  it can be quickly seen that xDSL is physically a pure access transmission technology that is only used in the local loop network area between the fiber termination point and the network termination at the end customer.

The basic elements of the xDSL standard are:
*the network termination  $\rm (NT)$,

*a subscriber line$\rm$,  and

*the line termination  $\rm (LT)$.

There is a lot of freedom for network operators in implementing this reference model in practice. What all previous implementations have in common is that they use existing metallic subscriber lines.
 
{{GraueBox|TEXT=
$\text{Examples 1:}$ 
In an example,  the configuration most frequently encountered in Germany is shown according to the graphic.  Note:

[[File:EN_Bei_T_2_2_S1b_v8.png|right|frame|Reference model according to  "$\text{1TR112 U-R2-V7.0 DTAG}$"]]

*In all xDSL variants deployed today,  the data service converted in the modems is combined with the telephone service.  This allows transmission over the existing telephone network.

*The  "splitter"  separate the signal on both sides of the subscriber line into the classes  "broadband"  and  "narrowband" .

*An important interface is designated  $\text{U-R2}$  $($red marking$)$.
#This was standardized in Germany in 2001 by Deutsche Telekom AG in order to be able to use any modems on the subscriber side.
#This means that the customer is no longer dependent on his provider's xDSL modem.}}

==Overview and common features of all xDSL systems==
 
The technical realization of an  $\rm xDSL$  system involves many system components,  which can be distributed over several localities.  There is a wide range of realization options.  To summarize  $($from the perspective of 2009$)$:
#The systems for  $\rm ADSL$  and  $\rm ADSL$  shown below represent the most common implementation.  Data transport at the protocol level is based on the  $\rm ATM$  technology  $($"Asynchronous Transfer Mode"$)$.
#Despite a large data overhead,  ATM still offers advantages over  "Ethernet"  in terms of guaranteed quality of service  $\rm (QoS)$,  i.e. effective bit rate,  low delay and jitter.
#Ethernet,  on the other hand,  enables very high data transmission rates,  especially through the  »'''10 Gbit/s Ethernet'''«  and »'''100 Gbit/s Ethernet'''«  $($"Metro Ethernet"$)$  variants.  ATM,  on the other hand,  is more suitable for lower data rates.
#In 2009 there were numerous discussions about whether ATM should be replaced by  "10 Gbit/s Ethernet"  in the course of  "Next Generation Network".  However,  upgrading the backbone from ATM to Ethernet represents a not inconsiderable investment.

{{BlaueBox|TEXT=
$\text{Summary:}$ 
As mentioned in the chapter  [[Examples_of_Communication_Systems/General_Description_of_DSL|"General Description of DSL"]],  the most commonly deployed xDSL variants
#$\rm ADSL$, 
#$\rm ADSL2$  resp.  $\rm ADSL2+$,
#$\rm VDSL(1)$ 
#$\rm VDSL(2)$

are defined in such a way that simultaneous operation of  »'''POTS'''«  $($"Plain Old Telephone Service"$)$  or  »'''ISDN'''«  $($"Integrated Services Digital Network"$)$  on the same line is possible at any time.  This is the basis of the further descriptions.}}

==ADSL – Asymmetric Digital Subscriber Line ==
 
[[File:EN_Bei_T_2_2_S4.png|right|frame|ADSL connection from the end customer to the local exchange '''KORREKTUR''']]

The physical network termination  $\rm (NT)$  is in the ADSL modem  $($"ADSL Transmission Unit Central Office",  $\text{ATU-C)}$  in the local exchange.  Before that,  in the  "splitter"  the low-frequency telephony spectrum is separated from the higher-frequency ADSL spectrum by low-pass and high-pass filtering.

The graphic shows an ADSL connection from the end customer to the local exchange,  which is described very briefly below.  The reverse data connection paths are in each case mirror-inverted.

*The splitter forwards the telephone signals to the ISDN/POTS exchange and the ADSL signals to the  "digital subscriber line access multiplexer"  $\rm (DSLAM)$,  in which the  "$\text{ATU-C}$"  is implemented as a plug-in card.

*The DSLAM bundles many ADSL connections and forwards the data after decoding at the ATM level via optical fiber to the  "ATM Service Access Multiplexer".  This sends the data from all DSLAMs over the backbone to the  "Broadband Remote Access Server"  $\rm (BBRAS)$.

*The BBRAS terminates the point-to-point protocol data link and forwards the IP packets via routers to the destination.  The backbone consists of optical components based on  $\rm SDH$  standard  $($"Synchronous Digital Hierarchy"$)$.

*The splitter connected to the  "telecommunications connection unit"  $\rm (TAE)$  separates the signals. 

*The telephone signals are routed to the telephony terminals,  the ADSL signals to the modem  $($"ADSL Transmission Unit Remote",  $\text{ATU-R)}$.  The modem decodes and forwards the binary data to the connected terminals.

*During initialization of the ADSL connection,  $\text{ATU-C}$  and  $\text{ATU-R}$  perform a so-called  "training"  in which relevant system parameters  $($data rate,  fast and interleaved mode,  ...$)$  are determined depending on the line conditions.

*The parameters negotiated in this process are retained until the next check and synchronization.  For the transmission of administrative data  $($"overhead"$)$,  $32$  kbit per frame are statically reserved in the ADSL systems.

==ADSL2 and ADSL2plus==
 
{{BlaueBox|TEXT=
$\text{These two system variants are further developments of ADSL:}$ 
*The enhanced system variant  "Asymmetric Digital Subscriber Line Transceivers 2"  $\rm (ADSL2)$  was specified in 2002 with ITU recommendations  $\rm G.992.3$,  $\rm G.992.4$.

*2003 followed by the ITU recommendation  $\rm G.992.5$:  "Extended-bandwidth ADSL2"  $\rm (ADSL2+)$.}}

Compared to ADSL,  the following changes occurred:
#In ADSL2,  the  "Seamless Rate Adaption"  $\rm (SRA)$  was included in the standard.  This allows transmission parameters to be changed during operation with time-variant channel quality without loss of synchronization.
#For this purpose,  ATU-C and ATU-R periodically check the signal–to–noise ratio  $\rm (SNR)$  of the transmission channels.  If a channel in use deteriorates,  the receiver notifies the transmitter of the new data rate and transmission level.  After a subsequent  "sync flag"  the parameters are adopted.
#ADSL2 systems also offer a wide range of diagnostic options even without the modems having been synchronized,  a feature that is particularly important for troubleshooting,  error analysis and error correction.
#In addition,  ADSL2 provides the ability to reduce transmit levels when SNR is sufficient,  thereby minimizing crosstalk and increasing throughput in the trunk cable.  This  "power cutback"  can be initiated not only by the DSLAM,  but also by the ATU-R.
#In ADSL2,  the number of overhead bits is no longer fixed,  but can vary between  $4$  and  $32$  kbit.  This increase in the user data bit rate of up to  $28$  kbit/s per data frame is all the more important the longer the distance between the modem and the DSLAM.

{{BlaueBox|TEXT=
$\text{As a result from 2009:}$ 
*$\text{ADSL2}$  systems achieve a transmission rate of more than  $8$  Mbit/s  $($up to  $12$  Mbit/s$)$  downstream and more than  $800$  kbit/s  $($up to  $3.5$  Mbit/s$)$  upstream.

*With  $\text{ADSL2+}$,  the transmission rate in the downstream is doubled again; the maximum rate is theoretically  $25$  Mbit/s.}}

==VDSL – Very–high–speed Digital Subscriber Line ==
 
In terms of the basic structure of their components,  VDSL systems are identical to ADSL systems,  with the only exception that the relocation of the splitter and the DSLAM from the local exchange to a cable branch makes the last section between the network operator and the customer,  the so-called  "last mile".
[[File:EN_Bei_T_2_2_S5_v8.png|right|frame|Modeling of a VDSL connection from the end customer to the local exchange '''KORREKTUR''']]

This measure was necessary because VDSL can only exploit its advantage  – the greater transmission speed –  over very short distances due to the attenuation of the higher frequencies,  which increases sharply with line length.

DSLAM and BBRAS are still connected via  [https://en.wikipedia.org/wiki/STM-1 $\text{STM-1}$]  interfaces.  Therefore,  the route between the local exchange and the cable branch must now also be laid with optical fiber.

A distinction is made between two alternative VDSL variants:
*$\rm VDSL(1)$  is based on  [[Modulation_Methods/Quadrature_Amplitude_Modulation|$\rm QAM$]]  $($"Quadrature Amplitude Modulation"$)$,  which is predominantly deployed in Asia,

*$\rm VDSL(2)$  is based on  [[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|$\rm DMT$]]  $($"Discrete Multitone Transmission"$)$.

:VDSL(1) systems were never deployed in Germany because of their inadequate ability to provide audio/video,  telephony and Internet  $($"triple play"$)$  with sufficient quality of service.

:Instead,  the VDSL(2) standard was established immediately:  Because of higher performance and greater range,  the better quality of service as well as the reusability of the ADSL(2+) infrastructure.

{{BlaueBox|TEXT=
$\text{Summary:}$ 
Following a few characteristics of the  $\rm VDSL(2)$  system:
#VDSL(2) has achieved since 2006 a maximum transmission rate of  $50$  to  $100$  Mbit/s,  depending on the standard used.
#The specified VDSL(2) transmission bandwidth of  $30$  MHz was considered the maximum reasonable bandwidth in 2009.
#With complementary measures such as  "Dynamic Spectrum Management"  and "Advanced Codes"  total transmission rates of up to  $280$ Mbit/s  were expected for short line lengths  $($up to  $300$  meters$)$.}}

==DSL Internet access from perspective of the communication protocols ==
 
Some xDSL modems offer an  Ethernet interface for connecting the data terminals and a transparent connection to the remote terminal,  based on the  "Internet Protocol"  $\rm (IP)$.  The following should be noted:
[[File:EN_Bei_T_2_2_S6b_v8.png|right|frame|Modeling of an xDSL connection through the use of an xDSL router]]

*This option is enabled by the  "LAN Emulation"  $\rm (RFC2684)$  and the  "ATM Adaption Layer Protocol"  $\rm (AAL5)$.  The Ethernet data stream must be converted to  $\rm ATM$  for this purpose.

*This eliminates the need to install ATM equipment and existing Ethernet hardware can be used,  greatly simplifying xDSL configuration at the customer site.

*The ATM connection extends at least as far as the  "Broadband Remote Access Server"  $\rm (BBRAS)$  and is converted there or continued directly,  depending on the backbone data transmission system.

The graph on the right shows the communication in an Internet connection according to the OSI model, where  "xDSL"  $($brown background$)$  is used only
*between the  "TU-R"  on the customer side  $($here realized with an "xDSL router"$)$

*and the  "TU-C"  on the provider side.
 
== Components of DSL Internet access ==
 
Finally,  necessary components for a DSL connection are listed.  The graph shows examples of these,  mostly from  "Deutsche Telekom".

[[File:EN_Bei_T_2_2_S7.png|right|frame|Necessary components to establish an xDSL connection]]

$\rm NTBA$:  The commonly used term is  "Network Termination for ISDN Basic Rate Access".  Tasks of the NTBA are:
*With help of a fork circuit and echo cancellation,  the two-wire  $\rm U_{K0}$  interface on the provider side is converted into the four-wire  $\rm S_0$ interface on the subscriber side.

*In addition,  the NTBA manages the ISDN code conversion from the  "Modified Monitored Sum 4B3T"  code  $\rm (U_{K0}$ bus$)$  to the modified AMI code  $\rm (S_{0}$ bus$)$.

$\text{xTU-R}$:  The abbreviation stands for  "xDSL Transceive Unit - Remote"'  and denotes the subscriber-side xDSL unit.
*Due to the widespread use of Ethernet,  today's xDSL modems and routers usually have only one Ethernet port for connecting data terminal equipment.  Originally,  they were used for subscriber-side connection of ATM data terminals.

*Thus,  this unit must also perform the function of a  "Layer 2 bridge"  in order to be able to transmit Ethernet over ATM to the  "Broadband Remote Access Server"  (BBRAS) for termination.

$\text{xDSL modem}$:  With this functional unit,  the data connection from/to the data terminal is initialized by a  "point-to-point protocol"  $\rm (PPP)$  via a  "PPP over Ethernet connection"  and terminated by the  "BBRAS".  Only data terminal devices that can separately establish a data connection via PPP are eligible.

$\text{xDSL router}$:  This initializes the data connection via PPP and enforces the addresses at IP level.  This allows multiple terminals to be connected and allows internal data exchange between them without having to dial into them separately.

$\rm Splitter$:  This is basically a combination of high-pass and low-pass with three interfaces, which handles the separation of the high-frequency xDSL data signals  $($above  $138$ kHz$)$  from the low-frequency POTS or ISDN telephone signals  $($below  $120$ kHz$)$,  or their combination.
*This  "broadband access unit"  is nothing other than a crossover unit.

*The sum of the signals is present on the subscriber line side,  while the xDSL data and the POTS/ISDN signals are separated from each other by a splitter on both the customer and provider sides.

$\text{xTU-C}$:  The abbreviation stands for  "xDSL Transceive Unit - Central office".  It is the provider-side xDSL unit and is usually implemented as a printed-circuit board insertion for the DSLAM.  The xTU-C terminates the physical retail xDSL subscriber lines,  modulates the ATM bit stream on the subscriber side,  and demodulates the xDSL signal on the provider side.

$\text{DSLAM}$:  The abbreviation stands for  "Digital Subscriber Line Access Multiplexer".  In its simplest form,  it terminates the physical subscriber lines with its xTU-C line cards.  In an extended form,  an  "ATM Service Access Multiplexer"  is also integrated in the DSLAM.
*The task of the DSLAM is to bundle the ATM bit streams of the subscriber lines and to forward them in concentrated form in the multiplexing process via an STM-1 fiber interface into the provider network.

*"STM" is an  "synchronous digital hierarchy"  $($SDH$)$  transmission standard for multiplexing optical channels and stands for  "Synchronous Transport Module".  STM-1 allows a bit rate of up to  $155.52$  Mbit/s,  STM-64 up to almost  $10$  Gbit/s.

==Exercises for the chapter ==
 
[[Exercise_2.2:_xDSL_Variants|Exercise 2.2: xDSL Variants]]

[[Exercise_2.2Z:_DSL_Internet_Connection|Exercise 2.2Z: DSL Internet Connection]]

{{Display}}

Examples of Communication Systems/xDSL Systems

2023-03-21T19:05:51Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=General Description of DSL
|Nächste Seite=xDSL as Transmission Technology
}}

==Reference models==
 
[[File:EN_Bei_T_2_2_S1_v3.png|right|frame|xDSL reference model of the ITU]]

Based on the following general ITU reference model,  it can be quickly seen that xDSL is physically a pure access transmission technology that is only used in the local loop network area between the fiber termination point and the network termination at the end customer.

The basic elements of the xDSL standard are:
*the network termination  $\rm (NT)$,

*a subscriber line$\rm$,  and

*the line termination  $\rm (LT)$.

There is a lot of freedom for network operators in implementing this reference model in practice. What all previous implementations have in common is that they use existing metallic subscriber lines.
 
{{GraueBox|TEXT=
$\text{Examples 1:}$ 
In an example,  the configuration most frequently encountered in Germany is shown according to the graphic.  Note:

[[File:EN_Bei_T_2_2_S1b_v8.png|right|frame|Reference model according to  "$\text{1TR112 U-R2-V7.0 DTAG}$"]]

*In all xDSL variants deployed today,  the data service converted in the modems is combined with the telephone service.  This allows transmission over the existing telephone network.

*The  "splitter"  separate the signal on both sides of the subscriber line into the classes  "broadband"  and  "narrowband" .

*An important interface is designated  $\text{U-R2}$  $($red marking$)$.
#This was standardized in Germany in 2001 by Deutsche Telekom AG in order to be able to use any modems on the subscriber side.
#This means that the customer is no longer dependent on his provider's xDSL modem.}}

==Overview and common features of all xDSL systems==
 
The technical realization of an  $\rm xDSL$  system involves many system components,  which can be distributed over several localities.  There is a wide range of realization options.  To summarize  $($from the perspective of 2009$)$:
#The systems for  $\rm ADSL$  and  $\rm ADSL$  shown below represent the most common implementation.  Data transport at the protocol level is based on the  $\rm ATM$  technology  $($"Asynchronous Transfer Mode"$)$.
#Despite a large data overhead,  ATM still offers advantages over  "Ethernet"  in terms of guaranteed quality of service  $\rm (QoS)$,  i.e. effective bit rate,  low delay and jitter.
#Ethernet,  on the other hand,  enables very high data transmission rates,  especially through the  »'''10 Gbit/s Ethernet'''«  and »'''100 Gbit/s Ethernet'''«  $($"Metro Ethernet"$)$  variants.  ATM,  on the other hand,  is more suitable for lower data rates.
#In 2009 there were numerous discussions about whether ATM should be replaced by  "10 Gbit/s Ethernet"  in the course of  "Next Generation Network".  However,  upgrading the backbone from ATM to Ethernet represents a not inconsiderable investment.

{{BlaueBox|TEXT=
$\text{Summary:}$ 
As mentioned in the chapter  [[Examples_of_Communication_Systems/General_Description_of_DSL|"General Description of DSL"]],  the most commonly deployed xDSL variants
#$\rm ADSL$, 
#$\rm ADSL2$  resp.  $\rm ADSL2+$,
#$\rm VDSL(1)$ 
#$\rm VDSL(2)$

are defined in such a way that simultaneous operation of  »'''POTS'''«  $($"Plain Old Telephone Service"$)$  or  »'''ISDN'''«  $($"Integrated Services Digital Network"$)$  on the same line is possible at any time.  This is the basis of the further descriptions.}}

==ADSL – Asymmetric Digital Subscriber Line ==
 
[[File:EN_Bei_T_2_2_S4.png|right|frame|ADSL connection from the end customer to the local exchange '''KORREKTUR''']]

The physical network termination  $\rm (NT)$  is in the ADSL modem  $($"ADSL Transmission Unit Central Office",  $\text{ATU-C)}$  in the local exchange.  Before that,  in the  "splitter"  the low-frequency telephony spectrum is separated from the higher-frequency ADSL spectrum by low-pass and high-pass filtering.

The graphic shows an ADSL connection from the end customer to the local exchange,  which is described very briefly below.  The reverse data connection paths are in each case mirror-inverted.

*The splitter forwards the telephone signals to the ISDN/POTS exchange and the ADSL signals to the  "digital subscriber line access multiplexer"  $\rm (DSLAM)$,  in which the  "$\text{ATU-C}$"  is implemented as a plug-in card.

*The DSLAM bundles many ADSL connections and forwards the data after decoding at the ATM level via optical fiber to the  "ATM Service Access Multiplexer".  This sends the data from all DSLAMs over the backbone to the  "Broadband Remote Access Server"  $\rm (BBRAS)$.

*The BBRAS terminates the point-to-point protocol data link and forwards the IP packets via routers to the destination.  The backbone consists of optical components based on  $\rm SDH$  standard  $($"Synchronous Digital Hierarchy"$)$.

*The splitter connected to the  "telecommunications connection unit"  $\rm (TAE)$  separates the signals. 

*The telephone signals are routed to the telephony terminals,  the ADSL signals to the modem  $($"ADSL Transmission Unit Remote",  $\text{ATU-R)}$.  The modem decodes and forwards the binary data to the connected terminals.

*During initialization of the ADSL connection,  $\text{ATU-C}$  and  $\text{ATU-R}$  perform a so-called  "training"  in which relevant system parameters  $($data rate,  fast and interleaved mode,  ...$)$  are determined depending on the line conditions.

*The parameters negotiated in this process are retained until the next check and synchronization.  For the transmission of administrative data  $($"overhead"$)$,  $32$  kbit per frame are statically reserved in the ADSL systems.

==ADSL2 and ADSL2plus==
 
{{BlaueBox|TEXT=
$\text{These two system variants are further developments of ADSL:}$ 
*The enhanced system variant  "Asymmetric Digital Subscriber Line Transceivers 2"  $\rm (ADSL2)$  was specified in 2002 with ITU recommendations  $\rm G.992.3$,  $\rm G.992.4$.

*2003 followed by the ITU recommendation  $\rm G.992.5$:  "Extended-bandwidth ADSL2"  $\rm (ADSL2+)$.}}

Compared to ADSL,  the following changes occurred:
#In ADSL2,  the  "Seamless Rate Adaption"  $\rm (SRA)$  was included in the standard.  This allows transmission parameters to be changed during operation with time-variant channel quality without loss of synchronization.
#For this purpose,  ATU-C and ATU-R periodically check the signal–to–noise ratio  $\rm (SNR)$  of the transmission channels.  If a channel in use deteriorates,  the receiver notifies the transmitter of the new data rate and transmission level.  After a subsequent  "sync flag"  the parameters are adopted.
#ADSL2 systems also offer a wide range of diagnostic options even without the modems having been synchronized,  a feature that is particularly important for troubleshooting,  error analysis and error correction.
#In addition,  ADSL2 provides the ability to reduce transmit levels when SNR is sufficient,  thereby minimizing crosstalk and increasing throughput in the trunk cable.  This  "power cutback"  can be initiated not only by the DSLAM,  but also by the ATU-R.
#In ADSL2,  the number of overhead bits is no longer fixed,  but can vary between  $4$  and  $32$  kbit.  This increase in the user data bit rate of up to  $28$  kbit/s per data frame is all the more important the longer the distance between the modem and the DSLAM.

{{BlaueBox|TEXT=
$\text{As a result from 2009:}$ 
*$\text{ADSL2}$  systems achieve a transmission rate of more than  $8$  Mbit/s  $($up to  $12$  Mbit/s$)$  downstream and more than  $800$  kbit/s  $($up to  $3.5$  Mbit/s$)$  upstream.

*With  $\text{ADSL2+}$,  the transmission rate in the downstream is doubled again; the maximum rate is theoretically  $25$  Mbit/s.}}

==VDSL – Very–high–speed Digital Subscriber Line ==
 
In terms of the basic structure of their components,  VDSL systems are identical to ADSL systems,  with the only exception that the relocation of the splitter and the DSLAM from the local exchange to a cable branch makes the last section between the network operator and the customer,  the so-called  "last mile".
[[File:EN_Bei_T_2_2_S5_v8.png|right|frame|Modeling of a VDSL connection from the end customer to the local exchange '''KORREKTUR''']]

This measure was necessary because VDSL can only exploit its advantage  – the greater transmission speed –  over very short distances due to the attenuation of the higher frequencies,  which increases sharply with line length.

DSLAM and BBRAS are still connected via  [https://en.wikipedia.org/wiki/STM-1 $\text{STM-1}$]  interfaces.  Therefore,  the route between the local exchange and the cable branch must now also be laid with optical fiber.

A distinction is made between two alternative VDSL variants:
*$\rm VDSL(1)$  is based on  [[Modulation_Methods/Quadrature_Amplitude_Modulation|$\rm QAM$]]  $($"Quadrature Amplitude Modulation"$)$,  which is predominantly deployed in Asia,

*$\rm VDSL(2)$  is based on  [[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|$\rm DMT$]]  $($"Discrete Multitone Transmission"$)$.

:VDSL(1) systems were never deployed in Germany because of their inadequate ability to provide audio/video,  telephony and Internet  $($"triple play"$)$  with sufficient quality of service.

:Instead,  the VDSL(2) standard was established immediately:  Because of higher performance and greater range,  the better quality of service as well as the reusability of the ADSL(2+) infrastructure.

{{BlaueBox|TEXT=
$\text{Summary:}$ 
Following a few characteristics of the  $\rm VDSL(2)$  system:
#VDSL(2) has achieved since 2006 a maximum transmission rate of  $50$  to  $100$  Mbit/s,  depending on the standard used.
#The specified VDSL(2) transmission bandwidth of  $30$nbsp; MHz was considered the maximum reasonable bandwidth in 2009.
#With complementary measures such as  "Dynamic Spectrum Management"  and "Advanced Codes"  total transmission rates of up to  $280$ Mbit/s  were expected for short line lengths  $($up to  $300$  meters$)$.}}

==DSL Internet access from perspective of the communication protocols ==
 
Some xDSL modems offer an  Ethernet interface for connecting the data terminals and a transparent connection to the remote terminal,  based on the  "Internet Protocol"  $\rm (IP)$.  The following should be noted:
[[File:EN_Bei_T_2_2_S6b_v8.png|right|frame|Modeling of an xDSL connection through the use of an xDSL router]]

*This option is enabled by the  "LAN Emulation"  $\rm (RFC2684)$  and the  "ATM Adaption Layer Protocol"  $\rm (AAL5)$.  The Ethernet data stream must be converted to  $\rm ATM$  for this purpose.

*This eliminates the need to install ATM equipment and existing Ethernet hardware can be used,  greatly simplifying xDSL configuration at the customer site.

*The ATM connection extends at least as far as the  "Broadband Remote Access Server"  $\rm (BBRAS)$  and is converted there or continued directly,  depending on the backbone data transmission system.

The graph on the right shows the communication in an Internet connection according to the OSI model, where  "xDSL"  $($brown background$)$  is used only
*between the  "TU-R"  on the customer side  $($here realized with an "xDSL router"$)$

*and the  "TU-C"  on the provider side.
 
== Components of DSL Internet access ==
 
Finally,  necessary components for a DSL connection are listed.  The graph shows examples of these,  mostly from  "Deutsche Telekom".

[[File:EN_Bei_T_2_2_S7.png|right|frame|Necessary components to establish an xDSL connection]]

$\rm NTBA$:  The commonly used term is  "Network Termination for ISDN Basic Rate Access".  Tasks of the NTBA are:
*With help of a fork circuit and echo cancellation,  the two-wire  $\rm U_{K0}$  interface on the provider side is converted into the four-wire  $\rm S_0$ interface on the subscriber side.

*In addition,  the NTBA manages the ISDN code conversion from the  "Modified Monitored Sum 4B3T"  code  $\rm (U_{K0}$ bus$)$  to the modified AMI code  $\rm (S_{0}$ bus$)$.

$\text{xTU-R}$:  The abbreviation stands for  "xDSL Transceive Unit - Remote"'  and denotes the subscriber-side xDSL unit.
*Due to the widespread use of Ethernet,  today's xDSL modems and routers usually have only one Ethernet port for connecting data terminal equipment.  Originally,  they were used for subscriber-side connection of ATM data terminals.

*Thus,  this unit must also perform the function of a  "Layer 2 bridge"  in order to be able to transmit Ethernet over ATM to the  "Broadband Remote Access Server"  (BBRAS) for termination.

$\text{xDSL modem}$:  With this functional unit,  the data connection from/to the data terminal is initialized by a  "point-to-point protocol"  $\rm (PPP)$  via a  "PPP over Ethernet connection"  and terminated by the  "BBRAS".  Only data terminal devices that can separately establish a data connection via PPP are eligible.

$\text{xDSL router}$:  This initializes the data connection via PPP and enforces the addresses at IP level.  This allows multiple terminals to be connected and allows internal data exchange between them without having to dial into them separately.

$\rm Splitter$:  This is basically a combination of high-pass and low-pass with three interfaces, which handles the separation of the high-frequency xDSL data signals  $($above  $138$ kHz$)$  from the low-frequency POTS or ISDN telephone signals  $($below  $120$ kHz$)$,  or their combination.
*This  "broadband access unit"  is nothing other than a crossover unit.

*The sum of the signals is present on the subscriber line side,  while the xDSL data and the POTS/ISDN signals are separated from each other by a splitter on both the customer and provider sides.

$\text{xTU-C}$:  The abbreviation stands for  "xDSL Transceive Unit - Central office".  It is the provider-side xDSL unit and is usually implemented as a printed-circuit board insertion for the DSLAM.  The xTU-C terminates the physical retail xDSL subscriber lines,  modulates the ATM bit stream on the subscriber side,  and demodulates the xDSL signal on the provider side.

$\text{DSLAM}$:  The abbreviation stands for  "Digital Subscriber Line Access Multiplexer".  In its simplest form,  it terminates the physical subscriber lines with its xTU-C line cards.  In an extended form,  an  "ATM Service Access Multiplexer"  is also integrated in the DSLAM.
*The task of the DSLAM is to bundle the ATM bit streams of the subscriber lines and to forward them in concentrated form in the multiplexing process via an STM-1 fiber interface into the provider network.

*"STM" is an  "synchronous digital hierarchy"  $($SDH$)$  transmission standard for multiplexing optical channels and stands for  "Synchronous Transport Module".  STM-1 allows a bit rate of up to  $155.52$  Mbit/s,  STM-64 up to almost  $10$  Gbit/s.

==Exercises for the chapter ==
 
[[Exercise_2.2:_xDSL_Variants|Exercise 2.2: xDSL Variants]]

[[Exercise_2.2Z:_DSL_Internet_Connection|Exercise 2.2Z: DSL Internet Connection]]

{{Display}}

Examples of Communication Systems/xDSL Systems

2023-03-21T18:57:48Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=General Description of DSL
|Nächste Seite=xDSL as Transmission Technology
}}

==Reference models==
 
[[File:EN_Bei_T_2_2_S1_v3.png|right|frame|xDSL reference model of the ITU]]

Based on the following general ITU reference model,  it can be quickly seen that xDSL is physically a pure access transmission technology that is only used in the local loop network area between the fiber termination point and the network termination at the end customer.

The basic elements of the xDSL standard are:
*the network termination  $\rm (NT)$,

*a subscriber line$\rm$,  and

*the line termination  $\rm (LT)$.

There is a lot of freedom for network operators in implementing this reference model in practice. What all previous implementations have in common is that they use existing metallic subscriber lines.
 
{{GraueBox|TEXT=
$\text{Examples 1:}$ 
In an example,  the configuration most frequently encountered in Germany is shown according to the graphic.  Note:

[[File:EN_Bei_T_2_2_S1b_v8.png|right|frame|Reference model according to  "$\text{1TR112 U-R2-V7.0 DTAG}$"]]

*In all xDSL variants deployed today,  the data service converted in the modems is combined with the telephone service.  This allows transmission over the existing telephone network.

*The  "splitter"  separate the signal on both sides of the subscriber line into the classes  "broadband"  and  "narrowband" .

*An important interface is designated  $\text{U-R2}$  $($red marking$)$.
#This was standardized in Germany in 2001 by Deutsche Telekom AG in order to be able to use any modems on the subscriber side.
#This means that the customer is no longer dependent on his provider's xDSL modem.}}

==Overview and common features of all xDSL systems==
 
The technical realization of an  $\rm xDSL$  system involves many system components,  which can be distributed over several localities.  There is a wide range of realization options.  To summarize  $($from the perspective of 2009$)$:
#The systems for  $\rm ADSL$  and  $\rm ADSL$  shown below represent the most common implementation.  Data transport at the protocol level is based on the  $\rm ATM$  technology  $($"Asynchronous Transfer Mode"$)$.
#Despite a large data overhead,  ATM still offers advantages over  "Ethernet"  in terms of guaranteed quality of service  $\rm (QoS)$,  i.e. effective bit rate,  low delay and jitter.
#Ethernet,  on the other hand,  enables very high data transmission rates,  especially through the  »'''10 Gbit/s Ethernet'''«  and »'''100 Gbit/s Ethernet'''«  $($"Metro Ethernet"$)$  variants.  ATM,  on the other hand,  is more suitable for lower data rates.
#In 2009 there were numerous discussions about whether ATM should be replaced by  "10 Gbit/s Ethernet"  in the course of  "Next Generation Network".  However,  upgrading the backbone from ATM to Ethernet represents a not inconsiderable investment.

{{BlaueBox|TEXT=
$\text{Summary:}$ 
As mentioned in the chapter  [[Examples_of_Communication_Systems/General_Description_of_DSL|"General Description of DSL"]],  the most commonly deployed xDSL variants
#$\rm ADSL$, 
#$\rm ADSL2$  resp.  $\rm ADSL2+$,
#$\rm VDSL(1)$ 
#$\rm VDSL(2)$

are defined in such a way that simultaneous operation of  »'''POTS'''«  $($"Plain Old Telephone Service"$)$  or  »'''ISDN'''«  $($"Integrated Services Digital Network"$)$  on the same line is possible at any time.  This is the basis of the further descriptions.}}

==ADSL – Asymmetric Digital Subscriber Line ==
 
[[File:EN_Bei_T_2_2_S4.png|right|frame|ADSL connection from the end customer to the local exchange '''KORREKTUR''']]

The physical network termination  $\rm (NT)$  is in the ADSL modem  $($"ADSL Transmission Unit Central Office",  $\text{ATU-C)}$  in the local exchange.  Before that,  in the  "splitter"  the low-frequency telephony spectrum is separated from the higher-frequency ADSL spectrum by low-pass and high-pass filtering.

The graphic shows an ADSL connection from the end customer to the local exchange,  which is described very briefly below.  The reverse data connection paths are in each case mirror-inverted.

*The splitter forwards the telephone signals to the ISDN/POTS exchange and the ADSL signals to the  "digital subscriber line access multiplexer"  $\rm (DSLAM)$,  in which the  "$\text{ATU-C}$"  is implemented as a plug-in card.

*The DSLAM bundles many ADSL connections and forwards the data after decoding at the ATM level via optical fiber to the  "ATM Service Access Multiplexer".  This sends the data from all DSLAMs over the backbone to the  "Broadband Remote Access Server"  $\rm (BBRAS)$.

*The BBRAS terminates the point-to-point protocol data link and forwards the IP packets via routers to the destination.  The backbone consists of optical components based on  $\rm SDH$  standard  $($"Synchronous Digital Hierarchy"$)$.

*The splitter connected to the  "telecommunications connection unit"  $\rm (TAE)$  separates the signals. 

*The telephone signals are routed to the telephony terminals,  the ADSL signals to the modem  $($"ADSL Transmission Unit Remote",  $\text{ATU-R)}$.  The modem decodes and forwards the binary data to the connected terminals.

*During initialization of the ADSL connection,  $\text{ATU-C}$  and  $\text{ATU-R}$  perform a so-called  "training"  in which relevant system parameters  $($data rate,  fast and interleaved mode,  ...$)$  are determined depending on the line conditions.

*The parameters negotiated in this process are retained until the next check and synchronization.  For the transmission of administrative data  $($"overhead"$)$,  $32$  kbit per frame are statically reserved in the ADSL systems.

==ADSL2 and ADSL2plus==
 
{{BlaueBox|TEXT=
$\text{These two system variants are further developments of ADSL:}$ 
*The enhanced system variant  "Asymmetric Digital Subscriber Line Transceivers 2"  $\rm (ADSL2)$  was specified in 2002 with ITU recommendations  $\rm G.992.3$,  $\rm G.992.4$.

*2003 followed by the ITU recommendation  $\rm G.992.5$:  "Extended-bandwidth ADSL2"  $\rm (ADSL2+)$.}}

Compared to ADSL,  the following changes occurred:
#In ADSL2,  the  "Seamless Rate Adaption"  $\rm (SRA)$  was included in the standard.  This allows transmission parameters to be changed during operation with time-variant channel quality without loss of synchronization.
#For this purpose,  ATU-C and ATU-R periodically check the signal–to–noise ratio  $\rm (SNR)$  of the transmission channels.  If a channel in use deteriorates,  the receiver notifies the transmitter of the new data rate and transmission level.  After a subsequent  "sync flag"  the parameters are adopted.
#ADSL2 systems also offer a wide range of diagnostic options even without the modems having been synchronized,  a feature that is particularly important for troubleshooting,  error analysis and error correction.
#In addition,  ADSL2 provides the ability to reduce transmit levels when SNR is sufficient,  thereby minimizing crosstalk and increasing throughput in the trunk cable.  This  "power cutback"  can be initiated not only by the DSLAM,  but also by the ATU-R.
#In ADSL2,  the number of overhead bits is no longer fixed,  but can vary between  $4$  and  $32$  kbit.  This increase in the user data bit rate of up to  $28$  kbit/s per data frame is all the more important the longer the distance between the modem and the DSLAM.

{{BlaueBox|TEXT=
$\text{As a result from 2009:}$ 
*$\text{ADSL2}$  systems achieve a transmission rate of more than  $8$  Mbit/s  $($up to  $12$  Mbit/s$)$  downstream and more than  $800$  kbit/s  $($up to  $3.5$  Mbit/s$)$  upstream.

*With  $\text{ADSL2+}$,  the transmission rate in the downstream is doubled again; the maximum rate is theoretically  $25$  Mbit/s.}}

==VDSL – Very–high–speed Digital Subscriber Line ==
 
In terms of the basic structure of their components,  VDSL systems are identical to ADSL systems,  with the only exception that the relocation of the splitter and the DSLAM from the local exchange to a cable branch makes the last section between the network operator and the customer,  the so-called  "last mile".
[[File:EN_Bei_T_2_2_S5_v8.png|right|frame|Modeling of a VDSL connection from the end customer to the local exchange]]

This measure was necessary because VDSL can only exploit its advantage  – the greater transmission speed –  over very short distances due to the attenuation of the higher frequencies,  which increases sharply with line length.

DSLAM and BBRAS are still connected via  [https://en.wikipedia.org/wiki/STM-1 $\text{STM-1}$]  interfaces.  Therefore,  the route between the local exchange and the cable branch must now also be laid with optical fiber.

A distinction is made between two alternative VDSL variants:
*$\rm VDSL(1)$  is based on  [[Modulation_Methods/Quadrature_Amplitude_Modulation|$\rm QAM$]]  $($"Quadrature Amplitude Modulation"$)$,  which is predominantly deployed in Asia,

*$\rm VDSL(2)$  is based on  [[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|$\rm DMT$]]  $($"Discrete Multitone Transmission"$)$.

:VDSL(1) systems were never deployed in Germany because of their inadequate ability to provide audio/video,  telephony and Internet  $($"triple play"$)$  with sufficient quality of service.

:Instead,  the VDSL(2) standard was established immediately:  Because of higher performance and greater range,  the better quality of service as well as the reusability of the ADSL(2+) infrastructure.

{{BlaueBox|TEXT=
$\text{Summary:}$ 
Following a few characteristics of the  $\rm VDSL(2)$  system:
#VDSL(2) has achieved since 2006 a maximum transmission rate of  $50$  to  $100$  Mbit/s,  depending on the standard used.
#The specified VDSL(2) transmission bandwidth of  $30$nbsp; MHz was considered the maximum reasonable bandwidth in 2009.
#With complementary measures such as  "Dynamic Spectrum Management"  and "Advanced Codes"  total transmission rates of up to  $280$ Mbit/s  were expected for short line lengths  $($up to  $300$  meters$)$.}}

==DSL Internet access from perspective of the communication protocols ==
 
Some xDSL modems offer an  Ethernet interface for connecting the data terminals and a transparent connection to the remote terminal,  based on the  "Internet Protocol"  $\rm (IP)$.  The following should be noted:
[[File:EN_Bei_T_2_2_S6b_v8.png|right|frame|Modeling of an xDSL connection through the use of an xDSL router]]

*This option is enabled by the  "LAN Emulation"  $\rm (RFC2684)$  and the  "ATM Adaption Layer Protocol"  $\rm (AAL5)$.  The Ethernet data stream must be converted to  $\rm ATM$  for this purpose.

*This eliminates the need to install ATM equipment and existing Ethernet hardware can be used,  greatly simplifying xDSL configuration at the customer site.

*The ATM connection extends at least as far as the  "Broadband Remote Access Server"  $\rm (BBRAS)$  and is converted there or continued directly,  depending on the backbone data transmission system.

The graph on the right shows the communication in an Internet connection according to the OSI model, where  "xDSL"  $($brown background$)$  is used only
*between the  "TU-R"  on the customer side  $($here realized with an "xDSL router"$)$

*and the  "TU-C"  on the provider side.
 
== Components of DSL Internet access ==
 
Finally,  necessary components for a DSL connection are listed.  The graph shows examples of these,  mostly from  "Deutsche Telekom".

[[File:EN_Bei_T_2_2_S7.png|right|frame|Necessary components to establish an xDSL connection]]

$\rm NTBA$:  The commonly used term is  "Network Termination for ISDN Basic Rate Access".  Tasks of the NTBA are:
*With help of a fork circuit and echo cancellation,  the two-wire  $\rm U_{K0}$  interface on the provider side is converted into the four-wire  $\rm S_0$ interface on the subscriber side.

*In addition,  the NTBA manages the ISDN code conversion from the  "Modified Monitored Sum 4B3T"  code  $\rm (U_{K0}$ bus$)$  to the modified AMI code  $\rm (S_{0}$ bus$)$.

$\text{xTU-R}$:  The abbreviation stands for  "xDSL Transceive Unit - Remote"'  and denotes the subscriber-side xDSL unit.
*Due to the widespread use of Ethernet,  today's xDSL modems and routers usually have only one Ethernet port for connecting data terminal equipment.  Originally,  they were used for subscriber-side connection of ATM data terminals.

*Thus,  this unit must also perform the function of a  "Layer 2 bridge"  in order to be able to transmit Ethernet over ATM to the  "Broadband Remote Access Server"  (BBRAS) for termination.

$\text{xDSL modem}$:  With this functional unit,  the data connection from/to the data terminal is initialized by a  "point-to-point protocol"  $\rm (PPP)$  via a  "PPP over Ethernet connection"  and terminated by the  "BBRAS".  Only data terminal devices that can separately establish a data connection via PPP are eligible.

$\text{xDSL router}$:  This initializes the data connection via PPP and enforces the addresses at IP level.  This allows multiple terminals to be connected and allows internal data exchange between them without having to dial into them separately.

$\rm Splitter$:  This is basically a combination of high-pass and low-pass with three interfaces, which handles the separation of the high-frequency xDSL data signals  $($above  $138$ kHz$)$  from the low-frequency POTS or ISDN telephone signals  $($below  $120$ kHz$)$,  or their combination.
*This  "broadband access unit"  is nothing other than a crossover unit.

*The sum of the signals is present on the subscriber line side,  while the xDSL data and the POTS/ISDN signals are separated from each other by a splitter on both the customer and provider sides.

$\text{xTU-C}$:  The abbreviation stands for  "xDSL Transceive Unit - Central office".  It is the provider-side xDSL unit and is usually implemented as a printed-circuit board insertion for the DSLAM.  The xTU-C terminates the physical retail xDSL subscriber lines,  modulates the ATM bit stream on the subscriber side,  and demodulates the xDSL signal on the provider side.

$\text{DSLAM}$:  The abbreviation stands for  "Digital Subscriber Line Access Multiplexer".  In its simplest form,  it terminates the physical subscriber lines with its xTU-C line cards.  In an extended form,  an  "ATM Service Access Multiplexer"  is also integrated in the DSLAM.
*The task of the DSLAM is to bundle the ATM bit streams of the subscriber lines and to forward them in concentrated form in the multiplexing process via an STM-1 fiber interface into the provider network.

*"STM" is an  "synchronous digital hierarchy"  $($SDH$)$  transmission standard for multiplexing optical channels and stands for  "Synchronous Transport Module".  STM-1 allows a bit rate of up to  $155.52$  Mbit/s,  STM-64 up to almost  $10$  Gbit/s.

==Exercises for the chapter ==
 
[[Exercise_2.2:_xDSL_Variants|Exercise 2.2: xDSL Variants]]

[[Exercise_2.2Z:_DSL_Internet_Connection|Exercise 2.2Z: DSL Internet Connection]]

{{Display}}

Examples of Communication Systems/xDSL Systems

2023-03-21T18:52:19Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=General Description of DSL
|Nächste Seite=xDSL as Transmission Technology
}}

==Reference models==
 
[[File:EN_Bei_T_2_2_S1_v3.png|right|frame|xDSL reference model of the ITU]]

Based on the following general ITU reference model,  it can be quickly seen that xDSL is physically a pure access transmission technology that is only used in the local loop network area between the fiber termination point and the network termination at the end customer.

The basic elements of the xDSL standard are:
*the network termination  $\rm (NT)$,

*a subscriber line$\rm$,  and

*the line termination  $\rm (LT)$.

There is a lot of freedom for network operators in implementing this reference model in practice. What all previous implementations have in common is that they use existing metallic subscriber lines.
 
{{GraueBox|TEXT=
$\text{Examples 1:}$ 
In an example,  the configuration most frequently encountered in Germany is shown according to the graphic.  Note:

[[File:EN_Bei_T_2_2_S1b_v8.png|right|frame|Reference model according to  "$\text{1TR112 U-R2-V7.0 DTAG}$"]]

*In all xDSL variants deployed today,  the data service converted in the modems is combined with the telephone service.  This allows transmission over the existing telephone network.

*The  "splitter"  separate the signal on both sides of the subscriber line into the classes  "broadband"  and  "narrowband" .

*An important interface is designated  $\text{U-R2}$  $($red marking$)$.
#This was standardized in Germany in 2001 by Deutsche Telekom AG in order to be able to use any modems on the subscriber side.
#This means that the customer is no longer dependent on his provider's xDSL modem.}}

==Overview and common features of all xDSL systems==
 
The technical realization of an  $\rm xDSL$  system involves many system components,  which can be distributed over several localities.  There is a wide range of realization options.  To summarize  $($from the perspective of 2009$)$:
#The systems for  $\rm ADSL$  and  $\rm ADSL$  shown below represent the most common implementation.  Data transport at the protocol level is based on the  $\rm ATM$  technology  $($"Asynchronous Transfer Mode"$)$.
#Despite a large data overhead,  ATM still offers advantages over  "Ethernet"  in terms of guaranteed quality of service  $\rm (QoS)$,  i.e. effective bit rate,  low delay and jitter.
#Ethernet,  on the other hand,  enables very high data transmission rates,  especially through the  »'''10 Gbit/s Ethernet'''«  and »'''100 Gbit/s Ethernet'''«  $($"Metro Ethernet"$)$  variants.  ATM,  on the other hand,  is more suitable for lower data rates.
#In 2009 there were numerous discussions about whether ATM should be replaced by  "10 Gbit/s Ethernet"  in the course of  "Next Generation Network".  However,  upgrading the backbone from ATM to Ethernet represents a not inconsiderable investment.

{{BlaueBox|TEXT=
$\text{Summary:}$ 
As mentioned in the chapter  [[Examples_of_Communication_Systems/General_Description_of_DSL|"General Description of DSL"]],  the most commonly deployed xDSL variants
#$\rm ADSL$, 
#$\rm ADSL2$  resp.  $\rm ADSL2+$,
#$\rm VDSL(1)$ 
#$\rm VDSL(2)$

are defined in such a way that simultaneous operation of  »'''POTS'''«  $($"Plain Old Telephone Service"$)$  or  »'''ISDN'''«  $($"Integrated Services Digital Network"$)$  on the same line is possible at any time.  This is the basis of the further descriptions.}}

==ADSL – Asymmetric Digital Subscriber Line ==
 
[[File:EN_Bei_T_2_2_S4.png|right|frame|ADSL connection from the end customer to the local exchange ]]

The physical network termination  $\rm (NT)$  is in the ADSL modem  $($"ADSL Transmission Unit Central Office",  $\text{ATU-C)}$  in the local exchange.  Before that,  in the  "splitter"  the low-frequency telephony spectrum is separated from the higher-frequency ADSL spectrum by low-pass and high-pass filtering.

The graphic shows an ADSL connection from the end customer to the local exchange,  which is described very briefly below.  The reverse data connection paths are in each case mirror-inverted.

*The splitter forwards the telephone signals to the ISDN/POTS exchange and the ADSL signals to the  "digital subscriber line access multiplexer"  $\rm (DSLAM)$,  in which the  "$\text{ATU-C}$"  is implemented as a plug-in card.

*The DSLAM bundles many ADSL connections and forwards the data after decoding at the ATM level via optical fiber to the  "ATM Service Access Multiplexer".  This sends the data from all DSLAMs over the backbone to the  "Broadband Remote Access Server"  $\rm (BBRAS)$.

*The BBRAS terminates the point-to-point protocol data link and forwards the IP packets via routers to the destination.  The backbone consists of optical components based on  $\rm SDH$  standard  $($"Synchronous Digital Hierarchy"$)$.

*The splitter connected to the  "telecommunications connection unit"  $\rm (TAE)$  separates the signals. 

*The telephone signals are routed to the telephony terminals,  the ADSL signals to the modem  $($"ADSL Transmission Unit Remote",  $\text{ATU-R)}$.  The modem decodes and forwards the binary data to the connected terminals.

*During initialization of the ADSL connection,  $\text{ATU-C}$  and  $\text{ATU-R}$  perform a so-called  "training"  in which relevant system parameters  $($data rate,  fast and interleaved mode,  ...$)$  are determined depending on the line conditions.

*The parameters negotiated in this process are retained until the next check and synchronization.  For the transmission of administrative data  $($"overhead"$)$,  $32$  kbit per frame are statically reserved in the ADSL systems.

==ADSL2 and ADSL2plus==
 
{{BlaueBox|TEXT=
$\text{These two system variants are further developments of ADSL:}$ 
*The enhanced system variant  "Asymmetric Digital Subscriber Line Transceivers 2"  $\rm (ADSL2)$  was specified in 2002 with ITU recommendations  $\rm G.992.3$,  $\rm G.992.4$.

*2003 followed by the ITU recommendation  $\rm G.992.5$:  "Extended-bandwidth ADSL2"  $\rm (ADSL2+)$.}}

Compared to ADSL,  the following changes occurred:
#In ADSL2,  the  "Seamless Rate Adaption"  $\rm (SRA)$  was included in the standard.  This allows transmission parameters to be changed during operation with time-variant channel quality without loss of synchronization.
#For this purpose,  ATU-C and ATU-R periodically check the signal–to–noise ratio  $\rm (SNR)$  of the transmission channels.  If a channel in use deteriorates,  the receiver notifies the transmitter of the new data rate and transmission level.  After a subsequent  "sync flag"  the parameters are adopted.
#ADSL2 systems also offer a wide range of diagnostic options even without the modems having been synchronized,  a feature that is particularly important for troubleshooting,  error analysis and error correction.
#In addition,  ADSL2 provides the ability to reduce transmit levels when SNR is sufficient,  thereby minimizing crosstalk and increasing throughput in the trunk cable.  This  "power cutback"  can be initiated not only by the DSLAM,  but also by the ATU-R.
#In ADSL2,  the number of overhead bits is no longer fixed,  but can vary between  $4$  and  $32$  kbit.  This increase in the user data bit rate of up to  $28$  kbit/s per data frame is all the more important the longer the distance between the modem and the DSLAM.

{{BlaueBox|TEXT=
$\text{As a result from 2009:}$ 
*$\text{ADSL2}$  systems achieve a transmission rate of more than  $8$  Mbit/s  $($up to  $12$  Mbit/s$)$  downstream and more than  $800$  kbit/s  $($up to  $3.5$  Mbit/s$)$  upstream.

*With  $\text{ADSL2+}$,  the transmission rate in the downstream is doubled again; the maximum rate is theoretically  $25$  Mbit/s.}}

==VDSL – Very–high–speed Digital Subscriber Line ==
 
In terms of the basic structure of their components,  VDSL systems are identical to ADSL systems,  with the only exception that the relocation of the splitter and the DSLAM from the local exchange to a cable branch makes the last section between the network operator and the customer,  the so-called  "last mile".
[[File:EN_Bei_T_2_2_S5_v8.png|right|frame|Modeling of a VDSL connection from the end customer to the local exchange]]

This measure was necessary because VDSL can only exploit its advantage  – the greater transmission speed –  over very short distances due to the attenuation of the higher frequencies,  which increases sharply with line length.

DSLAM and BBRAS are still connected via  [https://en.wikipedia.org/wiki/STM-1 $\text{STM-1}$]  interfaces.  Therefore,  the route between the local exchange and the cable branch must now also be laid with optical fiber.

A distinction is made between two alternative VDSL variants:
*$\rm VDSL(1)$  is based on  [[Modulation_Methods/Quadrature_Amplitude_Modulation|$\rm QAM$]]  $($"Quadrature Amplitude Modulation"$)$,  which is predominantly deployed in Asia,

*$\rm VDSL(2)$  is based on  [[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|$\rm DMT$]]  $($"Discrete Multitone Transmission"$)$.

:VDSL(1) systems were never deployed in Germany because of their inadequate ability to provide audio/video,  telephony and Internet  $($"triple play"$)$  with sufficient quality of service.

:Instead,  the VDSL(2) standard was established immediately:  Because of higher performance and greater range,  the better quality of service as well as the reusability of the ADSL(2+) infrastructure.

{{BlaueBox|TEXT=
$\text{Summary:}$ 
Following a few characteristics of the  $\rm VDSL(2)$  system:
#VDSL(2) has achieved since 2006 a maximum transmission rate of  $50$  to  $100$  Mbit/s,  depending on the standard used.
#The specified VDSL(2) transmission bandwidth of  $30$nbsp; MHz was considered the maximum reasonable bandwidth in 2009.
#With complementary measures such as  "Dynamic Spectrum Management"  and "Advanced Codes"  total transmission rates of up to  $280$ Mbit/s  were expected for short line lengths  $($up to  $300$  meters$)$.}}

==DSL Internet access from perspective of the communication protocols ==
 
Some xDSL modems offer an  Ethernet interface for connecting the data terminals and a transparent connection to the remote terminal,  based on the  "Internet Protocol"  $\rm (IP)$.  The following should be noted:
[[File:EN_Bei_T_2_2_S6b_v8.png|right|frame|Modeling of an xDSL connection through the use of an xDSL router]]

*This option is enabled by the  "LAN Emulation"  $\rm (RFC2684)$  and the  "ATM Adaption Layer Protocol"  $\rm (AAL5)$.  The Ethernet data stream must be converted to  $\rm ATM$  for this purpose.

*This eliminates the need to install ATM equipment and existing Ethernet hardware can be used,  greatly simplifying xDSL configuration at the customer site.

*The ATM connection extends at least as far as the  "Broadband Remote Access Server"  $\rm (BBRAS)$  and is converted there or continued directly,  depending on the backbone data transmission system.

The graph on the right shows the communication in an Internet connection according to the OSI model, where  "xDSL"  $($brown background$)$  is used only
*between the  "TU-R"  on the customer side  $($here realized with an "xDSL router"$)$

*and the  "TU-C"  on the provider side.
 
== Components of DSL Internet access ==
 
Finally,  necessary components for a DSL connection are listed.  The graph shows examples of these,  mostly from  "Deutsche Telekom".

[[File:EN_Bei_T_2_2_S7.png|right|frame|Necessary components to establish an xDSL connection]]

$\rm NTBA$:  The commonly used term is  "Network Termination for ISDN Basic Rate Access".  Tasks of the NTBA are:
*With help of a fork circuit and echo cancellation,  the two-wire  $\rm U_{K0}$  interface on the provider side is converted into the four-wire  $\rm S_0$ interface on the subscriber side.

*In addition,  the NTBA manages the ISDN code conversion from the  "Modified Monitored Sum 4B3T"  code  $\rm (U_{K0}$ bus$)$  to the modified AMI code  $\rm (S_{0}$ bus$)$.

$\text{xTU-R}$:  The abbreviation stands for  "xDSL Transceive Unit - Remote"'  and denotes the subscriber-side xDSL unit.
*Due to the widespread use of Ethernet,  today's xDSL modems and routers usually have only one Ethernet port for connecting data terminal equipment.  Originally,  they were used for subscriber-side connection of ATM data terminals.

*Thus,  this unit must also perform the function of a  "Layer 2 bridge"  in order to be able to transmit Ethernet over ATM to the  "Broadband Remote Access Server"  (BBRAS) for termination.

$\text{xDSL modem}$:  With this functional unit,  the data connection from/to the data terminal is initialized by a  "point-to-point protocol"  $\rm (PPP)$  via a  "PPP over Ethernet connection"  and terminated by the  "BBRAS".  Only data terminal devices that can separately establish a data connection via PPP are eligible.

$\text{xDSL router}$:  This initializes the data connection via PPP and enforces the addresses at IP level.  This allows multiple terminals to be connected and allows internal data exchange between them without having to dial into them separately.

$\rm Splitter$:  This is basically a combination of high-pass and low-pass with three interfaces, which handles the separation of the high-frequency xDSL data signals  $($above  $138$ kHz$)$  from the low-frequency POTS or ISDN telephone signals  $($below  $120$ kHz$)$,  or their combination.
*This  "broadband access unit"  is nothing other than a crossover unit.

*The sum of the signals is present on the subscriber line side,  while the xDSL data and the POTS/ISDN signals are separated from each other by a splitter on both the customer and provider sides.

$\text{xTU-C}$:  The abbreviation stands for  "xDSL Transceive Unit - Central office".  It is the provider-side xDSL unit and is usually implemented as a printed-circuit board insertion for the DSLAM.  The xTU-C terminates the physical retail xDSL subscriber lines,  modulates the ATM bit stream on the subscriber side,  and demodulates the xDSL signal on the provider side.

$\text{DSLAM}$:  The abbreviation stands for  "Digital Subscriber Line Access Multiplexer".  In its simplest form,  it terminates the physical subscriber lines with its xTU-C line cards.  In an extended form,  an  "ATM Service Access Multiplexer"  is also integrated in the DSLAM.
*The task of the DSLAM is to bundle the ATM bit streams of the subscriber lines and to forward them in concentrated form in the multiplexing process via an STM-1 fiber interface into the provider network.

*"STM" is an  "synchronous digital hierarchy"  $($SDH$)$  transmission standard for multiplexing optical channels and stands for  "Synchronous Transport Module".  STM-1 allows a bit rate of up to  $155.52$  Mbit/s,  STM-64 up to almost  $10$  Gbit/s.

==Exercises for the chapter ==
 
[[Exercise_2.2:_xDSL_Variants|Exercise 2.2: xDSL Variants]]

[[Exercise_2.2Z:_DSL_Internet_Connection|Exercise 2.2Z: DSL Internet Connection]]

{{Display}}

Examples of Communication Systems/General Description of DSL

2023-03-21T18:45:04Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=Further Developments of ISDN
|Nächste Seite=xDSL_Systems
}}

== # OVERVIEW OF THE SECOND MAIN CHAPTER # ==
 

$\rm D$igital $\rm S$ubscriber $\rm L$ine  – in short  $\rm DSL$ –  literally means only  "digital subscriber line".  At the same time,  "DSL"  was a synonym for  "high-speed Internet access in the local loop to the end customer",  although "high-speed"  must be put into perspective today  $(2018)$.

xDSL has been significantly standardized by the standards committees  [https://en.wikipedia.org/wiki/American_National_Standards_Institute $\rm ANSI$]  $($USA$)$  and  [https://en.wikipedia.org/wiki/ETSI $\rm ETSI$]  $($Europe$)$  as well as the  [https://en.wikipedia.org/wiki/International_Telecommunication_Union $\rm ITU$]  $($worldwide$)$.  Due to different pre-existing technical conditions and preferences of developers and operators,  a large variety of nationally different versions of nominally identical xDSL standards resulted.  In the following,  we will restrict ourselves primarily to the German xDSL versions.

This chapter contains in detail:

#An  »overview of the historical development and standardization«  of xDSL,
#the  »differences between ADSL and VDSL«  as well as statistics on their penetration,
#a brief description of xDSL from a  »communications protocol perspective«,
#the bandwidth allocations for the two  »xDSL variants ADSL and VDSL«, 
#a detailed description of the  »DSL transmission methods QAM, CAP and DMT«,
#the problems of  »digital signal transmission over copper twisted pairs«  in general,
#the relationship between  »SNR,  range and transmission rate«,
#the  »error correction measures«  used to reduce the bit error rate.

==Network infrastructure for DSL==
 

We start as in the  [[Examples_of_Communication_Systems/General_Description_of_ISDN#Network_infrastructure_for_ISDN|"ISDN chapter"]]  with the network infrastructure.  DSL was intended to use the existing analog telephone network for cost reasons. 

The greatest cost factor of the entire infrastructure is the  »'''subscriber line area'''«  between a main distribution frame  $($e.g. "switching office"$)$  and the subscribers.
[[File:EN_LZI_T_4_3_S2_neu.png| right|frame|Structure of the local loop area]]

*In Germany,  this so-called  »'''last mile'''«  is shorter than  $4$  kilometers on average,  and in urban areas  $90\%$  of the time it is even shorter than  $2.8$  kilometers.

*Due to the topological conditions,  the telephone network is increasingly branching out in a star configuration toward the end customer.

*In order to avoid having to lay a separate copper cable to the local exchange for each subscriber,  splitters have been installed in between and the lines bundled in correspondingly large cables.

The  »'''local loop area'''»  is therefore usually made up as follows:

#   The  "main cable"  with up to  $2000$  pairs between the local exchange  (or the switching office)  frame and a cable branch,
#  the  "branch cable"  between the cable branch and the final branch, with up to  $300$  pairs and a maximum length of 500 meters,  which is significantly shorter than a main cable,
#  the  "house connection cable"  between the terminal box and the network termination box at the subscriber with two pairs of wires.

==xDSL types and terms==
 

{{BlaueBox|TEXT=
$\text{Motivation for Digital Subscriber Line}$ 

$\rm DSL$  $($"Digital Subscriber Line"$)$  arose from the need,  '''to provide low cost high rate digital data access to the end user'''.  During the design process,  it was necessary to take into account:
*As explained in the last section,  the  "last mile"  is the largest cost factor in a communications network.

*Considerations to replace the estimated  130  million kilometers of copper twisted pairs in the local loop network with fiber optic lines  $($fiber-to-the-home,  $\rm FttH)$  have failed to date due to the enormous costs of the mostly underground laying work.

*A viable solution was to offer a broadband connection with somewhat lower data rates than in a fiber optic network by using the existing telephone line network and by cleverly combining different transmission techniques and coding methods.

*The telephone service – either analog or digital  $\rm (ISDN)$  – should be able to operate simultaneously on the same network.}}

Before we turn to the historical DSL development up to the current state,  the various types of  "$\rm xDSL$"  must first be defined and some terms explained.  Note:
#Here, "$\rm x$"  is merely a placeholder that designates the various DSL standards.
#The technical features will be covered in depth in the next chapters.

'''Part of the xDSL standard:'''
*$\text{ADSL}$  –  "Asymmetric Digital Subscriber Line: Asymmetric data transmission technology with data rates of  $8$  Mbit/s to the subscriber  $($"downstream"$)$  and  $1$  Mbit/s in the opposite direction  $($"upstream"$)$.

*$\text{ADSL2}$  and  $\text{ADSL2+}$: Extensions of ADSL with data rates of up to  $25$  Mbit/s  $($"downstream"$)$  and up to  $1$  Mbit/s  $($"upstream"$)$. The data rate is dynamically negotiated depending on the channel state.

*$\text{Re – ADSL2}$: Another extension of ADSL with about  $30\%$  range gain at a data rate of  $768$  kbit/s downstream.

*$\text{HDSL}$  –  "High Data Rate Digital Subscriber Line": Symmetrical data transmission technology  –  i.e. equal rates in downstream and upstream  –  with data rates between  $1.54$  Mbit/s and  $2.04$  Mbit/s. Note:  The name  "HDSL"  suggests higher data rates than ADSL;  however,  this is not the case.

*$\text{SDSL}$  –  "Symmetric Digital Subscriber Line": Symmetrical data transmission at rates of up to  $3$  Mbit/s.  With four-wire wiring  $($two copper twisted pairs$)$,  a maximum of  $4$  Mbit/s can be transmitted.  Alternatively,  the range can be increased at the expense of bandwidth.

*$\text{VDSL}$  –  "Very High Data Rate Digital Subscriber Line": A newer transmission technology based on QAM that operates in the asymmetrical variant with bit rates of  $25$  to  $50$  Mbit/s  downstream and  $5$ to  $10$  Mbit/s upstream.  The symmetrical variant has the same data transmission rates in upstream and downstream.

*$\text{VDSL2}$  –  "Very High Data Rate Digital Subscriber Line 2": Transmission technology with the currently (2009) highest total data rate of up to  $200$  Mbit/s.  The process is based on DMT  $($"Discrete Multitone Transmission"$)$.

*$\text{UDSL}$   or   $\text{UADSL}$    –  "Universal (Asymmetric) Digital Subscriber Line".

'''Non part of the xDSL standard:''' '''KORREKTUR: not part?'''
#There are also many products circulating under  "DSL"  that are not part of the xDSL standard.
#They are often only intended to make it clear that fast data access is involved. 

These include:
*$\text{cableDSL}$:   Brand name of the German company TELES AG, which offers high-speed Internet access via cable. The name was chosen for marketing reasons only.

*$\text{skyDSL}$:  Brand name for Internet access available throughout Europe via satellite with up to  $24$  Mbit/s downstream. The upstream here is via POTS  $($"Plain old telephone service"$)$  or  [[Examples_of_Communication_Systems/General_Description_of_ISDN|ISDN ]]  $($"Integrated Services Digital Network"$)$.

*$\text{T-DSL via satellite}$:   Brand name for a downstream Internet access of Telekom via satellite;  uses a conventional modem or an ISDN connection for transmission.

*$\text{WDSL}$  –  "Wireless Digital Subscriber Line":  Brand name of a German company that uses wireless technology to enable data rates of up to  $108$  Mbit/s in DSL-free areas.

*$\text{mvoxSatellit}$:   Brand name of an Internet access with  "WiMAX-like radio technology",  which like WDSL and PortableDSL is only an auxiliary construct for DSL-free areas.

==Historical development of xDSL standardizations ==
 

The need for digital subscriber lines to improve line utilization and increase customer convenience was recognized as early as the 1970s.  After the ISDN specification in the early 1980s,  the actual development of DSL began.

*This development was influenced by the findings of many groups located around the world.  Accordingly,  the standardization proceeded in an unstructured manner.

*From the list on the right,  it is clear that different committees around the world were in charge of the various standards.

*In the industry,  the technical realizations of the individual xDSL standards often deviated noticeably from the specification. 

*Some standards were started as projects even before the specification,  since the industry parties were also represented in the standardization committees.

The graph illustrates the relationships between milestones in the theoretical and practical design of transmission systems.

[[File:EN_Bei_2_1_s3_v4.png|right|frame|Milestones of the industrial xDSL development]]
$\text{Milestones of DSL development in short form:}$

'''1986'''   A first concept for  $\rm HDSL$  $($"'''H'''igh-bit-rate '''D'''igital '''S'''ubscriber '''L'''ine"$)$  is defined by AT&T,  Bell Laboratories and Bellcore.

'''1989'''   First HDSL prototypes appear;     ⇒     Bellcore meanwhile works on the conceptual definition of  $\rm ADSL$   $($"'''A'''symmetric '''D'''igital '''S'''ubscriber '''L'''ine"$)$.

'''1992'''   First publication of the  $\text{ANSI Technical Report E1T1/92-002R1}$: "High-bit-rate Digital Subscriber Line";    ⇒     The first ADSL prototypes appear.

'''1994'''   The  $\rm VDSL$  concept  $($"'''V'''ery-high-speed '''D'''igital '''S'''ubscriber '''L'''ine"$)$  is discussed for the first time.

'''1995'''   Publication of the  $\text{ETSI Technical Report ETR 152:}$  "High-bit-rate Digital Subscriber Line"  and  "Transmission Systems on Metallic Local Lines"; ⇒     Publication of  $\text{ADSL Standard ANSI T1.413}$:  $($"Asymmetric Digital Subscriber Line Metallic Interface"$)$;     ⇒     First field trials with ADSL in the USA .

'''1996'''   First publication of the  $\text{ETSI Technical Report ETR 328}$:  "Asymmetric Digital Subscriber Line"  and  "Transmission and Multiplexing".

'''04/1998''' First publication of  $\text{ETSI Technical Specification TS 101 270}$:  "Very-high-speed Digital Subscriber Line";     ⇒     Almost simultaneously, first publication of  $\text{ANSI Draft Technical Document T1E1.4/98-043R1}$:  "Very-high-speed Digital Subscriber Lines".

'''10/1998''' First publication of  $\text{ITU Recommendation G.991.1}$:  "High-bit-rate Digital Subscriber Line Transceivers";     ⇒     Almost simultaneously publication of  $\text{ETSI Technical Specification TS 101 135}$:  "High-bit-rate Digital Subscriber Line – Transmission Systems on Metallic Local Lines".

'''11/1998''' Publication of  $\text{ETSI Technical Specification TS 101 388 V1.1.1}$:  "Asymmetric Digital Subscriber Line – European Specific Requirements".

'''1999'''   In June,  publication of  $\text{ITU Recommendations G.992.1}$:  "Asymmetric Digital Subscriber Line – Transceivers"  and  $\text{G.992.2}$:  "Splitterless Asymmetric Digital Subscriber Line – Transceivers";     ⇒     On July 22, Deutsche Telekom AG offers ADSL in Germany for the first time  $\text{(T-DSL 768)}$.

'''2001'''   In February,  publication of  $\text{ITU Recommendation G.991.2}$:  "Single-pair High-speed Digital Subscriber Line Transceivers";     ⇒     In November,  publication of  $\text{ITU Recommendation G.993.1}$:  "Very-high-speed Digital Subscriber Line transceivers".

'''2002'''   First publications of  $\text{ITU Recommendations G.992.3}$:  "Asymmetric Digital Subscriber Line Transceivers 2"  $\rm (ADSL2)$  and  $\text{G.992.4}$:  "Splitterless ADSL2".

'''2003'''   First publication of  $\text{ITU Recommendation G.992.5}$:  "Asymmetric Digital Subscriber Line"  $\rm (ADSL)$  Transceivers"  and   "Extended-bandwidth ADSL2"  $\rm (ADSL2+)$.

'''02/2006'''   Publication of  $\text{ITU Recommendation G.993.2}$: "Very-high-speed Digital Subscriber Line Transceivers 2"  $\rm (VDSL2)$.

'''10/2006'''  Deutsche Telekom AG offers VDSL2 to end customers in selected cities for the first time.

== Development of ADSL and VDSL in Europe ==
 
From the above compilation it can be seen that the  »'''ADSL standardization'''«  was predominantly driven by  [https://en.wikipedia.org/wiki/American_National_Standards_Institute $\rm ANSI$]  $($"American National Standards Institute"$)$  and that  [https://en.wikipedia.org/wiki/ETSI $\rm ETSI$]  $($"European Telecommunications Standards Institute"$)$  followed shortly thereafter in each case:
#The first ADSL standard   "$\text{ANSI T1.413}$"   from 1995 was predominantly optimized for video-on-demand services,  which is also made clear by the ratio of the downstream and upstream data rates defined herein:  $1.5$  Mbit/s and  $16$  kbit/s,  $3$  Mbit/s and  $16$  kbit/s, and finally  $6$  Mbit/s and  $64$ kbit/s.
#The frequency range was originally defined in such a way that ADSL could only be used to operate an analog telephone on the access line.  ETSI published a technical report   "$\text{ETR 328}$"   in 1996 with only a few detailed changes and the possibility to transmit  $2048$  kbit/s.
#Since the second version of the ANSI standard also allowed only one additional analog telephone,  ETSI then defined an ADSL system that differed both in bit rates and in the possibility of using an ISDN basic access on the same twisted pair.
#The ANSI and ETSI standardization efforts of the previous years resulted in the  [https://en.wikipedia.org/wiki/International_Telecommunication_Union "ITU"] recommendation  "$\text{G.992.1}$"   in 1999,  which includes both standards and thus allows many options for the implementation.

However,  the many options led to major conceptual differences at the end of the 1990s  –  worldwide,  within Europe and also nationally  –  depending on the semiconductor manufacturer,  among other things.  Only a few systems,  modems,  and measuring devices interoperated with other manufacturers.

To counteract this proliferation,  The  "Deutsche Telekom AG"  passed the technical guideline  "$\text{1TR112}$"   at the end of 2001,  defining all the necessary interface parameters to ensure the interoperability of different manufacturer modems on the provider and customer side. 
*Due to Telekom's market power,  this became the quasi-standard for Germany.

*Furthermore,  only those ADSL variants were used in Germany that allowed simultaneous operation of ISDN at any time.  Thus,  when switching from POTS to ISDN,  it was not necessary to change the ADSL version as well.

The  »'''VDSL standardization'''«  relevant for Europe was decisively shaped by ETSI and often happened in parallel to the American activities.  Overall,  VDSL standardization proceeded in a more orderly fashion than ADSL.  The 3-step plan adopted by ETSI provided for:
*Stage 1:  Functional and electrical requirements for VDSL systems,

*Stage 2:  Transmission coding and access method requirements,

*Stage 3:  Interoperability requirements.

These efforts culminated in April 1998 in the publication of the ETSI Technical Specification  "$\text{TS 101 270-1}$",  which defines as modulation methods both  [[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|$\rm DMT$]]  $($"Discrete Multitone Transmission"$)$  and  [[Modulation_Methods/Quadrature_Amplitude_Modulation|$\rm QAM$]]  $($"Quadrature Amplitude Modulation"$)$.  The semiconductor manufacturers could not agree on a worldwide line code standard for a long time and there was even talk of the  "VDSL Line Code War".

In 2003,  at the so-called  "VDSL Olympics",  the decision was made in favor of DMT and against QAM or the slightly modified variant  [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Carrierless_Amplitude_Phase_Modulation|$\rm CAP$]]  $($"Carrierless Amplitude Phase Modulation"$)$,  namely
*because of the robustness of DMT against narrowband interference sources,

*although QAM/ CAP would allow a faster call setup.

==Exercises for the chapter ==
 
[[Exercise_2.1:_General_Description_of_xDSL|Exercise 2.1: General Description of xDSL]]

{{Display}}

Examples of Communication Systems/General Description of DSL

2023-03-21T18:38:03Z

Hwang:

{{Header
|Untermenü=DSL – Digital Subscriber Line
|Vorherige Seite=Further Developments of ISDN
|Nächste Seite=xDSL_Systems
}}

== # OVERVIEW OF THE SECOND MAIN CHAPTER # ==
 

$\rm D$igital $\rm S$ubscriber $\rm L$ine  – in short  $\rm DSL$ –  literally means only  "digital subscriber line".  At the same time,  "DSL"  was a synonym for  "high-speed Internet access in the local loop to the end customer",  although "high-speed"  must be put into perspective today  $(2018)$.

xDSL has been significantly standardized by the standards committees  [https://en.wikipedia.org/wiki/American_National_Standards_Institute $\rm ANSI$]  $($USA$)$  and  [https://en.wikipedia.org/wiki/ETSI $\rm ETSI$]  $($Europe$)$  as well as the  [https://en.wikipedia.org/wiki/International_Telecommunication_Union $\rm ITU$]  $($worldwide$)$.  Due to different pre-existing technical conditions and preferences of developers and operators,  a large variety of nationally different versions of nominally identical xDSL standards resulted.  In the following,  we will restrict ourselves primarily to the German xDSL versions.

This chapter contains in detail:

#An  »overview of the historical development and standardization«  of xDSL,
#the  »differences between ADSL and VDSL«  as well as statistics on their penetration,
#a brief description of xDSL from a  »communications protocol perspective«,
#the bandwidth allocations for the two  »xDSL variants ADSL and VDSL«, 
#a detailed description of the  »DSL transmission methods QAM, CAP and DMT«,
#the problems of  »digital signal transmission over copper twisted pairs«  in general,
#the relationship between  »SNR,  range and transmission rate«,
#the  »error correction measures«  used to reduce the bit error rate.

==Network infrastructure for DSL==
 

We start as in the  [[Examples_of_Communication_Systems/General_Description_of_ISDN#Network_infrastructure_for_ISDN|"ISDN chapter"]]  with the network infrastructure.  DSL was intended to use the existing analog telephone network for cost reasons. 

The greatest cost factor of the entire infrastructure is the  »'''subscriber line area'''«  between a main distribution frame  $($e.g. "switching office"$)$  and the subscribers.
[[File:EN_LZI_T_4_3_S2_neu.png| right|frame|Structure of the local loop area]]

*In Germany,  this so-called  »'''last mile'''«  is shorter than  $4$  kilometers on average,  and in urban areas  $90\%$  of the time it is even shorter than  $2.8$  kilometers.

*Due to the topological conditions,  the telephone network is increasingly branching out in a star configuration toward the end customer.

*In order to avoid having to lay a separate copper cable to the local exchange for each subscriber,  splitters have been installed in between and the lines bundled in correspondingly large cables.

The  »'''local loop area'''»  is therefore usually made up as follows:

#   The  "main cable"  with up to  $2000$  pairs between the local exchange  (or the switching office)  frame and a cable branch,
#  the  "branch cable"  between the cable branch and the final branch, with up to  $300$  pairs and a maximum length of 500 meters,  which is significantly shorter than a main cable,
#  the  "house connection cable"  between the terminal box and the network termination box at the subscriber with two pairs of wires.

==xDSL types and terms==
 

{{BlaueBox|TEXT=
$\text{Motivation for Digital Subscriber Line}$ 

$\rm DSL$  $($"Digital Subscriber Line"$)$  arose from the need,  '''to provide low cost high rate digital data access to the end user'''.  During the design process,  it was necessary to take into account:
*As explained in the last section,  the  "last mile"  is the largest cost factor in a communications network.

*Considerations to replace the estimated  130  million kilometers of copper twisted pairs in the local loop network with fiber optic lines  $($fiber-to-the-home,  $\rm FttH)$  have failed to date due to the enormous costs of the mostly underground laying work.

*A viable solution was to offer a broadband connection with somewhat lower data rates than in a fiber optic network by using the existing telephone line network and by cleverly combining different transmission techniques and coding methods.

*The telephone service – either analog or digital  $\rm (ISDN)$  – should be able to operate simultaneously on the same network.}}

Before we turn to the historical DSL development up to the current state,  the various types of  "$\rm xDSL$"  must first be defined and some terms explained.  Note:
#Here, "$\rm x$"  is merely a placeholder that designates the various DSL standards.
#The technical features will be covered in depth in the next chapters.

'''Part of the xDSL standard:'''
*$\text{ADSL}$  –  "Asymmetric Digital Subscriber Line: Asymmetric data transmission technology with data rates of  $8$  Mbit/s to the subscriber  $($"downstream"$)$  and  $1$  Mbit/s in the opposite direction  $($"upstream"$)$.

*$\text{ADSL2}$  and  $\text{ADSL2+}$: Extensions of ADSL with data rates of up to  $25$  Mbit/s  $($"downstream"$)$  and up to  $1$  Mbit/s  $($"upstream"$)$. The data rate is dynamically negotiated depending on the channel state.

*$\text{Re – ADSL2}$: Another extension of ADSL with about  $30\%$  range gain at a data rate of  $768$  kbit/s downstream.

*$\text{HDSL}$  –  "High Data Rate Digital Subscriber Line": Symmetrical data transmission technology  –  i.e. equal rates in downstream and upstream  –  with data rates between  $1.54$  Mbit/s and  $2.04$  Mbit/s. Note:  The name  "HDSL"  suggests higher data rates than ADSL;  however,  this is not the case.

*$\text{SDSL}$  –  "Symmetric Digital Subscriber Line": Symmetrical data transmission at rates of up to  $3$  Mbit/s.  With four-wire wiring  $($two copper twisted pairs$)$,  a maximum of  $4$  Mbit/s can be transmitted.  Alternatively,  the range can be increased at the expense of bandwidth.

*$\text{VDSL}$  –  "Very High Data Rate Digital Subscriber Line": A newer transmission technology based on QAM that operates in the asymmetrical variant with bit rates of  $25$  to  $50$  Mbit/s  downstream and  $5$ to  $10$  Mbit/s upstream.  The symmetrical variant has the same data transmission rates in upstream and downstream.

*$\text{VDSL2}$  –  "Very High Data Rate Digital Subscriber Line 2": Transmission technology with the currently (2009) highest total data rate of up to  $200$  Mbit/s.  The process is based on DMT  $($"Discrete Multitone Transmission"$)$.

*$\text{UDSL}$   or   $\text{UADSL}$    –  "Universal (Asymmetric) Digital Subscriber Line".

'''Non part of the xDSL standard:''' '''KORREKTUR: not part?'''
#There are also many products circulating under  "DSL"  that are not part of the xDSL standard.
#They are often only intended to make it clear that fast data access is involved. 

These include:
*$\text{cableDSL}$:   Brand name of the German company TELES AG, which offers high-speed Internet access via cable. The name was chosen for marketing reasons only.

*$\text{skyDSL}$:  Brand name for Internet access available throughout Europe via satellite with up to  $24$  Mbit/s downstream. The upstream here is via POTS  $($"Plain old telephone service"$)$  or  [[Examples_of_Communication_Systems/General_Description_of_ISDN|ISDN ]]  $($"Integrated Services Digital Network"$)$.

*$\text{T-DSL via satellite}$:   Brand name for a downstream Internet access of Telekom via satellite;  uses a conventional modem or an ISDN connection for transmission.

*$\text{WDSL}$  –  "Wireless Digital Subscriber Line":  Brand name of a German company that uses wireless technology to enable data rates of up to  $108$  Mbit/s in DSL-free areas.

*$\text{mvoxSatellit}$:   Brand name of an Internet access with  "WiMAX-like radio technology",  which like WDSL and PortableDSL is only an auxiliary construct for DSL-free areas.

==Historical development of xDSL standardizations ==
 

The need for digital subscriber lines to improve line utilization and increase customer convenience was recognized as early as the 1970s.  After the ISDN specification in the early 1980s,  the actual development of DSL began.

*This development was influenced by the findings of many groups located around the world.  Accordingly,  the standardization proceeded in an unstructured manner.

*From the list on the right,  it is clear that different committees around the world were in charge of the various standards.

*In the industry,  the technical realizations of the individual xDSL standards often deviated noticeably from the specification. 

*Some standards were started as projects even before the specification,  since the industry parties were also represented in the standardization committees.

The graph illustrates the relationships between milestones in the theoretical and practical design of transmission systems.

[[File:EN_Bei_2_1_s3_v4.png|right|frame|Milestones of the industrial xDSL development]]
$\text{Milestones of DSL development in short form:}$

'''1986'''   A first concept for  $\rm HDSL$  $($"'''H'''igh-bit-rate '''D'''igital '''S'''ubscriber '''L'''ine"$)$  is defined by AT&T,  Bell Laboratories and Bellcore.

'''1989'''   First HDSL prototypes appear;     ⇒     Bellcore meanwhile works on the conceptual definition of  $\rm ADSL$   $($"'''A'''symmetric '''D'''igital '''S'''ubscriber '''L'''ine"$)$.

'''1992'''   First publication of the  $\text{ANSI Technical Report E1T1/92-002R1}$: "High-bit-rate Digital Subscriber Line";    ⇒     The first ADSL prototypes appear.

'''1994'''   The  $\rm VDSL$  concept  $($"'''V'''ery-high-speed '''D'''igital '''S'''ubscriber '''L'''ine"$)$  is discussed for the first time.

'''1995'''   Publication of the  $\text{ETSI Technical Report ETR 152:}$  "High-bit-rate Digital Subscriber Line"  and  "Transmission Systems on Metallic Local Lines"; ⇒     Publication of  $\text{ADSL Standard ANSI T1.413}$:  $($"Asymmetric Digital Subscriber Line Metallic Interface"$)$;     ⇒     First field trials with ADSL in the USA .

'''1996'''   First publication of the  $\text{ETSI Technical Report ETR 328}$:  "Asymmetric Digital Subscriber Line"  and  "Transmission and Multiplexing".

'''04/1998''' First publication of  $\text{ETSI Technical Specification TS 101 270}$:  "Very-high-speed Digital Subscriber Line";     ⇒     Almost simultaneously, first publication of  $\text{ANSI Draft Technical Document T1E1.4/98-043R1}$:  "Very-high-speed Digital Subscriber Lines".

'''10/1998''' First publication of  $\text{ITU Recommendation G.991.1}$:  "High-bit-rate Digital Subscriber Line Transceivers";     ⇒     Almost simultaneously publication of  $\text{ETSI Technical Specification TS 101 135}$:  "High-bit-rate Digital Subscriber Line – Transmission Systems on Metallic Local Lines".

'''11/1998''' Publication of  $\text{ETSI Technical Specification TS 101 388 V1.1.1}$:  "Asymmetric Digital Subscriber Line – European Specific Requirements".

'''1999'''   In June,  publication of  $\text{ITU Recommendations G.992.1}$:  "Asymmetric Digital Subscriber Line – Transceivers"  and  $\text{G.992.2}$:  "Splitterless Asymmetric Digital Subscriber Line – Transceivers";     ⇒     On July 22, Deutsche Telekom AG offers ADSL in Germany for the first time  $\text{(T-DSL 768)}$.

'''2001'''   In February,  publication of  $\text{ITU Recommendation G.991.2}$:  "Single-pair High-speed Digital Subscriber Line Transceivers";     ⇒     In November,  publication of  $\text{ITU Recommendation G.993.1}$:  "Very-high-speed Digital Subscriber Line transceivers".

'''2002'''   First publications of  $\text{ITU Recommendations G.992.3}$:  "Asymmetric Digital Subscriber Line Transceivers 2"  $\rm (ADSL2)$  and  $\text{G.992.4}$:  "Splitterless ADSL2".

'''2003'''   First publication of  $\text{ITU Recommendation G.992.5}$:  "Asymmetric Digital Subscriber Line"  $\rm (ADSL)$  Transceivers"  and   "Extended-bandwidth ADSL2"  $\rm (ADSL2+)$.

'''02/2006'''   Publication of  $\text{ITU Recommendation G.993.2}$: "Very-high-speed Digital Subscriber Line Transceivers 2"  $\rm (VDSL2)$.

'''10/2006'''  Deutsche Telekom AG offers VDSL2 to end customers in selected cities for the first time.

== Development of ADSL and VDSL in Europe ==
 
From the above compilation it can be seen that the  »'''ADSL standardization'''«  was predominantly driven by  [https://en.wikipedia.org/wiki/American_National_Standards_Institute $\rm ANSI$]  $($"American National Standards Institute"$)$  and that  [https://en.wikipedia.org/wiki/ETSI $\rm ETSI$]  $($"European Telecommunications Standards Institute"$)$  followed shortly thereafter in each case:
#The first ADSL standard   "$\text{ANSI T1.413}$"   from 1995 was predominantly optimized for video-on-demand services,  which is also made clear by the ratio of the downstream and upstream data rates defined herein:  $1.5$  Mbit/s and  $16$  kbit/s,  $3$  Mbit/s and  $16$  kbit/s, and finally  $6$  Mbit/s and  $64$ kbit/s.
#The frequency range was originally defined in such a way that ADSL could only be used to operate an analog telephone on the access line.  ETSI published a technical report   "$\text{ETR 328}$"   in 1996 with only a few detailed changes and the possibility to transmit  $2048$  kbit/s.
#Since the second version of the ANSI standard also allowed only one additional analog telephone,  ETSI then defined an ADSL system that differed both in bit rates and in the possibility of using an ISDN basic access on the same twisted pair.
#The ANSI and ETSI standardization efforts of the previous years resulted in the  [https://en.wikipedia.org/wiki/International_Telecommunication_Union "ITU"] recommendation  "$\text{G.992.1}$"   in 1999,  which includes both standards and thus allows many options for the implementation.

However,  the many options led to major conceptual differences at the end of the 1990s  –  worldwide,  within Europe and also nationally  –  depending on the semiconductor manufacturer,  among other things.  Only a few systems,  modems,  and measuring devices interoperated with other manufacturers.

To counteract this proliferation,  The  "Deutsche Telekom AG"  passed the technical guideline  "$\text{1TR112}$"   at the end of 2001,  defining all the necessary interface parameters to ensure the interoperability of different manufacturer modems on the provider and customer side. 
*Due to Telekom's market power,  this became the quasi-standard for Germany.

*Furthermore,  only those ADSL variants were used in Germany that allowed simultaneous operation of ISDN at any time.  Thus,  when switching from POTS to ISDN,  it was not necessary to change the ADSL version as well.

The  »'''VDSL standardization'''«  relevant for Europe was decisively shaped by ETSI and often happened in parallel to the American activities.  Overall,  VDSL standardization proceeded in a more orderly fashion than ADSL.  The 3-step plan adopted by ETSI provided for:
*Stage 1:  Functional and electrical requirements for VDSL systems,

*Stage 2:  Transmission coding and access method requirements,

*Stage 3:  Interoperability requirements.

These efforts culminated in April 1998 in the publication of the ETSI Technical Specification  "$\text{TS 101 270-1}$",  which defines as modulation methods both  [[Modulation_Methods/Further_OFDM_Applications#A_brief_description_of_DSL_-_Digital_Subscriber_Line|$\rm DMT$]]  $($"Discrete Multitone Transmission"$)$  and  [[Modulation_Methods/Quadrature_Amplitude_Modulation|$\rm QAM$]]  $($"Quadrature Amplitude Modulation"$)$.  The semiconductor manufacturers could not agree on a worldwide line code standard for a long time and there was even talk of the  "VDSL Line Code War".

In 2003,  at the so-called  "VDSL Olympics",  the decision was made in favor of DMT and against QAM or the slightly modified variant  [[Examples_of_Communication_Systems/xDSL_as_Transmission_Technology#Carrierless_Amplitude_Phase_Modulation_.28CAP.29|$\rm CAP$]]  $($"Carrierless Amplitude Phase Modulation"$)$,  namely
*because of the robustness of DMT against narrowband interference sources,

*although QAM/ CAP would allow a faster call setup.

==Exercises for the chapter ==
 
[[Exercise_2.1:_General_Description_of_xDSL|Exercise 2.1: General Description of xDSL]]

{{Display}}

Applets:Principle of Pseudo-Ternary Coding

2023-03-21T18:12:50Z

Hwang:

{{LntAppletLinkEnDe|pseudoternarycodes_en|pseudoternarycodes}}

==Applet Description==
 
Das Applet behandelt die Eigenschaften der bekanntesten Pseudoternärcodes, nämlich:
#  Bipolarcode erster Ordnung bzw.  $\rm AMI$–Code  (von: ''Alternate Mark Inversion''),  gekennzeichnet durch die Parameter  $N_{\rm C} = 1, \ K_{\rm C} = +1$,
#  Duobinärcode,  $(\rm DUOB)$,  Codeparameter:  $N_{\rm C} = 1, \ K_{\rm C} = -1$,
#  Bipolarcode zweiter Ordnung  $(\rm BIP2)$,  Codeparameter:  $N_{\rm C} = 2, \ K_{\rm C} = +1$.

Am Eingang liegt die redundanzfreie binäre bipolare Quellensymbolfolge  $\langle \hspace{0.05cm}q_\nu \hspace{0.05cm}\rangle \ \in \{+1, -1\}$   ⇒   Rechtecksignal  $q(t)$  an.  Verdeutlicht wird die Generierung
*der binär–vorcodierten Folge  $\langle \hspace{0.05cm}b_\nu \hspace{0.05cm}\rangle \ \in \{+1, -1\}$,  dargestellt durch das ebenfalls redundanzfreie binäre bipolare Rechtecksignal  $b(t)$,
*der pseudoternären Codefolge  $\langle \hspace{0.05cm}c_\nu \hspace{0.05cm}\rangle \ \in \{+1,\ 0, -1\}$,  dargestellt durch das redundante ternäre bipolare Rechtecksignal  $c(t)$,
*das gleichermaßen redundante ternäre Sendesignal  $s(t)$, gekennzeichnet durch die Amplitudenkoeffizienten  $a_\nu $,  und den (Sende–) Grundimpuls  $g(t)$:
:$$s(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g( t - \nu \cdot T)\hspace{0.05cm}.$$

Der Grundimpuls  $g(t)$  –  im Applet  "Rechteck",  "Nyquist" und  "Wurzel–Nyquist"  –  bestimmt nicht nur die Form des Sendesignals, sondern auch den Verlauf
* der Autokorrelationsfunktion  $\rm (AKF)$  $\varphi_s (\tau)$  und
* des zugehörigen Leistungsdichtespektrums  $\rm (LDS)$  ${\it \Phi}_s (f)$.

Das Applet zeigt auch, dass das gesamte Leistungsdichtespektrum  ${\it \Phi}_s (f)$ aufgeteilt werden kann in den Anteil  ${\it \Phi}_a (f)$, der die statistischen Bindungen der Amplitudenkoeffizienten  $a_\nu$   berücksichtigt, und das Energiedichtespektrum $ {\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = |G(f)|^2 $, gekennzeichnet durch die Impulsform  $g(t)$.

''Anmerkung'':   Im Applet wird kein Unterschied zwischen den Codersymbolen  $c_\nu \in \{+1,\ 0, -1\}$  und den Amplitudenkoeffizienten  $a_\nu \in \{+1,\ 0, -1\}$  gemacht.  Dabei sollte nicht vergessen werden, dass die  $a_\nu$  stets Zahlenwerte sind, während für die Codersymbole auch die Notation  $c_\nu \in \{\text{Plus},\ \text{Null},\ \text{Minus}\}$  zulässig wäre.

==Theoretical Background==

=== Allgemeine Beschreibung der Pseudoternärcodes ===

Bei der symbolweisen Codierung wird mit jedem ankommenden Quellensymbol  $q_\nu$  ein Codesymbol  $c_\nu$  erzeugt, das außer vom aktuellen Eingangssymbol  $q_\nu$  auch von den  $N_{\rm C}$  vorangegangenen Symbolen  $q_{\nu-1}$, ... , $q_{\nu-N_{\rm C}} $  abhängt.  $N_{\rm C}$  bezeichnet man als die ''Ordnung''  des Codes.

[[File:P_ID1343__Dig_T_2_4_S1_v1.png|right|frame|Blockschaltbild und Ersatzschaltbild eines Pseudoternärcodierers|class=fit]]

Typisch für eine symbolweise Codierung ist, dass
*die Symboldauer  $T$  des Codersignals (und des Sendesignals) mit der Bitdauer  $T_{\rm B}$  des binären Quellensignals übereinstimmt, und
*Codierung und Decodierung nicht zu größeren Zeitverzögerungen führen, die bei Verwendung von Blockcodes unvermeidbar sind. 

Besondere Bedeutung besitzen ''Pseudomehrstufencodes''  – besser bekannt unter der englischen Bezeichnung ''Partial Response Codes''. 

*Im Folgenden werden ausschließlich ''Pseudoternärcodes''   ⇒   Stufenzahl  $M = 3$  betrachtet, die durch das Blockschaltbild entsprechend der linken Grafik beschreibbar sind. 
*In der rechten Grafik ist ein Ersatzschaltbild angegeben, das für eine Analyse dieser Codes sehr gut geeignet ist.

Man erkennt aus den beiden Darstellungen:
*Der Pseudoternärcodierer kann in den nichtlinearen Vorcodierer und ein lineares Codiernetzwerk aufgespalten werden, wenn man – wie im rechten Ersatzschaltbild dargestellt – die Verzögerung um  $N_{\rm C} \cdot T$  und die Gewichtung mit  $K_{\rm C}$  zur Verdeutlichung zweimal zeichnet. 

*Der ''nichtlineare Vorcodierer''  gewinnt durch eine Modulo–2–Addition  ("Antivalenz")  zwischen den Symbolen  $q_\nu$  und  $K_{\rm C} \cdot b_{\nu-N_{\rm C}} $  die vorcodierten Symbole  $b_\nu$, die ebenfalls binär sind:
:$$q_\nu \in \{-1, +1\},\hspace{0.1cm} K_{\rm C} \in \{-1,
+1\}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}b_\nu \in \{-1,
+1\}\hspace{0.05cm}.$$
*Die Symbole  $b_\nu$  sind wie die Quellensymbole  $q_\nu$  statistisch voneinander unabhängig.  Der Vorcodierer fügt also keine Redundanz hinzu.  Er gestattet aber eine einfachere Realisierung des Decoders und verhindert eine Fehlerfortpflanzung nach einem Übertragungsfehler. 

*Die eigentliche Umcodierung von binär  $(M_q = 2)$  auf ternär  $(M = M_c = 3)$  bewirkt das ''lineare Codiernetzwerk''  durch die herkömmliche Subtraktion
:$$c(t) ={1}/{2} \cdot \big [b(t) - K_{\rm C} \cdot b(t- N_{\rm
C}\cdot T)\big] \in \{-1, \ 0, +1\}\hspace{0.05cm},$$

:das durch folgende  [[Lineare_zeitinvariante_Systeme/Systembeschreibung_im_Zeitbereich#Impulsantwort|Impulsantwort]]  bzw.  [[Lineare_zeitinvariante_Systeme/Systembeschreibung_im_Frequenzbereich#.C3.9Cbertragungsfunktion_-_Frequenzgang|Übertragungsfunktion]]  bezüglich dem Eingangssignal  $b(t)$  und dem Ausgangssignal  $c(t)$  beschrieben werden kann:
:$$h_{\rm C}(t) = {1}/{2} \cdot \big [\delta(t) - K_{\rm C} \cdot \delta(t- N_{\rm
C}\cdot T)\big] \ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ H_{\rm C}(f) ={1}/{2} \cdot \left [1 - K_{\rm C} \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}N_{\rm C}\hspace{0.05cm}\cdot \hspace{0.05cm}T}\right]\hspace{0.05cm}. $$

*Die relative Coderedundanz ist für alle Pseudoternärcodes gleich. Setzt man in die  [[Digitalsignalübertragung/Redundanzfreie_Codierung#Blockweise_Codierung_vs._symbolweise_Codierung|allgemeine Definitionsgleichung]]  $M_q=2$,  $M_c=3$  und  $T_c =T_q$  ein, so erhält man
:$$r_c = 1- \frac{R_q}{R_c} = 1- \frac{T_c}{T_q} \cdot \frac{{\rm log_2}\hspace{0.05cm} (M_q)}{{\rm log_2} \hspace{0.05cm}(M_c)} = 1- \frac{T_c}{T_q \cdot {\rm log_2} \hspace{0.05cm}(M_c)}\hspace{0.5cm}\Rightarrow \hspace{0.5cm} r_c = 1 -1/\log_2\hspace{0.05cm}(3) \approx 36.9 \%\hspace{0.05cm}.$$

{{BlaueBox|TEXT=
Das  $\text{Sendesignal aller Pseudoternärcodes}$  wird im Folgenden stets wie folgt dargestellt:
:$$s(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g( t - \nu \cdot T)\hspace{0.05cm}.$$
*Die Eigenschaft des aktuellen Pseudoternärcodes spiegelt sich in den statistischen Bindungen zwischen den  $a_\nu$  wider.  In allen Fällen gilt  $a_\nu \in \{-1, \ 0, +1\}$.
*Der Sendegrundimpuls  $g(t)$  stellt zum einen die erforderliche Energie bereit, hat aber auch Einfluss auf die statistischen Bindungen innerhalb des Signals.
*Im Programm ausgewählt werden kann neben dem NRZ–Rechteckimpuls  $g_{\rm R}(t)$: 
:*der Nyquistimpuls   ⇒   Impulsantwort des Cosinus–Rolloff–Tiefpasses mit Rolloff–Faktor $r$:
:$$g_{\rm Nyq}(t)={\rm const.} \cdot \frac{\cos(\pi \cdot r\cdot t/T)}{1-(2\cdot r\cdot t/T)^2} \cdot {\rm si}(\pi \cdot t/T) \ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ G_{\rm Nyq}(f),$$

:*der Wurzel–Nyquistimpuls   ⇒   Impulsantwort des Wurzel–Cosinus–Rolloff–Tiefpasses mit Rolloff–Faktor $r$:
:$$g_{\sqrt{\rm Nyq} }(t)\ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ G_{\sqrt{\rm Nyq} }(f)={\rm const.} \cdot \sqrt{G_{\rm Nyq}(f)} .$$ }}
 

=== Eigenschaften des AMI-Codes===

Die Pseudoternärcodes unterscheiden sich in den Parametern  $N_{\rm C}$  und  $K_{\rm C}$.  Der bekannteste Vertreter ist der  '''Bipolarcode erster Ordnung'''  mit den Codeparametern  $N_{\rm C} = 1$  und  $K_{\rm C} = 1$, der auch unter der Bezeichnung  '''AMI–Code'''  (von: ''Alternate Mark Inversion'') bekannt ist.

[[File:P_ID1346__Dig_T_2_4_S2a_v1.png|right|frame|AMI– und HDB3–Codierung, jeweils dargestellt mit Rechtecksignalen|class=fit]]
Dieser wird zum Beispiel bei  [[Beispiele_von_Nachrichtensystemen/Allgemeine_Beschreibung_von_ISDN|ISDN]]  (''Integrated Services Digital Networks'') auf der so genannten  $S_0$–Schnittstelle eingesetzt.

Die Grafik zeigt oben das binäre Quellensignal  $q(t)$. Im zweiten und dritten Diagramm sind dargestellt:
* das ebenfalls binäre Signal  $b(t)$  nach dem Vorcodierer, und
* das Codersignal  $c(t) = s(t)$  des AMI–Codes.

Man erkennt das einfache AMI–Codierprinzip:
*Jeder Binärwert "-1" von  $q(t)$   ⇒   Symbol  $\rm L$  wird durch den ternären Koeffizienten  $a_\nu = 0$  codiert. 
*Der Binärwert "+1" von  $q(t)$   ⇒   Symbol  $\rm H$  wird alternierend mit  $a_\nu = +1$  und  $a_\nu = -1$  dargestellt. 

Damit wird sichergestellt, dass im AMI–codierten Signal keine langen "+1"– und auch keine keine langen "–1"–Sequenzen enthalten sind, was bei einem gleichsignalfreien Kanal zu Problemen führen würde.

Dagegen ist das Auftreten langer Nullfolgen durchaus möglich, bei denen über einen längeren Zeitraum keine Taktinformation übertragen wird.

Um dieses zweite Problem zu vermeiden, wurden einige modifizierte AMI–Codes entwickelt, zum Beispiel der ''B6ZS–Code''  und der ''HDB3''–Code:
*Beim '''HDB3–Code'''  (grüne Kurve in obiger Grafik) werden vier aufeinanderfolgende Nullen im AMI–codierten Signal durch eine Teilsequenz ersetzt, die die AMI–Codierregel verletzt. 

*Im grau hinterlegten Bereich ist dies die Folge "+ 0 0 +", da das letzte Symbol vor der Ersetzung ein "Minus" war. 

*Damit ist beim HDB3–Code die Anzahl aufeinanderfolgender Nullen auf  $3$  begrenzt und beim  [https://www.itwissen.info/B6ZS-bipolar-with-six-zero-substitution-B6ZS-Codierung.html B6ZS–Code]  auf  $5$.
*Der Decoder erkennt diese Codeverletzung und ersetzt "+ 0 0 +" wieder durch "0 0 0 0". 
 
=== Zur AKF–Berechnung eines Digitalsignals ===
In der Versuchsdurchführung werden einige Größen und Zusamenhänge verwendet, die hier kurz eräutert werden sollen:

*  Das (zeitlich unbegrenzte) Digitalsignal beinhaltet sowohl die Quellenstatistik $($Amplitudenkoeffizienten  $a_\nu$)  als auch die Sendeimpulsform  $g(t)$:
:$$s(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g ( t - \nu \cdot T)\hspace{0.05cm}.$$

*  Ist  $s(t)$  die Musterfunktion eines stationären und ergodischen Zufallsprozesses, so gilt für die  [[Stochastische_Signaltheorie/Autokorrelationsfunktion_(AKF)#Zufallsprozesse_.281.29|Autokorrelationsfunktion]]  $\rm (AKF)$:
:$$\varphi_s(\tau) = {\rm E}\big [s(t) \cdot s(t + \tau)\big ] = \sum_{\lambda = -\infty}^{+\infty}{1}/{T}
\cdot \varphi_a(\lambda)\cdot\varphi^{^{\bullet} }_{gs}(\tau -
\lambda \cdot T)\hspace{0.05cm}.$$

*  Diese Gleichung beschreibt die Faltung der diskreten AKF  $\varphi_a(\lambda) = {\rm E}\big [ a_\nu \cdot a_{\nu + \lambda}\big]$  der Amplitudenkoeffizienten mit der Energie–AKF des Grundimpulses:

:$$\varphi^{^{\bullet} }_{g}(\tau) =
\int_{-\infty}^{+\infty} g ( t ) \cdot g ( t +
\tau)\,{\rm d} t \hspace{0.05cm}.$$

*Der Punkt soll darauf hinweisen, dass  $\varphi^{^{\bullet} }_{g}(\tau)$  die Einheit einer Energie besitzt, während  $\varphi_s(\tau)$  eine Leistung angibt und  $\varphi_a(\lambda)$  dimensionslos ist.
 
=== Zur LDS-Berechnung eines Digitalsignals ===
Die Entsprechungsgröße zur AKF ist im Frequenzbereich das [[Stochastische_Signaltheorie/Leistungsdichtespektrum_(LDS)#Theorem_von_Wiener-Chintchine|Leistungsdichtespektrum]]  $\rm (LDS)$  ${\it \Phi}_s(f)$, das mit  $\varphi_s(\tau)$  über das Fourierintegral fest verknüpft ist: 
:$$\varphi_s(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}
{\it \Phi}_s(f) = \int_{-\infty}^{+\infty} \varphi_s(\tau) \cdot
{\rm e}^{- {\rm j}\hspace{0.05cm} 2 \pi f \hspace{0.02cm} \tau}
\,{\rm d} \tau \hspace{0.05cm}.$$
*Das Leistungsdichtespektrum  ${\it \Phi}_s(f)$  kann unter Berücksichtigung der Dimensionsbereinigung  $(1/T)$  als Produkt zweier Funktionen dargestellt werden:
:$${\it \Phi}_s(f) = {\it \Phi}_a(f) \cdot {1}/{T} \cdot
|G_s(f)|^2 \hspace{0.05cm}.$$
*Der erste Term  ${\it \Phi}_a(f)$  ist dimensionslos und beschreibt die spektrale Formung des Sendesignals durch die statistischen Bindungen der Quelle: 
:$$\varphi_a(\lambda) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}_a(f) = \sum_{\lambda =
-\infty}^{+\infty}\varphi_a(\lambda)\cdot {\rm e}^{- {\rm
j}\hspace{0.05cm} 2 \pi f \hspace{0.02cm} \lambda \hspace{0.02cm}T} =
\varphi_a(0) + 2 \cdot \sum_{\lambda =
1}^{\infty}\varphi_a(\lambda)\cdot\cos ( 2 \pi f
\lambda T) \hspace{0.05cm}.$$
*${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$  berücksichtigt die spektrale Formung durch  $g(t)$. Je schmaler dieser ist, desto breiter ist  $\vert G(f) \vert^2$  und um so größer ist damit der Bandbreitenbedarf:
:$$\varphi^{^{\hspace{0.05cm}\bullet}}_{g}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}
{\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = |G(f)|^2
\hspace{0.05cm}.$$
*Das Energiedichtespektrum ${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$  hat die Einheit  $\rm Ws/Hz$  und das Leistungsdichtespektrum  ${\it \Phi_{s}}(f)$  nach der Division durch den Symbolabstand  $T$  die Einheit  $\rm W/Hz$.

=== Leistungsdichtespektrum des AMI-Codes===

Der Frequenzgang des linearen Codiernetzwerks eines Pseudoternärcodes lautet allgemein:
:$$H_{\rm C}(f) = {1}/{2} \cdot \big [1 - K_{\rm C} \cdot {\rm
e}^{-{\rm j}\hspace{0.03cm}\cdot \hspace{0.03cm}
2\pi\hspace{0.03cm}\cdot \hspace{0.03cm}f \hspace{0.03cm}\cdot
\hspace{0.03cm} N_{\rm C}\hspace{0.03cm}\cdot \hspace{0.03cm}T}
\big] ={1}/{2} \cdot \big [1 - K \cdot {\rm
e}^{-{\rm j}\hspace{0.03cm}\cdot \hspace{0.03cm}
\alpha}
\big ]\hspace{0.05cm}.$$

Damit ergibt sich für das Leistungsdichtespektrum (LDS) der Amplitudenkoeffizienten  $(K$  und  $\alpha$  sind Abkürzungen entsprechend obiger Gleichung$)$:
:$$ {\it \Phi}_a(f) = | H_{\rm C}(f)|^2 = \frac{\big [1 - K \cos
(\alpha) + {\rm j}\cdot K \sin (\alpha) \big ] \big [1 - K \cos
(\alpha) - {\rm j}\cdot K \sin (\alpha) \big ] }{4} = \text{...} = {1}/{4} \cdot \big [2 - 2 \cdot K \cdot \cos
(\alpha) \big ] $$
[[File:P_ID1347__Dig_T_2_4_S2b_v2.png|right|frame|Leistungsdichtespektrum des AMI-Codes|class=fit]]
:$$ \Rightarrow \hspace{0.3cm}{\it \Phi}_a(f) = | H_{\rm C}(f)|^2 = {1}/{2} \cdot \big [1 - K_{\rm C} \cdot \cos
(2\pi f N_{\rm C} T)\big ]
\hspace{0.4cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.4cm}
\varphi_a(\lambda \cdot T)\hspace{0.05cm}.$$

Insbesondere erhält man für das Leistungsdichtespektrum (LDS) des AMI–Codes $(N_{\rm C} = K_{\rm C} = 1)$:
:$${\it \Phi}_a(f) = {1}/{2} \cdot \big [1 - \cos
(2\pi f T)\big ] = \sin^2
(\pi f T)\hspace{0.05cm}.$$

Die Grafik zeigt
*das LDS  ${\it \Phi}_a(f)$  der Amplitudenkoeffizienten (rote Kurve), und 
*das LDS  ${\it \Phi}_s(f)$  des gesamten Sendesignals (blau), gültig für NRZ–Rechteckimpulse. 

Man erkennt aus dieser Darstellung
*die Gleichsignalfreiheit des AMI–Codes, da  ${\it \Phi}_a(f = 0) = {\it \Phi}_s(f = 0) = 0$  ist, 
*die Leistung  $P_{\rm S} = s_0^2/2$  des AMI–codierten Sendesignals (Integral über  ${\it \Phi}_s(f)$  von  $- \infty$  bis  $+\infty$).

''Hinweise:''
*Das LDS von HDB3– und B6ZS–Code weicht von dem des AMI–Codes nur unwesentlich ab. 
*Die hier behandelte Thematik können Sie sich mit dem interaktiven Applet  [[Applets:Pseudoternaercodierung|Signale, AKF und LDS der Pseudoternärcodes]]  verdeutlichen.

=== Eigenschaften des Duobinärcodes ===

[[File:P_ID1348__Dig_T_2_4_S3b_v1.png|right|frame|Leistungsdichtespektrum des Duobinärcodes|right|class=fit]]
Der '''Duobinärcode''' ist durch die Codeparameter  $N_{\rm C} = 1$  und  $K_{\rm C} = -1$  festgelegt. Damit ergibt sich für das Leistungsdichtespektrum (LDS) der Amplitudenkoeffizienten bzw. für das LDS des Sendesignals:

:$${\it \Phi}_a(f) ={1}/{2} \cdot \big [1 + \cos
(2\pi f T)\big ] = \cos^2
(\pi f T)\hspace{0.05cm},$$
:$$ {\it \Phi}_s(f) = s_0^2 \cdot T \cdot \cos^2
(\pi f T)\cdot {\rm si}^2
(\pi f T)= s_0^2 \cdot T \cdot {\rm si}^2
(2 \pi f T) \hspace{0.05cm}.$$

Die Grafik zeigt das Leistungsdichtespektrum
*der Amplitudenkoeffizienten   ⇒   ${\it \Phi}_a(f)$  als rote Kurve, 
*des gesamten Sendsignals   ⇒   ${\it \Phi}_s(f)$  als blaue Kurve. 

In der zweiten Grafik sind die Signale  $q(t)$,  $b(t)$  und  $c(t) = s(t)$  skizziert. Wir verweisen hier wieder auf das Applet  [[Applets:Pseudoternaercodierung|Signale, AKF und LDS der Pseudoternärcodes]], das auch die Eigenschaften des Duobinärcodes verdeutlicht.
[[File:P_ID1349__Dig_T_2_4_S3a_v2.png|left|frame|Signale bei Duobinärcodierung|class=fit]]
 Aus diesen Darstellungen geht hervor:
*Beim Duobinärcode können beliebig viele Symbole mit gleicher Polarität ("+1" bzw. "–1") direkt aufeinanderfolgen.
*Deshalb gilt  ${\it \Phi}_a(f = 0)=1$  und  ${\it \Phi}_s(f = 0) = 1/2 \cdot s_0^2 \cdot T$. 
*Dagegen tritt beim Duobinärcode die alternierende Folge " ... , +1, –1, +1, –1, +1, ... " nicht auf, die hinsichtlich Impulsinterferenzen besonders störend ist.
*Deshalb gilt beim Duobinärcode:  ${\it \Phi}_s(f = 1/(2T) = 0$. 
*Das Leistungsdichtespektrum  ${\it \Phi}_s(f)$  des pseudoternären Duobinärcodes ist identisch mit dem LDS bei redundanzfreier Binärcodierung mit halber Rate $($Symboldauer  $2T)$. 
 

==Exercises==

 
*First, select the number  $(1,\ 2, \text{...} \ )$  of the task to be processed.  The number  "$0$"  corresponds to a "Reset":  Same setting as at program start.
*A task description is displayed.  The parameter values are adjusted.  Solution after pressing "Show Solution".

{{BlueBox|TEXT=
'''(1)'''  Consider and interpret the binary pre–coding of the  $\text{AMI}$  code using the source symbol sequence  $\rm C$  assuming  $b_0 = +1$. }}
*The modulo–2 addition can also be taken as "antivalence".  It holds  $b_{\nu} = +1$,  if  $q_{\nu}$  and  $b_{\nu - 1}$  differ, otherwise set  $b_{\nu} = -1$ :
:  $b_1 = (q_1 = +1)\ {\rm XOR}\ (b_0= +1) = -1,\ \ b_2 = (q_2 = -1)\ {\rm XOR}\ (b_1= -1) = -1,\ \ b_3 = (q_3 = -1)\ {\rm XOR}\ (b_2= -1) = -1,$
:  $b_4 = (q_4 = +1)\ {\rm XOR}\ (b_3= -1) = +1,\ \ b_5 = (q_5 = +1)\ {\rm XOR}\ (b_4= +1) = -1,\ \ b_6 = (q_6 = +1)\ {\rm XOR}\ (b_5= -1) = +1,\ \ b_7 = b_8 = \text{...} = -1.$
*With the initial condition  $b_0 = -1$  we get the negated sequence:  $b_4 = b_6 =-1$.  All others  $b_\nu = +1$.

{{BlueBox|TEXT=
'''(2)'''  Let  $b_0 = +1$.  Consider the AMI encoded sequence  $\langle c_\nu \rangle$  of the source symbol sequence  $\rm C$  and give their amplitude coefficients  $a_\nu$.  }}

*It holds:  $a_1= 0.5 \cdot (b_1-b_0) = -1$,  $a_2= 0.5 \cdot (b_2-b_1) =0$,  $a_3= 0. 5 \cdot (b_3-b_2) =0$,  $a_4= +1$,  $a_5= -1$,  $a_6= +1$,  $a_7= -1$,  $a_8= a_9 = \text{...} = 0$.
*In contrast to the pre–coding, the conventional addition (subtraction) is to be applied here and not the modulo–2 addition.

{{BlueBox|TEXT=
'''(3)'''  Now consider the AMI coding for several random sequences.  What rules can be derived from these experiments for the amplitude coefficients  $a_\nu$ ?}}

*Each binary value  "–1"  of   $q(t)$   ⇒   symbol  $\rm L$  is encoded by the ternary coefficient  $a_\nu = 0$.  Any number of  $a_\nu = 0$  can be consecutive.
*The binary value  "+1"  of   $q(t)$   ⇒   symbol  $\rm H$  is represented alternatively with   $a_\nu = +1$  and  $a_\nu = -1$ , starting with  $a_\nu = -1$,  if  $b_0 = +1$.
*From the source symbol sequence  $\rm A$   ⇒   $\langle \hspace{0.05cm}q_\nu \equiv +1 \hspace{0.05cm}\rangle$  the code symbol sequence  $+1, -1, +1, -1, \text{...}$ . Long sequences  $\langle \hspace{0.05cm}a_\nu \equiv +1 \hspace{0.05cm}\rangle$  or   $\langle \hspace{0.05cm}a_\nu \equiv -1 \hspace{0.05cm}\rangle$  are shot out.

{{BlueBox|TEXT=
'''(4)'''  Continue with  AMI  coding.  Interpret the autocorrelation function  $\varphi_a(\lambda)$  of the amplitude coefficients and the power density spectrum  ${\it \Phi}_a(f)$. }}
*The discrete ACF  $\varphi_a(\lambda)$  of the amplitude coefficients is only defined for integer  $\lambda$  values.   With AMI coding  $(N_{\rm C}=1)$  holds:   For  $|\lambda| > 1$   ⇒   all  $\varphi_a(\lambda)= 0$.
*$\varphi_a(\lambda = 0)$  is equal to the root mean square of the amplitude coefficients   ⇒   $\varphi_a(\lambda = 0) = {\rm Pr}(a_\nu = +1) \cdot (+1)^2 + {\rm Pr}(a_\nu = -1) \cdot (-1)^2 = 0.5.$
*Only the combinations  $(+1, -1)$  and  $(-1, +1)$  contribute to the expected value  ${\rm E}\big [a_\nu \cdot a_{\nu+1}\big]$   Result:  $\varphi_a(\lambda = \pm 1)={\rm E}\big [a_\nu \cdot a_{\nu+1}\big]=-0.25.$
*The power density spectrum  ${\it \Phi}_a(f)$  is the Fourier transform of the discrete ACF  $\varphi_a(\lambda)$.  Result:  ${\it \Phi}_a(f) = {1}/{2} \cdot \big [1 - \cos (2\pi f T)\big ] = \sin^2
(\pi f T)\hspace{0.05cm}.$
*From   ${\it \Phi}_a(f = 0) = 0$  follows:   The AMI code is especially interesting for channels over which no DC component can be transmitted.

{{BlueBox|TEXT=
'''(5)'''  We consider further AMI coding and rectangular pulses.  Interpret the ACF  $\varphi_s(\tau)$  of the transmission signal and the PDS  ${\it \Phi}_s(f)$. }}
*$\varphi_s(\tau)$  results from the convolution of the discrete AKF  $\varphi_a(\lambda)$  with  $\varphi^{^{\hspace{0.05cm}\bullet}}_{g}(\tau)$.   For rectangular pulses  $($duration $T)$:  The energy–AKF  $\varphi^{^{\hspace{0.05cm}\bullet}}_{g}(\tau)$  is a triangle of duration  $2T$.
*It holds  $\varphi_s(\tau = 0)= \varphi_a(\lambda = 0) =0.5, \ \varphi_s(\pm T)= \varphi_a( 1) =-0. 25,\ , \ \varphi_s( \pm 2T)= \varphi_a(2) =0.$  Between these discrete values, $\varphi_{s}(\tau)$  is always linear.
*The PDS  ${\it \Phi}_s(f)$  is obtained from  ${\it \Phi}_a(f) = \sin^2(\pi f T)$  by multiplying with  ${\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = {\rm sinc}^2(f T).$  This does not change anything at the zeros of  ${\it \Phi}_a(f)$ .

{{BlueBox|TEXT=
'''(6)'''  What changes with respect to  $s(t)$,  $\varphi_s(\tau)$  and  ${\it \Phi}_s(f)$  with the  $\rm Nyquist$  pulse?  Vary the roll–off factor here in the range  $0 \le r \le 1$.}}

*A single Nyquist pulse can be represented with the source symbol sequence  $\rm B$  in the  $s(t)$  range.  You can see equidistant zero crossings in the distance  $T$.
*Also, for any AMI random sequence, the signal values  $s(t=\nu \cdot T)$  for each  $r$  correspond exactly to their nominal positions.  Outside these points, there are deviations.
*In the special case  $r=0$  the energy–LDS  ${\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f)$  is constant in the range  $|f|<1/2T$.  Accordingly, the energy–ACF  ${\it \varphi}^{^{\hspace{0.08cm}\bullet}}_{g}(\tau)$  has a  $\rm sinc$  shape.
*On the other hand, for larger  $r$  the zeros of  ${\it \varphi}^{^{\hspace{0.08cm}\bullet}}_{g}(\tau)$  are no longer equidistant, since although  $G(f)$  satisfies the first Nyquist criterion, it does not  ${\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f)= [G(f)]^2$.
*The main advantage of the Nyquist pulse is the much smaller bandwidth.  Here only the frequency range  $|f| < (1+r)/(2T)$  has to be provided.

{{BlueBox|TEXT=
'''(7)'''  Repeat the last experiment using the  $\text{Root raised cosine}$  pulse instead of the Nyquist pulse.  Interpret the results. }}

*In the special case  $r=0$  the results are as in  '''(6)'''. ${\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f)$  is constant in the range  $|f|<1/2T$  and outside zero;  ${\it \varphi}^{^{\hspace{0.08cm}\bullet}}_{g}(\tau)$  has a  $\rm sinc$  shape.
*Also for larger  $r$  the zeros of  ${\it \varphi}^{^{\hspace{0.08cm}\bullet}}_{g}(\tau)$  are eqidistant  (but not  $\rm sinc$  shaped)   ⇒   ${\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f)= [G(f)]^2$  satisfies the first Nyquist criterion.
*On the other hand, $G(f)$  does not satisfy the first Nyquist criterion  $($except for  $r=0)$.  Intersymbol interference occurs already at the transmitter   ⇒   signal  $s(t)$.
*But this is also not a fundamental problem.  By using an identically shaped reception filter like  $G(f)$  intersymbol interference at the decider is avoided.

{{BlueBox|TEXT=
'''(8)'''  Consider and check the pre–coding  $(b_\nu)$  and the amplitude coefficients  $(a_\nu)$  with the  $\rm duobinary$  coding   $($source symbol sequence  $\rm C$,  $b_0 = +1)$. }}
*$b_1 = (q_1 = +1)\ {\rm XOR}\ (\overline{b_0}= -1) = +1,\ \ b_2 = (q_2 = -1)\ {\rm XOR}\ (\overline{b_1}= -1) = -1,\ \ b_3 = \text{...} =b_7 = +1,$  $b_8 = b_{10} = \text{...} =-1$,  $b_9 =b_{11} = \text{...}= +1$.
*$a_1= 0.5 \cdot (b_1+b_0) = +1$,  $a_2= 0.5 \cdot (b_2+b_1) =0$,  $a_3= 0.5 \cdot (b_3+b_2) = 0$,  $a_4= \text{...}= a_7=+1$,  $a_8=a_9= \text{...}= 0$.
*With the starting condition  $b_0 = -1$  we get again the negated sequence:     $a_1= -1$,  $a_2= a_3= 0$,  $a_4= \text{...}= a_7=-1$,  $a_8=a_9= \text{...}= 0$.

{{BlueBox|TEXT=
'''(9)'''  Now consider the Duobinary coding for several random sequences.  What rules can be derived from these experiments for the amplitude coefficients  $a_\nu$?}}

*The discrete ACF values are  $\varphi_a(\lambda = 0) = +0.5$,  $\varphi_a(\lambda = 1) = +0.25$,  $\varphi_a(\lambda = 2) = 0$   ⇒   ${\it \Phi}_a(f) = {1}/{2} \cdot \big [1 + \cos (2\pi f T)\big ] = \cos^2
(\pi f T)\hspace{0.05cm}.$
*Unlike the AMI coding, the encoded sequences  $\langle \hspace{0.05cm}a_\nu \equiv +1 \hspace{0.05cm}\rangle$  and   $\langle \hspace{0.05cm}a_\nu \equiv -1 \hspace{0.05cm}\rangle$  are possible here   ⇒   For the Duobinary code holds  ${\it \Phi}_a(f= 0) = 1 \ (\ne 0).$
*As with the AMI code, the "long zero sequence"   ⇒   $\langle \hspace{0.05cm}a_\nu \equiv 0 \hspace{0.05cm}\rangle$  is possible, which can again lead to synchronization problems.
*Excluded are the combinations  $a_\nu = +1, \ a_{\nu+1} = -1$  and   $a_\nu = -1, \ a_{\nu+1} = +1$,  recognizable by the PDS value  ${\it \Phi}_a(f= 1/(2T)) = 0.$
*Such direct transitions  $a_\nu = +1$   ⇒   $a_{\nu+1} = -1$  resp.   $a_\nu = -1$   ⇒   $a_{\nu+1} = +1$  lead to large intersymbol interference and thus to a higher error rate.

{{BlueBox|TEXT=
'''(10)'''  Compare the coding results of second order bipolar code  $\rm (BIP2)$  and first order bipolar code  $\rm (AMI)$  for different source symbol sequences. }}
*For a single rectangular pulse   ⇒   source symbol sequence  $\rm B$  both codes result in the same encoded sequence and the same encoded signal  $c(t)$   ⇒   also an isolated pulse.
*The "Permanent–One sequence"  $\rm A$  now results  $\langle c_\nu \rangle = \langle -1, -1, +1, +1, -1, -1, +1, +1, \text{. ..}\rangle $  instead of   $\langle c_\nu \rangle = \langle -1, +1, -1, +1, -1, +1, \text{...}\rangle $.
*The simple decoding algorithm of the AMI code  $($the ternary  $0$  becomes the binary  $-1$,  the ternary  $\pm 1$  becomes the binary  $+1)$  cannot be applied to  $\rm BIP2$ .

{{BlueBox|TEXT=
'''(11)'''  View and interpret the various ACF and LDS graphs of the  $\rm BIP2$  compared to the  $\rm AMI$  code.}}
*For  $\rm AMI$:   $\varphi_a(\lambda = \pm 1) = -0.25, \ \varphi_a(\lambda = \pm 2) = 0$.   For  $\rm BIP2$:   $\varphi_a(\lambda = \pm 1) = 0, \ \varphi_a(\lambda = \pm 2) = -0.25$.  In both cases:  $\varphi_a(\lambda = 0) = 0.5$.
*From the  $\rm AMI$ power density spectrum  ${\it \Phi}_a(f) = \sin^2 (\pi \cdot f T)$  follows for  $\rm BIP2$:  ${\it \Phi}_a(f) = \sin^2 (2\pi \cdot f T)$  by compression with respect to the  $f$–axis.
*Zero at  $f=0$:  At most two  $+1$  directly follow each other, and also at most only two  $-1$.  In the AMI code  $+1$  and  $-1$  occur only in isolation.
*Next zero at  $f=1/(2T)$:  The infinitely long  $(+1, -1)$  sequence is excluded in  $\rm BIP2$  as in the  $\rm Duobinary$  code.
*Consider and interpret also the functions  $\varphi_s(\tau)$  and  ${\it \Phi}_s(f)$  for the pulses "rectangle", "Nyquist" and "Root raised cosine".

==Applet Manual==
[[File:BS_Pseudoternär.png|right|600px|frame|Bildschirmabzug (deutsche Version, heller Hintergrund)]]
 
    '''(A)'''     Theme (veränderbare grafische Oberflächengestaltung)
:* Dark:   schwarzer Hintergrund  (wird von den Autoren empfohlen)
:* Bright:   weißer Hintergrund  (empfohlen für Beamer und Ausdrucke)
:* Deuteranopia:   für Nutzer mit ausgeprägter Grün–Sehschwäche
:* Protanopia:   für Nutzer mit ausgeprägter Rot–Sehschwäche

    '''(B)'''     Zugrundeliegendes Blockschaltbild

    '''(C)'''     Auswahl des Pseudoternörcodes:
                    AMI–Code, Duobinärcode, Bipolarcode 2. Ordnung

    '''(D)'''     Auswahl des Grundimpulses  $g(t)$:
                    Rechteckimpuls, Nyquistimpuls, Wurzel–Nyquistimpuls

    '''(E)'''     Rolloff–Faktor (Frequenzbereich) für "Nyquist" und "Wurzel–Nyquist"

    '''(F)'''     Einstellung von  $3 \cdot 4 = 12$  Bit der Quellensymbolfolge

    '''(G)'''     Auswahl einer drei voreingestellten Quellensymbolfolgen

    '''(H)'''     Zufällige binäre Quellensymbolfolge

    '''( I )'''     Schrittweise Verdeutlichung der Pseudoternärcodierung

    '''(J)'''     Ergebnis der Pseudoternärcodierung:  Signale  $q(t)$,  $b(t)$,  $c(t)$,  $s(t)$

    '''(K)'''     Löschen der Signalverläufe im Grafikbereich  $\rm M$

    '''(L)'''     Skizzen für Autokorrelationsfunktion & Leistungsdichtespektrum

    '''(M)'''     Grafikbereich:  Quellensignal  $q(t)$, Signal  $b(t)$  nach Vorcodierung,                    Codersignal  $c(t)$  mit Rechtecken, Sendesignal  $s(t)$  gemäß  $g(t)$

    '''(N)'''     Bereich für Übungen:  Aufgabenauswahl, Fragen, Musterlösung
 
==About the Authors==
This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
*The first version was created in 2010 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Stefan_M.C3.BCller_.28Diplomarbeit_LB_2010.29|Stefan Müller]]    as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2020 the program was redesigned by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  as part of her bachelor thesis  (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) via "HTML5".
*Last revision and English version 2021 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  in the context of a working student activity. 

The conversion of this applet to HTML 5 was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ "Studienzuschüsse"]  (Faculty EI of the TU Munich).  We thank.
 

==Once again: Open Applet in new Tab==

{{LntAppletLinkEnDe|pseudoternarycodes_en|pseudoternarycodes}}

Applets:Sampling of Analog Signals and Signal Reconstruction

2023-03-21T18:12:33Z

Hwang:

{{LntAppletLinkEnDe|sampling_en|sampling}}

== Applet Description==
 
The applet deals with the system components  "sampling"  and  "signal reconstruction", two components that are of great importance for understanding the  [[Modulation_Methods/Pulscodemodulation|Pulscodemodulation]]  $({\rm PCM})$  for example.   The upper graphic shows the model on which this applet is based.  Below it are the samples  $x(\nu \cdot T_{\rm A})$  of the time continuous signal  $x(t)$. The (infinite) sum over all these samples is called the sampled signal  $x_{\rm A}(t)$.

[[File:EN_Abtastung_1.png|center|frame|Top:    Underlying model for sampling and signal reconstruction Bottom:   Example for time discretization of the continuous–time signal  $x(t)$]]

*At the transmitter, the time discrete (sampled) signal  $x_{\rm A}(t)$  is obtained from the continuous–time signal  $x(t)$.  This process is called  '''sampling'''   or  '''A/D conversion'''.
*The corresponding program parameter for the transmitter is the sampling rate  $f_{\rm A}= 1/T_{\rm A}$.  The lower graphic shows the sampling distance  $T_{\rm A}$ .
*In the receiver, the discrete-time received signal  $y_{\rm A}(t)$  is used to generate the continuous-time sink signal  $y(t)$    ⇒   '''signal reconstruction'''  or  '''D/A conversion'''  corresponding to the receiver frequency response  $H_{\rm E}(f)$.

The applet does not consider the PCM blocks  "Quantization"and  "encoding/decoding".   The digital transmission channel is assumed to be ideal. 

[[File:Abtastung_2_neu.png|right|frame|Receiver frequency response  $H_{\rm E}(f)$]]

The following consequences result from this:
*In the program simplifying  $y_{\rm A}(t) = x_{\rm A}(t)$  is set.
* With suitable system parameters, the error signal   $\varepsilon(t) = y(t)-x(t)\equiv 0$  is therefore also possible.

The sampling theorem and the signal reconstruction can be better explained in the frequency domain.  Therefore all spectral functions are displayed in the program;

             $X(f)\ \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\,\ x(t)$,  $X_{\rm A}(f)\ \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\,\ x_{\rm A}(t)$,  $Y(f)\ \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\,\ y(t)$,  $E(f)\ \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\,\ \varepsilon(t).$ 

Parameters for the receiver frequency response  $H_{\rm E}(f)$  are the cut–off frequency and the rolloff factor  (see lower graph):
:$$f_{\rm G} = \frac{f_2 +f_1}{2},\hspace{1cm}r = \frac{f_2 -f_1}{f_2 +f_1}.$$

''Notes:''

'''(1)'''   All signal values are normalized to  $\pm 1$.

'''(2)'''   The power calculation is done by integration over the respective period duration  $T_0$:
:$$P_x = \frac{1}{T_0} \cdot \int_0^{T_0} x^2(t)\ {\rm d}t,\hspace{0.8cm}P_\varepsilon = \frac{1}{T_0} \cdot \int_0^{T_0} \varepsilon^2(t).$$

'''(3)'''   The signal power  $P_x$  and the distortion power  $P_\varepsilon$  are also output in normalized form, which implicitly assumes the reference resistance  $R = 1\, \rm \Omega$ ;

'''(4)'''   From these the signal–distortion–distance  $10 \cdot \lg \ (P_x/P_\varepsilon)$  can be calculated.

'''(5)'''   Does the spectral function  $X(f)$  for positive frequencies consists of  $I$  Diraclines with the (possibly complex) weights  $X_1$, ... , $X_I$,           so applies to the transmission power taking into account the mirror-image lines at the negative frequencies:

:$$P_x = 2 \cdot \sum_{i=1}^I |X_k|^2.$$

'''(6)'''   Correspondingly, the following applies to the distortion power if the spectral function  $E(f)$  in the range  $f>0$  has  $J$  Diraclines with weights  $E_1$, ... , $E_J$:

:$$P_\varepsilon = 2 \cdot \sum_{j=1}^J |E_j|^2.$$

==Theoretical Background==

===Description of sampling in the time domain===

[[File:P_ID1120__Sig_T_5_1_S1_neu.png|center|frame|Zur Zeitdiskretisierung des zeitkontinuierlichen Signals  $x(t)$]]

Im Folgenden verwenden wir für die Beschreibung der Abtastung folgende Nomenklatur:
*Das zeitkontinuierliche Signal sei  $x(t)$.
*Das in äquidistanten Abständen  $T_{\rm A}$  abgetastete zeitdiskretisierte Signal sei  $x_{\rm A}(t)$.
*Außerhalb der Abtastzeitpunkte  $\nu \cdot T_{\rm A}$  gilt stets  $x_{\rm A}(t) \equiv 0$.
*Die Laufvariable  $\nu$  sei  [[Signal_Representation/Zum_Rechnen_mit_komplexen_Zahlen#Reelle_Zahlenmengen|ganzzahlig]]:     $\nu \in \mathbb{Z} = \{\hspace{0.05cm} \text{...}\hspace{0.05cm} , –3, –2, –1, \hspace{0.2cm}0, +1, +2, +3, \text{...} \hspace{0.05cm}\} $.
*Dagegen ergibt sich zu den äquidistanten Abtastzeitpunkten mit der Konstanten  $K$:

:$$x_{\rm A}(\nu \cdot T_{\rm A}) = K \cdot x(\nu \cdot T_{\rm A})\hspace{0.05cm}.$$

Die Konstante hängt von der Art der Zeitdiskretisierung ab. Für die obige Skizze gilt  $K = 1$.
 
===Description of sampling with Dirac pulse (Ist das richtig?)===

Im Folgenden gehen wir von einer geringfügig anderen Beschreibungsform aus.  Die folgenden Seiten werden zeigen, dass diese gewöhnungsbedürftigen Gleichungen durchaus zu sinnvollen Ergebnissen führen, wenn man sie konsequent anwendet.

{{BlaueBox|TEXT=
$\text{Definitionen:}$ 

* Unter  '''Abtastung'''  verstehen wir hier die Multiplikation des zeitkontinuierlichen Signals  $x(t)$  mit einem  '''Diracpuls''':

:$$x_{\rm A}(t) = x(t) \cdot p_{\delta}(t)\hspace{0.05cm}.$$

*Der  '''Diracpuls (im Zeitbereich)'''  besteht aus unendlich vielen Diracimpulsen, jeweils im gleichen Abstand  $T_{\rm A}$  und alle mit gleichem Impulsgewicht  $T_{\rm A}$:

:$$p_{\delta}(t) = \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot
\delta(t- \nu \cdot T_{\rm A}
)\hspace{0.05cm}.$$}}

Aufgrund dieser Definition ergeben sich für das abgetastete Signal folgende Eigenschaften:
:$$x_{\rm A}(t) = \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot x(\nu \cdot T_{\rm A})\cdot
\delta (t- \nu \cdot T_{\rm A}
)\hspace{0.05cm}.$$

*Das abgetastete Signal zum betrachteten Zeitpunkt  $(\nu \cdot T_{\rm A})$  ist gleich  $T_{\rm A} \cdot x(\nu \cdot T_{\rm A}) · \delta (0)$.
*Da  $\delta (t)$  zur Zeit  $t = 0$  unendlich ist, sind eigentlich alle Signalwerte  $x_{\rm A}(\nu \cdot T_{\rm A})$  ebenfalls unendlich groß und auch der oben eingeführte Faktor  $K$.
*Zwei Abtastwerte  $x_{\rm A}(\nu_1 \cdot T_{\rm A})$  und  $x_{\rm A}(\nu_2 \cdot T_{\rm A})$  unterscheiden sich jedoch im gleichen Verhältnis wie die Signalwerte  $x(\nu_1 \cdot T_{\rm A})$  und  $x(\nu_2 \cdot T_{\rm A})$.
*Die Abtastwerte von  $x(t)$  erscheinen in den Impulsgewichten der Diracfunktionen:
*Die zusätzliche Multiplikation mit  $T_{\rm A}$  ist erforderlich, damit  $x(t)$  und  $x_{\rm A}(t)$  gleiche Einheit besitzen.  Beachten Sie hierbei, dass  $\delta (t)$  selbst die Einheit „1/s” aufweist.

===Description of sampling in the frequency domain===

Zum Spektrum des abgetasteten Signals  $x_{\rm A}(t)$  kommt man durch Anwendung des  [[Signal_Representation/The_Convolution_Theorem_and_Operation#Faltung_im_Frequenzbereich|Faltungssatzes]]. Dieser besagt, dass der Multiplikation im Zeitbereich die Faltung im Spektralbereich entspricht:

:$$x_{\rm A}(t) = x(t) \cdot p_{\delta}(t)\hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm}
X_{\rm A}(f) = X(f) \star P_{\delta}(f)\hspace{0.05cm}.$$

Entwickelt man den  Diracpuls  $p_{\delta}(t)$   (im Zeitbereich)   in eine  [[Signal_Representation/Fourierreihe|Fourierreihe]]  und transformiert diese unter Anwendung des  [[Signal_Representation/Fourier_Transform_Laws#Verschiebungssatz|Verschiebungssatzes]]  in den Frequenzbereich, so ergibt sich mit dem Abstand  $f_{\rm A} = 1/T_{\rm A}$  zweier benachbarter Diraclinien im Frequenzbereich folgende Korrespondenz   ⇒   [[Signal_Representation/Time_Discrete_Signal_Representation#Diracpuls_im_Zeit-_und_im_Frequenzbereich|Beweis]]:

:$$p_{\delta}(t) = \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot
\delta(t- \nu \cdot T_{\rm A}
)\hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} P_{\delta}(f) = \sum_{\mu = - \infty }^{+\infty} \delta
(f- \mu \cdot f_{\rm A} ).$$

[[File:P_ID1121__Sig_T_5_1_S3_NEU.png|right|frame|Diracpuls im Zeit- und Frequenzbereich mit  $T_{\rm A} = 50\ {\rm µs}$  und  $f_{\rm A} = 1/T_{\rm A} = 20\ \text{kHz}$]]
Das Ergebnis besagt:
*Der Diracpuls  $p_{\delta}(t)$  im Zeitbereich besteht aus unendlich vielen Diracimpulsen, jeweils im gleichen Abstand  $T_{\rm A}$  und alle mit gleichem Impulsgewicht  $T_{\rm A}$.
*Die Fouriertransformierte von  $p_{\delta}(t)$  ergibt wiederum einen Diracpuls, aber nun im Frequenzbereich   ⇒   $P_{\delta}(f)$.
*Auch  $P_{\delta}(f)$  besteht aus unendlich vielen Diracimpulsen, nun im jeweiligen Abstand  $f_{\rm A} = 1/T_{\rm A}$  und alle mit dem Impulsgewicht  $1$.
*Die Abstände der Diraclinien in Zeit– und Frequenzbereich folgen demnach dem  [[Signal_Representation/Fourier_Transform_Laws#Reziprozit.C3.A4tsgesetz_von_Zeitdauer_und_Bandbreite|Reziprozitätsgesetz]]:   $T_{\rm A} \cdot f_{\rm A} = 1 \hspace{0.05cm}.$

Daraus folgt:   Aus dem Spektrum  $X(f)$  wird durch Faltung mit der um  $\mu \cdot f_{\rm A}$  verschobenen Diraclinie:

:$$X(f) \star \delta
(f- \mu \cdot f_{\rm A}
)= X (f- \mu \cdot f_{\rm A}
)\hspace{0.05cm}.$$

Wendet man dieses Ergebnis auf alle Diraclinien des Diracpulses an, so erhält man schließlich:

:$$X_{\rm A}(f) = X(f) \star \sum_{\mu = - \infty }^{+\infty} \delta
(f- \mu \cdot f_{\rm A}
) = \sum_{\mu = - \infty }^{+\infty} X (f- \mu \cdot f_{\rm A}
)\hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Fazit:}$  Die Abtastung des analogen Zeitsignals  $x(t)$  in äquidistanten Abständen  $T_{\rm A}$  führt im Spektralbereich zu einer  '''periodischen Fortsetzung'''  von  $X(f)$  mit dem Frequenzabstand  $f_{\rm A} = 1/T_{\rm A}$.}}

[[File:P_ID1122__Sig_T_5_1_S4_neu.png|right|frame|Spektrum des abgetasteten Signals]]
{{GraueBox|TEXT=
$\text{Beispiel 1:}$ 
Die obere Grafik zeigt  '''(schematisch!)'''  das Spektrum  $X(f)$  eines Analogsignals  $x(t)$, das Frequenzen bis  $5 \text{ kHz}$  beinhaltet.

Tastet man das Signal mit der Abtastrate  $f_{\rm A}\,\text{ = 20 kHz}$, also im jeweiligen Abstand  $T_{\rm A}\, = {\rm 50 \, µs}$  ab, so erhält man das unten skizzierte periodische Spektrum  $X_{\rm A}(f)$.
*Da die Diracfunktionen unendlich schmal sind, beinhaltet das abgetastete Signal  $x_{\rm A}(t)$  auch beliebig hochfrequente Anteile.
*Dementsprechend ist die Spektralfunktion  $X_{\rm A}(f)$  des abgetasteten Signals bis ins Unendliche ausgedehnt.}}

===Signal reconstruction===

[[File:P_ID1123__Sig_T_5_1_S5a_neu.png|right|frame|Gemeinsames Modell von "Signalabtastung" und "Signalrekonstruktion"]]
Die Signalabtastung ist bei einem digitalen Übertragungssystem kein Selbstzweck, sondern sie muss irgendwann wieder rückgängig gemacht werden.  Betrachten wir zum Beispiel das folgende System:
*Das Analogsignal  $x(t)$  mit der Bandbreite  $B_{\rm NF}$  wird wie oben beschrieben abgetastet.
*Am Ausgang eines idealen Übertragungssystems liegt das ebenfalls zeitdiskrete Signal  $y_{\rm A}(t) = x_{\rm A}(t)$  vor.
*Die Frage ist nun, wie der Block   '''Signalrekonstruktion'''   zu gestalten ist, damit auch  $y(t) = x(t)$  gilt.

[[File:P_ID1124__Sig_T_5_1_S5b_neu.png|right|frame|Frequenzbereichsdarstellung der "Signalrekonstruktion"]]
 Die Lösung ist einfach, wenn man die Spektralfunktionen betrachtet:  

Man erhält aus  $Y_{\rm A}(f)$  das Spektrum  $Y(f) = X(f)$  durch ein Tiefpass Filter mit dem  [[Linear_and_Time_Invariant_Systems/Systembeschreibung_im_Frequenzbereich#.C3.9Cbertragungsfunktion_-_Frequenzgang|Frequenzgang]]  $H_{\rm E}(f)$, der 

*die tiefen Frequenzen unverfälscht durchlässt:
:$$H_{\rm E}(f) = 1 \hspace{0.3cm}{\rm{f\ddot{u}r}} \hspace{0.3cm} |f| \le B_{\rm
NF}\hspace{0.05cm},$$
*die hohen Frequenzen vollständig unterdrückt:
:$$H_{\rm E}(f) = 0 \hspace{0.3cm}{\rm{f\ddot{u}r}} \hspace{0.3cm} |f| \ge f_{\rm A} - B_{\rm
NF}\hspace{0.05cm}.$$

Weiter ist aus der nebenstehenden Grafik zu erkennen:   Solange die beiden oben genannten Bedingungen erfüllt sind, kann  $H_{\rm E}(f)$  im Bereich von  $B_{\rm NF}$  bis  $f_{\rm A}–B_{\rm NF}$  beliebig geformt sein kann,
*beispielsweise linear abfallend (gestrichelter Verlauf)
*oder auch rechteckförmig,

===The Sampling Theorem===

Die vollständige Rekonstruktion des Analogsignals  $y(t)$  aus dem abgetasteten Signal  $y_{\rm A}(t) = x_{\rm A}(t)$  ist nur möglich, wenn die Abtastrate  $f_{\rm A}$  entsprechend der Bandbreite  $B_{\rm NF}$  des Nachrichtensignals richtig gewählt wurde.

Aus der obigen Grafik erkennt man, dass folgende Bedingung erfüllt sein muss:   $f_{\rm A} - B_{\rm NF} > B_{\rm NF} \hspace{0.3cm}\Rightarrow \hspace{0.3cm}f_{\rm A} > 2 \cdot B_{\rm NF}\hspace{0.05cm}.$

{{BlaueBox|TEXT=
$\text{Abtasttheorem:}$  Besitzt ein Analogsignal  $x(t)$  nur Spektralanteile im Bereich  $\vert f \vert < B_{\rm NF}$, so kann dieses aus seinem abgetasteten Signal  $x_{\rm A}(t)$  nur dann vollständig rekonstruiert werden, wenn die Abtastrate hinreichend groß ist:
:$$f_{\rm A} ≥ 2 \cdot B_{\rm NF}.$$

Für den Abstand zweier Abtastwerte muss demnach gelten:

:$$T_{\rm A} \le \frac{1}{ 2 \cdot B_{\rm NF} }\hspace{0.05cm}.$$}}

Wird bei der Abtastung der größtmögliche Wert   ⇒   $T_{\rm A} = 1/(2B_{\rm NF})$  herangezogen,
*so muss zur Signalrekonstruktion des Analogsignals aus seinen Abtastwerten
*ein idealer, rechteckförmiger Tiefpass mit der Grenzfrequenz  $f_{\rm G} = f_{\rm A}/2 = 1/(2T_{\rm A})$  verwendet werden.

{{GraueBox|TEXT=
$\text{Beispiel 2:}$  Die Grafik zeigt oben das auf  $\pm\text{ 5 kHz}$  begrenzte Spektrum  $X(f)$  eines Analogsignals, unten das Spektrum  $X_{\rm A}(f)$  des im Abstand  $T_{\rm A} =\,\text{ 100 µs}$  abgetasteten Signals   ⇒   $f_{\rm A}=\,\text{ 10 kHz}$.
[[File:P_ID1125__Sig_T_5_1_S6_neu.png|right|frame|Abtasttheorem im Frequenzbereich]]
Zusätzlich eingezeichnet ist der Frequenzgang  $H_{\rm E}(f)$  des tiefpassartigen Empfangsfilters zur Signalrekonstruktion, dessen Grenzfrequenz exakt  $f_{\rm G} = f_{\rm A}/2 = 5\,\text{ kHz}$  betragen muss.

*Mit jedem anderen  $f_{\rm G}$–Wert ergäbe sich  $Y(f) \neq X(f)$.
*Bei  $f_{\rm G} < 5\,\text{ kHz}$  fehlen die oberen  $X(f)$–Anteile.
* Bei  $f_{\rm G} > 5\,\text{ kHz}$  kommt es aufgrund von Faltungsprodukten zu unerwünschten Spektralanteilen in  $Y(f)$.

Wäre am Sender die Abtastung mit einer Abtastrate  $f_{\rm A} < 10\ \text{ kHz}$  erfolgt   ⇒   $T_{\rm A} >100 \ {\rm µ s}$, so wäre das Analogsignal  $y(t) = x(t)$  aus den Abtastwerten  $y_{\rm A}(t)$  auf keinen Fall rekonstruierbar.}}
 
==Exercises==

*First, select the number  $(1,\ 2, \text{...} \ )$  of the task to be processed.  The number  $0$  corresponds to a "Reset":  Same setting as at program start.
*A task description is displayed.  The parameter values are adjusted.  Solution after pressing "Show Solution".
*All signal values are to be understood as normalized to  $\pm 1$.  Powers are normalized values, too.

{{BlaueBox|TEXT=
'''(1)'''  Source signal:  $x(t) = A \cdot \cos (2\pi \cdot f_0 \cdot t -\varphi)$  with  $f_0 = \text{4 kHz}$.   Sampling with  $f_{\rm A} = \text{10 kHz}$.  Rectanglular low-pass;  cut-off frequency:  $f_{\rm G} = \text{5 kHz}$.             Interpret the shown graphics and evaluate the present signal reconstruction for all permitted parameter values of $A$  and $\varphi$. }}

* The spectrum  $X(f)$  consists of two Dirac functions at  $\pm \text{4 kHz}$, each with impulse weight  $0.5$.
* By the periodic continuation  $X_{\rm A}(f)$  has lines of equal height at  $\pm \text{4 kHz}$,  $\pm \text{6 kHz}$,  $\pm \text{14 kHz}$,  $\pm \text{16 kHz}$,  $\pm \text{24 kHz}$,  $\pm \text{26 kHz}$,  etc.
* The rectangular low-pass with the cut-off frequency  $f_{\rm G} = \text{5 kHz}$  removes all lines except the two at  $\pm \text{4 kHz}$  ⇒  $Y(f) =X(f)$  ⇒  $y(t) =x(t)$  ⇒   $P_\varepsilon = 0$.
* The signal reconstruction works here perfectly  $(P_\varepsilon = 0)$  for all amplitudes $A$  and any phases $\varphi$.

{{BlaueBox|TEXT=
'''(2)'''  Continue with  $A=1$,  $f_0 = \text{4 kHz}$,  $\varphi=0$,  $f_{\rm A} = \text{10 kHz}$,  $f_{\rm G} = \text{5 kHz}$.   What is the influence of the rolloff–factors  $r=0.2$,  $r=0.5$  and   $r=1$?           Specify the power values  $P_x$  and  $P_\varepsilon$ .   For which  $r$–values is  $P_\varepsilon= 0$?  Do these results also apply to other  $A$  and  $\varphi$? }}

:* With  $|X(f = \pm \text{4 kHz})|=0.5$  the signal power is  $P_x = 2\cdot 0.5^2 = 0.5$.  The distortion power  $P_\varepsilon$  depends significantly on the rolloff–factor  $r$ .
:* $P_\varepsilon$  is zero for  $r \le 0.2$.  Then the  $X_{\rm A}(f)$ line at  $f_0 = \text{4 kHz}$  is not changed by the low-pass and the unwanted  line at  $\text{6 kHz}$  is fully suppressed.
:* $r = 0.5$ :  $Y(f = \text{4 kHz}) = 0.35$,  $Y(f = \text{6 kHz}) = 0.15$  ⇒   $|E(f = \text{4 kHz})| = |E(f = \text{6 kHz})|= 0. 15$  ⇒  $P_\varepsilon = 0.09$  ⇒  $10 \cdot \lg \ (P_x/P_\varepsilon)=7.45\ \rm dB$.
:*$r = 1.0$ :  $Y(f = \text{4 kHz}) = 0.3$,  $Y(f = \text{6 kHz}) = 0.2$  ⇒   $|E(f = \text{4 kHz})| = |E(f = \text{6 kHz})|= 0. 2$  ⇒  $P_\varepsilon = 0.16$  ⇒  $10 \cdot \lg \ (P_x/P_\varepsilon)=4.95\ \rm dB$.
:* For all  $r$  the distortion power $P_\varepsilon$  is independent of  $\varphi$.   The amplitude  $A$  affects  $P_x$  and  $P_\varepsilon$  in the same way   ⇒   the quotient is independent of  $A$.

{{BlaueBox|TEXT=
'''(3)'''  Now apply  $A=1$,  $f_0 = \text{5 kHz}$,  $\varphi=0$,  $f_{\rm A} = \text{10 kHz}$,  $f_{\rm G} = \text{5 kHz}$,  $r=0$  $($rectangular low–pass$)$.   Interpret the result of the signal reconstruction.}}

:*  $X(f)$  consists of two Dirac delta lines at  $\pm \text{5 kHz}$  $($weight  $0.5)$.  By periodic continuation  $X_{\rm A}(f)$  has lines at  $\pm \text{5 kHz}$,  $\pm \text{15 kHz}$,  $\pm \text{25 kHz}$,  etc.
:*  The  rectanglular low-pass  removes the lines at  $\pm \text{15 kHz}$,  $\pm \text{25 kHz}$.  The lines at  $\pm \text{5 kHz}$  are halved because of  $H_{\rm E}(\pm f_{\rm G}) = H_{\rm E}(\pm \text{5 kHz}) = 0.5$.
:*   ⇒   $\text{Weights of }X(f = \pm \text{5 kHz})$:  $0.5$   #   $\text{Weights of }X(f_{\rm A} = \pm \text{5 kHz})$:  $1. 0$;     #   $\text{Weights of }Y(f = \pm \text{5 kHz})$:  $0.5$   ⇒   $Y(f)=X(f)$.
:* So the signal reconstruction works perfectly here too  $(P_\varepsilon = 0)$.  The same is true for the phase  $\varphi=180^\circ$   ⇒   $x(t) = -A \cdot \cos (2\pi \cdot f_0 \cdot t)$.

{{BlaueBox|TEXT=
'''(4)'''  The settings of  $(3)$  continue to apply except for  $\varphi=30^\circ$.  Interpret the differences from the setting  $(3)$   ⇒   $\varphi=0^\circ$.}}

:* Phase relations are lost.  The sink signal  $y(t)$  is cosine-shaped  $(\varphi_y=0^\circ)$  with by the factor  $\cos(\varphi_x)$  smaller amplitude than the source signal  $x(t)$.
:* Justification in the frequency domain:  In the periodic continuation of  $X(f)$  ⇒  $X_{\rm A}(f)$  only the real parts are to be added.  The imaginary parts cancel out.
:* The Dirac delta line of  $X(f)$  at frequency  $f_0$   ⇒   $X(f_0)$  is complex,   $Y(f_0)$  is real, and  $E(f_0)$  is imaginary   ⇒   $\varepsilon(t)$  is minus–sinusoidal   ⇒   $P_\varepsilon = 0. 125$.

{{BlueBox|TEXT=
'''(5)'''  Illustrate again the result of  $(4)$  compared to the settings  $f_0 = \text{5 kHz}$,  $\varphi=30^\circ$,  $f_{\rm A} = \text{11 kHz}$,  $f_{\rm G} = \text{5.5 kHz}$.}}
:* With this setting, the spectrum  $X_{\rm A}(f)$  also has a positive imaginary part at  $\text{5 kHz}$  and a negative imaginary part of the same magnitude at  $\text{6 kHz}$.
:* The rectangular low-pass with cutoff frequency  $\text{5.5 kHz}$  removes this second component.  Thus, with the new setting  $Y(f) =X(f)$   ⇒   $P_\varepsilon = 0$.
:* Any $f_0$ oscillation of arbitrary phase is error-free reconstructible from its samples if  $f_{\rm A} = 2 \cdot f_{\rm 0} + \mu, \ f_{\rm G}= f_{\rm A}/2$  $($any small $\mu>0)$.
:* For value–continuous spectrum with   $X(|f|> f_0) \equiv 0$  ⇒   $\big[$no diraclines at $\pm f_0 \big ]$  the sampling rate  $f_{\rm A} = 2 \cdot f_{\rm 0}$  is sufficient in principle.

{{BlaueBox|TEXT=
'''(5)'''  Verdeutlichen Sie sich nochmals das Ergebnis von  '''(4)'''  im Vergleich zu den Einstellungen  $f_0 = \text{5 kHz}$,  $\varphi=30^\circ$,  $f_{\rm A} = \text{11 kHz}$,  $f_{\rm G} = \text{5.5 kHz}$.}}
:* Bei dieser Einstellung hat das  $X_{\rm A}(f)$–Spektrum auch einen positiven Imaginärteil bei  $\text{5 kHz}$  und einen negativen Imaginärteil gleicher Höhe bei  $\text{6 kHz}$.
:* Der Rechteck–Tiefpass mit der Grenzfrequenz  $\text{5.5 kHz}$  entfernt diesen zweiten Anteil.  Somit ist bei dieser Einstellung  $Y(f) =X(f)$   ⇒   $P_\varepsilon = 0$.
:* Jede  $f_0$–Schwingung beliebiger Phase ist fehlerfrei aus seinen Abtastwerten rekonstruierbar, falls  $f_{\rm A} = 2 \cdot f_{\rm 0} + \mu, \ f_{\rm G}= f_{\rm A}/2$  $($beliebig kleines $\mu>0)$.
:* Bei wertkontinuierlichem Spektrum mit   $X(|f|> f_0) \equiv 0$  ⇒   $\big[$keine Diraclinien bei $\pm f_0 \big ]$ genügt grundsätzlich die Abtastrate  $f_{\rm A} = 2 \cdot f_{\rm 0}$.

{{BlueBox|TEXT=
'''(6)'''  The settings of  $(3)$  and  $(4)$  continue to apply except for  $\varphi=90^\circ$.  Interpret the plots in the time and frequency domain.}}
:* The source signal is sampled exactly at its zero crossings   ⇒   $x_{\rm A}(t) \equiv 0$   ⇒     $y(t) \equiv 0$   ⇒   $\varepsilon(t)=-x(t)$   ⇒   $P_\varepsilon = P_x$   ⇒   $10 \cdot \lg \ (P_x/P_\varepsilon)=0\ \rm dB$.
:* Description in the frequency domain:   As in  $(4)$  the imaginary parts of  $X_{\rm A}(f)$  cancel out.  Also the real parts of  $X_{\rm A}(f)$  are zero because of the sinusoid.

{{BlueBox|TEXT=
'''(7)'''  Now consider the  $\text {Source Signal 2}$.  Let the other parameters be  $f_{\rm A} = \text{5 kHz}$,  $f_{\rm G} = \text{2.5 kHz}$,  $r=0$.  Interpret the results.}}
:* The source signal has spectral components up to  $\pm \text{2 kHz}$.  The signal power is $P_x = 2 \cdot \big[0.1^2 + 0.25^2+0.15^2\big]= 0.19 $. 
:* With the sampling rate  $f_{\rm A} = \text{5 kHz}$  and the receiver parameters  $f_{\rm G} = \text{2.5 kHz}$  and  $r=0$, the signal reconstruction works perfectly:  $P_\varepsilon = 0$.
:* Likewise with the trapezoidal low–pass with  $f_{\rm G} = \text{2.5 kHz}$, if for the rolloff factor holds:  $r \le 0.2$.

{{BlueBox|TEXT=
'''(8)'''  What happens if the cutoff frequency  $f_{\rm G} = \text{1.5 kHz}$  of the rectangular low–pass filter is too small?  In particular, interpret the error signal  $\varepsilon(t)=y(t)-x(t)$.}}
:* The error signal  $\varepsilon(t)=-0.3 \cdot \cos(2\pi \cdot \text{2 kHz} \cdot t -60^\circ)=0. 3 \cdot \cos(2\pi \cdot \text{2 kHz} \cdot t +120^\circ)$  is equal to the (negated) signal component at  $\text{2 kHz}$. 
:* The distortion power is  $P_\varepsilon=2 \cdot 0.15^2= 0.045$  and the signal–to–distortion ratio  $10 \cdot \lg \ (P_x/P_\varepsilon)=10 \cdot \lg \ (0.19/0.045)= 6.26\ \rm dB$.

{{BlueBox|TEXT=
'''(9)'''  What happens if the cutoff frequency  $f_{\rm G} = \text{3.5 kHz}$  of the rectangular low–pass filter is too large?  In particular, interpret the error signal  $\varepsilon(t)=y(t)-x(t)$.}}
:* The error signal  $\varepsilon(t)=0.3 \cdot \cos(2\pi \cdot \text{3 kHz} \cdot t +60^\circ)$  is now equal to the  $\text{3 kHz}$  portion of the sink signal  $y(t)$  not removed by the low-pass filter.
:* Compared to the subtask  $(8)$  the frequency changes from  $\text{2 kHz}$  to  $\text{3 kHz}$  and also the phase relationship.
:* The amplitude of this  $\text{3 kHz}$ error signal is equal to the amplitude of the  $\text{2 kHz}$ portion of  $x(t)$.  Again  $P_\varepsilon= 0.045$,  $10 \cdot \lg \ (P_x/P_\varepsilon)= 6.26\ \rm dB$.

{{BlueBox|TEXT=
'''(10)'''  Finally, we consider the  $\text {source signal 4}$  $($portions until  $\pm \text{4 kHz})$, as well as  $f_{\rm A} = \text{5 kHz}$,  $f_{\rm G} = \text{2. 5 kHz}$,  $0 \le r\le 1$.  Interpretation of results.}}
:* Up to  $r=0.2$  the signal reconstruction works perfectly  $(P_\varepsilon = 0)$.  If one increases  $r$, then  $P_\varepsilon$  increases continuously and  $10 \cdot \lg \ (P_x/P_\varepsilon)$  decreases.
:* With  $r=1$  the signal frequencies  $\text{0.5 kHz}$,  ...,  $\text{4 kHz}$  are attenuated, the more the higher the frequency is,  for example  $H_{\rm E}(f=\text{4 kHz}) = 0.6$.
:* Similarly,  $Y(f)$  also includes components at frequencies  $\text{6 kHz}$,  $\text{7 kHz}$,  $\text{8 kHz}$,  $\text{9 kHz}$  and  $\text{9.5 kHz}$ due to periodic continuation.
:* At the sampling times  $t\hspace{0.05cm}' = n \cdot T_{\rm A}$, the signals  $x(t\hspace{0.05cm}')$  and  $y(t\hspace{0.05cm}')$  agree exactly  ⇒   $\varepsilon(t\hspace{0.05cm}') = 0$.  In between, not  ⇒   small distortion power   $P_\varepsilon = 0.008$.

==Applet Manual==
 
[[File:Anleitung_Auge.png|right|600px]]
    '''(A)'''     Auswahl:   Codierung                    (binär,  quaternär,  AMI–Code,  Duobinärcode)

    '''(B)'''     Auswahl:   Detektionsgrundimpuls                     (nach Gauß–TP,  CRO–Nyquist,  nach Spalt–TP}

    '''(C)'''     Prametereingabe zu  '''(B)'''                    (Grenzfrequenz,  Rolloff–Faktor,  Rechteckdauer)

    '''(D)'''     Steuerung der Augendiagrammdarstellung                    (Start,  Pause/Weiter,  Einzelschritt,  Gesamt,  Reset)

    '''(E)'''     Geschwindigkeit der Augendiagrammdarstellung

    '''(F)'''     Darstellung:  Detektionsgrundimpuls  $g_d(t)$

    '''(G)'''     Darstellung:  Detektionsnutzsignal  $d_{\rm S}(t - \nu \cdot T)$

    '''(H)'''     Darstellung:  Augendiagramm im Bereich  $\pm T$

    '''( I )'''     Numerikausgabe:  $ö_{\rm norm}$  (normierte Augenöffnung)

    '''(J)'''     Prametereingabe  $10 \cdot \lg \ E_{\rm B}/N_0$  für  '''(K)'''

    '''(K)'''     Numerikausgabe:  $\sigma_{\rm norm}$  (normierter Rauscheffektivwert)

    '''(L)'''     Numerikausgabe:  $p_{\rm U}$  (ungünstigste Fehlerwahrscheinlichkeit)

    '''(M)'''     Bereich für die Versuchsdurchführung:   Aufgabenauswahl

    '''(N)'''     Bereich für die Versuchsdurchführung:   Aufgabenstellung

    '''(O)'''     Bereich für die Versuchsdurchführung:   Musterlösung einblenden
 
==About the Authors==
This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
*The first version was created in 2008 by [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]]  as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).

*Last revision and English version 2020/2021 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  in the context of a working student activity. 

The conversion of this applet to HTML 5 was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ Studienzuschüsse]  ("study grants")  of the TUM Faculty EI.  We thank.

==Once again:  Open Applet in new Tab==
{{LntAppletLinkEnDe|sampling_en|sampling}}

Applets:Capacity of Memoryless Digital Channels

2023-03-21T18:10:23Z

Hwang:

{{LntAppletLinkEnDe|transinformation_en|transinformation}}

==Applet Description==
 
In this applet,  binary  $(M=2)$  and ternary  $(M=3)$  channel models without memory are considered with  $M$  possible inputs  $(X)$  and  $M$  possible outputs  $(Y)$.  Such a channel is completely determined by the  "probability mass function"  $P_X(X)$  and the matrix  $P_{\hspace{0.01cm}Y\hspace{0.03cm} \vert \hspace{0.01cm}X}(Y\hspace{0.03cm} \vert \hspace{0.03cm} X)$  of the  "transition probabilities".

For these binary and ternary systems,  the following information-theoretic descriptive quantities are derived and clarified:
*the  "source entropy"   $H(X)$  and the  "sink entropy"   $H(Y)$,

*the  "equivocation"   $H(X|Y)$  and the   "irrelevance"   $H(Y|X)$,

*the  "joint entropy"   $H(XY)$  as well as the "mutual information"  $I(X; Y)$,

*the  "channel capacity"   as the decisive parameter of digital channel models without memory:
:$$C = \max_{P_X(X)} \hspace{0.15cm} I(X;Y) \hspace{0.05cm}.$$

These information-theoretical quantities can be calculated both in analytic–closed form or determined simulatively by evaluation of source and sink symbol sequence.

==Theoretical Background==
 
===Underlying model of digital signal transmission ===
 
The set of possible  »'''source symbols'''«  is characterized by the discrete random variable  $X$. 
*In the binary case   ⇒   $M_X= |X| = 2$  holds  $X = \{\hspace{0.05cm}{\rm A}, \hspace{0.15cm} {\rm B} \hspace{0.05cm}\}$  with the probability mass function   $($ $\rm PMF)$   $P_X(X)= \big (p_{\rm A},\hspace{0.15cm}p_{\rm B}\big)$  and the source symbol probabilities  $p_{\rm A}$  and  $p_{\rm B}=1- p_{\rm A}$.

*Accordingly, for a ternary source  ⇒   $M_X= |X| = 3$:     $X = \{\hspace{0.05cm}{\rm A}, \hspace{0.15cm} {\rm B}, \hspace{0.15cm} {\rm C} \hspace{0.05cm}\}$,     $P_X(X)= \big (p_{\rm A},\hspace{0.15cm}p_{\rm B},\hspace{0.15cm}p_{\rm C}\big)$,     $p_{\rm C}=1- p_{\rm A}-p_{\rm B}$.

The set of possible  »'''sink symbols'''«  is characterized by the discrete random variable  $Y$.  These come from the same symbol set as the source symbols   ⇒   $M_Y=M_X = M$.  To simplify the following description, we denote them with lowercase letters, for example, for  $M=3$:    $Y = \{\hspace{0.05cm}{\rm a}, \hspace{0.15cm} {\rm b}, \hspace{0.15cm} {\rm c} \hspace{0.05cm}\}$.

The relationship between the random variables  $X$  and  $Y$  is given by a   »'''discrete memoryless channel model'''«  $($ $\rm DMC)$.  The left graph shows this for  $M=2$  and the right graph for  $M=3$.

[[File:Transinf_1_neu.png|center|frame|  $M=2$  (left) and for  $M=3$  (right).     Please note:  In the right graph not all transitions are labeled]]

The following description applies to the simpler case  $M=2$.  For the calculation of all information theoretic quantities in the next section we need besides  $P_X(X)$  and  $P_Y(Y)$  the two-dimensional probability functions  $($each a  $2\times2$–matrix$)$  of all
#  [[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Conditional_probability|"conditional probabilities"]]   ⇒   $P_{\hspace{0.01cm}Y\hspace{0.03cm} \vert \hspace{0.01cm}X}(Y\hspace{0.03cm} \vert \hspace{0.03cm} X)$   ⇒   given by the DMC model;
#  [[Information_Theory/Some_Preliminary_Remarks_on_Two-Dimensional_Random_Variables#Joint_probability_and_joint_entropy|"joint probabilities"]]  ⇒   $P_{XY}(X,\hspace{0.1cm}Y)$;
#  [[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Inference_probability|"inference probabilities"]]   ⇒   $P_{\hspace{0.01cm}X\hspace{0.03cm} \vert \hspace{0.03cm}Y}(X\hspace{0.03cm} \vert \hspace{0.03cm} Y)$.

{{GraueBox|TEXT=
[[File:Transinf_2.png|right|frame|Considered model of the binary channel]]
$\text{Example 1}$:  We consider the sketched binary channel.
* Let the falsification probabilities be:

:$$\begin{align*}p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}A} & = {\rm Pr}(Y\hspace{-0.1cm} = {\rm a}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm A}) = 0.95\hspace{0.05cm},\hspace{0.8cm}p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}A} = {\rm Pr}(Y\hspace{-0.1cm} = {\rm b}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm A}) = 0.05\hspace{0.05cm},\\
p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}B} & = {\rm Pr}(Y\hspace{-0.1cm} = {\rm a}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm B}) = 0.40\hspace{0.05cm},\hspace{0.8cm}p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}B} = {\rm Pr}(Y\hspace{-0.1cm} = {\rm b}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm B}) = 0.60\end{align*}$$

:$$\Rightarrow \hspace{0.3cm} P_{\hspace{0.01cm}Y\hspace{0.05cm} \vert \hspace{0.05cm}X}(Y\hspace{0.05cm} \vert \hspace{0.05cm} X) =
\begin{pmatrix}
0.95 & 0.05\\
0.4 & 0.6
\end{pmatrix} \hspace{0.05cm}.$$

*Furthermore, we assume source symbols that are not equally probable:

:$$P_X(X) = \big ( p_{\rm A},\ p_{\rm B} \big )=
\big ( 0.1,\ 0.9 \big )
\hspace{0.05cm}.$$

*Thus, for the probability function of the sink we get:

:$$P_Y(Y) = \big [ {\rm Pr}( Y\hspace{-0.1cm} = {\rm a})\hspace{0.05cm}, \ {\rm Pr}( Y \hspace{-0.1cm}= {\rm b}) \big ] = \big ( 0.1\hspace{0.05cm},\ 0.9 \big ) \cdot
\begin{pmatrix}
0.95 & 0.05\\
0.4 & 0.6
\end{pmatrix} $$

:$$\Rightarrow \hspace{0.3cm} {\rm Pr}( Y \hspace{-0.1cm}= {\rm a}) =
0.1 \cdot 0.95 + 0.9 \cdot 0.4 = 0.455\hspace{0.05cm},\hspace{1.0cm}
{\rm Pr}( Y \hspace{-0.1cm}= {\rm b}) = 1 - {\rm Pr}( Y \hspace{-0.1cm}= {\rm a}) = 0.545.$$

*The joint probabilities  $p_{\mu \kappa} = \text{Pr}\big[(X = μ) ∩ (Y = κ)\big]$  between source and sink are:

:$$\begin{align*}p_{\rm Aa} & = p_{\rm a} \cdot p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}A} = 0.095\hspace{0.05cm},\hspace{0.5cm}p_{\rm Ab} = p_{\rm b} \cdot p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}A} = 0.005\hspace{0.05cm},\\
p_{\rm Ba} & = p_{\rm a} \cdot p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}B} = 0.360\hspace{0.05cm},
\hspace{0.5cm}p_{\rm Bb} = p_{\rm b} \cdot p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}B} = 0.540\hspace{0.05cm}.
\end{align*}$$

:$$\Rightarrow \hspace{0.3cm} P_{XY}(X,\hspace{0.1cm}Y) =
\begin{pmatrix}
0.095 & 0.005\\
0.36 & 0.54
\end{pmatrix} \hspace{0.05cm}.$$

* For the inference probabilities one obtains:

:$$\begin{align*}p_{\rm A\hspace{0.03cm}\vert \hspace{0.03cm}a} & = p_{\rm Aa}/p_{\rm a} = 0.095/0.455 = 0.2088\hspace{0.05cm},\hspace{0.5cm}p_{\rm A\hspace{0.03cm}\vert \hspace{0.03cm}b} = p_{\rm Ab}/p_{\rm b} = 0.005/0.545 = 0.0092\hspace{0.05cm},\\
p_{\rm B\hspace{0.03cm}\vert \hspace{0.03cm}a} & = p_{\rm Ba}/p_{\rm a} = 0.36/0.455 = 0.7912\hspace{0.05cm},\hspace{0.5cm}p_{\rm B\hspace{0.03cm}\vert \hspace{0.03cm}b} = p_{\rm Bb}/p_{\rm b} = 0.54/0.545 = 0.9908\hspace{0.05cm}
\end{align*}$$

:$$\Rightarrow \hspace{0.3cm} P_{\hspace{0.01cm}X\hspace{0.05cm} \vert \hspace{0.05cm}Y}(X\hspace{0.05cm} \vert \hspace{0.05cm} Y) =
\begin{pmatrix}
0.2088 & 0.0092\\
0.7912 & 0.9908
\end{pmatrix} \hspace{0.05cm}.$$ }}
 
===Definition and interpretation of various entropy functions ===
 
In the  [[Information_Theory/Verschiedene_Entropien_zweidimensionaler_Zufallsgrößen|"theory section"]],  all entropies relevant for two-dimensional random quantities are defined, which also apply to digital signal transmission.  In addition, you will find two diagrams illustrating the relationship between the individual entropies. 
*For digital signal transmission the right representation is appropriate, where the direction from source  $X$  to the sink  $Y$  is recognizable. 
*We now interpret the individual information-theoretical quantities on the basis of this diagram.

[[File:EN_Inf_T_3_3_S2_vers2.png|EN_Inf_T_4_2_S2.png|center|frame|Two information-theoretic models for digital signal transmission.
     Please note:  In the right graph  $H_{XY}$  cannot be represented]]

*The  »'''source entropy'''«  $H(X)$  denotes the average information content of the source symbol sequence.  With the symbol set size  $|X|$  applies:

:$$H(X) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{1}{P_X(X)}\right ] \hspace{0.1cm}
= -{\rm E} \big [ {\rm log}_2 \hspace{0.1cm}{P_X(X)}\big ] \hspace{0.2cm}
=\hspace{0.2cm} \sum_{\mu = 1}^{|X|}
P_X(x_{\mu}) \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{P_X(x_{\mu})} \hspace{0.05cm}.$$

*The  »'''equivocation'''«  $H(X|Y)$  indicates the average information content that an observer who knows exactly about the sink  $Y$  gains by observing the source  $X$ :

:$$H(X|Y) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}X\hspace{-0.01cm}|\hspace{-0.01cm}Y}(X\hspace{-0.01cm} |\hspace{0.03cm} Y)}\right ] \hspace{0.2cm}=\hspace{0.2cm} \sum_{\mu = 1}^{|X|} \sum_{\kappa = 1}^{|Y|}
P_{XY}(x_{\mu},\hspace{0.05cm}y_{\kappa}) \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}X\hspace{-0.01cm}|\hspace{0.03cm}Y}
(\hspace{0.05cm}x_{\mu}\hspace{0.03cm} |\hspace{0.05cm} y_{\kappa})}
\hspace{0.05cm}.$$

*The equivocation is the portion of the source entropy  $H(X)$  that is lost due to channel interference  (for digital channel: transmission errors).  The  »'''mutual information'''«  $I(X; Y)$  remains, which reaches the sink:

:$$I(X;Y) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{P_{XY}(X, Y)}{P_X(X) \cdot P_Y(Y)}\right ] \hspace{0.2cm}=\hspace{0.2cm} \sum_{\mu = 1}^{|X|} \sum_{\kappa = 1}^{|Y|}
P_{XY}(x_{\mu},\hspace{0.05cm}y_{\kappa}) \cdot {\rm log}_2 \hspace{0.1cm} \frac{P_{XY}(x_{\mu},\hspace{0.05cm}y_{\kappa})}{P_{\hspace{0.05cm}X}(\hspace{0.05cm}x_{\mu}) \cdot P_{\hspace{0.05cm}Y}(\hspace{0.05cm}y_{\kappa})}
\hspace{0.05cm} = H(X) - H(X|Y) \hspace{0.05cm}.$$

*The  »'''irrelevance'''«  $H(Y|X)$  indicates the average information content that an observer who knows exactly about the source  $X$  gains by observing the sink  $Y$:

:$$H(Y|X) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}Y\hspace{-0.01cm}|\hspace{-0.01cm}X}(Y\hspace{-0.01cm} |\hspace{0.03cm} X)}\right ] \hspace{0.2cm}=\hspace{0.2cm} \sum_{\mu = 1}^{|X|} \sum_{\kappa = 1}^{|Y|}
P_{XY}(x_{\mu},\hspace{0.05cm}y_{\kappa}) \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}Y\hspace{-0.01cm}|\hspace{0.03cm}X}
(\hspace{0.05cm}y_{\kappa}\hspace{0.03cm} |\hspace{0.05cm} x_{\mu})}
\hspace{0.05cm}.$$

*The  »'''sink entropy'''«  $H(Y)$, the mean information content of the sink.  $H(Y)$  is the sum of the useful mutual information  $I(X; Y)$  and the useless irrelevance  $H(Y|X)$, which comes exclusively from channel errors:

:$$H(Y) = {\rm E} \left [ {\rm log}_2 \hspace{0.1cm} \frac{1}{P_Y(Y)}\right ] \hspace{0.1cm}
= -{\rm E} \big [ {\rm log}_2 \hspace{0.1cm}{P_Y(Y)}\big ] \hspace{0.2cm} =I(X;Y) + H(Y|X)
\hspace{0.05cm}.$$

*The  »'''joint entropy'''«  $H(XY)$  is the average information content of the 2D random quantity  $XY$.  It also describes an upper bound for the sum of source entropy and sink entropy:

:$$H(XY) = {\rm E} \left [ {\rm log} \hspace{0.1cm} \frac{1}{P_{XY}(X, Y)}\right ] = \sum_{\mu = 1}^{|X|} \hspace{0.1cm} \sum_{\kappa = 1}^{|Y|} \hspace{0.1cm}
P_{XY}(x_{\mu}\hspace{0.05cm}, y_{\kappa}) \cdot {\rm log} \hspace{0.1cm} \frac{1}{P_{XY}(x_{\mu}\hspace{0.05cm}, y_{\kappa})}\le H(X) + H(Y) \hspace{0.05cm}.$$

{{GraueBox|TEXT=
[[File:Transinf_2.png|right|frame|Considered model of the binary channel]]
$\text{Example 2}$:  The same requirements as for  [[Applets:Capacity_of_Memoryless_Digital_Channels#Underlying_model_of_digital_signal_transmission|$\text{Example 1}$]] apply: 

'''(1)'''  The source symbols are not equally probable:
:$$P_X(X) = \big ( p_{\rm A},\ p_{\rm B} \big )=
\big ( 0.1,\ 0.9 \big )
\hspace{0.05cm}.$$
'''(2)'''  Let the falsification probabilities be:
:$$\begin{align*}p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}A} & = {\rm Pr}(Y\hspace{-0.1cm} = {\rm a}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm A}) = 0.95\hspace{0.05cm},\hspace{0.8cm}p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}A} = {\rm Pr}(Y\hspace{-0.1cm} = {\rm b}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm A}) = 0.05\hspace{0.05cm},\\
p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}B} & = {\rm Pr}(Y\hspace{-0.1cm} = {\rm a}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm B}) = 0.40\hspace{0.05cm},\hspace{0.8cm}p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}B} = {\rm Pr}(Y\hspace{-0.1cm} = {\rm b}\hspace{0.05cm}\vert X \hspace{-0.1cm}= {\rm B}) = 0.60\end{align*}$$

:$$\Rightarrow \hspace{0.3cm} P_{\hspace{0.01cm}Y\hspace{0.05cm} \vert \hspace{0.05cm}X}(Y\hspace{0.05cm} \vert \hspace{0.05cm} X) =
\begin{pmatrix}
0.95 & 0.05\\
0.4 & 0.6
\end{pmatrix} \hspace{0.05cm}.$$

[[File:Inf_T_1_1_S4_vers2.png|frame|Binary entropy function as a function of  $p$|right]]
*Because of condition  '''(1)'''  we obtain for the source entropy with the  [[Information_Theory/Discrete_Memoryless_Sources#Binary_entropy_function|"binary entropy function"]]  $H_{\rm bin}(p)$: 

:$$H(X) = p_{\rm A} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm A}\hspace{0.1cm} } + p_{\rm B} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{p_{\rm B} }= H_{\rm bin} (p_{\rm A}) = H_{\rm bin} (0.1)= 0.469 \ {\rm bit}
\hspace{0.05cm};$$

::$$H_{\rm bin} (p) = p \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p\hspace{0.1cm} } + (1 - p) \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{1 - p} \hspace{0.5cm}{\rm (unit\hspace{-0.15cm}: \hspace{0.15cm}bit\hspace{0.15cm}or\hspace{0.15cm}bit/symbol)}
\hspace{0.05cm}.$$

* Correspondingly, for the sink entropy with PMF  $P_Y(Y) = \big ( p_{\rm a},\ p_{\rm b} \big )=
\big ( 0.455,\ 0.545 \big )$:
:$$H(Y) = H_{\rm bin} (0.455)= 0.994 \ {\rm bit}
\hspace{0.05cm}.$$
*Next, we calculate the joint entropy:
:$$H(XY) = p_{\rm Aa} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm Aa}\hspace{0.1cm} }+ p_{\rm Ab} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm Ab}\hspace{0.1cm} }+p_{\rm Ba} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm Ba}\hspace{0.1cm} }+ p_{\rm Bb} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm Bb}\hspace{0.1cm} }$$
:$$\Rightarrow \hspace{0.3cm}H(XY) = 0.095 \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{0.095 } +0.005 \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{0.005 }+0.36 \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{0.36 }+0.54 \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{0.54 }= 1.371 \ {\rm bit}
\hspace{0.05cm}.$$

According to the upper left diagram, the remaining information-theoretic quantities are thus also computable:
[[File:Transinf_4.png|right|frame|Information-theoretic model for  $\text{Example 2}$]]

*the  »'''equivocation'''«:

:$$H(X \vert Y) \hspace{-0.01cm} =\hspace{-0.01cm} H(XY) \hspace{-0.01cm} -\hspace{-0.01cm} H(Y) \hspace{-0.01cm} = \hspace{-0.01cm} 1.371\hspace{-0.01cm} -\hspace{-0.01cm} 0.994\hspace{-0.01cm} =\hspace{-0.01cm} 0.377\ {\rm bit}
\hspace{0.05cm},$$

*the »'''irrelevance'''«:

:$$H(Y \vert X) = H(XY) - H(X) = 1.371 - 0.994 = 0.902\ {\rm bit}
\hspace{0.05cm}.$$

*the  »'''mutual information'''« :

:$$I(X;Y) = H(X) + H(Y) - H(XY) = 0.469 + 0.994 - 1.371 = 0.092\ {\rm bit}
\hspace{0.05cm},$$

The results are summarized in the graph.

Note:  Equivocation and irrelevance could also be computed (but with extra effort) directly from the corresponding probability functions, for example:

:$$H(Y \vert X) = \hspace{-0.2cm} \sum_{(x, y) \hspace{0.05cm}\in \hspace{0.05cm}XY} \hspace{-0.2cm} P_{XY}(x,\hspace{0.05cm}y) \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{P_{\hspace{0.05cm}Y\hspace{-0.01cm}\vert \hspace{0.03cm}X}
(\hspace{0.05cm}y\hspace{0.03cm} \vert \hspace{0.05cm} x)}= p_{\rm Aa} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}A} } +
p_{\rm Ab} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}A} } +
p_{\rm Ba} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}B} } +
p_{\rm Bb} \cdot {\rm log}_2 \hspace{0.1cm} \frac{1}{p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}B} } = 0.902 \ {\rm bit} \hspace{0.05cm}.$$}}

{{GraueBox|TEXT=
[[File:Transinf_3.png|right|frame|Considered model of the ternary channel:     Red transitions represent  $p_{\rm a\hspace{0.03cm}\vert \hspace{0.03cm}A} = p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}B} = p_{\rm c\hspace{0.03cm}\vert \hspace{0.03cm}C} = q$  and     blue ones for  $p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}A} = p_{\rm c\hspace{0.03cm}\vert \hspace{0.03cm}A} =\text{...}= p_{\rm b\hspace{0.03cm}\vert \hspace{0.03cm}C}= (1-q)/2$]]
$\text{Example 3}$:  Now we consider a transmission system with  $M_X = M_Y = M=3$. 

'''(1)'''  Let the source symbols be equally probable:
:$$P_X(X) = \big ( p_{\rm A},\ p_{\rm B},\ p_{\rm C} \big )=
\big ( 1/3,\ 1/3,\ 1/3 \big )\hspace{0.30cm}\Rightarrow\hspace{0.30cm}H(X)={\rm log_2}\hspace{0.1cm}3 \approx 1.585 \ {\rm bit}
\hspace{0.05cm}.$$
'''(2)'''  The channel model is symmetric   ⇒   the sink symbols are also equally probable:
:$$P_Y(Y) = \big ( p_{\rm a},\ p_{\rm b},\ p_{\rm c} \big )=
\big ( 1/3,\ 1/3,\ 1/3 \big )\hspace{0.30cm}\Rightarrow\hspace{0.30cm}H(Y)={\rm log_2}\hspace{0.1cm}3 \approx 1.585 \ {\rm bit}
\hspace{0.05cm}.$$
'''(3)'''  The joint probabilities are obtained as follows:
:$$p_{\rm Aa}= p_{\rm Bb}= p_{\rm Cc}= q/M,$$
:$$p_{\rm Ab}= p_{\rm Ac}= p_{\rm Ba}= p_{\rm Bc} = p_{\rm Ca}= p_{\rm Cb} = (1-q)/(2M)$$
:$$\Rightarrow\hspace{0.30cm}H(XY) = 3 \cdot p_{\rm Aa} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm Aa}\hspace{0.1cm} }+6 \cdot p_{\rm Ab} \cdot {\rm log_2}\hspace{0.1cm}\frac{1}{\hspace{0.1cm}p_{\rm Ab}\hspace{0.1cm} }= \
\text{...} \ = q \cdot {\rm log_2}\hspace{0.1cm}\frac{M}{q }+ (1-q) \cdot {\rm log_2}\hspace{0.1cm}\frac{M}{(1-q)/2 }.$$
[[File:Transinf_10.png|right|frame|Some results for  $\text{Example 3}$]]
'''(4)'''  For the mutual information we get after some transformations considering the equation 
:$$I(X;Y) = H(X) + H(Y) - H(XY)\text{:}$$
:$$\Rightarrow \hspace{0.3cm} I(X;Y) = {\rm log_2}\ (M) - (1-q) -H_{\rm bin}(q).$$
* For error-free ternary transfer  $(q=1)$  holds  $I(X;Y) = H(X) = H(Y)={\rm log_2}\hspace{0.1cm}3$.

* With  $q=0.8$  the mutual information already decreases to  $I(X;Y) = 0.663$,  with  $q=0.5$  to  $0.085$  bit.

*The worst case from the point of view of information theory is  $q=1/3$  ⇒   $I(X;Y) = 0$.

*On the other hand, the worst case from the point of view of transmission theory is  $q=0$  ⇒   "not a single transmission symbol arrives correctly"  is not so bad from the point of view of information theory.

* In order to be able to use this good result, however, channel coding is required at the transmitting end. }}
 
===Definition and meaning of channel capacity ===
 
If one calculates the mutual information  $I(X, Y)$  as explained in  $\text{Example 2}$,  then this depends not only on the discrete memoryless channel  $\rm (DMC)$,  but also on the source statistic   ⇒   $P_X(X)$.  Ergo:   '''The mutual information'''  $I(X, Y)$ ''' is not a pure channel characteristic'''.

{{BlaueBox|TEXT=
$\text{Definition:}$  The  »'''channel capacity'''«  introduced by  [https://en.wikipedia.org/wiki/Claude_Shannon $\text{Claude E. Shannon}$]  according to his standard work  [Sha48]<ref name = ''Sha48''>Shannon, C.E.:  A Mathematical Theory of Communication.[[File:Transinf_1_neu.png|center|frame|  $M=2$  (left) and for  $M=3$  (right).     Please note:  In the right graph not all transitions are labeled]] In: Bell Syst. Techn. J. 27 (1948), S. 379-423 und S. 623-656.</ref>:

:$$C = \max_{P_X(X)} \hspace{0.15cm} I(X;Y) \hspace{0.05cm}.$$

*The additional unit  "bit/use"  is often added.  Since according to this definition the best possible source statistics are always the basis:

:⇒   $C$  depends only on the channel properties   ⇒   $P_{Y \vert X}(Y \vert X)$   but not on the source statistics   ⇒   $P_X(X)$.  }}

Shannon needed the quantity  $C$  to formulate the  "Channel Coding Theorem"  –  one of the highlights of the information theory he founded.

{{BlaueBox|TEXT=
$\text{Shannon's Channel Coding Theorem:}$ 
*For every transmission channel with channel capacity  $C > 0$,  there exists  $($at least$)$  one  $(k,\ n)$  block code,  whose  $($block$)$  error probability approaches zero  as long as the code rate  $R = k/n$  is less than or equal to the channel capacity:  
:$$R ≤ C.$$
* The prerequisite for this,  however,  is that the following applies to the block length of this code:  
:$$n → ∞.$$

$\text{Reverse of Shannon's channel coding theorem:}$ 

:If the rate  $R$  of the  $(n$,  $k)$ block code used is larger than the channel capacity  $C$,  then an arbitrarily small block error probability can never be achieved.}}

{{GraueBox|TEXT=
[[File:Transinf_9.png|right|frame|Information-theoretic quantities for different  $p_{\rm A}$  and  $p_{\rm B}= 1- p_{\rm A}$ ]]
$\text{Example 4}$:  We consider the same discrete memoryless channel as in  $\text{Example 2}$. 
*The symbol probabilities  $p_{\rm A} = 0.1$  and  $p_{\rm B}= 1- p_{\rm A}=0.9$  were assumed. 

*The mutual information is  $I(X;Y)= 0.092$  bit/channel use   ⇒   first row,  see fourth column in the table.

The  »'''channel capacity'''«  is the mutual information  $I(X, Y)$  with best possible symbol probabilities     $p_{\rm A} = 0.55$  and  $p_{\rm B}= 1- p_{\rm A}=0.45$:
:$$C = \max_{P_X(X)} \hspace{0.15cm} I(X;Y) = 0.284 \ \rm bit/channel \hspace{0.05cm} access \hspace{0.05cm}.$$

From the table you can see further  $($we do without the additional unit "bit/channel use" in the following$)$:
#The parameter  $p_{\rm A} = 0.1$  was chosen very unfavorably,  because with the present channel the symbol  $\rm A$  is more falsified than  $\rm B$. 
#Already with  $p_{\rm A} = 0.9$  the mutual information results in a somewhat better value:  $I(X; Y)=0.130$.
#For the same reason  $p_{\rm A} = 0.55$,  $p_{\rm B} = 0.45$  gives a slightly better result than equally probable symbols  $(p_{\rm A} = p_{\rm B} =0.5)$.
#The more asymmetric the channel is,  the more the optimal probability function  $P_X(X)$  deviates from the uniform distribution.  Conversely:  If the channel is symmetric,  the uniform distribution is always obtained.}}

The ternary channel of  $\text{Example 3}$  is symmetric.  Therefore here  $P_X(X) = \big ( 1/3,\ 1/3,\ 1/3 \big )$  is optimal for each  $q$–value,  and the mutual information  $I(X;Y)$  given in the result table is at the same time the channel capacity  $C$.

==Exercises==
 
*First, select the number  $(1,\ 2, \text{...} \ )$  of the task to be processed.  The number  "$0$"  corresponds to a "Reset":  Same setting as at program start.
*A task description is displayed.  The parameter values are adjusted.  Solution after pressing "Show Solution".
*Source symbols are denoted by uppercase letters  (binary:  $\rm A$,  $\rm B$),  sink symbols by lowercase letters  ($\rm a$,  $\rm b$).  Error-free transmission:  $\rm A \rightarrow a$.
*For all entropy values, the unit "bit/use" would have to be added.

{{BlueBox|TEXT=
'''(1)'''  Let  $p_{\rm A} = p_{\rm B} = 0.5$  and  $p_{\rm b \vert A} = p_{\rm a \vert B} = 0.1$.  What is the channel model?  What are the entropies  $H(X), \, H(Y)$  and the mutual information  $I(X;\, Y)$?}}

:*  Considered is the BSC model  (Binary Symmetric Channel).  Because of  $p_{\rm A} = p_{\rm B} = 0.5$  it holds for the entropies:  $H(X) = H(Y) = 1$.
:*  Because of  $p_{\rm b \vert A} = p_{\rm a \vert B} = 0.1$  eqivocation and irrelevance are also equal:  $H(X \vert Y) = H(Y \vert X) = H_{\rm bin}(p_{\rm b \vert A}) = H_{\rm bin}(0.1) =0.469$.
:*  The mutual information is  $I(X;\, Y) = H(X) - H(X \vert Y)= 1-H_{\rm bin}(p_{\rm b \vert A}) = 0.531$  and the joint entropy is  $H(XY) =1.469$.

{{BlueBox|TEXT=
'''(2)'''  Let further  $p_{\rm b \vert A} = p_{\rm a \vert B} = 0.1$, but now the symbol probability is  $p_{\rm A} = 0. 9$.  What is the capacity  $C_{\rm BSC}$  of the BSC channel with  $p_{\rm b \vert A} = p_{\rm a \vert B}$?        
Which  $p_{\rm b \vert A} = p_{\rm a \vert B}$  leads to the largest possible channel capacity and which  $p_{\rm b \vert A} = p_{\rm a \vert B}$  leads to the channel capacity  $C_{\rm BSC}=0$?}}

:*  The capacity  $C_{\rm BSC}$  is equal to the maximum mutual information  $I(X;\, Y)$  considering the optimal symbol probabilities.
:*  Due to the symmetry of the BSC model  equally probable symbols  $(p_{\rm A} = p_{\rm B} = 0.5)$  lead to the optimum   ⇒   $C_{\rm BSC}=0.531$.
:*  The best is the  "ideal channel"  $(p_{\rm b \vert A} = p_{\rm a \vert B} = 0)$  ⇒   $C_{\rm BSC}=1$.   The worst BSC channel results with  $p_{\rm b \vert A} = p_{\rm a \vert B} = 0.5$   ⇒   $C_{\rm BSC}=0$.
:*  But also with   $p_{\rm b \vert A} = p_{\rm a \vert B} = 1$  we get  $C_{\rm BSC}=1$.  Here all symbols are inverted, which is information theoretically the same as  $\langle Y_n \rangle \equiv \langle X_n \rangle$.

{{BlueBox|TEXT=
'''(3)'''  Let  $p_{\rm A} = p_{\rm B} = 0.5$,  $p_{\rm b \vert A} = 0.05$  and  $ p_{\rm a \vert B} = 0.4$.   Interpret the results in comparison to the experiment  $(1)$  and to the  $\text{example 2}$  in the theory section.}}

:*  Unlike the experiment  $(1)$  no BSC channel is present here.  Rather, the channel considered here is asymmetric:  $p_{\rm b \vert A} \ne p_{\rm a \vert B}$.
:*  According to  $\text{Example 2}$  it holds for  $p_{\rm A} = 0.1,\ p_{\rm B} = 0.9$:     $H(X)= 0.469$,  $H(Y)= 0.994$,  $H(X \vert Y)=0.377$,  $H(Y \vert X)=0.902$,  $I(X;\vert Y)=0.092$.
:*  Now it holds  $p_{\rm A} = p_{\rm B} = 0.5$  and we get  $H(X)=1,000$,  $H(Y)=0.910$,  $H(X \vert Y)=0.719$,  $H(Y \vert X)=0.629$,  $I(X;\ Y)=0.281$.
:*  All output values depend significantly on  $p_{\rm A}$  and  $p_{\rm B}=1-p_{\rm A}$  except for the conditional probabilities  ${\rm Pr}(Y \vert X)\in \{\hspace{0.05cm}0.95,\ 0.05,\ 0.4,\ 0.6\hspace{0.05cm} \}$.

{{BlueBox|TEXT=
'''(4)'''  Let further  $p_{\rm A} = p_{\rm B}$,  $p_{\rm b \vert A} = 0.05$,  $ p_{\rm a \vert B} = 0.4$.  What differences do you see in terms of analytical calculation and "simulation"  $(N=10000)$.}}

:*  The joint probabilities are  $p_{\rm Aa} =0.475$,  $p_{\rm Ab} =0.025$,  $p_{\rm Ba} =0.200$, $p_{\rm Bb} =0.300$.  Simulation:  Approximation by relative frequencies:
:*  For example, for  $N=10000$:  $h_{\rm Aa} =0.4778$,  $h_{\rm Ab} =0.0264$,  $h_{\rm Ba} =0.2039$, $h_{\rm Bb} =0.2919$.  After pressing  "New sequence"  slightly different values.
:*  For all subsequent calculations, no principal difference between theory and simulation, except  $p \to h$.  Examples: 
:*  $p_{\rm A} = 0.5 \to h_{\rm A}=h_{\rm Aa} + h_{\rm Ab} =0.5042$,  $p_b = 0.325 \to h_{\rm b}=h_{\rm Ab} + h_{\rm Bb} =0. 318$,  $p_{b|A} = 0.05 \to h_{\rm b|A}=h_{\rm Ab}/h_{\rm A} =0.0264/0.5042= 0.0524$, 
:*  $p_{\rm A|b} = 0.0769 \to h_{\rm A|b}=h_{\rm Ab}/h_{\rm b} =0.0264/0.318= 0.0830$.  Thus, this simulation yields  $I_{\rm Sim}(X;\ Y)=0.269$  instead of  $I(X;\ Y)=0.281$.

{{BlueBox|TEXT=
'''(5)'''  Setting according to  $(4)$.  How does  $I_{\rm Sim}(X;\ Y)$  differ from  $I(X;\ Y) = 0.281$  for  $N=10^3$,  $10^4$,  $10^5$ ?  In each case, averaging over ten realizations. }}
:*  $N=10^3$:   $0.232 \le I_{\rm Sim} \le 0.295$,   mean:  $0.263$   #   $N=10^4$:   $0.267 \le I_{\rm Sim} \le 0.293$,   mean:  $0.279$   #   $N=10^5$:   $0.280 \le I_{\rm Sim} \le 0.285$   mean:  $0.282$.
:*  With  $N=10^6$  for this channel, the simulation result differs from the theoretical value by less than  $\pm 0.001$. 

{{BlueBox|TEXT=
'''(6)'''  What is the capacity  $C_6$  of this channel with  $p_{\rm b \vert A} = 0.05$,  $ p_{\rm a \vert B} = 0.4$?  Is the error probability  $0$  possible with the code rate  $R=0.3$? }}

:*  $C_6=0.284$  is the maximum of  $I(X;\ Y)$  for   $p_{\rm A} =0.55$  ⇒  $p_{\rm B} =0. 45$.  Simulation over  ten times  $N=10^5$:  $0.281 \le I_{\rm Sim}(X;\ Y) \le 0.289$.
:*  With the code rate  $R=0.3 > C_6$  an arbitrarily small block error probability is not achievable even with the best possible coding.

{{BlueBox|TEXT=
'''(7)'''  Now let  $p_{\rm A} = p_{\rm B}$,  $p_{\rm b \vert A} = 0$,  $ p_{\rm a \vert B} = 0.5$.   What property does this asymmetric channel exhibit?  What values result for  $H(X)$,  $H(X \vert Y)$,  $I(X;\ Y)$ ? }}

:*  The symbol  $\rm A$  is never falsified, the symbol  $\rm B$  with (information theoretically) maximum falsification probability  $ p_{\rm a \vert B} = 0.5$
:*  The total falsification probability is  $ {\rm Pr} (Y_n \ne X_n)= p_{\rm A} \cdot p_{\rm b \vert A} + p_{\rm B} \cdot p_{\rm a \vert B}= 0.25$  ⇒   about  $25\%$  of the output sink symbols are "purple".
:*  Joint probabilities:  $p_{\rm Aa}= 1/2,\ p_{\rm Ab}= 0,\ p_{\rm Ba}= p_{\rm Bb}= 1/4$,    Inference probabilities:   $p_{\rm A \vert a}= 1,\ p_{\rm B \vert a}= 0,\ p_{\rm A \vert b}= 1/3,\ p_{\rm B \vert b}= 2/3$.
:*  From this we get for equivocation  $H(X \vert Y)=0.689$;   with source entropy  $H(X)= 1$   ⇒   $I(X;\vert Y)=H(X)-H(X \vert Y)=0.311$.

{{BlueBox|TEXT=
'''(8)'''  What is the capacity  $C_8$  of this channel with  $p_{\rm b \vert A} = 0.05$,  $ p_{\rm a \vert B} = 035$?  Is the error probability  $0$  possible with the code rate  $R=0.3$? }}

:*  $C_8=0.326$  is the maximum of  $I(X;\ Y)$  for   $p_{\rm A} =0.55$.  Thus, because of  $C_8 >R=0.3 $  an arbitrarily small block error probability is achievable.
:*  The only difference compared to  $(6)$   ⇒   $C_6=0.284 < 0.3$  is the slightly smaller falsification probability  $ p_{\rm a \vert B} = 0.35$  instead of  $ p_{\rm a \vert B} = 0.4$.

{{BlueBox|TEXT=
'''(9)'''  We consider the ideal ternary channel:  $p_{\rm a \vert A} = p_{\rm b \vert B}=p_{\rm c \vert C}=1$.  What is its capacity  $C_9$?  What is the maximum mutual information displayed by the program? }}

:*  Due to the symmetry of the channel model, equally probable symbols  $(p_{\rm A} = p_{\rm B}=p_{\rm C}=1/3)$  lead to the channel capacity:  $C_9 = \log_2\ (3) = 1.585$.
:*  Since in the program all parameter values can only be entered with a resolution of  $0.05$ , for  $I(X;\ Y)$  this maximum value is not reached.
:*  Possible approximations:  $p_{\rm A} = p_{\rm B}= 0.3, \ p_{\rm C}=0.4$  ⇒   $I(X;\ Y)= 1. 571$     #     $p_{\rm A} = p_{\rm B}= 0.35, \ p_{\rm C}=0.3$  ⇒   $I(X;\ Y)= 1.581$.

{{BlueBox|TEXT=
'''(10)'''  Let the source symbols be (nearly) equally probable.  Interpret the other settings and the results. }}

:*  The falsification probabilities are  $p_{\rm b \vert A} = p_{\rm c \vert B}=p_{\rm a \vert C}=1$   ⇒   no single sink symbol is equal to the source symbol.
:*  This cyclic mapping has no effect on the channel capacity:  $C_{10} = C_9 = 1.585$.  The program returns  ${\rm Max}\big[I(X;\ Y)\big]= 1.581$.

{{BlueBox|TEXT=
'''(11)'''  We consider up to and including  $(13)$  the same ternary source.   What results are obtained for  $p_{\rm b \vert A} = p_{\rm c \vert B}=p_{\rm a \vert C}=0.2$  and  $p_{\rm c \vert A} = p_{\rm a \vert B}=p_{\rm b \vert C}=0$? }}
:*  Each symbol can only be falsified into one of the two possible other symbols.  From  $p_{\rm b \vert A} = p_{\rm c \vert B}=p_{\rm a \vert C}=0.2$  it follows  $p_{\rm a \vert A} = p_{\rm b \vert B}=p_{\rm c \vert C}=0.8$.
:*  This gives us for the maximum mutual information  ${\rm Max}\big[I(X;\ Y)\big]= 0.861$  and for the channel capacity a slightly larger value:  $C_{11} \gnapprox 0.861$.

{{BlueBox|TEXT=
'''(12)'''  How do the results change if each symbol is   $80\%$  transferred correctly and   $10\%$  falsified each in one of the other two symbols? }}
:*  Although the probability of correct transmission is with  $80\%$  as large as in  '''(11)''', here the channel capacity  $C_{12} \gnapprox 0.661$ is smaller.
:*  If one knows for the channel  $(11)$  that  $X = \rm A$  has been falsified, one also knows  $Y = \rm b$.  But not for channel  $(12)$  ⇒   the channel is less favorable.

{{BlueBox|TEXT=
'''(13)'''  Let the falsification probabilities now be  $p_{\rm b \vert A} = p_{\rm c \vert A} = p_{\rm a \vert B} = p_{\rm c \vert B}=p_{\rm a \vert C}=p_{\rm b \vert C}=0.5$.  Interpret this redundancy-free ternary channel. }}
:*  No single sink symbol is equal to its associated source symbol; with respect to the other two symbols, a  $50\hspace{-0.1cm}:\hspace{-0.1cm}50$  decision must be made.
:*  Nevertheless, here the channel capacity is  $C_{13} \gnapprox 0.584$  only slightly smaller than in the previous experiment:  $C_{12} \gnapprox 0.661$.
:*  The channel capacity  $C=0$  results for the redundancy-free ternary channel exactly for the case where all nine falsification probabilities are equal to  $1/3$ .

{{BlueBox|TEXT=
'''(14)'''  What is the capacity $C_{14}$ of the ternary channel with  $p_{\rm b \vert A} = p_{\rm a \vert B}= 0$  and  $p_{\rm c \vert A} = p_{\rm c \vert B} = p_{\rm a \vert C}=p_{\rm b \vert C}=0. 1$  ⇒   $p_{\rm a \vert A} = p_{\rm b \vert B}=0.9$,   $p_{\rm c \vert C} =0.8$? }}
:*  With the default  $p_{\rm A}=p_{\rm B}=0.2$   ⇒   $p_{\rm C}=0.6$  we get  $I(X;\ Y)= 0.738$.  Now we are looking for "better" symbol probabilities.
:*  From the symmetry of the channel, it is obvious that  $p_{\rm A}=p_{\rm B}$  is optimal.  The channel capacity  $C_{14}=0.995$  is obtained for  $p_{\rm A}=p_{\rm B}=0.4$   ⇒   $p_{\rm C}=0.2$.
:*  Example:  Ternary transfer if the middle symbol  $C$  can be distorted in two directions, but the outer symbols can only be distorted in one direction at a time.
 

==Applet Manual==
 
[[File:Anleitung_transinformation.png|left|600px|frame|Screenshot of the German version]]
 
    '''(A)'''     Select whether  "analytically"  or  "by simulation"

    '''(B)'''     Setting of the parameter  $N$  for the simulation

    '''(C)'''     Option to select "binary source"  or  "ternary source"

    '''(D)'''     Setting of the symbol probabilities

    '''(E)'''     Setting of the transition probabilities

    '''(F)'''     Numerical output of different probabilities

    '''(G)'''     Two diagrams with the information theoretic quantities

    '''(H)'''     Output of an exemplary source symbol sequence

    '''(I)'''     Associated simulated sink symbol sequence

    '''(J)'''     Exercise area: Selection, questions, sample solutions
 

==About the Authors==
 
This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].
*The first version was created in 2010 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Martin_V.C3.B6lkl_.28Diplomarbeit_LB_2010.29|Martin Völkl]]  as part of his diploma thesis with “FlashMX – Actionscript”.  Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|Günter Söder]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Dr.-Ing._Klaus_Eichin_.28at_LNT_from_1972-2011.29|Klaus Eichin]].

*In 2020 the program was redesigned via HTML5/JavaScript by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Veronika_Hofmann_.28Ingenieurspraxis_Math_2020.29|Veronika Hofmann]]  (Ingenieurspraxis Mathematik, Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Benedikt_Leible.2C_M.Sc._.28at_L.C3.9CT_since_2017.29|Benedikt Leible]]  and  [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29|Tasnád Kernetzky]].

*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  in the context of a working student activity. 

*The conversion of this applet was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ $\text{Studienzuschüsse}$]  $($TUM Department of Electrical and Computer Engineering$)$.  $\text{Many thanks}$.

==Once again: Open Applet in new Tab==

{{LntAppletLinkEnDe|transinformation_en|transinformation}}

==References==

Applets:Eye Pattern and Worst-Case Error Probability

2023-03-21T18:07:49Z

Hwang:

{{LntAppletLinkEnDe|eyeDiagram_en|eyeDiagram}}

==Applet Description==
 
The applet illustrates the eye pattern for different encodings 
* binary  (redundancy-free), 
*quaternary  (redundancy-free),
*pseudo–ternary:  (AMI and duobinary) 

and for various reception concepts 
*Matched Filter receiver, 
*CRO Nyquist system, 
*Gaussian low-pass filter.

The last reception concept leads to intersymbol interference, that is:  Neighboring symbols interfere with each other in symbol decision.

Such intersymbol interferences and their influence on the error probability can be captured and quantified very easily by the "eye pattern".  But also for the other two (without intersymbol interference) systems important insights can be gained from the graphs.

Furthermore, the most unfavorable ("worst case") error probability  
:$$p_{\rm U} = {\rm Q}\left[ö_{\rm norm}/\sigma_{\rm norm} \right ]$$

is output, which for binary Nyquist systems is identical to the mean error probability  $p_{\rm M}$  and represents a suitable upper bound for the other system variants:  $p_{\rm U} \ge p_{\rm M}$.

In the  $p_{\rm U}$–equation mean:
*${\rm Q}(x)$  is the  [[Applets:Complementary_Gaussian_Error_Functions|Complementary Gaussian Error Function]].  The normalized eye opening can have values between  $0 \le ö_{\rm norm} \le 1$  .
*The maximum value  $(ö_{\rm norm} = 1)$  applies to the binary Nyquist system and  $ö_{\rm norm}=0$  represents a "closed eye".
*The normalized detection noise rms value  $\sigma_{\rm norm}$  depends on the adjustable parameter  $10 \cdot \lg \ E_{\rm B}/N_0$  but also on the coding and the receiver concept.

==Theoretical Background==
 
 
=== System description and prerequisites===

The binary baseband transmission model outlined below applies to this applet. First, the following prerequisites apply:
*The transmission is binary, bipolar, and redundancy-free with bit rate  $R_{\rm B} = 1/T$, where  $T$  is the symbol duration.
*The transmitted signal  $s(t)$  is equal to  $ \pm s_0$   at all times  $t$  ⇒   The basic transmission pulse  $g_s(t)$  is NRZ–rectangular with amplitude  $s_0$  and pulse duration  $T$.

*Let the received signal be  $r(t) = s(t) + n(t)$, where the AWGN term  $n(t)$  is characterized by the (one-sided) noise power density  $N_0$. 
*Let the channel frequency response be best possible (ideal) and need not be considered further:  $H_{\rm K}(f) =1$.
*The receiver filter with the impulse response  $h_{\rm E}(t)$  forms the detection signal  $d(t) = d_{\rm S}(t)+ d_{\rm N}(t)$ from  $r(t)$. 
* This is evaluated by the decision with the decision threshold  $E = 0$  at the equidistant times  $\nu \cdot T$. 
*A distinction is made between the signal component  $d_{\rm S}(t)$  – originating from  $s(t)$  – and the noise component  $d_{\rm N}(t)$,  whose cause is the AWGN noise  $n(t)$. 
*$d_{\rm S}(t)$  can be represented as a weighted sum of weighted basic detection pulses  $T$,  each shifted by  $g_d(t) = g_s(t) \star h_{\rm E}(t)$. 

*To calculate the (average) error probability, one further needs the variance  $\sigma_d^2 = {\rm E}\big[d_{\rm N}(t)^2\big]$  of the detection noise component (for AWGN noise).

===Optimal intersymbol interference-free system – matched filter receiver===

The minimum error probability results for the case considered here  $H_{\rm K}(f) =1$  with the matched filter receiver, i.e. when  $h_{\rm E}(t)$  is equal in shape to the NRZ basic transmission pulse  $g_s(t)$.  The rectangular impulse response  $h_{\rm E}(t)$  then has duration  $T_{\rm E} = T$  and height  $1/T$.

[[File:Auge_1neu.png|center|frame|Binary baseband transmission system;  the sketch for  $h_{\rm E}(t)$  applies only to the matched filter receiver ]]
[[File:EN_Dig_T_1_4_S1_v2.png|center|frame|Binary baseband transmission system;  the sketch for  $h_{\rm E}(t)$  applies only to the matched filter receiver ]]

*The basic detection pulse  $g_d(t)$  is triangular with maximum  $s_0$  at  $t=0$ ;  $g_d(t)=0$  for  $|t| \ge T$. Due to this tight temporal constraint, there is no intersymbol interference   ⇒   $d_{\rm S}(t = \nu \cdot T) = \pm s_0$   ⇒   the distance of all useful samples from the threshold  $E = 0$  is always  $|d_{\rm S}(t = \nu \cdot T)| = s_0$.
*The detection noise power for this constellation is:
:$$\sigma_d^2 = N_0/2 \cdot \int_{-\infty}^{+\infty} |h_{\rm E}(t)|^2 {\rm d}t = N_0/(2T)=\sigma_{\rm MF}^2.$$
*For the (average) error probability, using the  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability|Complementary Gaussian Error Function]]  ${\rm Q}(x)$ :
:$$p_{\rm M} = {\rm Q}\left[\sqrt{{s_0^2}/{\sigma_d^2}}\right ] = {\rm Q}\left[\sqrt{{2 \cdot s_0^2 \cdot T}/{N_0}}\right ] = {\rm Q}\left[\sqrt{2 \cdot E_{\rm B}/ N_0}\right ].$$

The applet considers this case with the settings  "after gap–low-pass"  as well as  $T_{\rm E}/T = 1$. The output values are with regard to later constellations
*the normalized eye opening  $ö_{\rm norm} =1$   ⇒   this is the maximum possible value,
*the normalized detection noise rms value (equal to the square root of the detection noise power)  $\sigma_{\rm norm} =\sqrt{1/(2 \cdot E_{\rm B}/ N_0)}$  as well as
*the worst-case error probability  $p_{\rm U} = {\rm Q}\left[ö_{\rm norm}/\sigma_{\rm norm} \right ]$   ⇒   for intersymbol interference-free systems,   $p_{\rm M}$  and   $p_{\rm U}$  agree.

$\text{Differences in the multi-level systems}$
*There are  $M\hspace{-0.1cm}-\hspace{-0.1cm}1$ eyes and just as many thresholds   ⇒   $ö_{\rm norm} =1/(M\hspace{-0.1cm}-\hspace{-0.1cm}1)$  ⇒   $M=4$:  quaternary system,  $M=3$:  AMI code, duobinary code.
*The normalized detection noise rms value  $\sigma_{\rm norm}$  is smaller by a factor of  $\sqrt{5/9} \approx 0.745$  for the quaternary system than for the binary system.
*For the AMI code and the duobinary code, this improvement factor, which goes back to the smaller  $E_{\rm B}/ N_0$,  has the value  $\sqrt{1/2} \approx 0.707$.

 
===Nyquist system with cosine rolloff overall frequency response===

[[File:Auge_2_neu.png|right|frame|Cosine rolloff overall frequency response ]]

We assume that the overall frequency response between the Dirac-shaped source to the decision has the shape of a  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Raised-cosine_low-pass_filter|cosine rolloff low-pass]]    ⇒   $H_{\rm S}(f)\cdot H_{\rm E}(f) = H_{\rm CRO}(f)$ .
*The rolloff of  $H_{\rm CRO}(f)$  is point symmetric about the Nyquist frequency  $1/(2T)$. The larger the rolloff factor  $r_{ \hspace {-0.05cm}f}$,  ithe flatter the Nyquist slope.
*The basic detection pulse  $g_d(t) = s_0 \cdot T \cdot {\mathcal F}^{-1}\big[H_{\rm CRO}(f)\big]$  has zeros at times  $\nu \cdot T$  independent of  $r_{ \hspace {-0.05cm}f}$.  There are further zero crossings depending on  $r_{ \hspace {-0.05cm}f}$.  For the pulse holds:
:$$g_d(t) = s_0 \hspace{-0.05cm}\cdot\hspace{-0.05cm} {\rm sinc}( t/T )\hspace{-0.05cm}\cdot\hspace{-0.05cm}\frac {\cos(\pi \cdot r_{\hspace{-0.05cm}f} \cdot t/T )}{1 - (2 \cdot
r_{\hspace{-0.05cm}f} \cdot t/T)^2}.$$
*It follows:  As with the matched filter receiver, the eye is maximally open   ⇒   $ö_{\rm norm} =1$.

EN_Dig_T_1_4_S6.png File:Auge_3.png
[[File:EN_Dig_T_1_4_S6.png|right|frame|To optimize the rolloff factor]]
Let us now consider the noise power before the decision. For this holds:

:$$\sigma_d^2 = N_0/2 \cdot \int_{-\infty}^{+\infty} |H_{\rm E}(f)|^2 {\rm d}f = N_0/2 \cdot \int_{-\infty}^{+\infty} \frac{|H_{\rm CRO}(f)|^2}{|H_{\rm S}(f)|^2} {\rm d}f.$$

The graph shows the power transfer function  $|H_{\rm E}(f)|^2$  for three different rolloff factors

* $r_{ \hspace {-0.05cm}f}=0$   ⇒   green curve,
* $r_{ \hspace {-0.05cm}f}=1$   ⇒   red curve,
* $r_{ \hspace {-0.05cm}f}=0.8$   ⇒   blue curve.

The areas under these curves are each a measure of the noise power  $\sigma_d^2$.  The rectangle with a gray background marks the smallest value  $\sigma_d^2 =\sigma_{\rm MF}^2$, which also resulted with the matched filter receiver.
 
One can see from this plot:
*The rolloff factor  $r_{\hspace{-0.05cm}f} = 0$  (rectangular frequency response) leads to  $\sigma_d^2 =K \cdot \sigma_{\rm MF}^2$  with  $K \approx 1.5$ despite the very narrow receiver filter, since  $|H_{\rm E}(f)|^2$  increases steeply as  $f$  increases. The reason for this noise power increase is the  $\rm sinc^2(f T)$  function in the denominator, which is required to compensate for the  $|H_{\rm S}(f)|^2$–decay. 
* Since the area under the red curve is smaller than that under the green curve,  $r_{\hspace{-0.05cm}f} = 1$  leads to a smaller noise power despite a spectrum twice as wide:  $K \approx 1.23$.  For  $r_{\hspace{-0.05cm}f} \approx 0.8$, a slightly better value results. For this, the best possible compromise between bandwidth and excess noise is achieved.
*The normalized detection noise rms value is thus for the rolloff factor  $r_{ \hspace {-0.05cm}f}$:   $\sigma_{\rm norm} =\sqrt{K(r_f)/(2 \cdot E_{\rm B}/ N_0)}$. 
*Again, the worst-case error probability  $p_{\rm U} = {\rm Q}\left[ö_{\rm norm}/\sigma_{\rm norm} \right ]$   coincides exactly with the mean error probability  $p_{\rm M}$. 

$\text{Differences in the multi-level systems}$

All remarks in section $2.2$ apply in the same way to the "Nyquist system with cosine rolloff total frequency response".

===Intersymbol interference system with Gaussian receiver filter===

[[File:Auge_4.png|right|frame|System with Gaussian receiver filter]]

We start from the block diagram sketched on the right. Further it shall be valid:
*Rectangular NRZ basic transmission pulse  $g_s(t)$  with height  $s_0$  and duration  $T$:
:$$H_{\rm S}(f) = {\rm sinc}(f T).$$
*Gaussian receiver filter with cutoff frequency  $f_{\rm G}$:
:$$H_{\rm E}(f) = H_{\rm G}(f) = {\rm e}^{- \pi \hspace{0.05cm}\cdot \hspace{0.03cm} f^2/(2\hspace{0.05cm}\cdot \hspace{0.03cm}f_{\rm G})^2 } \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ
\hspace{0.2cm}h_{\rm E}(t) = h_{\rm G}(t) = {\rm e}^{- \pi \cdot (2\hspace{0.05cm}\cdot \hspace{0.03cm}
f_{\rm G}\hspace{0.05cm}\cdot \hspace{0.02cm} t)^2}
\hspace{0.05cm}.$$

Based on the assumptions made here, the following applies to the basic detection pulse:

[[File:Auge_5_neu.png|right|frame|Frequency response and impulse response of the receiver filter]]
:$$g_d(t) = s_0 \cdot T \cdot \big [h_{\rm S}(t) \star h_{\rm G}(t)\big ] = 2 f_{\rm G} \cdot s_0 \cdot \int_{t-T/2}^{t+T/2}
{\rm e}^{- \pi \hspace{0.05cm}\cdot\hspace{0.05cm} (2 \hspace{0.05cm}\cdot\hspace{0.02cm}
f_{\rm G}\hspace{0.05cm}\cdot\hspace{0.02cm} \tau )^2} \,{\rm d} \tau \hspace{0.05cm}.$$

The integration leads to the result:

:$$g_d(t) = s_0 \cdot \big [ {\rm Q} \left ( 2 \cdot \sqrt {2 \pi}
\cdot f_{\rm G}\cdot ( t - {T}/{2})\right )- {\rm Q} \left (
2 \cdot \sqrt {2 \pi} \cdot f_{\rm G}\cdot ( t + {T}/{2}
)\right ) \big ],$$

using the complementary Gaussian error function

:$${\rm Q} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it
x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d {\it u}
\hspace{0.05cm}.$$

The module  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|Complementary Gaussian Error Functions]]  provides the numerical values of  ${\rm Q} (x)$. 
*This basic detection pulse causes  [[Digital_Signal_Transmission/Causes_and_Effects_of_Intersymbol_Interference#Definition_of_the_term_.22Intersymbol_Interference.22|intersymbol interference]].
*This is understood to mean that the symbol decision is influenced by the spurs of neighboring pulses. While in intersymbol interference free transmission systems each symbol is falsified with the same probability – namely the mean error probability  $p_{\rm M}$  – there are favorable symbol combinations with the falsification probability  ${\rm Pr}(v_{\nu} \ne q_{\nu}) < p_{\rm M}$.
*In contrast, other symbol combinations increase the falsification probability significantly.

[[File:Auge_6.png|right|frame|Binary eye $($Gaussian low-pass,  $f_{\rm G}/R_{\rm B} = 0.35)$.]]
The intersymbol interferences can be captured and analyzed very easily by the so-called  '''eye diagram'''.  These are the focus of this applet. All important information can be found  [[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference#Definition_and_statements_of_the_eye_diagram|here]].
*The eye diagram is created by drawing all sections of the detection useful signal  $d_{\rm S}(t)$  of length  $2T$  on top of each other. You can visualize the formation in the program with "single step".

* A measure for the strength of the intersymbol interference is the ''vertical eye opening''. For the symmetric binary case, with  $g_\nu = g_d(\pm \nu \cdot T)$  and appropriate normalization:
:$$ ö_{\rm norm} = g_0 -2 \cdot (|g_1| + |g_2| + \text{...}).$$
* With larger cutoff frequency, the pulses interfere less and  $ ö_{\rm norm}$  increases continuously. At the same time, with larger  $f_{\rm G}/R_{\rm B}$,  the (normalized) detection noise rms value also becomes larger:
:$$ \sigma_{\rm norm} = \sqrt{\frac{f_{\rm G}/R_{\rm B}}{\sqrt{2} \cdot E_{\rm B}/N_{\rm 0}}}.$$
*The worst-case error probability  $p_{\rm U} = {\rm Q}\left[ö_{\rm norm}/\sigma_{\rm norm} \right ]$   ⇒   "Worst Case" is usually significantly higher than the mean error probability  $p_{\rm M}$.

$\text{Differences in the redundancy-free quaternary system}$
*For  $M=4$,  other basic pulse values result. ''Example'':     With  $M=4, \ f_{\rm G}/R_{\rm B}=0.4$  basic pulse values  $g_0 = 0.955, \ g_1 = 0.022$  are identical with  $M=2, \ f_{\rm G}/R_{\rm B}=0.8$.
* There are now three eye openings and just as many thresholds.  The equation for the normalized eye opening is now:   $ ö_{\rm norm} = g_0/3 -2 \cdot (|g_1| + |g_2| + \text{...}).$
*The normalized detection noise rms  $\sigma_{\rm norm}$  is again a factor of  $\sqrt{5/9} \approx 0.745$  smaller for the quaternary system than for the binary system.

===Pseudo-ternary codes===

In symbolwise coding, each incoming source symbol  $q_\nu$  generates an encoder symbol  $c_\nu$  that depends not only on the current input symbol  $q_\nu$  but also on the  $N_{\rm C}$  preceding symbols  $q_{\nu-1}$, ... , $q_{\nu-N_{\rm C}} $.   $N_{\rm C}$  is referred to as the ''order''  of the code.  It is typical for a symbolwise coding that
[[File:Dig_T_2_4_S1_v1_neu.png|right|frame|Block diagram and equivalent circuit of a pseudo-ternary encoder|class=fit]]

*the symbol duration  $T$  of the encoded signal (and of the transmitted signal) coincides with the bit duration  $T_{\rm B}$  of the binary source signal, and
*coding and decoding do not lead to major time delays, which are unavoidable when block codes are used. 

Special importance has ''pseudo-ternary codes''   ⇒   level number  $M = 3$, which can be described by the block diagram according to the left graphic. In the right graphic an equivalent circuit is given, which is very suitable for an analysis of these codes. More details can be found in the  [[Digital_Signal_Transmission/Symbolwise_Coding_with_Pseudo-Ternary_Codes|$\rm LNTwww$ theory section]].  Conclusion:

*Recoding from binary  $(M_q = 2)$  to ternary  $(M = M_c = 3)$:
:$$q_\nu \in \{-1, +1\},\hspace{0.5cm} c_\nu \in \{-1, \ 0, +1\}\hspace{0.05cm}.$$

*The relative code redundancy is the same for all pseudo-ternary codes:
:$$ r_c = 1 -1/\log_2\hspace{0.05cm}(3) \approx 36.9 \%\hspace{0.05cm}.$$

Based on the code parameter  $K_{\rm C}$,  different first-order pseudo-ternary codes  $(N_{\rm C} = 1)$  are characterized.

[[File:Auge_16.png|right|frame|Signals in AMI coding|class=fit]]
$\Rightarrow \ \ K_{\rm C} = 1\text{: AMI code}$  (from:   ''Alternate Mark Inversion'')

The graph shows the binary source signal  $q(t)$ at the top. Below are shown:
* the likewise binary signal  $b(t)$  after the pre-encoder, and
* the encoded signal  $c(t) = s(t)$  of the AMI code.

One can see the simple AMI coding principle:
*Each binary value  "–1"  of $q(t)$   ⇒   symbol  $\rm L$  is encoded by the ternary amplitude coefficient  $a_\nu = 0$.  
*The binary value  "+1"  of  $q(t)$   ⇒   symbol  $\rm H$  is alternately represented by  $a_\nu = +1$  and  $a_\nu = -1$.  

This ensures that there are no long  "+1"–  or  "–1" sequences in the AMI-encoded signal, which would be problematic for an equal-signal-free channel. 
 
[[File:Auge_16a.png|left|frame|class=fit]]

The eye diagram is shown on the left.
::* There are two eye openings and two thresholds.
::* The normalized eye opening is  $ö_{\rm norm}= 1/2 \cdot (g_0 -3 \cdot g_1)$, where  $g_0 = g_d(t=0)$  denotes the main value of the basic detection pulse and  $g_1 = g_d(t=\pm T)$  denotes the relevant precursors and postcursors that vertically limit the eye.

::* The normalized eye opening is thus significantly smaller than for the comparable binary system   ⇒   $ö_{\rm norm}= g_0 -2 \cdot g_1$.
::* The normalized noise rms  $\sigma_{\rm norm}$  is smaller than for the comparable binary system by a factor of  $\sqrt{1/2} \approx 0.707$. 
 
[[File:Auge_17.png|right|frame|Signals in duobinary coding|class=fit]]

$\Rightarrow \ \ K_{\rm C} = -1\text{: duobinary code}$ 

From the right graph with the signal curves one recognizes:
*Here, any number of symbols of the same polarity  ("+1" or "–1")  can directly follow each other   ⇒   the duobinary code is not free of equal signals. 
*In contrast, the alternating sequence  " ... , +1, –1, +1, –1, +1, ... "  does not occur, which is particularly disturbing with regard to intersymbol interference.
* Also the duobinary encoded sequence consists to 50% of zeros. The enhancement factor due to the smaller  $E_{\rm B}/ N_0$  is equal to  $\sqrt{1/2} \approx 0.707$, as in the AMI code.

[[File:Auge_17a.png|left|frame|class=fit]]
 
The eye diagram is shown on the left.
::* There are again two "eyes" and two thresholds.
::* The eye opening is   $ö_{\rm norm}= 1/2 \cdot (g_0 - g_1)$.
*$ö_{\rm norm}$  is thus larger than in the AMI code and also as in the comparable binary system.
*A disadvantage compared to the AMI code, however, is that it is not equal-signal-free.

==Exercises==

* First select the number  $(1,\ 2, \text{...})$  of the exercise.  The number  $0$  corresponds to a "Reset":  Same setting as at program start.
*A task description is displayed.  The parameter values are adjusted.  Solution after pressing "Show solution". 

{{BlueBox|TEXT=
'''(1)'''  Explain the occurrence of the eye pattern for  $M=2 \text{, Gaussian low-pass, }f_{\rm G}/R_{\rm B} = 0.48$. For this, select "step–by–step". }}

::* The eye pattern is obtained by dividing the "useful" signal  $d_{\rm S}(t)$  (without noise) into pieces of duration  $2T$  and drawing these pieces on top of each other.
::* In  $d_{\rm S}(t)$  all  "five bit combinations"  must be contained   ⇒   at least  $2^5 = 32$  pieces   ⇒   at most  $32$  distinguishable lines.
::* The eye pattern evaluates the transient response of the signal.  The larger the (normalized) eye opening, the smaller are the intersymbol interferences.

{{BlueBox|TEXT=
'''(2)'''  Same setting as in  $(1)$. In addition,  $10 \cdot \lg \ E_{\rm B}/N_0 = 10 \ {\rm dB}$.  Evaluate the output characteristics  $ö_{\rm norm}$,  $\sigma_{\rm norm}$,  and  $p_{\rm U}$.}}

::* $ö_{\rm norm}= 0.542$  indicates that symbol detection is affected by adjacent pulses. For binary systems without intersymbol interference:  $ö_{\rm norm}= 1$.
::* The eye opening indicates only the signal  $d_{\rm S}(t)$  without noise.  The noise influence is captured by  $\sigma_{\rm norm}= 0.184$ . This value should be as small as possible.
::* The error probability  $p_{\rm U} = {\rm Q}(ö_{\rm norm}/\sigma_{\rm norm}\approx 0.16\%)$  refers solely to the "worst-case sequences", for Gaussian low–pass e.g.  $\text{...}\ , -1, -1, +1, -1, -1, \text{...}$.
::* Other sequences are less distorted   ⇒   the mean error probability  $p_{\rm M}$  is (usually) significantly smaller than $p_{\rm U}$  (describing the worst case).

{{BlueBox|TEXT=
'''(3)'''  The last settings remain.  With which  $f_{\rm G}/R_{\rm B}$  value does the worst case error probability  $p_{\rm U}$  become minimal?  Consider also the eye pattern.}}

::* The minimum value  $p_{\rm U, \ min} \approx 0.65 \cdot 10^{-4}$  is obtained for  $f_{\rm G}/R_{\rm B} \approx 0.8$, and this is almost independent of the setting of  $10 \cdot \lg \ E_{\rm B}/N_0$.
::* The normalized noise rms value does increase compared to the experiment  $(2)$  from  $\sigma_{\rm norm}= 0.168$  to  $\sigma_{\rm norm}= 0.238$.
::* However, this is more than compensated by the larger eye opening  $ö_{\rm norm}= 0.91$  compared to  $ö_{\rm norm}= 0.542$  $($magnification factor $\approx 1.68)$.

{{BlueBox|TEXT=
'''(4)'''  Which cutoff frequencies  $(f_{\rm G}/R_{\rm B})$  result in a completely inadequate error probability  $p_{\rm U} \approx 50\%$ ? Look at the eye pattern again  ("Overall view").}}

::* For  $f_{\rm G}/R_{\rm B}<0.28$  we get a "closed eye"  $(ö_{\rm norm}= 0)$  and thus a worst case error probability on the order of  $50\%$.
::* The decision on unfavorably framed bits must then be random, even with low noise  $(10 \cdot \lg \ E_{\rm B}/N_0 = 16 \ {\rm dB})$.

{{BlueBox|TEXT=
'''(5)'''  Now select the settings  $M=2 \text{, Matched Filter receiver, }T_{\rm E}/T = 1$,  $10 \cdot \lg \ E_{\rm B}/N_0 = 10 \ {\rm dB}$  and  "Overall view". Interpret the results. }}

::* The basic detection impulse  $g_d(t)$  is triangular and the eye is "fully open".  Consequently, the normalized eye opening is  $ö_{\rm norm}= 1.$
::* From  $10 \cdot \lg \ E_{\rm B}/N_0 = 10 \ {\rm dB}$  it follows $E_{\rm B}/N_0 = 10$   ⇒   $\sigma_{\rm norm} =\sqrt{1/(2\cdot E_{\rm B}/ N_0)} = \sqrt{0.05} \approx 0.224 $  ⇒   $p_{\rm U} = {\rm Q}(4.47) \approx 3.9 \cdot 10^{-6}.$
::* This  $p_{\rm U}$ value is by a factor  $15$  better than in  $(3)$.   But:  For  $H_{\rm K}(f) \ne 1$  this so–called "Matched Filter Receiver" is not applicable.

{{BlueBox|TEXT=
'''(6)'''  Same settings as in  $(5)$.  Now vary  $T_{\rm E}/T$  in the range between  $0.5$  and  $1.5$.  Interpret the results.}}

::* For  $T_{\rm E}/T < 1$ ,  $ö_{\rm norm}= 1$  still holds.  But  $\sigma_{\rm norm}$  becomes larger, for example  $\sigma_{\rm norm} = 0.316$  for  $T_{\rm E}/T =0.5$   ⇒   the filter is too broadband!
::* $T_{\rm E}/T > 1$  results in a smaller  $\sigma_{\rm norm}$  compared to  $(5)$.  But the "eye" is no longer open, e.g.  $T_{\rm E}/T =1.25$:   $ö_{\rm norm}= g_0 - 2 \cdot g_1 = 0.6$.

{{BlueBox|TEXT=
'''(7)'''  Now select the settings  $M=2 \text{, CRO Nyquist system, }r_f = 0.2$  and  "Overall view". Interpret the eye pattern, also for other  $r_f$ values. }}

::* Unlike  $(6)$  here the basic detection impulse is not zero for  $|t|>T$,  but  $g_d(t)$  has equidistant zero crossings:  $g_0 = 1, \ g_1 = g_2 = 0$   ⇒   '''Nyquist system'''.
::* All  $32$  eye lines pass through only two points at  $t=0$.  The vertical eye opening is maximum for all  $r_f$    ⇒    $ö_{\rm norm}= 1$.
::* In contrast, the horizontal eye opening increases with  $r_f$  and is for  $r_f = 1$  maximum equal to  $T$   ⇒   the phase jitter has no influence in this case.

{{BlueBox|TEXT=
'''(8)'''  Same setting as in  $(7)$.  Now vary  $r_f$  with respect to minimum error probability.  Interpret the results.}}
::*$ö_{\rm norm}= 1$  always holds.  In contrast,  $\sigma_{\rm norm}$  shows a slight dependence on  $r_f$.  The minimum  $\sigma_{\rm norm}=0.236$  results for  $r_f = 0.9$   ⇒   $p_{\rm U} \approx 1.1 \cdot 10^{-5}.$
::* Compared to the best possible case according to  $(5)$   ⇒   "Matched Filter Receiver"  $p_{\rm U}$  is three times larger, although  $\sigma_{\rm norm}$  is only larger by about  $5\%$.
::* The larger  $\sigma_{\rm norm}$ value is due to the exaggeration of the noise PDS to compensate for the drop through the transmitter frequency response  $H_{\rm S}(f)$.

{{BlueBox|TEXT=
'''(9)'''  Select the settings  $M=4 \text{, Matched Filter receiver, }T_{\rm E}/T = 1$,  $10 \cdot \lg \ E_{\rm B}/N_0 = 10 \ {\rm dB}$  and  $12 \ {\rm dB}$.  Interpret the results. }}

::* Now there are three eye openings.  Compared to  $(5)$   $ö_{\rm norm}$  is thus smaller by a factor of  $3$.  $\sigma_{\rm norm}$  on the other hand, only by a factor of  $\sqrt{5/9)} \approx 0.75$.
::* For  $10 \cdot \lg \ E_{\rm B}/N_0 = 10 \ {\rm dB}$  the  (worst–case)  error probability is  $p_{\rm U} \approx 2.27\%$  and for  $10 \cdot \lg \ E_{\rm B}/N_0 = 12 \ {\rm dB}$  approx.  $0.59\%$.

{{BlueBox|TEXT=
'''(10)'''  For the remaining tasks, always  $10 \cdot \lg \ E_{\rm B}/N_0 = 12 \ {\rm dB}$. Consider the eye pattern  ("overall view")  for  $M=4 \text{, CRO Nyquist system, }r_f = 0.5$. }}

::* In the analyzed  $d_{\rm S}(t)$  region all  "five symbol combinations"  must be contained   ⇒   minimum  $4^5 = 1024$  parts   ⇒   maximum  $1024$  distinguishable lines.
::* All  $1024$  eye lines pass through only four points at  $t=0$ :   $ö_{\rm norm}= 0.333$.  $\sigma_{\rm norm} = 0.143$  is slightly larger than in  $(9)$  ⇒   $p_{\rm U} \approx 1\%$.

{{BlueBox|TEXT=
'''(11)'''  Select the settings  $M=4 \text{, Gaussian low-pass, }f_{\rm G}/R_{\rm B} = 0.48$  and vary  $f_{\rm G}/R_{\rm B}$.  Interpret the results. }}

::* $f_{\rm G}/R_{\rm B}=0.48$  leads to the minimum error probability  $p_{\rm U} \approx 0.21\%$.  $\text{Compromise between}$  $ö_{\rm norm}= 0.312$  and  $\sigma_{\rm norm}= 0.109$.
::* If the cutoff frequency is too small, intersymbol interference dominates.  Example:  $f_{\rm G}/R_{\rm B}= 0.3$:  $ö_{\rm norm}= 0.157; $ $\sigma_{\rm norm}= 0.086$  ⇒    $p_{\rm U} \approx 3.5\%$.
::* If the cutoff frequency is too high, noise dominates.  Example:  $f_{\rm G}/R_{\rm B}= 1.0$:  $ö_{\rm norm}= 0.333; $ $\sigma_{\rm norm}= 0.157$  ⇒    $p_{\rm U} \approx 1.7\%$.
::* From the comparison with  $(9)$  one can see:  $\text{With quaternary coding it is more convenient to allow intersymbol interference}$.

{{BlueBox|TEXT=
'''(12)'''  What differences does the eye pattern show for  $M=3 \text{ (AMI code), Gaussian low-pass, }f_{\rm G}/R_{\rm B} = 0.48$  compared to the binary system  $(1)$? Interpretation. }}
::* The basic detection impulse  $g_d(t)$  is the same in both cases.  The sample values are respectively  $g_0 = 0.771, \ g_1 = 0.114$.
::* With the AMI code, there are two eye openings with each  $ö_{\rm norm}= 1/2 \cdot (g_0 -3 \cdot g_1) = 0.214$.  With the binary code:  $ö_{\rm norm}= g_0 -2 \cdot g_1 = 0.543$.
::* The AMI sequence consists of  $50\%$  zeros.  The symbols  $+1$  and  $-1$  alternate   ⇒   there is no long  $+1$  sequence and no long  $-1$  sequence.
::* Therein lies the only advantage of the AMI code:  This can also be applied to a channel with  $H_{\rm K}(f= 0)=0$  ⇒   a DC signal is suppressed.

{{BlueBox|TEXT=
'''(13)'''  Same setting as in  $(12)$.  Select additionally  $10 \cdot \lg \ E_{\rm B}/N_0 = 12 \ {\rm dB}$.  Analyze the worst-case error probability of the AMI code. }}
::* Despite smaller  $\sigma_{\rm norm} = 0.103$  the AMI code has higher error probability  $p_{\rm U} \approx 2\%$  than the binary code:  $\sigma_{\rm norm} = 0.146, \ p_{\rm U} \approx \cdot 10^{-4}.$
::* $f_{\rm G}/R_{\rm B}<0.34$  results in a closed eye  $(ö_{\rm norm}= 0)$  ⇒    $p_{\rm U} =50\%$.  With binary coding:  For  $f_{\rm G}/R_{\rm B}>0.34$  the eye is open.

{{BlueBox|TEXT=
'''(14)'''  What differences does the eye pattern show for  $M=3 \text{ (Duobinary code), Gaussian low-pass, }f_{\rm G}/R_{\rm B} = 0.30$  compared to the binary system  '''(1)'''? }}
::* With redundancy-free binary code:  $ö_{\rm norm}= 0.096, \ \sigma_{\rm norm} = 0.116 \ p_{\rm U} \approx 20\% $.   With Duobinary code:  $ö_{\rm norm}= 0.167, \ \sigma_{\rm norm} = 0.082 \ p_{\rm U} \approx 2\% $.
::*In particular, with small  $f_{\rm G}/R_{\rm B}$  the Duobinary code gives good results, since the transitions from  $+1$  to  $-1$  (and vice versa) are absent in the eye pattern.
::*Even with  $f_{\rm G}/R_{\rm B}=0.2$  the eye is open.  But in contrast to AMI  the Duobinary code is not applicable with a DC-free channel   ⇒   $H_{\rm K}(f= 0)=0$.

==Applet Manual==
 
[[File:Anleitung_Auge.png|right|600px|frame|Screenshot of the German version]]
    '''(A)'''     Selection:   Coding                    (binary,  quaternary,  AMI code,  duobinary code)

    '''(B)'''     Selection:   Basic detection pulse                     (according to Gauss–TP,  CRO–Nyquist,  according to gap–TP}

    '''(C)'''     Parameter input for  '''(B)'''                    (cutoff frequency,  rolloff factor,  rectangular duration)

    '''(D)'''     Control of the eye diagram display                    (start,  pause/continue,  single step,  total,  reset)

    '''(E)'''     Speed of the eye diagram display

    '''(F)'''     Display:  basic detection pulse  $g_d(t)$

    '''(G)'''     Display:  detection useful signal  $d_{\rm S}(t - \nu \cdot T)$

    '''(H)'''     Display:  eye diagram in the range  $\pm T$

    '''( I )'''     Numerical output:  $ö_{\rm norm}$  (normalized eye opening)

    '''(J)'''     Parameter input  $10 \cdot \lg \ E_{\rm B}/N_0$  for  '''(K)'''

    '''(K)'''     Numerical output:  $\sigma_{\rm norm}$  (normalized noise rms)

    '''(L)'''     Numerical output:  $p_{\rm U}$  (worst-case error probability)

    '''(M)'''     Range for experimental performance:   task selection

    '''(N)'''     Range for experimental performance:   task description

    '''(O)'''     Range for experimental performance:   Show sample solution
 
==About the Authors==

This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
*The first version was created in 2008 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|Thomas Großer]]  as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).

*Last revision and English version 2020/2021 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  in the context of a working student activity. 

The conversion of this applet to HTML 5 was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ Studienzuschüsse]  ("study grants")  of the TUM Faculty EI.  We thank.

==Once again: Open Applet in new Tab==
{{LntAppletLinkEnDe|eyeDiagram_en|eyeDiagram}}

Applets:Principle of 4B3T Coding

2023-03-21T18:07:40Z

Hwang:

{{LntAppletLinkEnDe|4B3T_en|4B3T}}

== Applet Description==
 
Das Applet verdeutlicht das Prinzip der  $\rm 4B3T$–Codierung.  Hierbei wird jeweils ein Block von vier Binärsymbolen durch eine Sequenz aus drei Ternärsymbolen ersetzt.  Daraus ergibt sich eine relative Coderedundanz von knapp  $16\%$,  die dazu verwendet wird, um Gleichsignalfreiheit zu erzielen.

Die Umcodierung der sechzehn möglichen Binärblöcke in die entsprechenden Ternärblöcke könnte prinzipiell nach einer festen Codetabelle erfolgen. Um die spektralen Eigenschaften dieser Codes weiter zu verbessern, werden bei den 4B3T–Codes aber stets mehrere Codetabellen verwendet, die nach der "laufenden digitalen Summe"  $($englisch:  ''Running Digital Sum'',  kurz  $\rm RDS)$  blockweise ausgewählt werden.

Im Applet sind im unteren Bereich die entsprechenden Codetabellen angegeben, und zwar alternativ für
* den $\rm MS43$–Code (von:   $\rm M$onitored $\rm S$um $\rm 4$B$\rm 3$T–Code), und
* den $\rm MMS43$–Code (von:  $\rm M$odified $\rm MS43$).

Eingabeparameter sind neben dem gewünschten Code (MS43 oder MMS43) der RDS–Startwert  $\rm RDS_0$  sowie zwölf binäre Quellensymbole  $q_\nu \in \{0,\ 1\}$,  entweder per Hand, per Voreinstellung  $($Quellensymbolfolge  $\rm A$,  $\rm B$,  $\rm C)$ oder per Zufallsgenerator. 

Vom Programm angeboten werden zwei verschiedene Modi:

*Im Modus "Schritt" werden die drei Blöcke sukzessive abgearbeitet (jeweils Festlegung der drei Ternärsymbole, Aktualisierung des RDS–Wertes und damit Festlegung der Codetabelle für den nächsten Block.

*Im Modus "Gesamt" werden nur die Codierergebnisse angezeigt, aber gleichzeitig für die beiden möglichen Codes und jeweils für alle vier möglichen RDS–Startwerte.  Die Grafik und der RDS–Ausgabeblock rechts beziehen sich dabei auf die getroffenen Einstellungen.

==Theoretical Background==

===Klassifizierung verschiedener Codierverfahren ===
 
Wir betrachten das dargestellte digitale Übertragungsmodell.  Wie aus diesem Blockschaltbild zu erkennen ist, unterscheidet man je nach Zielrichtung zwischen drei verschiedenen Arten von Codierung, jeweils realisiert durch den sendeseitigen Codierer (Coder) und den zugehörigen Decodierer (Decoder) beim Empfänger:

[[File:P_ID2315__Inf_T_2_1_S1_neu.png|right|frame|Vereinfachtes Modell eines Nachrichtenübertragungssystems]]

*$\text{Quellencodierung:}$  Entfernen (unnötiger) Redundanz, um Daten möglichst effizient speichern oder übertragen zu können   ⇒   Datenkomprimierung.  Beispiel:  Differentielle Pulscodemodulation  $\rm (DPCM)$  in der Bildcodierung.

*$\text{Kanalcodierung:}$  Gezieltes Hinzufügen (sinnvoller) Redundanz, die man beim Empfänger zur Fehlererkennung oder zur Fehlererkennung nutzen kann.  Wichtigste Vertreter:  Blockcodes, Faltungscodes, Turbocodes.

*$\text{Leitungscodierung:}$  Umcodierung der Quellensymbole, um das Signal an die Spektraleigenschaften von Kanal und Empfangseinrichtungen anzupassen, etwa, um bei einem Kanal mit  $H_{\rm K}(f = 0) = 0$  ein gleichsignalfreies Sendesignal  $x(t)$  zu erreichen.  
 
Bei den Leitungscodes unterscheidet man weiter:
*$\text{Symbolweise Codierung:}$  Mit jedem ankommenden Binärsymbol  $q_ν$  wird ein mehrstufiges (zum Beispiel: ternäres) Codesymbol  $c_ν$  erzeugt, das auch von den vorherigen Binärsymbolen abhängt.  Die Symboldauern  $T_q$  und  $T_c$  sind hierbei identisch.  Beispiel:  Pseudoternärcodes (AMI–Code, Duobinärcode).

*$\text{Blockweise Codierung:}$  Ein Block aus  $m_q$  Binärsymbolen  $(M_q = 2)$  wird durch eine Sequenz aus  $m_c$  höherstufigen Symbolen  $(M_c > 2)$  ersetzt.  Ein Kennzeichen dieser Codeklasse ist  $T_c> T_q$.  Beispiele sind redundanzfreie Mehrstufencodes  $(M_c$ ist eine Zweierpotenz$)$  sowie die hier betrachteten $\text{4B3T-Codes}$.

=== Allgemeine Beschreibung der 4B3T–Codes ===

Der bekannteste Blockcode zur Leitungscodierung ist der  $\rm 4B3T–Code$  mit den Codeparametern
:$$m_q = 4,\hspace{0.2cm}M_q = 2,\hspace{0.2cm}m_c =
3,\hspace{0.2cm}M_c = 3\hspace{0.05cm},$$

der bereits in den 1970–er Jahren entwickelt wurde und beispielsweise bei  [[Beispiele_von_Nachrichtensystemen/Allgemeine_Beschreibung_von_ISDN| ISDN]]  (Integrated Services Digital Networks ) eingesetzt wird.

Ein 4B3T–Code besitzt folgende Eigenschaften:
*Wegen  $m_q \cdot T_q = m_c \cdot T_c$  ist die Symboldauer  $T=T_c$  des Codersignals um den Faktor  $4/3$  größer als die Bitdauer  $T_{\rm B}=T_q$  des binären Quellensignals. Daraus ergibt sich die günstige Eigenschaft, dass der Bandbreitenbedarf um ein Viertel geringer ist als bei redundanzfreier Binärübertragung.
*Die relative Redundanz der 4B3T–Codes ergibt sich zu

:$$r_c = 1- \frac{m_q \cdot {\rm log_2}\hspace{0.05cm} (M_q)}{m_c \cdot {\rm log_2} \hspace{0.05cm}(M_c)} = 1- \frac{4 \cdot {\rm log_2}\hspace{0.05cm} (2)}{3 \cdot {\rm log_2} \hspace{0.05cm}3}= 1- \frac{4 }{3 \cdot 1.585}\hspace{0.05cm}\approx{0.158}.$$
*Diese Redundanz von knapp  $16\%$  wird dazu verwendet, um Gleichsignalfreiheit zu erzielen.  Das 4B3T–codierte Signal kann somit ohne merkbare Beeinträchtigung auch über einen Kanal mit der Eigenschaft  $H_{\rm K}(f= 0) = 0$  übertragen werden.

Die Umcodierung der sechzehn möglichen Binärblöcke in die entsprechenden Ternärblöcke könnte prinzipiell nach einer festen Codetabelle vorgenommen werden.  Um die spektralen Eigenschaften dieser Codes weiter zu verbessern, werden bei den gebräuchlichen 4B3T–Codes, nämlich

*dem 4B3T–Code nach Jessop und Waters, 
*dem MS43–Code (von:  $\rm M$onitored $\rm S$um $\rm 4$B$\rm 3$T–Code), 
*dem FoMoT–Code (von:  $\rm Fo$ur $\rm Mo$de $\rm T$ernary), 

zwei oder mehrere Codetabellen verwendet, deren Auswahl von der "laufenden digitalen Summe" der Amplitudenkoeffizienten gesteuert wird.

=== Laufende digitale Summe ===

[[File:P_ID1334__Dig_T_2_3_S2.png|right|frame|Codetabellen für drei 4B3T-Codes|class=fit]]

Die ternären Amplitudenkoeffizienten seien $a_\nu \in \{ -1, \ 0, +1\}$. 

Nach der Übertragung von  l  Blöcken gilt für die "Laufende Digitale Summe"  $($englisch:  ''Running Digital Sum'',  kurz  $\rm RDS)$:
:$${\it \Sigma}_l = \sum_{\nu = 1}^{3 \hspace{0.05cm}\cdot
\hspace{0.05cm} l}\hspace{0.02cm} a_\nu \hspace{0.05cm}.$$

Die Auswahl der Tabelle zur Codierung des  $(l + 1)$–ten Blocks erfolgt abhängig vom aktuellen Wert  ${\it \Sigma}_l$.

In der Tabelle sind die Codierregeln für die drei oben genannten 4B3T–Codes angegeben. Zur Vereinfachung der Schreibweise steht "+" für den Amplitudenkoeffizienten "+1" und "–" für den Koeffizienten "–1". 

# Die zwei Codetabellen des Jessop–Waters–Codes sind so gewählt, dass die laufende digitale Summe  ${\it \Sigma}_l$  stets zwischen $0$ und $5$ liegt.
# Bei den beiden anderen Codes (MS43, FoMoT) erreicht man durch drei bzw. vier alternative Tabellen die Beschränkung der laufenden digitalen Summe auf den Wertebereich  $0 \le {\it \Sigma}_l \le 3$.

Im Applet werden betrachtet:
* der $\rm MS43$–Code (von:   $\rm M$onitored $\rm S$um $\rm 4$B$\rm 3$T–Code),
* der $\rm MMS43$–Code (von:  $\rm M$odified $\rm MS43$).

Zur Farbgebung der nebenstehenden Grafik:
# Graue Hintergung:  Der RDS–Wert bleibt gleich:  ${\it \Sigma}_{l+1} = {\it \Sigma}_l$.
# Rote Hintergung:  Der RDS–Wert wird größer:  ${\it \Sigma}_{l+1} > {\it \Sigma}_l$.
# Blaue Hintergung:  Der RDS–Wert wird kleiner:  ${\it \Sigma}_{l+1} < {\it \Sigma}_l$.
# Zu– bzw. Abnahme ist umso größer, je intensiver die Farben sind.

=== AKF und LDS der 4B3T–Codes===

[[File:P_ID1335__Dig_T_2_3_S3_v1.png|right|frame|Markovdiagramm zur Analyse des 4B3T-Codes (FoMoT)|class=fit]]
Die Vorgehensweise zur Berechnung von AKF und LDS wird hier nur stichpunktartig skizziert (im Applet wird hierauf nicht eingegangen):

'''(1)'''   Der Übergang der laufenden digitalen Summe von  ${\it \Sigma}_l$  nach  ${\it \Sigma}_{l+1}$  wird durch eine homogene stationäre Markovkette erster Ordnung mit sechs (Jessop–Waters) bzw. vier Zuständen (MS43, FoMoT) beschrieben. Für den FoMoT–Code gilt das rechts skizzierte Markovdiagramm. 

'''(2)'''   Die Werte an den Pfeilen kennzeichnen die Übergangswahrscheinlichkeiten  ${\rm Pr}({\it \Sigma}_{l+1}|{\it \Sigma}_{l})$, die sich aus den jeweiligen Codetabellen ergeben. Die Farben korrespondieren zu den Hinterlegungen der Tabelle auf der letzten Seite. Aufgrund der Symmetrie des FoMoT–Markovdiagramms sind die vier Wahrscheinlichkeiten alle gleich:
:$${\rm Pr}({\it \Sigma}_{l} = 0) = \text{...} = {\rm Pr}({\it \Sigma}_{l} = 3) = 1/4.$$

'''(3)'''   Die Autokorrelationsfunktion (AKF)  $\varphi_a(\lambda) = {\rm E}\big [a_\nu \cdot a_{\nu+\lambda}\big ]$  der Amplitudenkoeffizienten kann aus diesem Diagramm ermittelt werden. Einfacher als die analytische Berechnung, die eines sehr großen Rechenaufwands bedarf, ist die simulative Bestimmung der AKF–Werte mittels Computer. 

Durch Fouriertransformation der AKF kommt man zum Leistungsdichtespektrum (LDS)  ${\it \Phi}_a(f)$  der Amplitudenkoeffizienten entsprechend der folgenden Grafik aus [ST85]<ref name ='ST85'>Söder, G.; Tröndle, K.: Digitale Übertragungssysteme - Theorie, Optimierung & Dimensionierung der Basisbandsysteme. Berlin – Heidelberg: Springer, 1985.</ref>. Das skizzierte LDS wurde für den FoMoT–Code ermittelt, dessen Markovdiagramm oben dargestellt ist. Die Unterschiede der einzelnen 4B3T–Codes sind nicht sonderlich ausgeprägt. So gilt für den MS43–Code  ${\rm E}\big [a_\nu^2 \big ] \approx 0.65$  und für die beiden anderen 4B3T-Codes (Jessop/Waters, MS43)  ${\rm E}\big [a_\nu^2 \big ] \approx 0.69$. 
[[File:P_ID1336__Dig_T_2_3_S3b_v1.png|right|frame|Leistungsdichtespektrum (der Ampltudenkoeffizienten) von 4B3T im Vergleich zu redundanzfreier und AMI-Codierung|class=fit]]
Die Aussagen dieser Grafik kann man wie folgt zusammenfassen:

*Die Grafik zeigt das LDS  ${\it \Phi}_a(f)$  der Amplitudenkoeffizienten  $a_\nu$  des 4B3T-Codes   ⇒   rote Kurve.
*Das LDS  ${\it \Phi}_s(f)$  unter Einbeziehung des Sendegrundimpulses erhält man durch Multiplikation mit  $1/T \cdot |G(f)|^2$. Beispielsweise muss man  ${\it \Phi}_a(f)$  mit einer  $\rm si^2$–Funktion multiplizieren, wenn  $g(t)$  einen Rechteckimpuls beschreibt. 
*Bei redundanzfreier Binär– oder Ternärcodierung ergibt sich jeweils ein konstantes  ${\it \Phi}_a(f)$, dessen Höhe von der Stufenzahl  $M$  abhängt (unterschiedliche Signalleistung).
*Dagegen weist das 4B3T–Leistungsdichtespektrum Nullstellen bei  $f = 0$  und Vielfachen von  $f = 1/T$  auf. 
*Die Nullstelle bei  $f = 0$  hat den Vorteil, dass das 4B3T–Signal ohne große Einbußen auch über einen so genannten ''Telefonkanal''  übertragen werden kann, der aufgrund von Übertragern für ein Gleichsignal nicht geeignet ist. 
*Die Nullstelle bei  $f = 1/T$  hat den Nachteil, dass dadurch die Taktrückgewinnung am Empfänger erschwert wird. Außerhalb dieser Nullstellen weisen die 4B3T–Codes ein flacheres  ${\it \Phi}_a(f)$  auf als beispielsweise der  [[Digitalsignalübertragung/Symbolweise_Codierung_mit_Pseudoternärcodes#Eigenschaften_des_AMI-Codes|AMI–Code]]  (blaue Kurve), was von Vorteil ist. 
*Der Grund für den flacheren LDS–Verlauf bei mittleren Frequenzen sowie den steileren Abfall zu den Nullstellen hin ist, dass bei den 4B3T–Codes bis zu fünf  $+1$–  bzw.  $-1$–Koeffizienten aufeinanderfolgen können.  Beim AMI–Code treten diese Symbole nur isoliert auf. 

==Exercises==

*First, select the number  $(1,\ 2, \text{...} \ )$  of the task to be processed.  The number  $0$  corresponds to a "Reset":  Same setting as at program start.
*A task description is displayed.  The parameter values are adjusted.  Solution after pressing "Show Solution".
*Both the input signal  $x(t)$  and the filter impulse response  $h(t)$  are normalized, dimensionless and energy-limited ("time-limited pulses").
*All times, frequencies, and power values are to be understood normalized, too.

{{BlueBox|TEXT=
'''(1)'''  Illustrate the 4B3T coding of the source symbol sequence  $\rm A$   ⇒   $\langle q_\nu \rangle = \langle 0, 1, 0, 1; \ 1, 0, 1, 1; \ 0, 1, 1, 0 \rangle $  according to the  $\rm MS43$  code ("Block–by–Block").         Let the RDS initial value be  ${\it \Sigma}_0= 0$.   ''Note'':  The source symbol sequence is already divided by semicolons into subsequences of four bits each. }}
* Starting from the RDS initial value  ${\it \Sigma}_0= 0$  you recognize the following coding of the first four bits (first block):  $(0, 1, 0, 1)\ \rightarrow\ (+,\ 0 ,\ +) $   ⇒   ${\it \Sigma}_1= 2.$
*For the next four bits (second block), now assume  ${\it \Sigma}_1= 2$  $(1, 0, 1, 1)\ \rightarrow\ (+,\ 0 ,\ 0) $   ⇒   ${\it \Sigma}_2= 3.$
*The encoding of bits 9 to 12 (third block) results:  ${\it \Sigma}_2= 3$  to  $(0, 1, 1, 0,)\ \rightarrow\ (-,\ 0 ,\ 0) $   ⇒   ${\it \Sigma}_3= 2.$

{{BlueBox|TEXT=
'''(2)'''  Repeat this experiment with the other possible RDS initial values  ${\it \Sigma}_0= 1$,  ${\it \Sigma}_0= 2$  and  ${\it \Sigma}_0= 3.$  How do the coding results differ? }}

*${\it \Sigma}_0= 1$:     $(0, 1, 0, 1)\ \rightarrow\ (0,\ - ,\ 0) $   ⇒   ${\it \Sigma}_1= 0$:     $(1, 0, 1, 1)\ \rightarrow\ (+,\ 0 ,\ 0) $   ⇒   ${\it \Sigma}_2= 1$:     $(0, 1, 1, 0)\ \rightarrow\ (-,\ 0 ,\ 0) $   ⇒   ${\it \Sigma}_3= 0.$
*${\it \Sigma}_0= 2$:     $(0, 1, 0, 1)\ \rightarrow\ (0,\ - ,\ 0) $   ⇒   ${\it \Sigma}_1= 1$:     $(1, 0, 1, 1)\ \rightarrow\ (+,\ 0 ,\ 0) $   ⇒   ${\it \Sigma}_2= 2$:     $(0, 1, 1, 0)\ \rightarrow\ (-,\ 0 ,\ 0) $   ⇒   ${\it \Sigma}_3= 1.$
*${\it \Sigma}_0= 3$:     $(0, 1, 0, 1)\ \rightarrow\ (0,\ - ,\ 0) $   ⇒   ${\it \Sigma}_1= 2$:     $(1, 0, 1, 1)\ \rightarrow\ (+,\ 0 ,\ 0) $   ⇒   ${\it \Sigma}_2= 3$:     $(0, 1, 1, 0)\ \rightarrow\ (-,\ 0 ,\ 0) $   ⇒   ${\it \Sigma}_3= 2.$

{{BlueBox|TEXT=
'''(3)'''  How many different code tables does the  $\rm MS43$ code use? }}

*From the previous experiments, we can see that the MS43 code uses at least two tables, switching between them according to the current RDS value.
*From the table given in the applet, it can be seen that three tables are actually used.  The entries for  ${\it \Sigma}_l= 1$  and  ${\it \Sigma}_l= 2$  are in fact identical.

{{BlueBox|TEXT=
'''(4)'''  Interpret the results of 4B3T coding for the source symbol sequence  $\rm B$   ⇒   $\langle q_\nu \rangle = \langle 1, 1, 1, 0; \ 0, 0, 1, 0; \ 1, 1, 1 \rangle $  and the MS43 code.}}

*For this source symbol sequence, the RDS value is not changed.   For each starting value  $(0$,  $1$,  $2$  and  $3)$  holds  ${\it \Sigma}_0 = {\it \Sigma}_1 ={\it \Sigma}_2 ={\it \Sigma}_3 $,  for example:
*${\it \Sigma}_0= 1$:     $(1, 1, 1, 0)\ \rightarrow\ (0,\ - ,\ +) $   ⇒   ${\it \Sigma}_1= 1$:     $(0, 0, 1, 0)\ \rightarrow\ (+,\ 0 ,\ -) $   ⇒   ${\it \Sigma}_2= 1$:     $(1, 1, 1, 1)\ \rightarrow\ (-,\ 0 ,\ +) $   ⇒   ${\it \Sigma}_3= 1.$
*The reason for this is that with this source symbol sequence, each ternary–triple contains exactly one "plus" and one "minus" after encoding.

{{BlueBox|TEXT=
'''(5)'''  In contrast, how many different code tables does the modified MS43 code   ⇒   $\rm MMS43$ use? }}

*It can be seen from the table given in the applet that in the modified MS43 code all four tables are in fact different.  
*The entries for  ${\it \Sigma}_l= 1$  and  ${\it \Sigma}_l= 2$  are indeed largely the same.  They differ only for the binary sequences  $(0, 1, 1, 0)$  and  $(1, 0, 1, 0)$.
*The  $\rm MMS43$ code is used with  $\rm ISDN$  ("Integrated Services Digital Network")  on the local loop  $(U_{K0}$ interface$)$. 
*We do not know why the original MS43 code was modified during standardization.  We suspect a slightly more favorable power density spectrum.

{{BlueBox|TEXT=
'''(6)'''  Compare the  $\rm MS43$  and  $\rm MMS43$  results for the source symbol sequences  $\rm A$  and  $\rm B$  and any RDS initial values.  Select "Overall View".}}
*For source symbol sequence  $\rm A$  there are two different  $\rm MS43$  code symbol sequences and three different   $\rm MMS43$  code symbol sequences.
*For the source symbol sequence  $\rm B$  the  $\rm MS43$  code symbol sequences are the same for all possible RDS initial values.  For  $\rm MMS43$:  two different coding results.

{{BlueBox|TEXT=
'''(7)'''  Interpret the results for the sequence  $\rm C$   ⇒   $\langle q_\nu \rangle = \langle 0, 1, 1, 0; \ 0, 1, 1, 0; \ 0, 1, 1, 0 \rangle $  for both codes and all RDS initial values. Select "Overall View".}}
*The four input bits of each block are  $(0,\ 1,\ 1,\ 0)$.   With  $\rm MS43$  these are replaced by  $(0,\ +,\ +)$,  if  ${\it \Sigma}_l=0$;  resp.  $(-,\ 0,\ 0)$,  if  ${\it \Sigma}_l\ne0$.
*In the  $\rm MMS43$,  however, these are replaced by  $(-,\ +,\ +)$,  if  ${\it \Sigma}_l\le 1$;  resp.   $(-,\ -,\ +)$,  if  ${\it \Sigma}_l\ge 2$.  '''Only if you have enough time to spare:'''
*Try to make sense of this modification from  $\rm MS43$  to  $\rm MMS43$.  Our LNTww team did not succeed.

==Applet Manual==
 
[[Datei:Anleitung_4B3T.png|right|600px]]
 
    '''(A)'''     Auswahl eines von vier Quellensignalen

    '''(B)'''     Parameterwahl für Quellensignal  $1$  (Amplitude, Frequenz, Phase)

    '''(C)'''     Ausgabe der verwendeten Programmparameter

    '''(D)'''     Parameterwahl für Abtastung  $(f_{\rm G})$  und                 Signalrekonstruktion  $(f_{\rm A},\ r)$

    '''(E)'''     Skizze des Empfänger–Frequenzgangs  $H_{\rm E}(f)$

    '''(F)'''     Numerische Ausgabe  $(P_x, \ P_{\rm \varepsilon}, \ 10 \cdot \lg(P_x/ P_{\rm \varepsilon})$

    '''(G)'''     Darstellungsauswahl für Zeitbereich

    '''(H)'''     Grafikbereich für Zeitbereich

    '''( I )'''     Darstellungsauswahl für Frequenzbereich

    '''(J)'''     Grafikbereich für Frequenzbereich

    '''(K)'''     Bereich für Übungen:  Aufgabenauswahl, Fragen, Musterlösung
 
==About the Authors==

This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
*The first version was created in 2010 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Stefan_M.C3.BCller_.28Diplomarbeit_LB_2010.29|Stefan Müller]]  as part of his bachelor thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).

*Last revision and English version 2020/2021 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  in the context of a working student activity. 

The conversion of this applet to HTML 5 was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ Studienzuschüsse]  ("study grants")  of the TUM Faculty EI.  We thank.

==Once again:  Open Applet in new Tab==

{{LntAppletLinkEnDe|4B3T_en|4B3T}}

Applets:Matched Filter Properties

2023-03-21T18:07:13Z

Hwang:

{{LntAppletLinkEnDe|matchedFilter_en|matchedFilter}}

== Applet Description==
 
The applet is intended to illustrate the properties of the so-called "matched filter"  $({\rm MF})$.  This is used to optimally determine the presence (detection) of the amplitude and/or location of a known waveform in a highly noisy environment.  Or more generally speaking:  The matched filter – sometimes also referred to as "optimal filter"  or as "correlation filter"  – is used to detect the existence of the signal. 

[[File:EN_Sto_T_5_4_S1_neu2.png |right|frame| Block diagram of the  "matched filter receiver"]]

The graphic shows the so-called  '''matched filter receiver''':

*This can decide with the greatest possible certainty – in other words:   with the maximum signal–to–noise–ratio  $($ $\rm SNR)$  – whether a pulse-shaped signal component  $g(t)$  disturbed by additive noise  $n(t)$  is present or not.
*One application is radar technology, where the pulse shape  $g(t)$  is known, but not when the pulse was sent and with what strength and delay it arrives.
*However, the matched filter is also used as a receiver filter in digital transmission systems (or at least as part of them) to minimize the error probability of the system.

All parameters, times and frequencies are to be understood as normalized quantities and thus dimensionless.

* For the '''input pulse'''  $g(t)$,  "rectangular",  "Gaussian"  and  "exponential"  can be set, each described by the pulse amplitude  $A_g$,  the equivalent pulse duration  $\Delta t_g$  and the shift  $\tau_g$  with respect to the (with respect to time) symmetrical case.  Further information in the section  [[Applets:Matched_Filter_Properties#Further_information_about_the_considered_input_pulses|Further information about the considered input pulses]].
* For the '''receiver filter'''  you can choose between the alternatives  "slit lowpass",  "Gaussian lowpass",  "first order lowpass" and  "lowpass 4".  The respective impulse responses  $h(t)$,  are shown, characterized by their height  $A_h$,  the equivalent duration  $\Delta t_h$  and the shift  $\tau_h$.  Further information in the section  [[Applets:Matched_Filter_Properties#Further_information_on_the_considered_impulse_responses|Further information on the considered impulse responses]].
* Further input parameters are the detection time  $T_{\rm D}$  and the also normalized noise power density  $N_0$  at the receiver input.

The following are output as numerical values
*the energy  $E_g$  of the input pulse  $g(t)$,  useful signal value  $d_{\rm S} (T_{\rm D})$  at the filter output as well as the noise variance  $\sigma_d^2$  at the filter output,
*the signal–to–noise ratio  $\rm (SNR)$  $\rho_{d} (T_{\rm D})$  at the filter output and the corresponding dB specification  $10 \cdot \lg \ \rho_{d} (T_{\rm D})$,
*the maximum value for this is  $10 \cdot \lg \ \rho_{\rm MF}$. 

If the input configuration meets the matched filter conditions, then:   $10 \cdot \lg \ \rho_{d} (T_{\rm D,\ opt}) = 10 \cdot \lg \ \rho_{\rm MF}$.

==Theoretical Background==
 
===Detailed description of the underlying model===

The following conditions apply to the individual components:
*Let the useful component  $g(t)$  of the received signal  $r(t)=g(t)+n(t)$  be pulse-shaped and thus  "energy-limited". 
*That means:   The integral over  $\big [g(t)\big ]^2$  from  $–∞$  to  $+∞$  yields the finite value  $E_g$.
*Let the noise signal  $n(t)$  be  "white Gaussian noise"  with  (one–sided)  noise power density  $N_0$.
*The signal  $d(t)$  is additively composed of two components:  The component  $d_{\rm S}(t)$  is due to the  "$\rm S$"ignal  $g(t)$,  the component  $d_{\rm N}(t)$  is due to the  "$\rm N$"oise  $n(t)$.
*The receiver,  consisting of a linear filter   ⇒   frequency response  $H_{\rm MF}(f)$  and the  "decision maker",  is to be dimensioned so that the instantaneous S/N ratio at the output is maximized:
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {d_{\rm S} ^2 ( {T_{\rm D} } )} }{ {\sigma _d ^2 } }\mathop = \limits^{\rm{!} }\hspace{0.1cm} {\rm{Maximum} }.$$
*Here,  $σ_d^2$  denotes the  variance  ("power")  of the signal  $d_{\rm N}(t)$,  and  $T_{\rm D}$  denotes the (suitably chosen)  "detection time".

===Matched filter optimization===

Let be given an energy-limited useful signal  $g(t)$  with the corresponding spectrum  $G(f)$.
*Thus,  the filter output signal at detection time  $T_{\rm D}$  for any filter with impulse response  $h(t)$  and frequency response  $H(f) =\mathcal{ F}\{h(t)\}$  can be written as follows  (ignoring noise   ⇒   subscript  $\rm S$  for "signal"):
:$$d_{\rm S} ( {T_{\rm D} } ) = g(t) * h(t) = \int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e}}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} }\hspace{0.1cm} {\rm{d}}f} .$$
*The  "noise component"  $d_{\rm N}(t)$  of the filter output signal  (subscript  $\rm N$  for "noise")  stems solely from the white noise  $n(t)$  at the input of the receiver.  For its variance  (power)  applies independently of the detection time  $T_{\rm D}$:
:$$\sigma _d ^2 = \frac{ {N_0 } }{2} \cdot \int_{ - \infty }^{ + \infty } {\left| {H(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} .$$
*Thus,  the optimization problem at hand is:
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left| {\int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e} }^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} }\hspace{0.1cm} {\rm{d} }f} } \right|^2 } }{ {N_0 /2 \cdot \int_{ - \infty }^{ + \infty } {\left| {H(f)} \right|^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } } \stackrel{!}{=} {\rm{Maximum} }.$$

{{BlaueBox|TEXT=
$\text{Here first without proof:}$    One can show that this quotient becomes largest for the following frequency response  $H(f)$: 
:$$H(f) = H_{\rm MF} (f) = K_{\rm MF} \cdot G^{\star} (f) \cdot {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} } . $$
*Thus,  for the signal–to–noise power ratio at the matched filter output  $($independent of the dimensionally constant  $K_{\rm MF})$,  we obtain:
:$$\rho _d ( {T_{\rm D} } ) = { {2 \cdot E_g } }/{ {N_0 } }.$$

* $E_g$  denotes the energy of the input pulse,  which can be calculated using  [https://en.wikipedia.org/wiki/Parseval%27s_theorem Parseval's theorem]  in both the time and frequency domains:
:$$E_g = \int_{ - \infty }^{ + \infty } {g^2 (t)\hspace{0.1cm}{\rm{d} }t} = \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right\vert ^{\rm{2} }\hspace{0.1cm} {\rm d}f} .$$}}

{{GraueBox|TEXT=
$\text{Example 1:}$   A rectangular pulse  $g(t)$  with amplitude  $\rm 1\hspace{0.05cm}V$,  duration  $0.5\hspace{0.05cm} \rm ms$  and unknown position is to be found in a noisy environment.
*Thus the pulse energy  $E_g = \rm 5 · 10^{–4} \hspace{0.05cm}V^2s$.
*Let the noise power density be  $N_0 = \rm 10^{–6} \hspace{0.05cm}V^2/Hz$.

The best result   ⇒   the  '''maximum S/N ratio'''  is obtained with the matched filter:
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {2 \cdot E_g } }{ {N_0 } } =
\frac{ {2 \cdot 5 \cdot 10^{-4}\, {\rm V^2\,s} } }{ {10^{-6}\, {\rm V^2/Hz} } } = 1000
\hspace{0.3cm}\Rightarrow\hspace{0.3cm}
10 \cdot {\rm lg}\hspace{0.15cm}\rho _d ( {T_{\rm D} } ) = 30\,{\rm dB}.$$}}

The matched filter criterion given above is now derived step by step.  If you are not interested in this, please skip to the next section  [[Theory_of_Stochastic_Signals/Matched_Filter#Interpretation_of_the_matched_filter|"Interpretation of the matched filter"]].

{{BlaueBox|TEXT=
$\text{Derivation of the matched filter criterion:}$ 

$(1)$  The Schwarz inequality with the two  (generally complex)  functions  $A(f)$  and  $B(f)$:
:$$\left \vert {\int_a^b {A(f) \cdot B(f)\hspace{0.1cm}{\rm{d} }f} } \right \vert ^2 \le \int_a^b {\left \vert {A(f)} \right \vert^{\rm{2} } \hspace{0.1cm}{\rm{d} }f} \cdot \int_a^b {\left\vert {B(f)} \right \vert^{\rm{2} } \hspace{0.1cm}{\rm{d} }f} .$$
$(2)$  We now apply this equation to the signal–to–noise ratio:
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left \vert {\int_{ - \infty }^{ + \infty } {G(f) \cdot H(f) \cdot {\rm{e} }^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} } \hspace{0.1cm}{\rm{d} }f} } \right \vert^2 } }{ {N_0 /2 \cdot \int_{ - \infty }^{ + \infty } {\left \vert {H(f)} \right \vert^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } }.$$
$(3)$  Thus,  with  $A(f) = G(f)$  and  $B(f) = H(f) · {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} }$  the following bound is obtained:
:$$\rho_d ( {T_{\rm D} } ) \le \frac{1}{ {N_0 /2} } \cdot \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert^{\rm{2} } }\hspace{0.1cm}{\rm{d} }f .$$
$(4)$  We now tentatively set for the filter frequency response:
:$$H(f) = H_{\rm MF} (f) = K_{\rm MF} \cdot G^{\star} (f) \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} }.$$
$(5)$  Then,  from the above equation  $(2)$,  we obtain the following result:
:$$\rho _d ( {T_{\rm D} } ) = \frac{ {\left \vert K_{\rm MF}\cdot {\int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } \right \vert ^2 } }{ {N_0 /2 \cdot K_{\rm MF} ^2 \cdot \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} } } = \frac{1}{ {N_0 /2} } \cdot \int_{ - \infty }^{ + \infty } {\left \vert {G(f)} \right \vert ^{\rm{2} }\hspace{0.1cm} {\rm{d} }f} .$$

$\text{This means:}$
*With the approach  $(4)$  for the matched filter $H_{\rm MF}(f)$,  the maximum possible value is indeed obtained in the above estimation.
*No other filter  $H(f) ≠ H_{\rm MF}(f)$  can achieve a higher signal–to–noise power ratio.
*The matched filter is optimal with respect to the maximization criterion on which it is based.
<div align="right">'''q.e.d.'''</div>
}}

===Interpretation of the matched filter===

In the last section,  the frequency response of the matched filter was derived as follows:
:$$H_{\rm MF} (f) = K_{\rm MF} \cdot G^{\star} (f) \cdot {\rm{e} }^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm D} } .$$
By  [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|Fourier inverse transformation]]  the corresponding impulse response is obtained:
:$$h_{\rm MF} (t) = K_{\rm MF} \cdot g(T_{\rm D} - t).$$

These two functions can be interpreted as follows:
*The  "matched filter"  is matched by the term  $G^{\star}(f)$  to the spectrum of the pulse  $g(t)$  which is to be found  –  hence its name.
*The  "constant"  $K_{\rm MF}$  is necessary for dimensional reasons.
*If  $g(t)$  is a voltage pulse,  this constant has the unit "Hz/V".  The frequency response  $H_{\rm MF} (f)$  is therefore dimensionless.
*The  "impulse response"  $h_{\rm MF}(t)$  results from the useful signal  $g(t)$  by mirroring   ⇒   from  $g(t)$  becomes $g(–t)$ $]$   as well as a shift by  $T_{\rm D}$  to the right.
*The  "earliest detection time"  $T_{\rm D}$  follows for realizable systems from the condition  $h_{\rm MF}(t < 0)\equiv 0$   $($"causality",  see book  [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Causal_systems|Linear and Time-Invariant Systems]]$)$.
*The  "useful component"  $d_{\rm S} (t)$  of the filter output signal is equal in shape to the  [[Digital_Signal_Transmission/Grundlagen_der_codierten_Übertragung#AKF.E2.80.93Berechnung_eines_Digitalsignals|energy auto-correlation function]]   $\varphi^{^{\bullet} }_{g} (t )$  and shifted with respect to it by  $T_{\rm D}$.  It holds:
:$$d_{\rm S} (t) = g(t) * h_{\rm MF} (t) = K_{\rm MF} \cdot g(t) * g(T_{\rm D} - t) = K_{\rm MF} \cdot \varphi^{^{\bullet} }_{g} (t - T_{\rm D} ).$$

{{BlaueBox|TEXT=
$\text{Please note:}$ 
For an energy-limited signal  $g(t)$,  one can only specify the  '''energy ACF''': 
:$$\varphi^{^{\bullet} }_g (\tau ) = \int_{ - \infty }^{ + \infty } {g(t) \cdot g(t + \tau )\,{\rm{d} }t} .$$
Compared to the ACF definition of a power-limited signal  $x(t)$, viz.
:$$\varphi _x (\tau ) = \mathop {\lim }_{T_{\rm M} \to \infty } \frac{1}{ {T_{\rm M} } }\int_{ - T_{\rm M} /2}^{+T_{\rm M} /2} {x(t) \cdot x(t + \tau )\hspace{0.1cm}\,{\rm{d} }t} ,$$
the division by the measurement duration  $T_{\rm M}$  and the boundary transition  $T_{\rm M} → ∞$  are omitted in the calculation of the energy ACF.}}

{{GraueBox|TEXT=
$\text{Example 2:}$  We assume that the rectangular pulse is between   $\rm 2\hspace{0.08cm}ms$   and   $\rm 2.5\hspace{0.08cm}ms$   and the detection time  $T_{\rm D} =\rm 2\hspace{0.08cm}ms$  is desired.

Under these conditions:
*The matched filter impulse response  $h_{\rm MF}(t)$  must be constant in the range from   $t_1 (= 4 - 2.5) =\rm 1.5\hspace{0.08cm}ms$   to  $t_2 (= 4 - 2) =\rm 2\hspace{0.08cm}ms$. 
*For  $t < t_1$  as well as for  $t > t_2$  it must not have any components.
*The magnitude frequency response  $\vert H_{\rm MF}(f)\vert$  is  $\rm sinc$–shaped here.
*The magnitude of the impulse response  $h_{\rm MF}(t)$  is not important for the S/N ratio, because  $\rho _d ( {T_{\rm D} } )$  is independent of  $K_{\rm MF}$.}}

===Further information about the considered input pulses===
All data are without consideration of the delay  $\tau_g$.

  '''(1)  Rectangular Pulse''' 
*The pulse  $g(t)$  has constant height  $A_g$  in the range  $\pm \Delta t_g/2$  and is zero outside.
*The spectral function  $G(f)=A_g\cdot \Delta t_g \cdot {\rm si}(\pi\cdot \Delta t_g \cdot f)$  has zeros at equidistant distances $1/\Delta t_g$.
*The pulse energy is  $E_g=A_g^2\cdot \Delta t_g$.

  '''(2)  Gaussian Pulse''' 
*The pulse  $g(t)=A_g\cdot {\rm e}^{-\pi\cdot(t/\Delta t_g)^2}$  is infinitely extended.  The maximum is  $g(t= 0)=A_g$.
*The smaller the equivalent time duration  $\Delta t_g$,  the broader and lower the spectrum   $G(f)=A_g \cdot \Delta t_g \cdot {\rm e}^{-\pi\hspace{0.05cm}\cdot\hspace{0.05cm}(f\hspace{0.05cm}\cdot\hspace{0.05cm} \Delta t_g)^2}$. 
*The pulse energy is  $E_g=A_g^2\cdot \Delta t_g/\sqrt{2}$.

  '''(3)  Exponential Pulse''' 
*The pulse is identically zero for  $t<0$  and infinitely extended for positive times   ⇒   $g(t)=A_g\cdot {\rm e}^{-t/\Delta t_g}$.
*$g(t)$  is (strongly) asymmetric   ⇒   the spectrum   $G(f)=A_g \cdot \Delta t_g/( 1 + {\rm j} \cdot 2\pi \cdot f \cdot \Delta t_g)$  is complex-valued;
*The pulse energy is  $E_g=A_g^2\cdot \Delta t_g/2$.
 
===Further information on the considered impulse responses===

The different receiver filters  $H(f)$  are described by their impulse responses  $h(t)$. 

Similar to the input pulses  $g(t)$,  these are characterized by the pulse height  $A_h$, the equivalent pulse duration   $\Delta t_h$  and the delay  $\tau_h$  compared to the symmetrical case.  The following short descriptions are always valid for   $\tau_h= 0$.

  '''(1)  Slit lowpass'''  ⇒   "rectangular impulse response"
*The impulse response  $h(t)$  has constant height  $A_h$  in the range  $\pm \Delta t_h/2$  and is zero outside.
*The frequency response  $H(f)=K \cdot {\rm si}(\pi\cdot \Delta t_g \cdot f)$  has zeros at equidistant intervals $1/\Delta t_h$.
*For white noise, the noise variance at the filter output is:  $\sigma_d^2= N_0/2 \cdot A_h^2 \cdot \Delta t_h$.

  '''(2)  Gaussian lowpass'''  ⇒   "Gaussian impulse response"
*The impulse response  $h(t)=A_h\cdot {\rm e}^{-\pi\cdot(t/\Delta t_h)^2}$  ist is infinitely extended.  The maximum is  $h(t= 0)=A_h$.
*The smaller the equivalent time duration  $\Delta t_h$,  the broader and lower the frequency response  $H(f)=K \cdot {\rm e}^{-\pi\hspace{0.05cm}\cdot\hspace{0.05cm}(f\hspace{0.05cm}\cdot\hspace{0.05cm} \Delta t_h)^2}$. 
*For white noise, the noise variance at the filter output is:  $\sigma_d^2= N_0/2 \cdot A_h^2 \cdot \Delta t_h/\sqrt{2}$.

  '''(3)  First order lowpass'''  ⇒   "Exponentially decaying impulse response"
*The impulse response is identically zero for  $t<0$  and infinitely extended for positive times   ⇒   $h(t)=A_h\cdot {\rm e}^{-t/\Delta t_h}$.
*$h(t)$  is causal and (strongly) asymmetric.  The frequency response $H(f)=A_g \cdot \Delta t_g/( 1 + {\rm j} \cdot 2\pi \cdot f \cdot \Delta t_g)$  is complex-valued.
*For white noise, the noise variance at the filter output is:  $\sigma_d^2= N_0/4 \cdot A_h^2 \cdot \Delta t_h$.

  '''(4)  Lowpass 4"'''  ⇒   "Impulse response mirror image of  '''(3)'''"
*The impulse response is identically zero for  $t>0$  and infinitely extended for negative times   ⇒   $h(t)=A_h\cdot {\rm e}^{t/\Delta t_h}$  für  $t<0$.
*The frequency response $H(f)$  is conjugate complex to the frequency response of the first order lowpass filter.
*The noise variance at the filter output is exactly the same for white noise as for the first order lowpass:  $\sigma_d^2= N_0/4 \cdot A_h^2 \cdot \Delta t_h$.

==Exercises==

*First, select the number  $(1,\ 2, \text{...} \ )$  of the task to be processed.  The number  $0$  corresponds to a "Reset":  Same setting as at program start.
*A task description is displayed.  The parameter values are adjusted.  Solution after pressing "Show Solution".
*Both the input signal  $x(t)$  and the filter impulse response  $h(t)$  are normalized, dimensionless and energy-limited ("time-limited pulses").
*All times, frequencies, and power values are to be understood normalized, too.

{{BlueBox|TEXT=
'''(1)'''  Let the input pulse be Gaussian with  $A_g=1,\ \Delta t_g=1,\ \tau_g=1$.   Which setting leads to the "Matched Filter"?  What value has  $10 \cdot \lg \ \rho_{\rm MF}$  with  $N_0=0.01$? }}

:* The Matched Filter must also have a Gaussian shape and it must hold:  $\Delta t_h=\Delta t_g=1,\ \tau_h =\tau_g=1$   ⇒   $T_{\rm D} = \tau_h +\tau_g=2$.
:* The (instantaneous) signal-to-noise power ratio at the Matched Filter output is  $\rho _{\rm MF} = { {2 \cdot E_g } }/{ {N_0 } } \approx 141.4$  ⇒   $10 \cdot \lg \ \rho _{\rm MF} \approx 21.5$  dB.
:* With no other filter than the Matched Filter this  $\rm SNR$  (or an even better one)  can be achieved   ⇒   $10 \cdot \lg \ \rho _{d} \le 10 \cdot \lg \ \rho _{\rm MF}$.

{{BlueBox|TEXT=
'''(2)'''  The "Matched Filter" on rectangular input pulse with  $A_g=1,\ \Delta t_g=1,\ \tau_g=0$  is a rectangular-in-time low–pass   ⇒   rectangular impulse response.           What value has  $10 \cdot \lg \ \rho_{\rm MF}$  with  $N_0=0.01$?  Interpret all the graphs shown and the numerical results in different ways}}

:* The MF parameters are  $A_h=A_g=1, \ \Delta t_h=\Delta t_g=1,\ \tau_h =\tau_g=0$   ⇒   $T_{\rm D} = \tau_h +\tau_g=0$   ⇒   $\rho _{\rm MF} = 200$   ⇒   $10 \cdot \lg \ \rho _{\rm MF} \approx 23$  dB.
:* The pulse energy is the integral over  $g^2(t)$   ⇒   $E_g = A_g^2 \cdot \Delta t_g=1$   ⇒   $\rho _{\rm MF} = 2 \cdot E_g /N_0 =200$.   $T_{\text{D, opt} }=0$  is implicitly considered here.
:* Another equation is  $\rho_d (T_{\rm D}) =d_{\rm S}^2 (T_{\rm D})/\sigma_d^2$.  The noise variance can, for example, be calculated as the integral over  $h^2(t)$   ⇒   $\sigma_d^2= N_0 \cdot \Delta t_h/2 = 0.005$.
:* The useful detection signal  $d_{\rm S} (t)= g(t) * h(t)$  has a triangular shape with the maximum  $d_{\rm S} (T_{\rm D, \ opt} = 0 )= 1$   ⇒   $\rho_d (T_{\rm D, \ opt} = 0 ) = 200= \rho _{\rm MF}$.

{{BlueBox|TEXT=
'''(3)'''  The settings of  $(2)$  continue to apply, with the exception of  $N_0=0.02 $  instead of  $N_0=0.01$.  What changes can be seen? }}
:* The only difference is twice the noise variance  $\sigma_d^2= 0.01$   ⇒   $\rho_d (T_{\rm D, \ opt} = 0 ) = 100= \rho _{\rm MF}$   ⇒   $10 \cdot \lg \rho_{\rm MF} =20$ dB.

{{BlueBox|TEXT=
'''(4)'''  The settings of  $(3)$  continue to apply, except  $T_{\rm D} = 0.1 $  instead of  $T_{\rm D, \ opt} = 0$.  What is the effect of this non-optimal detection time? }}
:* Now the useful signal value  $d_{\rm S} (T_{\rm D} = 0.1 )= 0.9$  is smaller   ⇒   $\rho_d (T_{\rm D} = 0.1 ) =0.9^2/0.01= 81< \rho _{\rm MF}$.  There is a degradation of nearly  $1$ dB.
:* For the further tasks the optimal detection time  $T_{\rm D, \ opt}$  is assumed, if not explicitly stated otherwise.

{{BlueBox|TEXT=
'''(5)'''  The settings of  $(3)$  apply again except for a lower impulse response  $A_h = 0.8 $  instead of  $A_h = 1$.  Interpret the changes. }}
:* With  $A_h \ne A_g$  it is also a Matched Filter as long as  $h(t)$  is equal in shape to  $g(t)$    ⇒   $\rho _{\rm MF} = { {2 \cdot E_g } }/{ {N_0 } } =100$   ⇒   $10 \cdot \lg \rho _{\rm MF} =20$ dB.
:* The equation  $\rho_d (T_{\rm D}=0) =d_{\rm S}^2 (T_{\rm D}=0)/\sigma_d^2$  leads to the same result, since  ${d_{\rm S}}^2 (T_{\rm D})$  and  $\sigma_d^2$  are compared to  $(3)$  each reduced by a factor  $0. 8^2$.

{{BlueBox|TEXT=
'''(6)'''  Compared to  $(5)$  now the height of the input pulse  $g(t)$  is increased from  $A_g = 1$  to  $A_g = 1. 25$.  Does  $h(t)$  describe a Matched Filter?  What is the SNR  $\rho_{\rm MF}$? }}
:* Again, this is a Matched Filter, since  $h(t)$  and  $g(t)$  are equal in shape.  With  $E_g = 1.25^2$:     $\rho _{\rm MF} = { {2 \cdot 1.25^2 } }/{ 0.02 } =156.25$  ⇒  $10 \cdot \lg \rho_{\rm MF} \approx 21.9$ dB.
:* The higher value  $21.9$ dB compared to  $(5)$  is related to the fact that for the same noise variance  $\sigma_d^2= 0.0064$  the useful detection sample is again  ${d_{\rm S}} (T_{\rm D}) = 1$.

{{BlueBox|TEXT=
'''(7)'''  We continue from the rectangle–rectangle combination with  $A_h=A_g=1,\ \ \Delta t_h=\Delta t_g=1,\ \tau_h=\tau_g=0,\ N_0 =0.02,\ T_{\rm D}=0$.             Interpret the results after varying the equivalent pulse duration  $\Delta t_h$  of  $h(t)$  in the range  $0.6$ ... $1.4$.  Use the graph representation over  $\Delta t_h$. }}

:* As expected the optimum is obtained for the equivalent pulse duration  $\Delta t_h=\Delta t_g=1$.  Then  $10 \cdot \lg \ \rho_d (T_{\rm D, \ opt} = 0 ) =20$ dB  $\big(= 10 \cdot \lg \rho_{\rm MF}\big)$.
:* If  $\Delta t_h<\Delta t_g=1$, the useful detection signal is trapezoidal.  For  $\Delta t_h=0.6$:   $d_{\rm S} (T_{\rm D}=0)= 0.6$,  $\sigma_d^2\approx0.006$   ⇒   $10 \cdot \lg \ \rho_d (T_{\rm D, \ opt} = 0 ) \approx 17.8$ dB.
:* Also for  $\Delta t_h>1$  the useful detection signal is trapezoidal, but now still  $d_{\rm S} (T_{\rm D}=0)= 1$.  The noise variance  $\sigma_d^2$  increases continuously with  $\Delta t_h$.
:* For  $\Delta t_h=1.4$:     $\sigma_d^2=0.014$   ⇒   $10 \cdot \lg \ \rho_d (T_{\rm D, \ opt} = 0 ) \approx 18. 5$ dB.  Compared to the Matched Filter  $(\Delta t_h=1)$  the degradation is approx.  $1.5$ dB.

{{BlueBox|TEXT=
'''(8)'''  Now interpret the results for different  $\Delta t_g$  of the input pulse  $g(t)$  in the range  $0.6$ ... $1.4$.  Use the graph representation over  $\Delta t_g$. }}
:* Note:   The blue curve  $10 \cdot \lg \ \rho_d (T_{\rm D,\ opt} )$  is the difference between  $20\cdot \lg \ \big [{K \cdot d_{\rm S}} (T_{\rm D,\ opt}) \big ]$    (purple curve)  and  $20\cdot \lg \ \big [K \cdot \sigma_d \big ]$  (green curve).
:* For the considered parameter set  and  $K=10$  the "green term"  $20\cdot \lg \ \big [K \cdot \sigma_d \big ] = 0$ dB  for all  $\Delta t_g$   ⇒   the blue and the purple curves are identical.
:* The blue curve  $10 \cdot \lg \ \rho_d (T_{\rm D,\ opt} )$  increases from  $15.6$ dB  $($for  $\Delta t_g = 0. 6)$  to  $20$ dB  $($for  $\Delta t_g = 1)$  continuously and then remains constant for  $\Delta t_g > 1$.
:* But, the setting  $(\Delta t_g = 1.4,\ \Delta t_h = 1)$  does not yield a "Matched Filter".   Rather, with  $\Delta t_h = \Delta t_g = 1.4$:    $10 \cdot \lg \ \rho_{\rm MF}=10 \cdot \lg \ (2 \cdot E_g/N_0) \approx 21.5$ dB.
:* On the other hand the plot over  $\Delta t_h$  with the default setting  $(\Delta t_g = 1.4,\ \Delta t_h = 1)$  shows a monotonic increase of the blue curve   $10 \cdot \lg \ \rho_d (T_{\rm D,\ opt} )$.
:* For  $\Delta t_h = 0.6$  this gives  $10 \cdot \lg \ \rho_d (T_{\rm D,\ opt} )\approx 17.8$ dB,  and for  $\Delta t_h = 1. 4$  against  $10 \cdot \lg \ \rho_d (T_{\rm D,\ opt} )\approx 21.5$ dB  $=10 \cdot \lg \ \rho_{\rm MF}$.

{{BlueBox|TEXT=
'''(9)'''  We consider the exponential pulse  $g(t)$  and the first order low–pass, where  $A_h=A_g=1,\ \Delta t_h=\Delta t_g=1,\ \tau_h=\tau_g=0,\ N_0 =0.02,\ T_{\rm D}=1$.             Does this setting meet the Matched Filter criteria?  Justify your answers with as many arguments as possible. }}
:* No!   Here  $h(t)=g(t)$.  In a Matched Filter configuration, the impulse response should be  $h(t)={\rm const.}\cdot g(T_{\rm D}-t) $. 
:* The useful detection signal  $d_{\rm S}(t)$  does not have a symmetric shape around the maximum. For the Matched Filter,  $d_{\rm S}(T_{\rm D}-t) = d_{\rm S}(T_{\rm D}+t) $  would have to hold.
:* Despite  $\Delta t_h=\Delta t_g$  the SNR  $10 \cdot \lg \ \rho_d (T_{\rm D,\ opt}) \approx 14. 3$ dB   is now less than  $10 \cdot \lg \ \rho _{\rm MF} = 10 \cdot \lg \ 2 \cdot E_g/N_0 \approx 17$ dB.

{{BlueBox|TEXT=
'''(10)'''  With all other settings being the same, what changes with the "extremely acausal filter"?  Does the setting meet the Matched Filter criteria?  Reason. }}
:* Now here  $h(t)=g(-t)$  and the useful detection signal  $d_{\rm S}(t)$  is symmetric around  $t=0$.  It makes sense to choose  $T_{\rm D} = 0 $  here.
:* This gives  $10 \cdot \lg \ \rho_d (T_{\rm D,\ opt}) =10 \cdot \lg \ d_{\rm S}^2 (T_{\rm D,\ opt})/\sigma_d^2 = 17$ dB   –   the same value as for  $10 \cdot \lg \ \rho _{\rm MF} = 10 \cdot \lg \ 2 \cdot E_g/N_0 = 17$ dB.
:* The useful detection signal  $d_{\rm S}(t)$  is of the same shape as the energy ACF of the input pulse  $g(t)$.  The Matched Filter focuses the energy around the time  $T_{\rm D,\ opt}$.

{{BlueBox|TEXT=
'''(11)'''  With which rectangular pulse  $g(t)$  can one achieve the same  $\rho _{\rm MF}=50$  as in task  $(10)$?             With  $(A_h=A_g=1,\ \ \Delta t_h=\Delta t_g=0.5)$   or with   $(A_h=A_g=0.5,\ \ \Delta t_h=\Delta t_g=1)$ ? }}
:* From the equation  $\rho _{\rm MF} = 2 \cdot E_g/N_0$  it is already clear that the SNR depends only on the energy  $E_g$  of the input pulse and not on its shape.
:* The exponential pulse with   $(A_g=1,\ \Delta t_g=1)$   has the energy  $E_g=0.5$  ⇒   $\rho _{\rm MF}=50$.  As well as the rectangular pulse with  $(A_g=1,\ \Delta t_g=0.5)$.
:* In contrast, the rectangular pulse with  $(A_g=0.5,\ \Delta t_g=1)$  has a smaller energy   ⇒   $E_g=0. 25$   ⇒   $\rho _{\rm MF}=25$   ⇒   $10 \cdot \lg \ \rho _{\rm MF} = 14$ dB.

==Applet Manual==
 
[[File:Anleitung_abtast.png|right|600px]]
 
    '''(A)'''     Auswahl eines von vier Quellensignalen

    '''(B)'''     Parameterwahl für Quellensignal  $1$  (Amplitude, Frequenz, Phase)

    '''(C)'''     Ausgabe der verwendeten Programmparameter

    '''(D)'''     Parameterwahl für Abtastung  $(f_{\rm G})$  und                 Signalrekonstruktion  $(f_{\rm A},\ r)$

    '''(E)'''     Skizze des Empfänger–Frequenzgangs  $H_{\rm E}(f)$

    '''(F)'''     Numerische Ausgabe  $(P_x, \ P_{\rm \varepsilon}, \ 10 \cdot \lg(P_x/ P_{\rm \varepsilon})$

    '''(G)'''     Darstellungsauswahl für Zeitbereich

    '''(H)'''     Grafikbereich für Zeitbereich

    '''( I )'''     Darstellungsauswahl für Frequenzbereich

    '''(J)'''     Grafikbereich für Frequenzbereich

    '''(K)'''     Bereich für Übungen:  Aufgabenauswahl, Fragen, Musterlösung
 
==About the Authors==
This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
*The first version was created in 2006 by [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]]  as part of his bachelor thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).

*Last revision and English version 2020/2021 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  in the context of a working student activity. 

The conversion of this applet to HTML 5 was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ Studienzuschüsse]  ("study grants")  of the TUM Faculty EI.  We thank.

==Once again:  Open Applet in new Tab==

{{LntAppletLinkEnDe|matchedFilter_en|matchedFilter}}

Applets:Discrete Fouriertransform and Inverse

2023-03-21T18:07:02Z

Hwang:

{{LntAppletLinkEnDe|dft_en|dft}}

==Applet Description==
 
The conventional Fourier Transform  $\rm (FT)$  allows the calculation of the spectral function  $X(f)$  of a time continuous signal  $x(t)$. 

In contrast, the Discrete Fourier Transform  $\rm (DFT)$  is limited to a time discrete signal, represented by  $N$  time domain coefficients   $d(\nu)$  with indices  $\nu = 0, \text{...} , N\hspace{-0.1cm}-\hspace{-0.1cm}1$, which can be interpreted as equidistant samples of the time continuous signal  $x(t)$.

If the  [[Signal_Representation/Discrete-Time_Signal_Representation#Sampling_theorem|"sampling theorem"]]  is fulfilled, the DFT algorithm likewise allows the calculation of  $N$  frequency domain coefficients  $D(\mu)$  with indices  $\mu = 0, \text{...} , N\hspace{-0.1cm}-\hspace{-0.1cm}1$.  These are equidistant samples of the frequency continuous spectrum  $X(f)$.

*The applet illustrates the properties of the  $\text{DFT:}\hspace{0.3cm}d(\nu)\hspace{0.1cm} \Rightarrow \hspace{0.1cm} D(\mu)$  by using the example  $N=16$.  The default   $d(\nu)$ assignments for the DFT are:

:(a)  According to the input field,  (b)  Constant signal,  (c)  Complex exponential function (of time),  (d)  Harmonic oscillation (with  $($Phase  $\varphi = 45^\circ)$,
:(e)  Cosine signal (one period),  (f)  Sinusoidal signal (one period),  (g)  Cosine signal (two periods), (h)  Alternating time coefficients, (i)  Dirac delta impulse, 
: (j)  Rectangular pulse ,  (k)  Triangular pulse,  (l)  Gaussian pulse.

*Possible  $D(\mu)$ assignments for the Inverse Discrete Fourier Transform   ⇒   $\text{IDFT:}\hspace{0.3cm}D(\mu)\hspace{0.1cm} \Rightarrow \hspace{0.1cm} d(\nu)$  are:

:(A)  According to the input field,  (B)  Constant spectrum,  (C)  Complex exponential function (of frequency),  (D)  Equivalent to setting (d) in the time domain,
:(E)  Cosine spectrum (one frequency period),  (F)  Sinusoidal spectrum (one frequency period),  (G)  Cosine spectrum (two frequency periods), 
:(H)  Alternating spectral coefficients, (I)  Dirac spectrum,  (J)  Rectangular spectrum,  (K)  Triangular spectrum,  (L)  Gaussian spectrum.

The applet uses the framework  [https://en.wikipedia.org/wiki/Plotly Plot.ly].

==Theoretical Background==
 
===Arguments for the discrete realization of the Fourier transform===

The  '''Fourier transform'''  according to the conventional description for continuous-time signals has an infinitely high selectivity due to the unlimited extension of the integration interval and is therefore an ideal theoretical tool for spectral analysis.

If the spectral components  $X(f)$  of a time function  $x(t)$  are to be determined numerically, the general transformation equations

:$$\begin{align*}X(f) & = \int_{-\infty
}^{+\infty}x(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi f t}\hspace{0.1cm} {\rm d}t\hspace{0.5cm} \Rightarrow\hspace{0.5cm} \text{Inward transformation}\hspace{0.7cm} \Rightarrow\hspace{0.5cm} \text{First Fourier integral}
\hspace{0.05cm},\\
x(t) & = \int_{-\infty
}^{+\infty}\hspace{-0.15cm}X(f) \cdot {\rm e}^{\hspace{0.05cm}+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi f t}\hspace{0.1cm} {\rm d}f\hspace{0.35cm} \Rightarrow\hspace{0.5cm}
\text{Backward transformation}\hspace{0.4cm} \Rightarrow\hspace{0.5cm} \text{Second Fourier integral}
\hspace{0.05cm}\end{align*}$$

are unsuitable for two reasons:
*The equations apply exclusively to continuous-time signals. With digital computers or signal processors, however, only discrete-time signals can be processed.
*For a numerical evaluation of the two Fourier integrals it is necessary to limit the respective integration interval to a finite value.

{{BlaueBox|TEXT=
$\text{This results in the following consequence:}$ 

A  '''continuous signal'''  must undergo two processes before the numerical determination of its spectral properties, viz.
*  '''sampling'''  for discretization, and
*  '''windowing'''  to limit the integration interval.}}

In the following, starting from an aperiodic time function  $x(t)$  and the corresponding Fourier spectrum  $X(f)$,  a discrete-time and discrete-frequency description suitable for computer processing is presented.

===Time discretization – periodization in the frequency domain===

The following graphs uniformly show the time domain on the left and the frequency domain on the right. Without restriction of generality,  $x(t)$  and  $X(f)$  are real and Gaussian, respectively.

[[File:P_ID1132__Sig_T_5_1_S2_neu.png|center|frame|Discretization in the time domain – periodization in the frequency domain]]

One can describe the sampling of the time signal  $x(t)$  by multiplication with a Dirac pulse  $p_{\delta}(t)$.  This results in the time signal sampled at distance  $T_{\rm A}$ 

:$${\rm A}\{x(t)\} = \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot x(\nu \cdot T_{\rm A})\cdot
\delta (t- \nu \cdot T_{\rm A}
)\hspace{0.05cm}.$$

We now transform this sampled signal  $\text{A}\{ x(t)\}$  into the frequency domain. The multiplication of the Dirac pulse  $p_{\delta}(t)$  with  $x(t)$  corresponds in the frequency domain to the convolution of  $P_{\delta}(f)$  with  $X(f)$. The periodized spectrum  $\text{P}\{ X(f)\}$ is obtained, where  $f_{\rm P}$  is the frequency period of the function  $\text{P}\{ X(f)\}$: 

:$${\rm A}\{x(t)\} \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} {\rm P}\{X(f)\} = \sum_{\mu = - \infty }^{+\infty}
X (f- \mu \cdot f_{\rm P} )\hspace{0.5cm} {\rm with }\hspace{0.5cm}f_{\rm
P}= {1}/{T_{\rm A}}\hspace{0.05cm}.$$

*We call the sampled signal  $\text{A}\{ x(t)\}$.
*The  '''frequency period'''  is denoted by  $f_{\rm P}$ = $1/T_{\rm A}$. 

The graph above shows the functional relationship described here. It should be noted:
*The frequency period  $f_{\rm P}$  was deliberately chosen small here so that the overlap of the spectra to be summed can be clearly seen.
*In practice, due to the sampling theorem,  $f_{\rm P}$  should be at least twice as large as the largest frequency contained in the signal  $x(t)$. 
*If this is not fulfilled,  '''aliasing'''  must be expected.

===Frequency discretization – periodization in the time domain===

The discretization of  $X(f)$  can also be described by a multiplication with a Dirac pulse. The result is the spectrum sampled at distance  $f_{\rm A}$: 

:$${\rm A}\{X(f)\} = X(f) \cdot \sum_{\mu = - \infty }^{+\infty}
f_{\rm A} \cdot \delta (f- \mu \cdot f_{\rm A } ) = \sum_{\mu = - \infty }^{+\infty}
f_{\rm A} \cdot X(\mu \cdot f_{\rm A } ) \cdot\delta (f- \mu \cdot f_{\rm A } )\hspace{0.05cm}.$$

Transforming the frequency Dirac pulse used here $($with pulse weights  $f_{\rm A})$  into the time domain, we obtain with  $T_{\rm P} = 1/f_{\rm A}$:

:$$\sum_{\mu = - \infty }^{+\infty}
f_{\rm A} \cdot \delta (f- \mu \cdot f_{\rm A } ) \hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}
\sum_{\nu = - \infty }^{+\infty}
\delta (t- \nu \cdot T_{\rm P } ) \hspace{0.05cm}.$$

The multiplication with  $X(f)$  corresponds in the time domain to the convolution with  $x(t)$. The signal  $\text{P}\{ x(t)\}$ periodized at distance  $T_{\rm P}$  is obtained:

:$${\rm A}\{X(f)\} \hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}
{\rm P}\{x(t)\} = x(t) \star \sum_{\nu = - \infty }^{+\infty}
\delta (t- \nu \cdot T_{\rm P } )= \sum_{\nu = - \infty }^{+\infty}
x (t- \nu \cdot T_{\rm P } ) \hspace{0.05cm}.$$

[[File:P_ID1134__Sig_T_5_1_S3_neu.png|right|frame|Discretization in the frequency domain – periodization in the time domain]]
{{GraueBox|TEXT=
$\text{Example 1:}$ 
This relationship is illustrated in the graphic:
*Due to the coarse frequency rastering, this example results in a relatively small value for the time period  $T_{\rm P}$. 

*Therefore, the (blue) periodized time signal  $\text{P}\{ x(t)\}$  differs significantly from  $x(t)$ due to overlaps.}}

===Finite signal representation===

[[File:P_ID1135__Sig_T_5_1_S4_neu.png|right|frame|Finite signals of the Discrete Fourier Transform]]
One arrives at the so-called  ''finite signal representation'' 
*when both the time function  $x(t)$ and
*the spectral function  $X(f)$

are specified exclusively by their sample values.
 
The graph is to be interpreted as follows:
*In the left picture the function  $\text{A}\{ \text{P}\{ x(t)\}\}$ is drawn in blue. It is obtained by sampling the periodized time function  $\text{P}\{ x(t)\}$  with equidistant Dirac pulses in the distance  $T_{\rm A} = 1/f_{\rm P}$.
*In the right picture the function  $\text{P}\{ \text{A}\{ X(f)\}\}$ is drawn in green. This results from periodization $($with  $f_{\rm P})$  of the sampled spectral function  $\{ \text{A}\{ X(f)\}\}$.
*There is also a Fourier correspondence between the blue finite signal and the green finite signal, as follows:

:$${\rm A}\{{\rm P}\{x(t)\}\} \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} {\rm P}\{{\rm A}\{X(f)\}\} \hspace{0.05cm}.$$

{{BlaueBox|TEXT=
However, the Diraclines of the periodic continuation  $\text{P}\{ \text{A}\{ X(f)\}\}$  of the sampled spectral function fall into the same frequency grid as those of  $\text{A}\{ X(f)\}$ only if the frequency period  $f_{\rm P}$  is an integer multiple  $(N)$  of the frequency sampling interval  $f_{\rm A}$. 

*When using the finite signal representation, the following condition must always be fulfilled, where in practice a power of two is usually used for the natural number  $N$  (the graph above is based on the value  $N = 8$ ):

:$$f_{\rm P} = N \cdot f_{\rm A} \hspace{0.5cm} \Rightarrow\hspace{0.5cm} {1}/{T_{\rm A} }= N \cdot f_{\rm A} \hspace{0.5cm} \Rightarrow\hspace{0.5cm}
N \cdot f_{\rm A}\cdot T_{\rm A} = 1\hspace{0.05cm}.$$}}

If the condition  $N \cdot f_{\rm A} \cdot T_{\rm A} = 1$  is satisfied, the order of periodization and sampling can be interchanged. Thus:

:$${\rm A}\{{\rm P}\{x(t)\}\} = {\rm P}\{{\rm A}\{x(t)\}\}\hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm}
{\rm P}\{{\rm A}\{X(f)\}\} = {\rm A}\{{\rm P}\{X(f)\}\}\hspace{0.05cm}.$$

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
*The time function  $\text{P}\{ \text{A}\{ x(t)\}\}$  has the period  $T_{\rm P} = N \cdot T_{\rm A}$.
*The period in the frequency domain is  $f_{\rm P} = N \cdot f_{\rm A}$.
*For the description of the discretized time and frequency course  $N$  '''complex numerical values''' in the form of pulse weights are sufficient in each case.}}

{{GraueBox|TEXT=
$\text{Example 2:}$ 
A time-limited (pulse-like) signal  $x(t)$  is present in sampled form, where the distance between two samples is  $T_{\rm A} = 1\, {\rm µ s}$: 
*After a discrete Fourier transformation with  $N = 512$  the spectrum  $X(f)$  is available as samples with the distance  $f_{\rm A} = (N \cdot T_{\rm A})^{–1} \approx 1.953\,\text{kHz} $. 
*Increasing the DFT parameter to  $N= 2048$ results in a finer frequency grid with  $f_{\rm A} \approx 488\,\text{Hz}$.}}

===Discrete Fourier Transform===

From the conventional  "first Fourier integral"

:$$X(f) =\int_{-\infty
}^{+\infty}x(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} f \hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}t$$

discretization  $(\text{d}t \to T_{\rm A}$,  $t \to \nu \cdot T_{\rm A}$,  $f \to \mu \cdot f_{\rm A}$,  $T_{\rm A} \cdot f_{\rm A} = 1/N)$  yields the sampled and periodized spectral function

:$${\rm P}\{X(\mu \cdot f_{\rm A})\} = T_{\rm A} \cdot \sum_{\nu = 0 }^{N-1}
{\rm P}\{x(\nu \cdot T_{\rm A})\}\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm} \cdot \hspace{0.05cm}\nu \hspace{0.05cm}
\cdot \hspace{0.05cm}\mu /N} \hspace{0.05cm}.$$

It is taken into account that due to the discretization the periodized functions have to be used in each case.

For reasons of a simplified notation we now make the following substitutions:
*Let the  $N$  '''time-domain coefficients'''  be associated with the indexing variable  $\nu = 0$, ... , $N - 1$:
:$$d(\nu) =
{\rm P}\left\{x(t)\right\}{\big|}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm A}}\hspace{0.05cm}.$$
*Let the  $N$  '''frequency-domain coefficients'''  be associated with the indexing variable  $\mu = 0,$ ... , $N$ – 1:
:$$D(\mu) = f_{\rm A} \cdot
{\rm P}\left\{X(f)\right\}{\big|}_{f \hspace{0.05cm}= \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm A}}\hspace{0.05cm}.$$
*Abbreviation for the  '''complex rotation factor'''  depending on  $N$  is written:
:$$w = {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N}
= \cos \left( {2 \pi}/{N}\right)-{\rm j} \cdot \sin \left( {2 \pi}/{N}\right)
\hspace{0.05cm}.$$

[[File:P_ID2730__Sig_T_5_1_S5_neu.png|right|frame|On the definition of the Discrete Fourier Transform (DFT) with  $N=8$]]
{{BlaueBox|TEXT=
$\text{Definition:}$ 

The term  '''Discrete Fourier Transform'''  ('''DFT''' for short)  means the calculation of the  $N$  spectral coefficients  $D(\mu)$  from the  $N$  signal coefficients  $d(\nu)$:

:$$D(\mu) = \frac{1}{N} \cdot \sum_{\nu = 0 }^{N-1}
d(\nu)\cdot {w}^{\hspace{0.05cm}\nu \hspace{0.03cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}. $$

In the graph you can see in an example
*the  $N = 8$  signal coefficients  $d(\nu)$  at the blue filling,
*the  $N = 8$  spectral coefficients  $D(\mu)$  at the green filling.}}

===Inverse Discrete Fourier Transform===

The Inverse Discrete Fourier Transform (IDFT) describes the  "second Fourier integral"

:$$\begin{align*}x(t) & = \int_{-\infty
}^{+\infty}X(f) \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} f \hspace{0.05cm}\cdot \hspace{0.05cm}
t}\hspace{0.1cm} {\rm d}f\end{align*}$$

in discretized form:   $d(\nu) =
{\rm P}\left\{x(t)\right\}{\big|}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm
A}}\hspace{0.01cm}.$

[[File:P_ID2731__Sig_T_5_1_S6_neu.png|right|frame|For the definition of the IDFT with  $N=8$]]
{{BlaueBox|TEXT=
$\text{Definition:}$ 

The term  '''Inverse Discrete Fourier Transform'''  ('''IDFT''' for short)  refers to the calculation of the signal coefficients  $d(\nu)$  from the spectral coefficients  $D(\mu)$:

:$$d(\nu) = \sum_{\mu = 0 }^{N-1}
D(\mu) \cdot {w}^{-\nu \hspace{0.03cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}.$$

With the indexing variables  $\nu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1$  und  $\mu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1$  also holds here:
:$$d(\nu) =
{\rm P}\left\{x(t)\right\}{\big \vert}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm
A} }\hspace{0.01cm},$$

:$$D(\mu) = f_{\rm A} \cdot
{\rm P}\left\{X(f)\right\}{\big \vert}_{f \hspace{0.05cm}= \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm A} }
\hspace{0.01cm},$$

:$$w = {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N}
\hspace{0.01cm}.$$}}

A comparison between DFT and IDFT shows that exactly the same algorithm can be used. The only differences of the IDFT compared to the DFT are:
*The exponent of the rotation factor must be applied with a different sign.
*With the IDFT the division by  $N$ is omitted.

===Interpretation of DFT and IDFT===
 
The graph shows the discrete coefficients in the time and frequency domain together with the periodized continuous-time functions.

[[File:P_ID1136__Sig_T_5_1_S7_neu.png|center|frame|Time and frequency domain coefficients of the DFT]]

When using DFT or IDFT, it should be noted:
*According to the above definitions, the DFT coefficients  $d(ν)$  and  $D(\mu)$  always have the unit of the time function.
*Dividing  $D(\mu)$  by  $f_{\rm A}$ gives the spectral value  $X(\mu \cdot f_{\rm A})$.
*The spectral coefficients  $D(\mu)$  must always be set complex to be able to consider also odd time functions.
*In order to be able to transform band–pass signals in the equivalent lowndash;pass range, complex time coefficients  $d(\nu)$ are usually used.
*The basic interval for  $\nu$  and  $\mu$  is usually defined as the range from  $0$  to  $N - 1$, as in the above diagram.
*With the complex-valued number sequences  $\langle \hspace{0.1cm}d(\nu)\hspace{0.1cm}\rangle = \langle \hspace{0.1cm}d(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , d(N-1) \hspace{0.1cm}\rangle$   as well as   $\langle \hspace{0.1cm}D(\mu)\hspace{0.1cm}\rangle = \langle \hspace{0.1cm}D(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , D(N-1) \hspace{0.1cm}\rangle$  DFT and IDFT are symbolized similar to the conventional Fourier transform:
:$$\langle \hspace{0.1cm} D(\mu)\hspace{0.1cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.1cm} d(\nu) \hspace{0.1cm}\rangle \hspace{0.05cm}.$$
*If the time function  $x(t)$  is already limited to the range  $0 \le t \lt N \cdot T_{\rm A}$,  then the time coefficients output by the IDFT directly indicate the samples of the time function:   $d(\nu) = x(\nu \cdot T_{\rm A}).$
*If  $x(t)$  is shifted with respect to the basic interval, one has to choose the assignment between  $x(t)$  and the coefficients  $d(\nu)$  shown in  $\text{Example 3}$. 

{{GraueBox|TEXT=
$\text{Example 3:}$ 
The upper graph shows the asymmetric triangular pulse  $x(t)$ whose absolute width is smaller than  $T_{\rm P} = N \cdot T_{\rm A}$.

[[File:P_ID1139__Sig_T_5_1_S7b_neu.png|right|frame|For the assignment of the DFT coefficients with  $N=8$]]

The sketch below shows the assigned DFT coefficients valid for  $N = 8$

*For  $\nu = 0,\hspace{0.05cm}\text{...} \hspace{0.05cm} , N/2 = 4$,    $d(\nu) = x(\nu \cdot T_{\rm A})$ is valid:

:$$d(0) = x (0)\hspace{0.05cm}, \hspace{0.15cm}
d(1) = x (T_{\rm A})\hspace{0.05cm}, \hspace{0.15cm}
d(2) = x (2T_{\rm A})\hspace{0.05cm}, $$
:$$d(3) = x (3T_{\rm A})\hspace{0.05cm}, \hspace{0.15cm}
d(4) = x (4T_{\rm A})\hspace{0.05cm}.$$
*On the other hand, the coefficients  $d(5)$,  $d(6)$  and  d$(7)$  are to be set as follows:

:$$d(\nu) = x \big ((\nu\hspace{-0.05cm} - \hspace{-0.05cm} N ) \cdot T_{\rm A}\big ) $$

:$$ \Rightarrow \hspace{0.2cm}d(5) = x (-3T_{\rm A})\hspace{0.05cm}, \hspace{0.35cm}
d(6) = x (-2T_{\rm A})\hspace{0.05cm}, \hspace{0.35cm}
d(7) = x (-T_{\rm A})\hspace{0.05cm}.$$ }}

 

==Exercises==
 
[[File:Aufgaben_2D-Gauss.png|right]]
* First select the number (1,...) of the exercise. 
* A description of the exercise will be displayed.
*The parameter values are adjusted. 
*Solution after pressing "Show solution". 
*The number 0 corresponds to a "Reset":
:Same setting as at program start.
:Output of a "reset text" with further explanations about the applet.

{{BlaueBox|TEXT=
'''(1)'''  New setting:  DFT of signal  $\rm (b)$:  Constant signal.  Interpret the result in the frequency domain.  What is the analogon of the conventional Fourier transform?}}

:* All coefficients in the time domain are  $d(\nu)=1$.  Thus all  $D(\mu)=0$  with the exception of  $\textrm{Re}[D(0)]=1$. 
:* This corresponds to the conventional (time-continuous) Fourier Transform:  $x(t)=A\hspace{0.15cm} \circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm} X(f)=A\cdot \delta (f=0)$  with  $A=1$.

{{BlaueBox|TEXT=
'''(2)'''  Assume the obtained  $D(\mu)$  field and shift all coefficients one entry down.  Which time function does the IDFT provide?
}}

:* Now all  $D(\mu)=0$,  except for  $\textrm{Re}[D(1)]=1$.  The result in the time domain is a complex exponential function.
:* The real part of the  $d(\nu)$  field shows a cosine and the imaginary part a sine function.  For each function one can see one period respectively.

{{BlaueBox|TEXT=
'''(3)'''  Add the following coefficient to the current  $D(\mu)$ field:  $\textrm{Im}[D(1)]=1$.  What are the differences compared to  '''(2)'''  in the time domain?
 }}
:* On the one hand, a phase shift of two support values can now be detected for the real and the imaginary parts.  This corresponds to the phase  $\varphi = 45^\circ$.
:* On the other hand, the amplitudes of the real and the imaginary part were each increased by the factor  $\sqrt{2}$.

{{BlaueBox|TEXT=
'''(4)'''  Set the  $D(\mu)$ field  to zero except for  $\textrm{Re}[D(1)]=1$.  Which additional  $D(\mu)$  coefficient yields a real  $d(\nu)$  field?
}}

:* By trial and error, one can see that  $\textrm{Re}[D(15)]=1$  must apply additionally.  Then the  $d(\nu)$  field describes a cosine.
:* The following applies to the conventional (time continuous) Fourier Transform:  $x(t)=2\cdot \cos(2\pi \cdot f_0 \cdot t)\hspace{0.15cm}\circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}
X(f)=\delta (f-f_0)+\delta (f+f_0)$.
:* The entry  $D(1)$  is representative of the frequency  $f_0$  and due to the periodicity with  $N=16$  the frequency  $-f_0$  is expressed by  $D(15)=D(-1)$.

{{BlaueBox|TEXT=
'''(5)'''  According to the IDFT in the  $d(\nu)$  field, by which  $D(\mu)$  field does one obtain a real cosine function with the amplitude  $A=1$?}}

:* Like the conventional Fourier transform the discrete Fourier Transform is linear  ⇒   $D(1)=D(15)=0.5$.

{{BlaueBox|TEXT=
'''(6)'''  New setting:  DFT of signal  $\rm (e)$:  Cosine signal and subsequent signal shifts.  What are the effects of these shifts in the frequency domain?
}}

:* A shift in the time domain changes the cosine signal to a  "harmonic oscillation"  with arbitrary phase.
:* The  $D(\mu)$  field is still zero except for  $D(1)$  and  $D(15)$.  The absolute values  $|D(1)|$  and  $|D(15)|$  also remain the same.
:* The only change concerns the phase,  i.e. the different distribution of the absolute values between the real and imaginary part.

{{BlaueBox|TEXT=
'''(7)'''  New setting:  DFT of signal  $\rm (f)$:  Sinusoidal signal.  Interpret the result in the frequency domain.  What is the analogon of the conventional Fourier Transform?
}}

:* The sine signal results from the cosine signal by applying four time shifts.  Therefore all statements of  '''(6)'''  are still valid.
:* For the conventional (time continuous) Fourier transform it holds that  $x(t)= \sin(2\pi \cdot f_0 \cdot t)\hspace{0.15cm}\circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}
X(f)=j/2 \cdot [\delta (f+f_0)-\delta (f-f_0)]$.
:* The coefficient  $D(1)$   $\Rightarrow$  $($frequency:  $+f_0)$  is imaginary and has the imaginary part  $-0.5$.  Accordingly,  $\textrm{Im}[D(15)]=+0.5$   ⇒   $($frequency:  $-f_0)$  applies.

{{BlaueBox|TEXT=
'''(8)'''  New setting:  DFT of signal  $\rm (g)$:  Cosine signal (two periods).  Interpret the result in comparison to exercise  '''(5)'''.
}}

:* Here the time continuous Fourier transform reads  $x(t)=\cos(2\pi \cdot (2 f_0) \cdot t)\hspace{0.15cm}\circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}X(f)=0.5 \cdot \delta (f- 2 f_0)+0.5 \cdot \delta (f+ 2 f_0)$.
:* $D(2)$  is representative of the frequency  $2 f_0$.  Due to the periodicity,  $D(14)=D(-2)$:   $D(2)=D(14)=0.5$ is representative of the frequency  $-2 f_0$.

{{BlaueBox|TEXT=
'''(9)'''  Now examine the case DFT of a sinodial signal (two periods).  Which modifications do you need to make in the time domain?  Interpret the result.
}}

:* The desired signal can be obtained from the DFT of signal  $\rm (g)$:  Cosine signal (two periods) with two shifts.  With the result of  '''(7)''':  Four shifts.
:* The DFT result is accordingly  $\textrm{Im}[D(2)]=-0.5$  and  $\textrm{Im}[D(14)]=+0.5$.

{{BlaueBox|TEXT=
'''(10)'''  New setting: DFT of signal  $\rm (h)$:  Alternating time coefficients.  Interpret the DFT result.}}

:* Here, the time continuous Fourier transform is given by:  $x(t)=\cos(2\pi \cdot (8 f_0) \cdot t)\hspace{0.15cm}\circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}
X(f)=0.5 \cdot \delta (f- 8 f_0)+0.5 \cdot \delta (f+ 8 f_0)$.
:* $8 f_0$ is the highest frequency that can be displayed with  $N=16$  in the DFT.  There are only two sampled values per period, namely $+1$ and $-1$.
:* Difference to exercise  '''(5)''':  $D(1)=0.5$  now becomes  $D(8)=0.5$.  Likewise,  $D(15)=0.5$  is shifted to  $D(8)=0.5$.  Final result:  $D(8)=1$.

{{BlaueBox|TEXT=
'''(11)'''  What are the differences between the two settings DFT from signal  $\rm (i)$:  Dirac delta impulse and   IDFT from spectrum  $\rm (I)$:  Dirac spectrum?
}}

:* None! In the first case, all coefficients are  $D(\mu)=1$ (real);  in the second case, however, equivalently  $d(\nu)=1$ (real).

{{BlaueBox|TEXT=
'''(12)'''  Are there differences in shifting the real  "$1$"  in the according input fields by one place at a time, that is for  $d(\nu = 1)=1$  and  $D(\mu = 1)=1$?
}}

:* The first case  $\Rightarrow$  $\textrm{Re}[d(\nu = 1)]=1$  results in the complex exponential function in the frequency domain given by  $X(f)= \textrm{e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2 \pi\hspace{0.05cm}\cdot\hspace{0.05cm} f/f_0}$  with negative sign.
:* The second case  $\Rightarrow$  $\textrm{Re}[D(\mu = 1)]=1$ results in the complex exponential function in the time domain given by  $x(t)= \textrm{e}^{+{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2 \pi\hspace{0.05cm}\cdot\hspace{0.05cm} f_0\cdot t}$  with positive sign.
:* Note:  With  $\textrm{Re}[D(\mu=15)]=1$  the result in the time domain would also be a complex exponential function  $x(t)= \textrm{e}^{-{\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2 \pi\hspace{0.05cm}\cdot\hspace{0.05cm} f_0\hspace{0.05cm}\cdot\hspace{0.05cm} t}$ with negative sign.

{{BlaueBox|TEXT=
'''(13)'''  New setting: DFT of signal  $\rm (k)$:  Triangle pulse.  Interpret the  $d(\nu)$  assignment under the assumption  $T_\textrm{A} = 1$ ms.}}

:* Change the display to "absolute value".  $x(t)$ is symmetrical around  $t=0$  and extends from  $-8 \cdot T_\textrm{A} = -8$  ms to  $+8 \cdot T_\textrm{A}= +8$  ms.
:* $d(\nu)$  assignment:  $d(0)=x(0)=1$,  $d(1)=x(T_\textrm{A})=0.875$, ... ,  $d(8)=x(8 T_\textrm{A})=0$,  $d(9)=x(-7 T_\textrm{A})=0.125$, ... ,  $d(15)=x(-T_\textrm{A})=0.875$.

{{BlaueBox|TEXT=
'''(14)'''  Same setting as  '''(13)'''.  Interpret the DFT result, especially the coefficients  $D(0)$,  $D(1)$,  $D(2)$  and  $D(15)$.}}

:* In the frequency range  $D(0)$  stands for the frequency  $f=0$  and  $D(1)$  and  $D(15)$  for the frequencies  $\pm f_\textrm{A}$.  It holds that  $f_\textrm{A}= 1/ (N\cdot T_\textrm{A})=62.5$  Hz.
:* For the value of the continuous spectrum at $f=0$ the following applies:   $X(f=0)=D(0)/f_\textrm{A} = 0.5/ (0.0625$ kHz$)=8\cdot \textrm{kHz}^{-1}$.
:* The first zero of the  $\textrm{si}^2$–shaped spectrum  $X(f)$  occurs at  $2\cdot f_\textrm{A} = 125$ Hz.  The other zeros are equidistant.

{{BlaueBox|TEXT=
'''(15)'''  New setting: DFT of signal  $\rm (i)$:  Rectangular pulse.  Interpret the displayed results.}}
:* The set (symmetrical) rectangle extends over  $\pm 4 \cdot T_\textrm{A}$.  At the edges, the time coefficients are only half as large:  $d(4)=d(12)=0.5$.
:* The further statements of  '''(14)'''  also apply to this  $\textrm{si}$–shaped spectrum  $X(f)$.

{{BlaueBox|TEXT=
'''(16)'''  Same setting as for  '''(15)'''.  Which modifications need to be made in the  $d(\nu)$  field,
to have the duration of the rectangle   $\Rightarrow$   $\pm 2 \cdot T_\textrm{A}$.
}}
:* $d(0) = d(1) = d(15) =1, \ d(2) = d(14) = 0.5$. All other time coefficients zero  ⇒   first zero of the  ${\rm si}$ spectrum at  $4 \cdot f_{\rm A}= 250\text{ Hz}$.

{{BlaueBox|TEXT=
'''(17)'''  New setting:  IDFT of spectrum  $\rm (L)$:  Gaussian spectrum.  Interpret the result in the time domain.}}
:* Here, the time function  $x(t)$  is Gaussian with the maximum  $x(t=0)=4$.  For the spectrum the following applies:  $X(f=0)=D(0)/f_\textrm{A} = 16 \cdot \textrm{kHz}^{-1}$.
:* The equivalent duration of the pulse is  $\Delta t = X(f=0)/x(t=0)=4\text{ ms}$.  The inverse value gives the equivalent bandwidth  $\Delta f = 1/\Delta t = 250\text{ Hz}$.

==Applet Manual==
 
[[File:Anleitung_DFT_endgültig.png|left|600px|frame|Screenshot of the German version]]
    '''(A)'''     Time domain (input and result field)

    '''(B)'''     '''(A)''' representation numerical, graphical, magnitude

    '''(C)'''     frequency domain (input and result field)

    '''(D)'''     '''(C)''' representation numerical, graphical, magnitude

    '''(E)'''     Selection: DFT  $(t \to f)$  or IDFT  $(f \to t)$

    '''(F)'''     Given  $d(\nu)$ assignments (if DFT), or

                    Given  $D(\mu)$ assignments (if IDFT)

    '''(G)'''     Set input field to zero

    '''(H)'''     Move input field cyclically down (or up)

    '''( I )'''     Range for experiment execution:   exercise selection

    '''(J)'''     Range for experiment execution:   exercise definition

    '''(K)'''     Range for experiment execution:   show sample solution
 
*Given  $d(\nu)$ assignments (for DFT):

:(a)  corresponding number field,  (b)  DC signal,  (c)  Complex exponential function of time,  (d)  Harmonic oscillation  $($phase  $\varphi = 45^\circ)$,
:(e)  Cosine signal (one period),  (f)  Sine signal (one period),  (g)  Cosine signal (two periods), (h)  Alternating time coefficients,
:  (i)  Dirac pulse,  (j)  Rectangular pulse,  (k)  Triangular pulse,  (l)  Gaussian pulse.

*Given  $D(\mu)$ assignments (for IDFT):

:(A)  corresponding number field,  (B)  Constant spectrum,  (C)  Complex exponential function of frequency,  (D)  equivalent to setting (d) in time domain ,
:(E)  Cosine signal (one frequency period),  (F)  Sine signal (one frequency period),  (G)  Cosine signal (two frequency periods),  (H)  Alternating spectral coefficients,
:(I)  Dirac spectrum,  (J)  Rectangular spectrum,  (K)  Triangular spectrum,  (L)  Gaussian spectrum.

==About the Authors==

This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
*The first version was created in 2008 by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|»Thomas Großer«]]  as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).

*Last revision and English version 2020/2021 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|»Carolin Mirschina«]]  in the context of a working student activity. 

The conversion of this applet to HTML 5 was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ Studienzuschüsse]  ("study grants") of the TUM Faculty EI. We thank them.

==Once again: Open Applet in new Tab==

{{LntAppletLinkEnDe|dft_en|dft}}

Applets:Coherent and Non-Coherent On-Off Keying

2023-03-21T18:06:54Z

Hwang:

{{LntAppletLinkEnDe|on-off-keying_en|on-off-keying}}

==Applet Description==
 

Considered is the symbol error probability  $p_{\rm S}$  of   "On–off keying"   $\rm (OOK)$  in the presence of white noise,  characterized by the standard deviation  $\sigma_{\rm AWGN}$,  both in the case of  coherent demodulation  and in the case of  noncoherent demodulation.   Plotted for both cases are the probability density functions  $\rm (PDF)$  of the received signal  $r(t)$  for the possible transmitted symbols  $s_0$  and  $s_1 \equiv 0$. 
*In the coherent case, there are two Gaussian functions around  $s_0$  and  $s_1$.

*In the incoherent case,  there is a Rayleigh PDF for the symbol  $s_1 = 0$  and a Rice PDF for  $s_0 \ne 0$,  whose form also depends on the input parameter  $C_{\rm Rice}$.

The applet returns the composite probabilities  ${\rm Pr}(\boldsymbol{r_0} \cap \boldsymbol{s_1})$   ⇒   $($filled blue area in the PDF graph$)$  and  ${\rm Pr}(\boldsymbol{r_1} \cap \boldsymbol{s_0})$   ⇒   $($red area$)$  and as a final result: 
:$$p_{\rm S} = {\rm Pr}(\boldsymbol{r} \ne \boldsymbol{s})= {\rm Pr}(\boldsymbol{r_0} \cap \boldsymbol{s_1}) + {\rm Pr}(\boldsymbol{r_1} \cap \boldsymbol{s_0}). $$
*All these quantities also depend on the decision threshold  $G$  whose optimal value in each case is also determined.

*In addition,  the applet shows which error one makes when approximating the generally more complicated Rice PDF by the best possible Gaussian PDF.

==Theoretical Background==
 
===On–Off–Keying with coherent demodulation===
The simplest digital modulation method is  "On–off keying"  $\rm (OOK)$.  This method – also called  "Amplitude Shift Keying"  $\rm (2–ASK)$  – can be characterized as follows: 

[[File:EN_Dig_T_4_4_S3.png|right|frame|Signal space constellations for on-off keying|class=fit]]

*$\rm OOK$  is a binary and one-dimensional modulation method,  for example with  $s_{1} \equiv 0$  and
:*$\boldsymbol{s}_{0} = \{s_0,\ 0\}$  $($for cosinusoidal carrier,  left graph$)$  resp.

:*$\boldsymbol{s}_{0} = \{0,\ -s_0\}$  $($for sinusoidal carrier,  right graph$)$.

*With coherent demodulation,  the signal space constellation of the received signal is equal to that of the transmitted signal and again consists of the two points  $\boldsymbol{r}_0=\boldsymbol{s}_0$  and  $\boldsymbol{r}_1=\boldsymbol{s}_1$.  

*In this case,  the AWGN noise is one-dimensional with variance  $\sigma_{\rm AWGN}^2$  and one obtains  corresponding to the    [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Error_probability_for_symbols_with_equal_probability|"theory part"]]  for the  "symbol error probability":
:$$p_{\rm S} = {\rm Pr}(\boldsymbol{r}\ne \boldsymbol{s})= {\rm Q} \left ( \frac{s_0/2}{\sigma_{\rm AWGN}}\right )
= {\rm Q} \left ( \sqrt{ {E_{\rm S}}/{N_0}}\right ) \hspace{0.05cm}. $$

To this it should be noted:
#The function  ${\rm Q}(x)$  is called the  [[Applets:Complementary_Gaussian_Error_Functions|"Complementary Gaussian Error Function"]].
#The above equation applies to equally probable symbols with the decision threshold  $G$  midway between  $\boldsymbol{r}_0$  and  $\boldsymbol{r}_1$. 
#The distance of the two signal points from the decision threshold  $G$  is thus respectively  $\Delta G = s_0/2$  $($counter in the argument of the first  $\rm Q$–function$)$.
#$E_{\rm S}=s_0^2/2 \cdot T$  denotes for this case the  "average energy per symbol"  and  $N_0=2T \cdot \sigma_{\rm AWGN}^2$  is the  $($one-sided$)$  AWGN noise power density.

[[File:Applet_Bild3.png|right|frame| BER calculation for coherent demodulation]]
{{GraueBox|TEXT=
$\text{Example 1:}$  Let be  $\sigma_{\rm AWGN}= 0.8$  and  $s_{0} = 2$,  ⇒   $G=1$  $($these values are normalized to  $1\hspace{0.05cm} {\rm V})$.

The graph shows two  "half Gaussian functions"  around  $s_1=0$  $($blue curve$)$  and  $s_0=2$  $($red curve$)$.  The threshold value  $G$.  The shaded areas mark the symbol error probability.

*According to the first equation,  with  $\Delta G = s_{0} -G= G-s_1 = 1$:  
:$$p_{\rm S} = {\rm Q} ( 1/0.8 )= {\rm Q} ( 1.25 )\approx 10.56 \%.$$
*Similarly,  the second equation provides:  $E_{\rm S}/{N_0} = 1/4 \cdot s_0^2/\sigma_{\rm AWGN}^2 = 1.5615$:
:$$p_{\rm S} = {\rm Q} (\sqrt{1.5615} )\approx 10.56 \%.$$

Due to symmetry,  the threshold  $G=1$  is optimal.  In this case,  the red and blue shaded areas are equal   ⇒   the symbols  $\boldsymbol{s}_{0}$  and  $\boldsymbol{s}_{1}$  are falsified in the same way.

With  $G\ne 1$  there is a larger falsification probability.  For example,  with  $G=0.6$:
:$$p_{\rm S} = {\rm Pr}(\boldsymbol{r}\ne \boldsymbol{s}) = {\rm Pr}(\boldsymbol{r_0} \cap \boldsymbol{s_1})
+ {\rm Pr}(\boldsymbol{r_1} \cap \boldsymbol{s_0})= 1/2 \cdot {\rm Q} ( 0.75)+ 1/2 \cdot {\rm Q} ( 1.75)\approx 13.33\% .$$

Here the falsification probability for the symbol  $\boldsymbol{s}_{1}$   ⇒   blue filled area ${\rm Pr}(\boldsymbol{r_0} \cap \boldsymbol{s_1}) \approx 11. 33\%$  is much larger than that of the symbol  $\boldsymbol{s}_{0}$   ⇒   red filled area ${\rm Pr}(\boldsymbol{r_1} \cap \boldsymbol{s_0}) \approx 2\%$. }}

===On–Off–Keying with noncoherent demodulation===

The following diagram shows the structure  $($in the equivalent low-pass range$)$  of the optimal OOK receiver for incoherent demodulation.  See  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Non-coherent_demodulation_of_.22on.E2.80.93off_keying.22_.28OOK.29|Detailed description]].  According to this graph applies:

[[File:EN_Dig_T_4_5_S2b_neu.png|right|frame|Receiver for incoherent OOK demodulation  $($complex signals are labeled blue$)$|class=fit]]

*The input signal  $\boldsymbol{r}(t) = \boldsymbol{s}(t) \cdot {\rm e}^{\hspace{0.02cm}{\rm j}\hspace{0.03cm}\phi} + \boldsymbol{n}(t)$  at the receiver is generally complex because of the current phase angle  $\phi$  and because of the complex noise term  $\boldsymbol{n}(t)$.

*Now the correlation between the complex received signal  $\boldsymbol{r}(t)$  and a  [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#Basis_functions_of_complex_time_signals| complex basis function]]  $\boldsymbol{\xi}(t)$  is required. 

*The result is the  $($complex$)$  detected value  $\boldsymbol{r}$,  from which the magnitude  $y = |\boldsymbol{r}(t)|$  is formed as a real decision input. 

*If  $y \gt G$,  then the estimated value  $m_0$  for the symbol  $\boldsymbol{s}_{0}$  is output,  otherwise the estimated value  $m_1$  for the symbol  $\boldsymbol{s}_{1}$.

*Once again,  the mean symbol error probability can be represented as the sum of two composite probabilities:
:$$p_{\rm S} = {\rm Pr}(\boldsymbol{r}\ne \boldsymbol{s}) = {\rm Pr}(\boldsymbol{r_0} \cap \boldsymbol{s_1})
+ {\rm Pr}(\boldsymbol{r_1} \cap \boldsymbol{s_0}).$$

===Error probability calculation considering Rayleigh and Rice distribution===

To calculate the symbol error probability for incoherent demodulation,  we start from the following graph.  Shown is the received signal in the equivalent low-pass region in the complex plane.

[[File:Applet_Bild1.png|right|frame|Incoherent demodulation of On-Off-Keying|class=fit]]

#The point  $\boldsymbol{s_1}=0$  leads in the received signal again to  $\boldsymbol{r_1}=0$.
#In contrast,  $\boldsymbol{r}_0 = \boldsymbol{s}_0 \cdot {\rm e}^{\hspace{0.02cm}{\rm j}\hspace{0.03cm}\phi}$  can lie on any point of a circle with  radius  $1$  since the phase  $\phi$  is unknown. 
#The decision process taking into account that the AWGN noise is now to be interpreted in two dimensions,  as indicated by the arrows in the graph. 
#The decision region  $I_1$  for symbol  $\boldsymbol{s_1}$  is the blue filled circle with radius  $G$,  where the correct value of  $G$  remains to be determined.
#If the received value  $\boldsymbol{r}$ is outside this circle,  i.e. in the red highlighted area  $I_0$,  the decision is in favor of  $\boldsymbol{s_0}$. 

$\rm Rayleigh\ portion$ 

Considering the AWGN–noise,  $\boldsymbol{r_1}=\boldsymbol{s_1} + \boldsymbol{n_1}$.  The noise component  $\boldsymbol{n_1}$  has a  [[Theory_of_Stochastic_Signals/Further_Distributions#Rayleigh_PDF|Rayleigh distribution]]  $($amount of the two mean-free Gaussian components for  $I$  and  $Q)$.

*Their conditional PDF is with the rotationally symmetric noise component  $\eta$  with  $\sigma=\sigma_{\rm AWGN}$ :
:$$f_{y\hspace{0.05cm}\vert \hspace{0.05cm}\boldsymbol{s_1}} (\eta \hspace{0.05cm}\vert \hspace{0.05cm} \boldsymbol{s_1})=\frac{\eta}{\sigma^2}\cdot {\rm e}^{-\eta^2 / ( 2 \hspace{0.05cm}\cdot \hspace{0.05cm}\sigma^2) } = f_{\rm Rayleigh}(\eta) .$$
*Thus one obtains for the conditional probability
:$${\rm Pr}(\boldsymbol{r_0}|\boldsymbol{s_1}) = \int_{G}^{\infty}f_{\rm Rayleigh}(\eta) \,{\rm d} \eta
\hspace{0.05cm},$$
:and with the factor  $1/2$  because of the equally probable transmit symbols, the composite probability:
:$${\rm Pr}(\boldsymbol{r_0} \cap \boldsymbol{s_1}) = 1/2 \cdot {\rm Pr}(\boldsymbol{r_0}|\boldsymbol{s_1})= 1/2 \cdot \int_{G}^{\infty}f_{\rm Rayleigh}(\eta) \,{\rm d} \eta
\hspace{0.05cm}.$$

$\rm Rice\ portion$ 

The noise component  $\boldsymbol{n_0}$  has a  [[Theory_of_Stochastic_Signals/Further_Distributions#Rice_PDF|Rice distribution]] 
$($magnitude of Gaussian components with mean values  $m_x$  and  $m_y)$   ⇒   constant  $C=\sqrt{m_x^2 + m_y^2}$ $($Note:   In the applet, the constant  $C$  is denoted by  $C_{\rm Rice}$ $)$.

:$$f_{y\hspace{0.05cm}\vert \hspace{0.05cm}\boldsymbol{s_0}} (\eta \hspace{0.05cm}\vert \hspace{0.05cm} \boldsymbol{s_0})=\frac{\eta}{\sigma^2}\cdot{\rm e}^{-({C^2+\it \eta^{\rm 2} })/ ({\rm 2 \it \sigma^{\rm 2} })}\cdot {\rm I_0}(\frac{\it \eta\cdot C}{\sigma^{\rm 2} }) = f_{\rm Rice}(\eta) \hspace{1.4cm}{\rm with} \hspace{1.4cm} {\rm I_0}(\eta) = \sum_{k=0}^{\infty}\frac{(\eta/2)^{2k} }{k! \cdot {\rm \Gamma ({\it k}+1)} }.$$

This gives the second composite probability:

:$${\rm Pr}(\boldsymbol{r_1} \cap \boldsymbol{s_0}) = 1/2 \cdot \int_{0}^{G}f_{\rm Rice}(\eta) \,{\rm d} \eta
\hspace{0.05cm}.$$

[[File:P ID3148 Dig T 4 5 S2c version1.png|right|frame|Density functions for "OOK, non-coherent"]]
{{GraueBox|TEXT=
$\text{Example 2:}$  The graph shows the result of this equation for  $\sigma_{\rm AWGN} = 0.5$  and  $C_{\rm Rice} = 2$.  The decision threshold is at  $G \approx 1.25$.  One can see from this plot:

*The symbol error probability  $p_{\rm S}$  is the sum of the two colored areas.  As in Example 1 for the coherent case:
:$$p_{\rm S} = {\rm Pr}(\boldsymbol{r}\ne \boldsymbol{s}) = {\rm Pr}(\boldsymbol{r_0} \cap \boldsymbol{s_1})
+ {\rm Pr}(\boldsymbol{r_1} \cap \boldsymbol{s_0}).$$
*The area marked in blue gives the composite probability  ${\rm Pr}(\boldsymbol{r_0} \cap \boldsymbol{s_1}) \approx 2.2\%$  This is calculated as the integral over half the Rayleigh PDF in the range from  $G$  to  $\infty$.

*The red highlighted area gives the composite probability  ${\rm Pr}(\boldsymbol{r_1} \cap \boldsymbol{s_0}) \approx 2.4\%$  This is calculated as the integral over half the Rice PDF in the range from  $0$  to  $G$.

*Thus obtaining  $p_{\rm S} \approx 4.6\%$.  Note that the red and blue areas are not equal and that the optimal decision boundary  $G_{\rm opt}$  is obtained from the intersection of the two curves.

*The optimal decision threshold  $G_{\rm opt}$  is obtained as the intersection of the blue and red curves.}}

==Exercises==
 

*Select the number  $(1,\ 2$, ... $)$  of the task to be processed.  The number "0" corresponds to a "Reset":  Setting as at the program start.
*A task description is displayed.  Parameter values are adjusted.  Solution after pressing "Sample solution". 
*Always interpret the graphics and the numerical results.  The symbols  $s_0$  (adjustable) and  ${s}_{1}\equiv 0$  are equal probability.
*For space reasons, in some of the following questions and sample solutions we also use  $\sigma = \sigma_{\rm AWGN}$  and  $C = C_{\rm Rice}$. 

{{BlueBox|TEXT=
'''(1)'''   We consider  $\text{coherent}$  demodulation with  $\sigma_{\rm AWGN} = 0.5$  and  $s_0 = 2$.  What is the smallest possible value for the symbol error probability  $p_{\rm S}$? }}

*For coherent demodulation, the PDF of the reception signal is composed of two "half" Gaussian functions around  $s_0 = 2$  $($red$)$ and  $s_1 = 0$  $($blue$)$.
*Here the minimum  $p_{\rm S}$ value results with  $G=1$  and  $\Delta G = s_{0} -G= G-s_1 = 1$  to  $p_{\rm S}= {\rm Q} ( \Delta G/\sigma )={\rm Q} ( 1/0.5 )= {\rm Q} ( 2 )\approx 2.28 \%.$
*With  $G=1$  both symbols are falsified equally.   The blue area ${\rm Pr}(\boldsymbol{r_0} \cap \boldsymbol{s_1})$  is equal to the red area  ${\rm Pr}(\boldsymbol{r_1} \cap \boldsymbol{s_0})$.  Their sum gives  $p_{\rm S}$.
*With  $G=0.5$  the red area is almost zero.  Nevertheless   $p_{\rm S}\approx 8\%$  (sum of both areas)  is more than twice as large as with  $G_{\rm opt}=1$.

{{BlueBox|TEXT=
'''(2)'''   Now let  $\sigma = 0.75$.  With what  $s_0$  value does optimal $G$ give the same symbol error probability as in $(1)$?  Then what is the quotient  $E_{\rm S}/N_0$?}}

*In general  $p_{\rm S}= {\rm Q}\big ( (s_0/2) / \sigma \big )$.  If one increases  $\sigma$  from  $0. 5$  to   $0.75$, then  $s_0$  must also be increased   ⇒   $s_0 = 3$   ⇒   $p_{\rm S}= {\rm Q} ( 1.5/ 0.75 )= {\rm Q} ( 2 )$.
*Except  $p_{\rm S}= {\rm Q}\big ( (s_0/2) / \sigma \big )$  but also holds:  $p_{\rm S}= {\rm Q} ( \sqrt{E_{\rm S}/N_0} )$.  It follows:  $p_{\rm S}= {\rm Q}(2) ={\rm Q} ( \sqrt{E_{\rm S}/N_0})$   ⇒   $\sqrt{E_{\rm S}/N_0}= 2$   ⇒   $E_{\rm S}/N_0= 4$.
*For control:  $E_{\rm S}=s_0^2/2 \cdot T, \ N_0=2T \cdot \sigma^2$   ⇒   $E_{\rm S}/N_0 =s_0^2/(4 \cdot \sigma^2)= 3^2/(4 \cdot 0. 75^2)=4$.  The same  $E_{\rm S}/N_0 =4$  results for the problem  $(1)$.

{{BlueBox|TEXT=
'''(3)'''   Now consider  $\text{non–coherent}$  demodulation with  $\sigma_{\rm AWGN} = 0.75$,  $C_{\rm Rice} = 2.25$  and  $G=2$.  What is the symbol error probability  $p_{\rm S}$? }}

*For non–coherent demodulation, the PDF of the reception signal is composed of "half" a Rayleigh function $($blue$)$ and "half" a Rice function $($red$)$.
*${\rm Pr}(\boldsymbol{r_0} \cap \boldsymbol{s_1}) \approx 1.43\%$  gives the proportions of the blue curve above  $G =2$, and ${\rm Pr}(\boldsymbol{r_1} \cap \boldsymbol{s_0}) \approx 15. 18\%$  the proportions of the red curve below  $G =2$.
*With  $G=2$  the sum for the symbol error probability is  $p_{\rm S}\approx 16.61\%$ , and with  $G_{\rm opt}=1.58$  a slightly better value:  $p_{\rm S}\approx 12.25\%$.

{{BlueBox|TEXT=
'''(4)'''   Let  $X$  be a Rayleigh random variable in general and  $Y$  be a Rice random variable, each with above parameters.  How large are  ${\rm Pr}(X\le 2)$  and  ${\rm Pr}(Y\le 2)$ ?}}

* It holds  ${\rm Pr}(Y\le 2) = 2 \cdot {\rm Pr}(\boldsymbol{r_1} \cap \boldsymbol{s_0}) \approx 30.36\%$,  since in the applet the Rice PDF is represented by the factor  $1/2$.
*In the same way  ${\rm Pr}(X> 2) = 2 \cdot {\rm Pr}(\boldsymbol{r_0} \cap \boldsymbol{s_1}) \approx 2.86\%$   ⇒   ${\rm Pr}(X \le 2)= 1-0.0286 = 97.14\%$.

{{BlueBox|TEXT=
'''(5)'''   We consider the values  $\sigma_{\rm AWGN} = 0.75$,  $C_{\rm Rice} = 2.25$  and  $G=G_{\rm opt}=1. 58$.  How does  $p_{\rm S}$ change when "Rice" is replaced by "Gauss" as best as possible? }}

*After the exact calculation, using the optimal threshold  $G_{\rm opt}=1.58$:     ${\rm Pr}(\boldsymbol{r_0} \cap \boldsymbol{s_1}) \approx 5. 44\%$,  ${\rm Pr}(\boldsymbol{r_1} \cap \boldsymbol{s_0}) \approx 6.81\%$   ⇒   $p_{\rm S}\approx 12.25\%$.
*With the Gaussian approximation, for the same  $G$  the first term is not changed.  The second term increases to  ${\rm Pr}(\boldsymbol{r_1} \cap \boldsymbol{s_0}) \approx 9.29\%$   ⇒   $p_{\rm S}\approx 14.73\%$.
*The new optimization of the threshold  $G$  considering the Gaussian approximation leads to  $G_{\rm opt}=1.53$  and  $p_{\rm S}\approx 14.67\%$.
*The parameters of the Gaussian distribution are set as follows:  mean  $m_{\rm Gaussian}= C_{\rm Rice}=2.25$,  standard deviation  $\sigma_{\rm Gaussian}= \sigma_{\rm AWGN}=0.75$.

{{BlueBox|TEXT=
'''(6)'''   How do the results change from  $(5)$  with  $\sigma_{\rm AWGN} = 0. 5$,  $C_{\rm Rice} = 1.5$  and with  $\sigma_{\rm AWGN} = 1$,  $C_{\rm Rice} = 3$  respectively,  each with   $G=G_{\rm opt}$? }}

*With the optimal decision threshold  $G_{\rm opt}$, the probabilities are the same, both for the exact Rice distribution and with the Gaussian approximation.
*For all three parameter sets,  $E_{\rm S}/N_0= 2.25$.  This suggests:  The results with non–coherent demodulation depend on this characteristic value alone.

{{BlueBox|TEXT=
'''(7)'''   Let the setting continue to be  $\text{non–coherent/approximation}$  with  $C_{\rm Rice} = 3$,  $G=G_{\rm opt}$.  Vary the AWGN standard deviation in the range  $0.5 \le \sigma \le 1$.          Interpret the relative error   ⇒   $\rm (False - Correct)/Correct$  as a function of the quotient  $E_{\rm S}/N_0$.}}

*With  $\sigma =0.5$   ⇒   $E_{\rm S}/N_0 = 9$  one obtains  $p_{\rm S}^{\ \rm (exact)}\approx 0. 32\%$  and  $p_{\rm S}^{\ \rm (approximate)}\approx 0.38\%$.  The absolute error is  $0.06\%$  and the relative error  $18.75\%$.
*With  $\sigma =1$   ⇒   $E_{\rm S}/N_0 = 2.25$  one obtains  $p_{\rm S}^{\ \rm (exact)}\approx 12. 25\%$  and  $p_{\rm S}^{\ \rm (approximate)}\approx 14.67\%$.  The absolute error is  $2.42\%$  and the relative error  $19.75\%$.
* ⇒   The Gaussian approximation becomes better with larger  $E_{\rm S}/N_0$.  This statement can be seen more clearly from the absolute than from the relative error.

{{BlueBox|TEXT=
'''(8)'''   Now repeat the last experiment with  $\text{coherent}$  demodulation and  $s_0 = 3$,  $G=G_{\rm opt}$.  What conclusion does the comparison with  $(7)$ allow? }}

*The comparison of  $(7)$  and  $(8)$  shows:     For each  $E_{\rm S}/N_0$  there is a greater (worse) symbol error probability with non–coherent demodulation.
*For  $E_{\rm S}/N_0= 9$:     $p_{\rm S}^{\ \rm (coherent)}\approx 0.13\%$  and  $p_{\rm S}^{\ \rm (non–coherent)}\approx 0.32\%$.   And for  $E_{\rm S}/N_0= 2.25$:     $p_{\rm S}^{\ \rm (coherent)}\approx 6.68\%$  and  $p_{\rm S}^{\ \rm (non–coherent)}\approx 12.25\%$. 
*The simpler realization of the incoherent demodulator (no clock synchronization) causes a loss of quality   ⇒   greater error probability.

==Applet Manual==
 
[[File:Exercise_impuls.png |right|frame|Bildschirmabzug (englische Version, heller Hintergrund)]]
    '''(A)'''     Theme (veränderbare grafische Oberflächengestaltung)
:* Dark:   schwarzer Hintergrund  (wird von den Autoren empfohlen)
:* Bright:   weißer Hintergrund  (empfohlen für Beamer und Ausdrucke)
:* Deuteranopia:   für Nutzer mit ausgeprägter Grün–Sehschwäche
:* Protanopia:   für Nutzer mit ausgeprägter Rot–Sehschwäche

    '''(B)'''     Vorauswahl für die Impulsform  $x_1(t)$  (rote Kurve)

    '''(C)'''     Parameterfestlegung für  $x_1(t)$ 

    '''(D)'''     Numerikausgabe für  $x_1(t_*)$  und  $X_1(f_*)$

    '''(E)'''     Vorauswahl für die Impulsform  $x_2(t)$  (blaue Kurve)

    '''(F)'''     Parameterfestlegung für  $x_2(t)$ 

    '''(G)'''     Numerikausgabe für  $x_2(t_*)$  und  $X_2(f_*)$

    '''(H)'''     Einstellung der Zeit  $t_*$  für die Numerikausgabe

    '''(I)'''     Einstellung der Frequenz  $f_*$  für die Numerikausgabe

    '''(J)'''     Bereich der graphischen Darstellung im Zeitbereich

    '''(K)'''     Bereich der graphischen Darstellung im Frequenzbereich

    '''(L)'''     Auswahl der Aufgabe entsprechend der Aufgabennummer

    '''(M)'''     Aufgabenbeschreibung und Fragestellung

    '''(N)'''     Musterlösung anzeigen und verbergen

'''Details zu den obigen Punkten  (J ) und  (K)'''

Zoom–Funktionen:        "$+$" (Vergrößern),      "$-$" (Verkleinern),      "$\rm o$" (Zurücksetzen)

Verschiebe–Funktionen:     "$\leftarrow$"     "$\uparrow$"     "$\downarrow$"     "$\rightarrow$"         "$\leftarrow$"  bedeutet:     Bildausschnitt nach links, Ordinate nach rechts

Andere Möglichkeiten:

*Bei gedrückter Shifttaste und Scrollen kann im Koordinatensystem gezoomt werden.
*Bei gedrückter Shifttaste und gedrückter linker Maustaste kann das Koordinatensystem verschoben werden.
 

==About the Authors==
 
This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
*The first version was created in 2011 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Martin_V.C3.B6lkl_.28Diplomarbeit_LB_2010.29|Martin Völkl]]  as part of his diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]  and  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Dr.-Ing._Klaus_Eichin_.28am_LNT_von_1972-2011.29|Klaus Eichin]]).

*In 2020 the program was redesigned via HTML5/JavaScript by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  in the context of a working student activity  (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ).

*Last revision and English version 2021 by Carolin Mirschina. 

*The conversion of this applet was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ "Studienzuschüsse"]  (TUM Department of Electrical and Computer Engineering).  We thank.

==Once again: Open Applet in new Tab==

{{LntAppletLinkEnDe|on-off-keying_en|on-off-keying}}

Applets:Two-dimensional Gaussian Random Variables

2023-03-21T18:06:26Z

Hwang:

{{LntAppletLinkEnDe|gauss_en|gauss}}

==Applet Description==
 
The applet illustrates the properties of two-dimensional Gaussian random variables  $XY\hspace{-0.1cm}$, characterized by the standard deviations (rms)  $\sigma_X$  and  $\sigma_Y$  of their two components, and the correlation coefficient  $\rho_{XY}$ between them. The components are assumed to be zero mean:  $m_X = m_Y = 0$.

The applet shows
* the two-dimensional probability density function   ⇒   $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$  $f_{XY}(x, \hspace{0.1cm}y)$  in three-dimensional representation as well as in the form of contour lines,
* the corresponding marginal probability density function  ⇒   $\rm 1D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$  $f_{X}(x)$  of the random variable  $X$  as a blue curve; likewise  $f_{Y}(y)$  for the second random variable,
* the two-dimensional distribution function  ⇒   $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}CDF$  $F_{XY}(x, \hspace{0.1cm}y)$  as a 3D plot,
* the distribution function  ⇒   $\rm 1D\hspace{-0.1cm}-\hspace{-0.1cm}CDF$  $F_{X}(x)$  of the random variable  $X$; also  $F_{Y}(y)$  as a red curve.

The applet uses the framework  [https://en.wikipedia.org/wiki/Plotly Plot.ly]

==Theoretical Background==
 

===Joint probability density function   ⇒   2D–PDF===

We consider two continuous value random variables  $X$  and  $Y\hspace{-0.1cm}$, between which statistical dependencies may exist. To describe the interrelationships between these variables, it is convenient to combine the two components into a  '''two-dimensional random variable'''  $XY =(X, Y)$  . Then holds:

{{BlaueBox|TEXT=
$\text{Definition:}$ 
The  '''joint probability density function'''  is the probability density function (PDF) of the two-dimensional random variable  $XY$  at location  $(x, y)$:
:$$f_{XY}(x, \hspace{0.1cm}y) = \lim_{\left.{\delta x\rightarrow 0 \atop {\delta y\rightarrow 0} }\right. }\frac{ {\rm Pr}\big [ (x - {\rm \Delta} x/{\rm 2} \le X \le x + {\rm \Delta} x/{\rm 2}) \cap (y - {\rm \Delta} y/{\rm 2} \le Y \le y +{\rm \Delta}y/{\rm 2}) \big] }{ {\rm \Delta} \ x\cdot{\rm \Delta} y}.$$

*The joint probability density function, or in short  $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$  is an extension of the one-dimensional PDF.
*$∩$  denotes the logical AND operation.
*$X$  and  $Y$ denote the two random variables, and  $x \in X$  and   $y \in Y$ indicate realizations thereof.
*The nomenclature used for this applet thus differs slightly from the description in the [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Joint_probability_density_function|Theory section]].}}

Using this 2D–PDF  $f_{XY}(x, y)$  statistical dependencies within the two-dimensional random variable  $XY$  are also fully captured in contrast to the two one-dimensional density functions   ⇒   '''marginal probability density functions''':
:$$f_{X}(x) = \int _{-\infty}^{+\infty} f_{XY}(x,y) \,\,{\rm d}y ,$$
:$$f_{Y}(y) = \int_{-\infty}^{+\infty} f_{XY}(x,y) \,\,{\rm d}x .$$

These two marginal density functions  $f_X(x)$  and  $f_Y(y)$
*provide only statistical information about the individual components  $X$  and  $Y$, respectively,
*but not about the bindings between them.

As a quantitative measure of the linear statistical bindings  ⇒   '''correlation'''  one uses.
* the  '''covariance'''  $\mu_{XY}$, which is equal to the first-order common linear moment for mean-free components:
:$$\mu_{XY} = {\rm E}\big[X \cdot Y\big] = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} X \cdot Y \cdot f_{XY}(x,y) \,{\rm d}x \, {\rm d}y ,$$
*the  '''correlation coefficient'''  after normalization to the two rms values  $σ_X$  and $σ_Y$  of the two components:
:$$\rho_{XY}=\frac{\mu_{XY} }{\sigma_X \cdot \sigma_Y}.$$

{{BlaueBox|TEXT=
$\text{Properties of correlation coefficient:}$ 
*Because of normalization, $-1 \le ρ_{XY} ≤ +1$ always holds .
*If the two random variables  $X$  and  $Y$ are uncorrelated, then  $ρ_{XY} = 0$.
*For strict linear dependence between  $X$  and  $Y$,  $ρ_{XY}= ±1$   ⇒   complete correlation.
*A positive correlation coefficient means that when  $X$ is larger, on statistical average,  $Y$  is also larger than when  $X$ is smaller.
*In contrast, a negative correlation coefficient expresses that  $Y$  becomes smaller on average as  $X$  increases}}.
 

===2D–PDF for Gaussian random variables===

For the special case  '''Gaussian random variables'''  - the name goes back to the scientist  [https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss Carl Friedrich Gauss]  - we can further note:
*The joint PDF of a Gaussian 2D random variable  $XY$  with means  $m_X = 0$  and  $m_Y = 0$  and the correlation coefficient  $ρ = ρ_{XY}$  is:
: $$f_{XY}(x, y)=\frac{\rm 1}{\rm 2\it\pi \cdot \sigma_X \cdot \sigma_Y \cdot \sqrt{\rm 1-\rho^2}}\ \cdot\ \exp\Bigg[-\frac{\rm 1}{\rm 2 \cdot (1- \it\rho^{\rm 2} {\rm)}}\cdot(\frac {\it x^{\rm 2}}{\sigma_X^{\rm 2}}+\frac {\it y^{\rm 2}}{\sigma_Y^{\rm 2}}-\rm 2\it\rho\cdot\frac{x \cdot y}{\sigma_x \cdot \sigma_Y}\rm ) \rm \Bigg]\hspace{0.8cm}{\rm with}\hspace{0.5cm}-1 \le \rho \le +1.$$
*Replacing  $x$  by  $(x - m_X)$  and  $y$  by  $(y- m_Y)$, we obtain the more general PDF of a two-dimensional Gaussian random variable with mean.
*The marginal probability density functions  $f_{X}(x)$  and  $f_{Y}(y)$  of a 2D Gaussian random variable are also Gaussian with the standard deviations  $σ_X$  and  $σ_Y$, respectively.
*For uncorrelated components  $X$  and  $Y$, in the above equation  $ρ = 0$  must be substituted, and then the result is obtained:
:$$f_{XY}(x,y)=\frac{1}{\sqrt{2\pi}\cdot\sigma_{X}} \cdot\rm e^{-\it {x^{\rm 2}}\hspace{-0.08cm}/{\rm (}{\rm 2\hspace{0.05cm}\it\sigma_{X}^{\rm 2}} {\rm )}} \cdot\frac{1}{\sqrt{2\pi}\cdot\sigma_{\it Y}}\cdot e^{-\it {y^{\rm 2}}\hspace{-0.08cm}/{\rm (}{\rm 2\hspace{0.05cm}\it\sigma_{Y}^{\rm 2}} {\rm )}} = \it f_{X} \rm ( \it x \rm ) \cdot \it f_{Y} \rm ( \it y \rm ) .$$

{{BlaueBox|TEXT=
$\text{Conclusion:}$  In the special case of a 2D random variable with Gaussian PDF  $f_{XY}(x, y)$  it also follows directly from  ''uncorrelatedness''  the  ''statistical independence:''
:$$f_{XY}(x,y)= f_{X}(x) \cdot f_{Y}(y) . $$

Please note:
*For no other PDF can the  ''uncorrelatedness''  be used to infer  ''statistical independence''  .
*But one can always   ⇒   infer  ''uncorrelatedness'' from  ''statistical independence''  for any 2D-PDF  $f_{XY}(x, y)$  because:
*If two random variables  $X$  and  $Y$  are completely (statistically) independent of each other, then of course there are no ''linear''  dependencies between them   ⇒   they are then also uncorrelated  ⇒   $ρ = 0$. }}
 
===Contour lines for uncorrelated random variables===

[[File:Sto_App_Bild2.png |frame| Contour lines of 2D-PDF with uncorrelated variables | right]]
From the conditional equation  $f_{XY}(x, y) = {\rm const.}$  the contour lines of the PDF can be calculated.

If the components  $X$  and  $Y$ are uncorrelated  $(ρ_{XY} = 0)$, the equation obtained for the contour lines is:

:$$\frac{x^{\rm 2}}{\sigma_{X}^{\rm 2}}+\frac{y^{\rm 2}}{\sigma_{Y}^{\rm 2}} =\rm const.$$
In this case, the contour lines describe the following figures:
*'''Circles'''  (if  $σ_X = σ_Y$,   green curve), or
*'''Ellipses'''  (for  $σ_X ≠ σ_Y$,   blue curve) in alignment of the two axes.
 
===Regression line===

As  '''regression line'''  is called the straight line  $y = K(x)$  in the  $(x, y)$–plane through the "center" $(m_X, m_Y)$. This has the following properties:
[[File:Sto_App_Bild1a.png|frame| Gaussian 2D PDF (approximation with $N$ measurement points) and correlation line  $y = K(x)$]]

*The mean square error from this straight line - viewed in  $y$–direction and averaged over all  $N$  measurement points - is minimal:
:$$\overline{\varepsilon_y^{\rm 2} }=\frac{\rm 1}{N} \cdot \sum_{\nu=\rm 1}^{N}\; \;\big [y_\nu - K(x_{\nu})\big ]^{\rm 2}={\rm minimum}.$$
*The correlation straight line can be interpreted as a kind of "statistical symmetry axis". The equation of the straight line in the general case is:
:$$y=K(x)=\frac{\sigma_Y}{\sigma_X}\cdot\rho_{XY}\cdot(x - m_X)+m_Y.$$

*The angle that the correlation line makes to the  $x$–axis is:
:$$\theta={\rm arctan}(\frac{\sigma_{Y} }{\sigma_{X} }\cdot \rho_{XY}).$$

===Contour lines for correlated random variables===

For correlated components  $(ρ_{XY} ≠ 0)$  the contour lines of the PDF are (almost) always elliptic, so also for the special case  $σ_X = σ_Y$.

Exception:  $ρ_{XY}=\pm 1$   ⇒   Diracwand; see  [[Aufgaben:Exercise_4.4:_Two-dimensional_Gaussian_PDF|Exercise 4.4]]  in the book "Stochastic Signal Theory", subtask  ''(5)'''.
[[File:Sto_App_Bild3.png|right|frame|height lines of the two dimensional PDF with correlated quantities]]
Here, the determining equation of the PDF height lines is:

:$$f_{XY}(x, y) = {\rm const.} \hspace{0.5cm} \rightarrow \hspace{0.5cm} \frac{x^{\rm 2} }{\sigma_{X}^{\rm 2}}+\frac{y^{\rm 2} }{\sigma_{Y}^{\rm 2} }-{\rm 2}\cdot\rho_{XY}\cdot\frac{x\cdot y}{\sigma_X\cdot \sigma_Y}={\rm const.}$$
The graph shows a contour line in lighter blue for each of two different sets of parameters.

*The ellipse major axis is dashed in dark blue.
*The  [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Regression_line|regression line]]  $K(x)$  is drawn in red throughout.

Based on this plot, the following statements are possible:
*The ellipse shape depends not only on the correlation coefficient  $ρ_{XY}$  but also on the ratio of the two standard deviations  $σ_X$  and  $σ_Y$  .
*The angle of inclination  $α$  of the ellipse major axis (dashed straight line) with respect to the  $x$–axis also depends on  $σ_X$,  $σ_Y$  and  $ρ_{XY}$  :
:$$\alpha = {1}/{2} \cdot {\rm arctan } \big ( 2 \cdot \rho_{XY} \cdot \frac {\sigma_X \cdot \sigma_Y}{\sigma_X^2 - \sigma_Y^2} \big ).$$
*The (red) correlation line  $y = K(x)$  of a Gaussian 2D random variable always lies below the (blue dashed) ellipse major axis.
* $K(x)$  can be geometrically constructed from the intersection of the contour lines and their vertical tangents, as indicated in the sketch in green color.
 
===Two dimensional cumulative distribution function   ⇒   2D–CDF===

{{BlaueBox|TEXT=
$\text{Definition:}$  The  '''2D cumulative distribution function'''  like the 2D-CDF, is merely a useful extension of the  [[Theory_of_Stochastic_Signals/Cumulative_Distribution_Function#CDF_for_continuous-valued_random_variables|one-dimensional distribution function]]  (PDF):
:$$F_{XY}(x,y) = {\rm Pr}\big [(X \le x) \cap (Y \le y) \big ] .$$}}

The following similarities and differences between the "1D–CDF" and the" 2D–CDF" emerge:
*The functional relationship between "2D–PDF" and "2D–CDF" is given by the integration as in the one-dimensional case, but now in two dimensions. For continuous random variables, the following holds:
:$$F_{XY}(x,y)=\int_{-\infty}^{y} \int_{-\infty}^{x} f_{XY}(\xi,\eta) \,\,{\rm d}\xi \,\, {\rm d}\eta .$$
*Inversely, the probability density function can be given from the cumulative distribution function by partial differentiation to  $x$  and  $y$  :
:$$f_{XY}(x,y)=\frac{{\rm d}^{\rm 2} F_{XY}(\xi,\eta)}{{\rm d} \xi \,\, {\rm d} \eta}\Bigg|_{\left.{x=\xi \atop {y=\eta}}\right.}.$$
*In terms of the cumulative distribution function  $F_{XY}(x, y)$  the following limits apply:
:$$F_{XY}(-\infty,\ -\infty) = 0,\hspace{0.5cm}F_{XY}(x,\ +\infty)=F_{X}(x ),\hspace{0.5cm}
F_{XY}(+\infty,\ y)=F_{Y}(y ) ,\hspace{0.5cm}F_{XY}(+\infty,\ +\infty) = 1.$$
*In the limiting case $($infinitely large  $x$  and  $y)$  thus the value  $1$ is obtained for the "2D–CDF". From this we obtain the  '''normalization condition'''  for the two-dimensional probability density function:
:$$\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f_{XY}(x,y) \,\,{\rm d}x \,\,{\rm d}y=1 . $$

{{BlaueBox|TEXT=
$\text{Conclusion:}$  Note the significant difference between one-dimensional and two-dimensional random variables:
*For one-dimensional random variables, the area under the PDF always yields $1$.
*For two-dimensional random variables, the PDF volume always equals $1$.}}
 

==Exercises==
 
*Select the number  $(1,\ 2$, ... $)$  of the task to be processed.  The number "0" corresponds to a "Reset":  Setting as at the program start.
*A task description is displayed.  Parameter values are adjusted.  Solution after pressing "Sample solution". 
*In the task description, we use  $\rho$  instead of  $\rho_{XY}$.
*For the one-dimensional Gaussian PDF holds:  $f_{X}(x) = \sqrt{1/(2\pi \cdot \sigma_X^2)} \cdot {\rm e}^{-x^2/(2 \hspace{0.05cm}\cdot \hspace{0.05cm} \sigma_X^2)}$.

{{BlueBox|TEXT=
'''(1)'''  Get familiar with the program using the default  $(\sigma_X=1, \ \sigma_Y=0.5, \ \rho = 0.7)$.  Interpret the graphs for  $\rm PDF$  and  $\rm CDF$.}}

* $\rm PDF$  is a ridge with the maximum at  $x = 0, \ y = 0$.  The ridge is slightly twisted with respect to the  $x$–axis.
* $\rm CDF$  is obtained from  $\rm PDF$  by continuous integration in both directions.  The maximum $($near  $1)$  occurs at  $x=3, \ y=3$.

{{BlueBox|TEXT=
'''(2)'''  The new setting is  $\sigma_X= \sigma_Y=1, \ \rho = 0$.  What are the values for  $f_{XY}(0,\ 0)$  and  $F_{XY}(0,\ 0)$?  Interpret the results}}

* The PDF maximum is  $f_{XY}(0,\ 0) = 1/(2\pi)= 0.1592$, because of  $\sigma_X= \sigma_Y = 1, \ \rho = 0$.  The contour lines are circles.
* For the CDF value:  $F_{XY}(0,\ 0) = [{\rm Pr}(X \le 0)] \cdot [{\rm Pr}(Y \le 0)] = 0.25$.  Minor deviation due to numerical integration.

{{BlueBox|TEXT=
'''(3)'''  The settings of  $(2)$  continue to apply.  What are the values for  $f_{XY}(0,\ 1)$  and  $F_{XY}(0,\ 1)$?  Interpret the results.}}

* It holds  $f_{XY}(0,\ 1) = f_{X}(0) \cdot f_{Y}(1) = [ \sqrt{1/(2\pi)}] \cdot [\sqrt{1/(2\pi)} \cdot {\rm e}^{-0.5}] = 1/(2\pi) \cdot {\rm e}^{-0.5} = 0.0965$.
* The program returns  $F_{XY}(0,\ 1) = [{\rm Pr}(X \le 0)] \cdot [{\rm Pr}(Y \le 1)] = 0.4187$, i.e. a larger value than in  $(2)$,  since it integrates over a wider range.

{{BlueBox|TEXT=
'''(4)'''  The settings are kept.  What values are obtained for  $f_{XY}(1,\ 0)$  and  $F_{XY}(1,\ 0)$?  Interpret the results}}

* Due to rotational symmetry, same results as in  $(3)$.

{{BlueBox|TEXT=
'''(5)'''  Is the statement true: "Elliptic contour lines exist only for  $\rho \ne 0$".  Interpret the  $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$  and  $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}CDF$  for  $\sigma_X=1, \ \sigma_Y=0.5$  and  $\rho = 0$.}}

* No!  Also, for  $\ \rho = 0$  the contour lines are elliptical  (not circular)  if  $\sigma_X \ne \sigma_Y$.
* For $\sigma_X \gg \sigma_Y$  the  $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$  has the shape of an elongated ridge parallel to  $x$–axis, for $\sigma_X \ll \sigma_Y$  parallel to  $y$–axis.
* For $\sigma_X \gg \sigma_Y$  the slope of  $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}CDF$  in the direction of the  $y$–axis is much steeper than in the direction of the  $x$–axis.

{{BlueBox|TEXT=
'''(6)'''  Starting from  $\sigma_X=\sigma_Y=1\ \rho = 0.7$  vary the correlation coefficient  $\rho$.  What is the slope angle  $\alpha$  of the ellipse main axis?}}

* For  $\rho > 0$:   $\alpha = 45^\circ$.     For  $\rho < 0$:   $\alpha = -45^\circ$.  For  $\rho = 0$:  The contour lines are circular and thus there are no ellipses main axis.

{{BlueBox|TEXT=
'''(7)'''  Starting from  $\sigma_X=\sigma_Y=1\ \rho = 0.7$  vary the correlation coefficient  $\rho$.  What is the slope angle  $\theta$  of the correlation line  $K(x)$?}}

* For  $\sigma_X=\sigma_Y$:   $\theta={\rm arctan}\ (\rho)$.  The slope increases with increasing  $\rho > 0$.  In all cases,  $\theta < \alpha = 45^\circ$ holds. For  $\rho = 0.7$  this gives  $\theta = 35^\circ$.

{{BlueBox|TEXT=
'''(8)'''  Starting from  $\sigma_X=\sigma_Y=0.75, \ \rho = 0.7$  vary the parameters  $\sigma_Y$  and  $\rho $.  What statements hold for the angles  $\alpha$  and  $\theta$?}}

* For  $\sigma_Y<\sigma_X$:   $\alpha < 45^\circ$.     For  $\sigma_Y>\sigma_X$:   $\alpha > 45^\circ$.  For all settings:  '''The correlation line is below the ellipse main axis'''.

{{BlueBox|TEXT=
'''(9)'''  Assume  $\sigma_X= 1, \ \sigma_Y=0.75, \ \rho = 0.7$.  Vary  $\rho$.  How to construct the correlation line from the contour lines?}}

* The correlation line intersects all contour lines at that points where the tangent line is perpendicular to the contour line.

{{BlueBox|TEXT=
'''(10)'''  Now let be  $\sigma_X= \sigma_Y=1, \ \rho = 0.95$.  Interpret the  $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$.  Which statements are true for the limiting case  $\rho \to 1$ ?}}

* The  $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}WDF$  only has components near the ellipse main axis.  The correlation line is just below:  $\alpha = 45^\circ, \ \theta = 43.5^\circ$.
* In the limiting case  $\rho \to 1$  it holds  $\theta = \alpha = 45^\circ$.  Outside the correlation line, the  $\rm 2D\hspace{-0.1cm}-\hspace{-0.1cm}PDF$  would have no shares.  That is:
* Along the correlation line, there would be a  "Dirac wall"   ⇒   All values are infinitely large, nevertheless Gaussian weighted around the mean.

==Applet Manual==
 
[[File:Anleitung_2D-Gauss.png|left|500px|frame|Screen shot from the German version]]
 
    '''(A)'''     Parameter input via slider:  $\sigma_X$,  $\sigma_Y$ and  $\rho$.

    '''(B)''''     Selection:  Representation of PDF or CDF.

    '''(C)'''     Reset:  Setting as at program start.

    '''(D)'''     Display contour lines instead of one-dimensional PDF.

    '''(E)'''     Display range for two-dimensional PDF.

    '''(F)'''     Manipulation of the three-dimensional graph (zoom, rotate, ...)

    '''(G)'''     Display range for  "one-dimensional PDF"  or  "contour lines".

    '''(H)'''     Manipulation of the two-dimensional graphics ("one-dimensional PDF")

    '''( I )'''     Area for exercises: Task selection.

    '''(J)'''     Area for exercises: Task description

    '''(K)'''     Area for exercises: Show/hide solution

    '''( L)'''     Area for exercises: Output of the sample solution

Note:    Value output of the graphics  $($both 2D and 3D$)$  via mouse control.
 

==About the Authors==
 
This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
*The first version was created in 2003 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]]   as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2019 the program was redesigned by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  as part of her bachelor thesis  (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) via "HTML5".
*Last revision and English version 2021 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  in the context of a working student activity. 

The conversion of this applet to HTML 5 was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ "Studienzuschüsse"]  (Faculty EI of the TU Munich).  We thank.

==Once again: Open Applet in new Tab==

{{LntAppletLinkEnDe|gauss_en|gauss}}

Applets:Pulses and Spectra

2023-03-21T18:06:16Z

Hwang:

{{LntAppletLinkEnDe|pulsesAndSpectra_en|pulsesAndSpectra}}

==Applet Description==
 
Time-limited symmetric signals   ⇒   "pulses"  $x(t)$  and the corresponding spectral functions  $X(f)$  are considered, namely

*Gaussian pulse, 
*rectangular pulse,  
*triangular pulse, 
*trapezoidal pulse, 
*cosine roll-off pulse,
*cosine square pulse.

Further it is to be noted:
* The functions  $x(t)$  resp.  $X(f)$  are shown for up to two parameter sets in one diagram each.
* The red curves and numbers apply to the left parameter set, the blue ones to the right parameter set.
* The abscissas  $t$  (time) and  $f$  (frequency) as well as the ordinates  $x(t)$  (signal values) and  $X(f)$  (spectral values) are normalized.

==Theoretical Background==
 
===Relationship $x(t)\Leftrightarrow X(f)$===
*Der Zusammenhang zwischen der Zeitfunktion  $x(t)$  und dem Spektrum  $X(f)$  ist durch das  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_erste_Fourierintegral|erste Fourierintegral]]  gegeben:
:$$X(f)={\rm FT} [x(t)] = \int_{-\infty}^{+\infty}x(t)\cdot {\rm e}^{-{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}t\hspace{1cm}
\rm FT\hspace{-0.1cm}: \ Fouriertransformation.$$

*Um aus der Spektralfunktion  $X(f)$  die Zeitfunktion  $x(t)$  berechnen zu können, benötigt man das  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_zweite_Fourierintegral|zweite Fourierintegral]]:
:$$x(t)={\rm IFT} [X(f)] = \int_{-\infty}^{+\infty}X(f)\cdot {\rm e}^{+{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}f\hspace{1cm}
{\rm IFT}\hspace{-0.1cm}: \rm Inverse \ Fouriertransformation.$$

*In allen Beispielen verwenden wir reelle und gerade Funktionen.  Somit gilt:
:$$x(t)=\int_{-\infty}^{+\infty}X(f)\cdot \cos(2\pi ft) \hspace{0.15cm} {\rm d}f \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ X(f)=\int_{-\infty}^{+\infty}x(t)\cdot \cos(2\pi ft) \hspace{0.15cm} {\rm d}t .$$
*$x(t)$  und  $X(f)$  haben unterschiedliche Einheiten, beispielsweise  $x(t)$  in  $\rm V$,  $X(f)$  in  $\rm V/Hz$.
*Der Zusammenhang zwischen diesem Modul und dem ähnlich aufgebauten Applet  [[Applets:Frequenzgang_und_Impulsantwort|Frequenzgang & Impulsantwort]]  basiert auf dem  [[Signal_Representation/Fourier_Transform_Laws#Vertauschungssatz|Vertauschungssatz]].
*Alle Zeiten sind auf eine Zeit  $T$  normiert und alle Frequenzen auf  $1/T$   ⇒   die Spektralwerte  $X(f)$  müssen noch mit der Normierungszeit  $T$  multipliziert werden.

{{GraueBox|TEXT=
$\text{Beispiel:}$   Stellt man einen Rechteckimpuls mit Amplitude  $A_1 = 1$  und äquivalenter Impulsdauer  $\Delta t_1 = 1$  ein, so ist  $x_1(t)$  im Bereich  $-0.5 < t < +0.5$  gleich Eins und außerhalb dieses Bereichs gleich Null.  Die Spektralfunktion  $X_1(f)$  verläuft  $\rm si$–förmig mit  $X_1(f= 0) = 1$  und der ersten Nullstelle bei  $f=1$.

*Soll mit dieser Einstellung ein Rechteckimpuls mit  $A = K = 3 \ \rm V$  und  $\Delta t = T = 2 \ \rm ms$  nachgebildet werden, dann sind alle Signalwerte mit  $K = 3 \ \rm V$  und alle Spektralwerte mit  $K \cdot T = 0.006 \ \rm V/Hz$  zu multiplizieren.
*Der maximale Spektralwert ist dann  $X(f= 0) = 0.006 \ \rm V/Hz$  und die erste Nullstelle liegt bei  $f=1/T = 0.5 \ \rm kHz$.}}

===Gaussian Pulse ===

*Die Zeitfunktion des Gaußimpulses mit der Höhe  $K$  und der (äquivalenten) Dauer  $\Delta t$  lautet:
:$$x(t)=K\cdot {\rm e}^{-\pi\hspace{0.05cm}\cdot \hspace{0.05cm}(t/\Delta t)^2}.$$
*Die äquivalente Zeitdauer  $\Delta t$  ergibt sich aus dem flächengleichen Rechteck.
*Der Wert bei  $t = \Delta t/2$  ist um den Faktor  $0.456$  kleiner als der Wert bei  $t=0$.
*Für die Spektralfunktion erhält man gemäß der Fouriertransformation:
:$$X(f)=K\cdot \Delta t \cdot {\rm e}^{-\pi(f\hspace{0.05cm}\cdot \hspace{0.05cm} \Delta t)^2} .$$
*Je kleiner die äquivalente Zeitdauer  $\Delta t$  ist, um so breiter und niedriger ist das Spektrum   ⇒   [[Signal_Representation/Fourier_Transform_Laws#Reziprozit.C3.A4tsgesetz_von_Zeitdauer_und_Bandbreite|Reziprozitätsgesetz von Bandbreite und Impulsdauer]].
*Sowohl  $x(t)$  als auch  $X(f)$  sind zu keinem  $f$–  bzw.  $t$–Wert exakt gleich Null.
*Für praktische Anwendungen kann der Gaußimpuls jedoch in Zeit und Frequenz als begrenzt angenommen werden.  Zum Beispiel ist  $x(t)$  bereits bei  $t=1.5 \Delta t$  auf weniger als  $0.1\% $  des Maximums abgefallen.

===Rectangular Pulse ===
*Die Zeitfunktion des Rechteckimpulses mit der Höhe  $K$  und der (äquivalenten) Dauer  $\Delta t$  lautet:

:$$x(t) = \left\{ \begin{array}{l} \hspace{0.25cm}K \\ K /2 \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{20}c} {\rm{f\ddot{u}r}} \\ {\rm{f\ddot{u}r}} \\ {\rm{f\ddot{u}r}} \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.05cm} t\hspace{0.05cm} \right| < T/2,} \\ {\left| \hspace{0.05cm}t\hspace{0.05cm} \right| = T/2,} \\ {\left|\hspace{0.05cm} t \hspace{0.05cm} \right| > T/2.} \\ \end{array}$$

*Der  $\pm \Delta t/2$–Wert liegt mittig zwischen links- und rechtsseitigem Grenzwert.
*Für die Spektralfunktion erhält man entsprechend den Gesetzmäßigkeiten der Fouriertransformation (1. Fourierintegral):
:$$X(f)=K\cdot \Delta t \cdot {\rm si}(\pi\cdot \Delta t \cdot f) \quad \text{mit} \ {\rm si}(x)=\frac{\sin(x)}{x}.$$
*Der Spektralwert bei  $f=0$  ist gleich der Rechteckfläche der Zeitfunktion.
*Die Spektralfunktion besitzt Nullstellen in äquidistanten Abständen  $1/\Delta t$.
*Das Integral über der Spektralfunktion  $X(f)$  ist gleich dem Signalwert zum Zeitpunkt  $t=0$, also der Impulshöhe  $K$.

===Triangular Pulse===
*Die Zeitfunktion des Dreieckimpulses mit der Höhe  $K$  und der (äquivalenten) Dauer  $\Delta t$  lautet:

:$$x(t) = \left\{ \begin{array}{l} \hspace{0.25cm}K\cdot (1-|t|/{\Delta t}) \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{20}c} {\rm{f\ddot{u}r}} \\ {\rm{f\ddot{u}r}} \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.05cm} t\hspace{0.05cm} \right| < \Delta t,} \\ {\left| \hspace{0.05cm}t\hspace{0.05cm} \right| \ge \Delta t.} \\ \end{array}$$

*Die absolute Zeitdauer ist  $2 \cdot \Delta t$;  diese ist doppelt so groß als die des Rechtecks.
*Für die Spektralfunktion erhält man gemäß der Fouriertransformation:
:$$X(f)=K\cdot \Delta f \cdot {\rm si}^2(\pi\cdot \Delta t \cdot f) \quad \text{mit} \quad {\rm si}(x)=\frac{\sin(x)}{x}.$$
*Obige Zeitfunktion ist gleich der Faltung zweier Rechteckimpulse, jeweils mit Breite  $\Delta t$.
*Daraus folgt:  $X(f)$  beinhaltet anstelle der  ${\rm si}$-Funktion die  ${\rm si}^2$-Funktion.
*$X(f)$  weist somit ebenfalls Nullstellen im äquidistanten Abständen  $1/\Delta f$  auf.
*Der asymptotische Abfall von  $X(f)$  erfolgt hier mit  $1/f^2$, während zum Vergleich der Rechteckimpuls mit  $1/f$  abfällt.

===Trapezoidal Pulse ===
Die Zeitfunktion des Trapezimpulses mit der Höhe  $K$  und den Zeitparametern  $t_1$  und  $t_2$  lautet:
:$$x(t) = \left\{ \begin{array}{l} \hspace{0.25cm}K \\ K\cdot \frac{t_2-|t|}{t_2-t_1} \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{20}c} {\rm{f\ddot{u}r}}\quad \\ {\rm{f\ddot{u}r}}\quad \\ {\rm{f\ddot{u}r}} \quad \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.05cm} t\hspace{0.05cm} \right| \le t_1,} \\ {t_1\le \left| \hspace{0.05cm}t\hspace{0.05cm} \right| \le t_2,} \\ {\left|\hspace{0.05cm} t \hspace{0.05cm} \right| \ge t_2.} \\ \end{array}$$

*Für die äquivalente Impulsdauer (flächengleiches Rechteck) gilt:   $\Delta t = t_1+t_2$.
*Der Rolloff-Faktor (im Zeitbereich) kennzeichnet die Flankensteilheit:
:$$r=\frac{t_2-t_1}{t_2+t_1}.$$
*Der Sonderfall  $r=0$  entspricht dem Rechteckimpuls und der Sonderfall  $r=1$  dem Dreieckimpuls.
*Für die Spektralfunktion erhält man gemäß der Fouriertransformation:
:$$X(f)=K\cdot \Delta t \cdot {\rm si}(\pi\cdot \Delta t \cdot f)\cdot {\rm si}(\pi \cdot r \cdot \Delta t \cdot f) \quad \text{mit} \quad {\rm si}(x)=\frac{\sin(x)}{x}.$$
*Der asymptotische Abfall von  $X(f)$  liegt zwischen  $1/f$  $($für Rechteck,  $r=0)$  und  $1/f^2$  $($für Dreieck,  $r=1)$.

===Cosine roll-off Pulse ===
Die Zeitfunktion des Cosinus-Rolloff-Impulses mit der Höhe  $K$  und den Zeitparametern  $t_1$  und  $t_2$  lautet:

:$$x(t) = \left\{ \begin{array}{l} \hspace{0.25cm}K \\ K\cdot \cos^2\Big(\frac{|t|-t_1}{t_2-t_1}\cdot {\pi}/{2}\Big) \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{20}c} {\rm{f\ddot{u}r}}\quad \\ {\rm{f\ddot{u}r}}\quad \\ {\rm{f\ddot{u}r}}\quad \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.05cm} t\hspace{0.05cm} \right| \le t_1,} \\ {t_1\le \left| \hspace{0.05cm}t\hspace{0.05cm} \right| \le t_2,} \\ {\left|\hspace{0.05cm} t \hspace{0.05cm} \right| \ge t_2.} \\ \end{array}$$

*Für die äquivalente Impulsdauer (flächengleiches Rechteck) gilt:   $\Delta t = t_1+t_2$.
*Der Rolloff-Faktor (im Zeitbereich) kennzeichnet die Flankensteilheit:
:$$r=\frac{t_2-t_1}{t_2+t_1}.$$
*Der Sonderfall  $r=0$  entspricht dem Rechteckimpuls und der Sonderfall  $r=1$  dem Cosinus-Quadrat-Impuls.
*Für die Spektralfunktion erhält man gemäß der Fouriertransformation:
:$$X(f)=K\cdot \Delta t \cdot \frac{\cos(\pi \cdot r\cdot \Delta t \cdot f)}{1-(2\cdot r\cdot \Delta t \cdot f)^2} \cdot {\rm si}(\pi \cdot \Delta t \cdot f).$$
*Je größer der Rolloff-Faktor  $r$  ist, desto schneller nimmt  $X(f)$  asymptotisch mit  $f$  ab.

===Cosine square Pulse ===
*Dies ist ein Sonderfall des Cosinus-Rolloff-Impulses und ergibt sich für  $r=1 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}t_1=0, \ t_2= \Delta t$:

:$$x(t) = \left\{ \begin{array}{l} \hspace{0.25cm}K\cdot \cos^2\Big(\frac{|t|\hspace{0.05cm}\cdot \hspace{0.05cm} \pi}{2\hspace{0.05cm}\cdot \hspace{0.05cm} \Delta t}\Big) \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{20}c} {\rm{f\ddot{u}r}} \\ {\rm{f\ddot{u}r}} \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.05cm} t\hspace{0.05cm} \right| < \Delta t,} \\ {\left| \hspace{0.05cm}t\hspace{0.05cm} \right| \ge \Delta t.} \\ \end{array}$$

*Für die Spektralfunktion erhält man gemäß der Fouriertransformation:
:$$X(f)=K\cdot \Delta f \cdot \frac{\pi}{4}\cdot \big [{\rm si}(\pi(\Delta t\cdot f +0.5))+{\rm si}(\pi(\Delta t\cdot f -0.5))\big ]\cdot {\rm si}(\pi \cdot \Delta t \cdot f).$$
*Wegen der letzten  ${\rm si}$-Funktion ist  $X(f)=0$  für alle Vielfachen von  $F=1/\Delta t$.  Die äquidistanten Nulldurchgänge des Cos-Rolloff-Impulses bleiben erhalten.
*Aufgrund des Klammerausdrucks weist  $X(f)$  nun weitere Nulldurchgänge bei  $f=\pm1.5 F$,  $\pm2.5 F$,  $\pm3.5 F$, ... auf.
*Für die Frequenz  $f=\pm F/2$  erhält man die Spektralwerte  $K\cdot \Delta t/2$.
*Der asymptotische Abfall von  $X(f)$  verläuft in diesem Sonderfall mit  $1/f^3$.

==Exercises==
 

* First select the number  $(1,\text{...}, 7)$  of the exercise.  The number  $0$  corresponds to a "Reset":  Same setting as at program start.
*A task description is displayed.  The parameter values are adjusted.  Solution after pressing "Show solution". 
* "Red" refers to the first parameter set ⇒ $x_1(t) \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ X_1(f)$,  "Blue" refers to the second parameter set ⇒ $x_2(t) \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ X_2(f)$.
*Values with magnitude less than  $0.0005$  are output in the program as "zero".

{{BlaueBox|TEXT=
'''(1)'''   Compare the  red Gaussian pulse  $(A_1 = 1, \Delta t_1 = 1)$  with the  blue rectangular pulse  $(A_2 = 1, \Delta t_2 = 1)$ ⇒ default setting.
           What are the differences in the time and frequency domain?}}

* The Gaussian pulse theoretically reaches infinity in the time as well as in the frequency domain. 
* Practically  $x_1(t)$  for  $|t| > 1.5$  and  $X_1(f)$  for  $|f| > 1.5$  are almost zero. 
* The rectangle is strictly limited in time:  $x_2(|t| > 0.5) \equiv 0$.  $X_2(f)$  has shares in a much larger range than  $X_1(f)$. 
* It holds  $X_1(f = 0) = X_2(f = 0)$  since the integral over the Gaussian pulse  $x_1(t)$  is equal to the integral over the rectangular pulse  $x_2(t)$.

{{BlaueBox|TEXT=
'''(2)'''   Compare the  red Gaussian pulse  $(A_1 = 1,  \Delta t_1 = 1)$ with the  blue rectangular pulse  $(A_2 = 1,  \Delta t_2)$.           Vary the equivalent pulse duration  $\Delta t_2$  between  $0.5$  and  $2$.  Interpret the displayed graphs.}}

* One can recognize the reciprocity law of bandwidth and pulse duration.  The greater  $\Delta t_2$, the higher and narrower the spectral function  $X_2(f)$. 
* For each setting of  $\Delta t_2$,  $x_1(t=0)$  and  $x_2(t=0)$  are equal   ⇒   Also, the integrals over  $X_1(f)$  and  $X_2(f)$  are identical.

{{BlaueBox|TEXT=
'''(3)'''   Compare the  red Gaussian pulse  $(A_1 = 1,  \Delta t_1 = 1)$ with the  blue rectangular pulse  $(A_2 = 1,  \Delta t_2 = 0.5)$.           Vary  $\Delta t_2$  between  $0.05$  and  $2$.  Interpret the displayed graphs and extrapolate the result.}}

* The blue spectrum is now twice as wide as the red one, but only half as high.  First zero of  $X_1(f)$  at  $f = 1$, of  $X_2(f)$  at  $f = 2$. 
* Reduction of  $\Delta t_2$:  $X_2(f)$  lower and wider.  Very flat course at  $\Delta t_2 = 0.05$:  $X_2(f = 0)= 0.05$,  $X_2(f = \pm 3)= 0.048$. 
* If one choose  $\Delta t_2 = \varepsilon \to 0$  (not possible in the program),  the result would be the almost constant, very small spectrum  $X_2(f)=A \cdot \varepsilon \to 0$. 
* Increasing the amplitude to  $A=1/\varepsilon$  results in the constant spectral function  $X_2(f) = 1$  of the Dirac function  $\delta(t)$.  That means: 
* $\delta(t)$  is approximated by a rectangle  $($width  $\Delta t = \varepsilon \to 0$,  height  $A = 1/\varepsilon \to \infty)$.  The weight of the Dirac function is one:  $x(t) = 1 \cdot \delta (t)$.

{{BlaueBox|TEXT=
'''(4)'''   Compare the  rectangular pulse  $(A_1 = 1,   \Delta t_1 = 1)$  with the  triangular pulse  $(A_2 = 1,   \Delta t_2 = 1)$.  Interpret the spectral functions.}}

* The (normalized) spectrum of the rectangle  $x_1(t)$  with the (normalized) parameters  $A_1 = 1, \ \ \Delta t_1 = 1$  is:  $X_1(f)= {\rm si}(\pi\cdot f)= {\rm sinc}(f)$. 
* The convolution of the rectangle  $x_1(t)$  with itself gives the triangle  $x_2(t) = x_1(t) \star x_1(t)$.  By the convolution theorem:   $X_2(f) = X_1(f)^2 $. 
* By squaring the  ${\rm sinc}(f)$–shaped spectral function  $X_1(f)$  the zeros of  $X_2(f)$  remain unchanged.  But now it holds that: $X_2(f) \ge 0$.

{{BlaueBox|TEXT=
'''(5)'''   Compare the  trapezoidal pulse  $(A_1 = 1, \Delta t_1 = 1, r_1 = 0.5)$  with the 
triangular pulse $(A_2 = 1,   \Delta t_2 = 1)$.          Vary  $r_1$  between  $0$  and  $1$.  Interpret the spectral function  $X_1(f)$.}}

* The trapezoidal pulse with roll–off factor  $r_1= 0$  is identical to the rectangular pulse.  The "normalized spectrum" is  $X_1(f)= {\rm sinc}(f)$. 
* The trapezoidal pulse with roll–off factor  $r_1= 1$  is identical to the triangular pulse.  The "normalized spectrum" is  $X_1(f)= {\rm sinc}^2(f)$. 
* In both cases  $X_1(f)$  has equidistant zeros at  $\pm 1$,  $\pm 2$, ...  (none else);   $0 < r_1 < 1$:  depending on  $r_1$  further zeros.

{{BlaueBox|TEXT=
'''(6)'''   Compare this  trapezoidal pulse  with the cosine roll-off pulse 
$(A_2 = 1,\ \Delta t_2 = 1.0,\ r_2 = 0.5)$.          Vary  $r_2$  between  $0$  and  $1$.  Interpret the spectral function  $X_2(f)$  for  $r_2 = 0.7$.}}

* With the same  $r= 0.5$  the cosine roll-off pulse  $X_2(f)$ is for  $f > 1$  greater in magnitude than the trapezoidal pulse. 
* With the same roll-off factor  $(r_1 = r_2= 0.5)$  the drop of  $X_2(f)$  around the frequency  $f = 0.5$  is steeper than the drop of  $X_1(f)$. 
* With  $r_1 = 0.5$  and  $r_2 = 0.7$  $x_1(t) \approx x_2(t)$  is valid and therefore also  $X_1(f) \approx X_2(f)$.  Comparable edge steepness.

{{BlaueBox|TEXT=
'''(7)'''   Compare the  red trapezoidal pulse  $(A_1 = 1, \Delta t_1 = 1, \ r_1 = 1)$  with the  blue cosine roll-off pulse  $(A_2 = 1,\ \Delta t_2 = 1.0, \ r_2 = 1)$.           Interpret the time function  $x_2(t)$  and the spectral function  $X_2(f)$  system theoretically.}}

* $x_2(t) = \cos^2(|t|\cdot \pi/2) \ \ \ \text{for} \ |t| \le 1$  is the  cosine square pulse.  Zeros at  $f = \pm 1$,  $\pm 2$, ... 
* For the frequency  $f=\pm 0.5$  one obtains the spectral values  $X_2(f)=0.5$.  The asymptotic decline is shown here with  $1/f^3$.

==Applet Manual==
 
[[File:Exercise_impuls.png |right|frame|Screenshot]]
    '''(A)'''     Theme (changeable graphical user interface design)
:* Dark:   dark background  (recommended by the authors)
:* Bright:   white background  (recommended for beamers and printouts)
:* Deuteranopia:   for users with pronounced green visual impairment
:* Protanopia:   for users with pronounced red visual impairment

    '''(B)'''     Preselection for pulse shape  $x_1(t)$  (red curve)

    '''(C)'''     Parameter definition for  $x_1(t)$ 

    '''(D)'''     Numeric output for  $x_1(t_*)$  and  $X_1(f_*)$

    '''(E)'''     Preselection for pulse shape  $x_2(t)$  (blue curve)

    '''(F)'''     Parameter definition for  $x_2(t)$ 

    '''(G)'''     Numeric output for  $x_2(t_*)$  and  $X_2(f_*)$

    '''(H)'''     Setting the time  $t_*$  for the numeric output

    '''(I)'''     Setting the frequency  $f_*$  for the numeric output

    '''(J)'''     Graphic field for the time domain

    '''(K)'''     Graphic field for the frequency domain

    '''(L)'''     Selection of the exercise according to the numbers

    '''(M)'''     Task description and questions

    '''(N)'''     Show and hide sample solution

==About the Authors==
 
This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
*The first version was created in 2005 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]]  as part of her diploma thesis with “FlashMX – Actionscript”  (Supervisor:  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2017 the program was redesigned by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#David_Jobst_.28Ingenieurspraxis_Math_2017.29|David Jobst]]  (Ingenieurspraxis_Math, Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) via "HTML5".
*Last revision and English version 2020 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  in the context of a working student activity. 

==Once again: Open Applet in new Tab==
{{LntAppletLinkEnDe|pulsesAndSpectra_en|pulsesAndSpectra}}

Applets:Graphical Convolution

2023-03-21T18:06:08Z

Hwang:

{{LntAppletLinkEnDe|convolution_en|convolution}}

== Applet Description==
 
This applet illustrates the convolution operation in the time domain
*between an input pulse  $x(t)$   ⇒   rectangle, triangle, Gaussian, exponential function
*and the impulse response  $h(t)$  of an LTI system with low–pass character  ⇒   slit low–pass, first or second order low–pass, Gaussian low–pass.

For the output signal  $y(t)$  corresponding to the block diagram in  $\text{Example 1}$,  then, as stated in the chapter  [[Applets:Graphical_Convolution#Graphical_Convolution|"Graphical Convolution"]]: 
:$$y( t ) = x(t) * h( t ) = \int_{ - \infty }^{ + \infty } \hspace{-0.15cm}{x( \tau )} \cdot h( {t - \tau } )\hspace{0.1cm}{\rm d}\tau .$$

For causal systems   ⇒    $h(t) \equiv 0$  for  $t < 0$  (examples: slit low–pass as well as first and second order low–pass)   can be written for this also:

:$$y( t ) = \int_{ - \infty }^{ t } \hspace{-0.15cm}{x( \tau )} \cdot h( {t - \tau } )\hspace{0.1cm}{\rm d}\tau .$$

Please note:
*All quantities – also the time $t$ – are to be understood normalized (dimensionless).
*The time functions  $x(t)$,  $h(t)$  and  $y(t)$  cannot assume negative signal values in the program.
*The ''absolute duration''  of a pulse  $y(t)$  is the (continuous) time range for which  $y(t) > 0$. 
*The ''equivalent duration''  of a pulse can be calculated by the rectangle of equal area.

==Theoretical Background==

===Convolution in the time domain===

The  [[Signal_Representation/The_Convolution_Theorem_and_Operation|"convolution theorem"]]  is one of the most important laws of the Fourier transform. We first consider the convolution theorem in the time domain and assume that the spectra of two time functions  $x_1(t)$  and  $x_2(t)$  are known:

:$$X_1 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}x_1( t ),\quad X_2 ( f )\hspace{0.1cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.1cm}x_2 ( t ).$$

Then for the time function of the product  $X_1(f) \cdot X_2(f)$ holds:

:$$X_1 ( f ) \cdot X_2 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}\int_{ - \infty }^{ + \infty } {x_1 ( \tau )} \cdot x_2 ( {t - \tau } )\hspace{0.1cm}{\rm d}\tau.$$

Here  $\tau$  is a formal integration variable with the dimension of a time.

{{BlaueBox|TEXT=
$\text{Definition:}$  The above connection of the time function  $x_1(t)$  and  $x_2(t)$  is called  '''convolution'''  and represents this functional connection with a star:

:$$x_{\rm{1} } (t) * x_{\rm{2} } (t) = \int_{ - \infty }^{ + \infty } {x_1 ( \tau ) } \cdot x_2 ( {t - \tau } ) \hspace{0.1cm}{\rm d}\tau.$$

Thus, above Fourier correspondence can also be written as follows:

:$$X_1 ( f ) \cdot X_2 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}{ {x} }_{\rm{1} } ( t ) * { {x} }_{\rm{2} } (t ).$$

[[Signal_Representation/The_Convolution_Theorem_and_Operation#Proof_of_the_Convolution_Theorem|$\text{Proof}$]]}}

''Note'':   The convolution is  '''commutative'''   ⇒   The order of the operands is interchangeable:   ${ {x}}_{\rm{1}} ( t ) * { {x}}_{\rm{2}} (t ) ={ {x}}_{\rm{2}} ( t ) * { {x}}_{\rm{1}} (t ) $.

[[File:P_ID579__Sig_T_3_4_S1_neu.png|right|frame|For the calculation of signal and spectrum at the LTI output]]
{{GraueBox|TEXT=
$\text{Example 1:}$  Any linear time-invariant (LTI) system can be described both by the frequency response  $H(f)$  and by the impulse response  $h(t)$,  the relationship between these two system quantities also being given by the Fourier transform.

If a signal  $x(t)$  with the spectrum  $X(f)$  is applied to the input, the following applies to the spectrum of the output signal:

:$$Y(f) = X(f) \cdot H(f)\hspace{0.05cm}.$$

With the convolution theorem it is now possible to calculate the output signal also directly in the time domain:

:$$y( t ) = x(t) * h( t ) = \int_{ - \infty }^{ + \infty } \hspace{-0.15cm}{x( \tau )} \cdot h( {t - \tau } )\hspace{0.1cm}{\rm d}\tau = \int_{ - \infty }^{ + \infty } \hspace{-0.15cm} {h( \tau )} \cdot x( {t - \tau } )\hspace{0.1cm}{\rm d}\tau = h(t) * x( t ).$$

From this equation, it is again clear that the convolution operation is  ''commutative''.  }}

===Convolution in the frequency domain===

The duality between the time and frequency domains also allows us to make statements regarding the spectrum of a product signal:

:$$x_1 ( t ) \cdot x_2 ( t )\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_1 (f) * X_2 (f) = \int_{ - \infty }^{ + \infty } {X_1 ( \nu )} \cdot X_2 ( {f - \nu })\hspace{0.1cm}{\rm d}\nu.$$

This result can be proved similarly to the  [[Applets:Graphical_Convolution#Convolution_in_the_time_domain|"convolution theorem in the time domain"]].  However, the integration variable  $\nu$  now has the dimension of a frequency.

[[File:P_ID580__Sig_T_3_4_S2_neu.png|right|frame|Convolution in the frequency domain]]
{{GraueBox|TEXT=
$\text{Example 2:}$  The  [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Description_in_the_time_domain|"double-sideband amplitude modulation"]]  (DSB-AM) without carrier is described by the sketched model.
*In the time domain representation (blue), the modulated signal  $s(t)$  is the product of the message signal  $q(t)$  and the (normalized) carrier signal  $z(t)$.
*According to the convolution theorem, it follows for the frequency domain (red) that the output spectrum  $S(f)$  is equal to the convolution product of  $Q(f)$  and  $Z(f)$.  }}

===Convolution of a function with a Dirac function===

The convolution operation becomes very simple if one of the two operands is a  [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|"Dirac function"]].  This is equally true for convolution in the time and frequency domain.

As an example, we consider the convolution of a function  $x_1(t)$  with the function

:$$x_2 ( t ) = \alpha \cdot \delta ( {t - T} ) \quad \circ\,\!\!\!-\!\!\!-\!\!\!-\!\!\bullet \quad X_2 ( f )= \alpha \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.03cm}2\hspace{0.03cm}{\rm{\pi }}\hspace{0.01cm}f\hspace{0.01cm}T}.$$

Then for the spectral function of the signal  $y(t) = x_1(t) \ast x_2(t)$  holds:

:$$Y( f ) = X_1 ( f ) \cdot X_2 ( f ) = X_1 ( f ) \cdot \alpha \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.03cm}2\hspace{0.03cm}{\rm{\pi }}\hspace{0.01cm}f\hspace{0.01cm}T} .$$

The complex exponential function leads to a shift by  $T$   ⇒   [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|"shifting theorem"]], the factor  $\alpha$  to an attenuation  $(\alpha < 1)$  or an amplification  $(\alpha > 1)$. It follows that:

:$$x_1 (t) * x_2 (t) = \alpha \cdot x_1 ( {t - T} ).$$

{{BlaueBox|TEXT=
$\text{In words: }$  The convolution of an arbitrary function with a Dirac function at  $t = T$  results in the function shifted to the right by  $T$,  whereby the weighting of the Dirac function by the factor  $\alpha$  must still be taken into account.}}

{{GraueBox|TEXT=
$\text{Example 3:}$  A rectangular signal  $x(t)$  is delayed by a running time  $\tau = 3\,\text{ ms}$  and attenuated by a factor  $\alpha = 0.5$  by an LTI system.

[[File:P_ID522__Sig_T_3_4_S3_neu.png|center|frame|Convolution of a rectangle with a Dirac function]]

Shift and attenuation can be seen in the output signal  $y(t)$  as well as in the impulse response  $h(t)$.}}

===Graphical Convolution===

This applet assumes the following convolution operation:
[[File:P_ID2723__Sig_T_3_4_programm.png|right|frame|Screen capture of the program "Graphical Convolution" (former version)]]
:$$y(t) = x (t) * h (t) = \int_{ - \infty }^{ + \infty } {x ( \tau )} \cdot h ( {t - \tau } )\hspace{0.1cm}{\rm d}\tau.$$

The solution of the convolution integral is to be done graphically. It is assumed that  $x(t)$  and  $h(t)$  are continuous-time signals.

Then the following steps are required:
#    '''Change''' the  '''time variables'''  of the two functions:       $x(t) \to x(\tau)$,   $h(t) \to h(\tau)$.
#  '''Mirror''' second '''function''':   $h(\tau) \to h(-\tau)$.
#  '''Shift''' mirrored '''function''' by  $t$:   $h(-\tau) \to h(t-\tau)$.
#  '''Multiplication''' of the two functions  $x(\tau)$  and  $h(t-\tau)$.
#  '''Integration'''  over the product with respect to  $\tau$  in the limits from  $-\infty$  to  $+\infty$.

Since the convolution is commutative,  $h(\tau)$  can be mirrored instead of  $x(\tau)$. 

 
The accompanying graphic shows a screen shot of an older version of this applet.
 

[[File:P_ID582__Sig_T_3_4_S4_neu.png|right|frame|Example of a convolution operation: Jump function convolved with exponential function]]
{{GraueBox|TEXT=
$\text{Example 4:}$ 
The procedure of graphical convolution is now explained with a detailed example:
*Let a jump function  $x(t) = \gamma(t)$  be applied to the input of a filter.
*Let the impulse response of the RC low-pass filter be  $h( t ) = {1}/{T} \cdot {\rm{e} }^{ - t/T}.$

The graph shows the input signal  $x(\tau)$ in red, the impulse response  $h(\tau)$ in blue, and the output signal  $y(\tau)$ in gray.
The time axis has already been renamed  $\tau$. 

The output signal can be calculated, for example, according to the following equation:

:$$y(t) = h(t) * x(t) = \int_{ - \infty }^{ + \infty } {h( \tau )} \cdot x( {t - \tau } )\hspace{0.1cm}{\rm d}\tau.$$

A few more remarks on the graphical convolution:
*The output value at  $t = 0$  is obtained by mirroring the input signal  $x(\tau)$,  multiplying this mirrored signal  $x(-\tau)$  by the impulse response  $h(\tau)$,  and integrating over it.
*Since there is no time interval here where both the blue curve  $h(\tau)$  and at the same time the red dashed mirroring  $x(-\tau)$  is not equal to zero, it follows that  $y(t=0)=0$.
*For any other time  $t$,  the input signal must be shifted   ⇒   $x(t-\tau)$, for example corresponding to the green dashed curve for  $t=T$.
*Since in this example also  $x(t-\tau)$  can only take the values  $0$  or  $1$,  the integration  $($generally from  $\tau_1$  to  $\tau_2)$  becomes simple and we obtain with  $\tau_1 = 0$  and  $\tau_2 = t$ :
:$$y( t) = \int_0^{\hspace{0.05cm} t} {h( \tau)}\hspace{0.1cm} {\rm d}\tau = \frac{1}{T}\cdot\int_0^{\hspace{0.05cm} t} {{\rm{e}}^{ - \tau /T } }\hspace{0.1cm} {\rm d}\tau = 1 - {{\rm{e}}^{ - t /T } }.$$

The sketch is valid for  $t=T$  and leads to the initial value  $y(t=T) = 1 – 1/\text{e} \approx 0.632$.}}

==Exercises==

*First, select the number  $(1,\ 2, \text{...} \ )$  of the exercise to be processed.  The number  $0$  corresponds to a "Reset":  Same setting as at program start.
* An exercise description is displayed.  The parameter values are adjusted.  Solution after pressing "Show Solution".
*Both the input signal  $x(t)$  and the impulse response  $h(t)$  of the filter are are normalized, dimensionless and energy-limited ("time-limited pulses").

{{BlueBox|TEXT=
'''(1)'''   Select the following parameters:  $\text{Gaussian pulse: }A_x = 1, \ \Delta t_x= 1, \ \tau_x = 1;     \text{ Impulse response according to 2nd order low-pass: } \Delta t_h= 1$.
          Interpret the displayed graphs.  What is the maximum output value  $y_{\rm max}$?  At what time  $t_{\rm max}$  does  $y_{\rm max}$  occur? }}

* After renaming:  Input signal  $x(\tau)$   ⇒   red curve,   impulse response  $h(\tau)$   ⇒   blue curve,  after mirroring  $h(-\tau)$   ⇒   green curve.
* Shifting the green curve by  $t$  to the right, we get  $h(t-\tau)$.  The output signal  $y(t)$  is obtained by multiplication and integration with respect to  $\tau$:

:$$y (t) = \int_{ - \infty }^{ +\infty } {x ( \tau ) } \cdot h ( {t - \tau } ) \hspace{0.1cm}{\rm d}\tau = \int_{ - \infty }^{ t } {x ( \tau ) } \cdot h ( {t - \tau } ) \hspace{0.1cm}{\rm d}\tau .$$
* The output pulse  $y(t)$  is asymmetric in the present case;  the maximum output value  $y_{\rm max}\approx 0.67$  occurs at  $t_{\rm max}\approx 1.5$.

{{BlueBox|TEXT=
'''(2)'''   What changes if we increase the equivalent pulse duration of  $h(t)$  to  $\Delta t_h= 1.5$? }}

* $y_{\rm max}\approx 0.53$  now occurs at  $t_{\rm max}\approx 1.75$.  Due to the less favorable (wider) impulse response, the input pulse is more deformed.
* In a digital communication system, this would result in stronger  "intersymbol interferences".

{{BlueBox|TEXT=
'''(3)'''   Now select the symmetric  $\text{rectangular pulse: }A_x = 1, \ \Delta t_x= 1, \ \tau_x = 0$  and the  $\text{rectangular impulse response}$  of the low-pass filter:  $\Delta t_h= 1$.
          Interpret the convolution result.  What is the maximum output value  $y_{\rm max}$?  At what times is  $y(t)>0$?  Does  $h(t)$  describe a causal system? }}

* The convolution of two rectangles with respective durations  $1$  yields a triangle with absolute duration  $2$  ⇒   equivalent pulse duration  $\Delta t_y= 1$.
* $y(t)$  is different from zero in the range from  $-0.5$  to  $+1.5$ .  The pulse maximum  $y_{\rm max} = 1$  is at  $t_{\rm max} = +0.5$.
* $h(t)$  describes a causal system, since  $h(t) \equiv 0$  for  $t < 0$.  That means:  The "effect"  $y(t)$  does not come before the "cause"  $x(t)$.

{{BlueBox|TEXT=
'''(4)'''   What changes if we increase the equivalent pulse duration of  $h(t)$  to  $\Delta t_h= 2$ ? }}

* The convolution of two rectangles of different widths results in a trapezoid, here between  $-0.5$  and  $+2.5$ ⇒   equivalent pulse duration  $\Delta t_y= 2$.
* The maximum  $y_{\rm max} = 0.5$  occurs in the range  $0.5 \le t \le 1.5$.  Nothing changes with respect to causality.

{{BlueBox|TEXT=
'''(5)'''   Now select the (unsymmetrical)  $\text{rectangular pulse: }A_x = 1, \ \Delta t_x= 1, \ \tau_x = 0.5$  and the  $\text{impulse response of a 1st order low-pass: }\Delta t_h= 1$.
          Interpret the results.  What is the value of  $y_{\rm max}$?  At what times is  $y(t)>0$ ?  Does  $h(t)$  describe a causal system? }}

* $h(t)$  has an exponentially decreasing curve for  $t > 0$.  It always applies:   $y(t>0) > 0$,  but the signal values can become very small.
* $y_{\rm max} = 0.63$  occurs for  $t_{\rm max} = +1$.  For  $ t < t_{\rm max}$ the progression is exponentially increasing, for  $ t > t_{\rm max}$  exponentially decreasing.
* The 1st order low-pass can be realized with a resistor and a capacitor.  Any realizable system is causal per se.

{{BlueBox|TEXT=
'''(6)'''   Select as in  $(3)$  the  $\text{rectangular impulse response}$  of the low-pass filter:  $\Delta t_h= 1$.  With which  $x(t)$  results the same  $y(t)$  as for  $(5)$?}}

* The signal  $y(t)$  in  $(5)$  resulted as a convolution between the rectangular input  $x(t)$  and the exponential function  $h(t)$.
* Since the convolution operation is commutative, the same result is obtained with the exponential function  $x(t)$  and the rectangular function  $h(t)$.
* Thus, the correct setting for the input signal  $x(t)$  is the  $\text{exponential pulse }$with  $ A_x = 1, \ \Delta t_x= 1, \ \tau_x = 0$ .

{{BlueBox|TEXT=
'''(7)'''   For the remainder of the exercises, we consider the Gaussian low-pass.  The equivalent duration of the impulse response  $h(t)$  should first be  $\Delta t_h= 0.8$.
        Analyze and interpret this  "system"  in terms of causality and the resulting distortions for the rectangular pulse. }}

* The low-pass is not causal (and therefore non–realizable):  For  $t < 0$:  $h(t) \equiv 0$  does not hold. Suitable model if infinite delay is ignored.
* The larger  $\Delta t_h$  is, the wider the output pulse and the stronger the degradation of a digital system due to intersymbol interference.
* The frequency response  $H(f)$  is the Fourier transform of  $h(t)$. The larger  $\Delta t_h$  is, the smaller  $\Delta f_h = 1/\Delta t_h$   ⇒   The system is more narrowband.

{{BlueBox|TEXT=
'''(8)'''   Now select  $\text{Gaussian pulse: }A_x = 1, \ \Delta t_x= 1.5, \ \tau_x = 0$  and  $\text{Gaussian low-pass: }\Delta t_h= 2$.  What is the course of the output pulse  $y(t)$?
          What are the equivalent duration  $\Delta t_y$  of the output pulse and the maximum output value  $y_{\rm max}$?  At what time  $t_{\rm max}$  does  $y_{\rm max}$  occur? }}

* $y(t)$  is (exactly) Gaussian, too.   Mnemonic:  $\text{"Gaussian convolved with Gaussian gives always Gaussian"}$.
* Equivalent duration:  $\Delta t_y =\sqrt{\Delta t_x^2+ \Delta t_h^2} = 2.5$.  Pulse maximum  $($at  $t=0)$:  $y_{\rm max} = A_x \cdot \Delta t_x/\Delta t_y = 1 \cdot 1.5/2.5 = 0.6$.

{{BlueBox|TEXT=
'''(9)'''   Now select  $\text{Triangular pulse: }A_x = 1, \ \Delta t_x= 1.5, \ \tau_x = 0$  and  $\text{Gaussian low–pass: }\Delta t_h= 2$.  What is the course of the output pulse  $y(t)$?
        What is the equivalent duration  $\Delta t_y$  of the output pulse and the maximum output value  $y_{\rm max}$?  At what time  $t_{\rm max}$  does  $y_{\rm max}$  occur?}}

* $y(t)$  is nearly Gaussian, but not exactly.  Mnemonic:  $\text{"Gaussian convolved with non–Gaussian never gives exactly Gaussian"}$.
* The characteristics of the output pulse  $y(t)$  differ only slightly from  $(8)$:  $\Delta t_y \approx 2.551$,  $y_{\rm max} \approx 0.588$.

==Applet Manual==
 
[[File:Anleitung_Faltung_2.png|right|frame|Screen shot  (German version)]]

    '''(A)'''     Selection:   Shape of the input pulse  $x(t)$

    '''(B)'''     Parameter input for the input pulse  $x(t)$

    '''(C)'''     Selection:   Shape of the impulse response  $h(t)$  of the low-pass system

    '''(D)'''     Parameter input for the impulse response  $h(t)$

    '''(E)'''     Control panel  (Start;   Pause/Continue   ;  Step >   ;  Step <  ;  Reset)

    '''(F)'''     Output the initial value  $y(t)$  at the continuous time  $t$

    '''(G)'''     Maximum value  $y_{\rm max} = y(t_{\rm max})$  and equivalent width $\Delta\hspace{0.03cm} t_y$

                 After renaming the abscissa:   $t \ \to \ \tau$:

    '''(H)'''     Representation of  $x(\tau)$  ⇒   red static curve.

    '''(I)'''       Representation of  $h(\tau)$  ⇒  blue curve  and   $h(t-\tau)$  ⇒   green curve                 $($this is shifted to the right with the motion parameter   $t$ $)$

    '''(J)'''       Plot of  $x(\tau) \cdot h(t - \tau)$  ⇒   purple curve, dynamic with  $t$

    '''(K)'''      Successive representation of the output signal  $y(t)$  ⇒   brown curve

    '''(L)'''      Area for exercise execution:   Exercise selection.

    '''(M)'''      Exercise execution:   Area for exercise description

    '''(N)'''      Exercise execution:   Area for the sample solution
 
==About the Authors==
This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
*The first version was created in 2006 by [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]]  as part of his bachelor thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).

*Last revision and English version 2020/2021 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  in the context of a working student activity. 

The conversion of this applet to HTML 5 was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ Studienzuschüsse]  ("study grants")  of the TUM Faculty EI.  Many thanks.

==Once again:  Open Applet in new Tab==
{{LntAppletLinkEnDe|convolution_en|convolution}}

Applets:Complementary Gaussian Error Functions

2023-03-21T18:06:00Z

Hwang:

{{LntAppletLinkEnDe|qfunction_en|qfunction}}

==Applet Description==
 
This applet allows the calculation and graphical representation of the (complementary) Gaussian error functions  ${\rm Q}(x)$  and  $1/2\cdot {\rm erfc}(x)$, which are of great importance for error probability calculation.
*Both the abscissa and the function value can be represented either linearly or logarithmically.
*For both functions an upper bound  $\rm (UB)$  and a lower bound  $\rm (LB)$  are given.

==Theoretical Background==
 
In the study of digital transmission systems, it is often necessary to determine the probability that a (mean-free) Gaussian distributed random variable  $x$  with variance  $σ^2$  exceeds a given value  $x_0$.  For this probability holds:
:$${\rm Pr}(x > x_0)={\rm Q}(\frac{x_0}{\sigma}) = 1/2 \cdot {\rm erfc}(\frac{x_0}{\sqrt{2} \cdot \sigma}).$$

===The function ${\rm Q}(x )$===

The function  ${\rm Q}(x)$  is called the  '''complementary Gaussian error integral'''.  The following calculation rule applies:
:$${\rm Q}(x ) = \frac{1}{\sqrt{2\pi}}\int_{x}^{ +\infty}\hspace{-0.2cm}{\rm e}^{-u^{2}/\hspace{0.05cm} 2}\,{\rm d} u .$$
*This integral cannot be solved analytically and must be taken from tables if one does not have this applet available.
*Specially for larger  $x$  values  (i.e., for small error probabilities), the bounds given below provide a useful estimate for  ${\rm Q}(x)$, which can also be calculated without tables.
*An upper bound  $\rm (UB)$  of this function is:
:$${\rm Q}_{\rm UB}(x )=\text{Upper Bound }\big [{\rm Q}(x ) \big ] = \frac{ 1}{\sqrt{2\pi}\cdot x}\cdot {\rm e}^{- x^{2}/\hspace{0.05cm}2} > {\rm Q}(x).$$
*Correspondingly, for the lower bound  $\rm (LB)$:
:$${\rm Q}_{\rm LB}(x )=\text{Lower Bound }\big [{\rm Q}(x ) \big ] =\frac{1-1/x^2}{\sqrt{2\pi}\cdot x}\cdot {\rm e}^{-x^ 2/\hspace{0.05cm}2} ={\rm Q}_{\rm UB}(x ) \cdot (1-1/x^2)< {\rm Q}(x).$$

However, in many program libraries, the function  ${\rm Q}(x )$  cannot be found.
 

===The function $1/2 \cdot {\rm erfc}(x )$===

On the other hand, in almost all program libraries, you can find the  '''Complementary Gaussian Error Function''':
:$${\rm erfc}(x) = \frac{2}{\sqrt{\pi}}\int_{x}^{ +\infty}\hspace{-0.2cm}{\rm e}^{-u^{2}}\,{\rm d} u ,$$
which is related to  ${\rm Q}(x)$  as follows:   ${\rm Q}(x)=1/2\cdot {\rm erfc}(x/{\sqrt{2}}).$
*Since in almost all applications this function is used with the factor  $1/2$, in this applet exactly this function was realized:
:$$1/2 \cdot{\rm erfc}(x) = \frac{1}{\sqrt{\pi}}\int_{x}^{ +\infty}\hspace{-0.2cm}{\rm e}^{-u^{2}}\,{\rm d} u .$$

*Once again, an upper and lower bound can be specified for this function:
:$$\text{Upper Bound }\big [1/2 \cdot{\rm erfc}(x) \big ] = \frac{ 1}{\sqrt{\pi}\cdot 2x}\cdot {\rm e}^{- x^{2}} ,$$
:$$\text{Lower Bound }\big [1/2 \cdot{\rm erfc}(x) \big ] = \frac{ {1-1/(2x^2)}}{\sqrt{\pi}\cdot 2x}\cdot {\rm e}^{- x^{2}} .$$

===When which function offers advantages?===

{{GreyBox|TEXT=
$\text{Example 1:}$  We consider binary baseband transmission. Here, the bit error probability  $p_{\rm B} = {\rm Q}({s_0}/{\sigma_d})$, where the useful signal can take the values  $\pm s_0$  and the noise root mean square value  $\sigma_d$ .

It is assumed that tables are available listing the argument of the two Gaussian error functions at distance  $0.1$.  With  $s_0/\sigma_d = 4$  one obtains for the bit error probability according to the function  ${\rm Q}(x )$:
:$$p_{\rm B} = {\rm Q} (4) \approx 0.317 \cdot 10^{-4}\hspace{0.05cm}.$$
According to the second equation, we get:
:$$p_{\rm B} = {1}/{2} \cdot {\rm erfc} ( {4}/{\sqrt{2} })= {1}/{2} \cdot {\rm erfc} ( 2.828)\approx {1}/{2} \cdot {\rm erfc} ( 2.8)= 0.375 \cdot 10^{-4}\hspace{0.05cm}.$$
*The first value is more correct.  In the second method of calculation, one must round or – even better – interpolate, which is very difficult due to the strong nonlinearity of this function. 
*Accordingly, with the given numerical values, ${\rm Q}(x )$  is more suitable.  However, outside of exercise examples  $s_0/\sigma_d$  will usually have a "curvilinear" value.  In this case, of course,  ${\rm Q}(x)$  offers no advantage over  $1/2 \cdot{\rm erfc}(x)$. }}

{{GreyBox|TEXT=
$\text{Example 2:}$ 
With the energy per bit  $(E_{\rm B})$  and the noise power density  $(N_0)$  the bit error probability of ''Binary Phase Shift Keying''  (BPSK) is:
:$$p_{\rm B} = {\rm Q} \left ( \sqrt{ {2 E_{\rm B} }/{N_0} }\right ) = {1}/{2} \cdot { \rm erfc} \left ( \sqrt{ {E_{\rm B} }/{N_0} }\right ) \hspace{0.05cm}.$$
For the numerical values  $E_{\rm B} = 16 \rm mWs$  and  $N_0 = 1 \rm mW/Hz$  we obtain:
:$$p_{\rm B} = {\rm Q} \left (4 \cdot \sqrt{ 2} \right ) = {1}/{2} \cdot {\rm erfc} \left ( 4\right ) \hspace{0.05cm}.$$
*The first way leads to the result  $p_{\rm B} = {\rm Q} (5.657) \approx {\rm Q} (5.7) = 0.6 \cdot 10^{-8}\hspace{0.01cm}$, while   $1/2 \cdot{\rm erfc}(x)$  here the more correct value  $p_{\rm B} \approx 0.771 \cdot 10^{-8}$  yields.
*As in the first example, however, you can see:   The functions  ${\rm Q}(x)$  and  $1/2 \cdot{\rm erfc}(x)$  are basically equally well suited.
*Advantages or disadvantages of one or the other function arise only for concrete numerical values.}}
 

==Exercises==

* First select the number  $(1, 2, \text{...})$  of the exercise.  The number  $0$  corresponds to a "Reset":  Same setting as at program start.
*A task description is displayed.  The parameter values are adjusted.  Solution after pressing "Show solution". 

{{BlueBox|TEXT=
'''(1)'''   Find the values of the function  ${\rm Q}(x)$  for  $x=1$,  $x=2$,  $x=4$  and  $x=6$.  Interpret the graphs for linear and logarithmic ordinates.}}

*The applet returns the values  ${\rm Q}(1)=1.5866 \cdot 10^{-1}$,  ${\rm Q}(2)=2. 275 \cdot 10^{-2}$,  ${\rm Q}(4)=3.1671 \cdot 10^{-5}$  and  ${\rm Q}(6)=9.8659 \cdot 10^{-10}$.
*With linear ordinate, the values for  $x>3$  are indistinguishable from the zero line.  More interesting is the plot with logarithmic ordinate.

{{BlueBox|TEXT=
'''(2)'''   Evaluate the two bounds  ${\rm UB}(x )=\text{Upper Bound }\big [{\rm Q}(x ) \big ]$  and  ${\rm LB}(x )=\text{Lower Bound }\big [{\rm Q}(x ) \big ]$  for the  ${\rm Q}$  function. }}

*For  $x \ge 2$  the upper bound is only slightly above  ${\rm Q}(x)$  and the lower bound is only slightly below  ${\rm Q}(x)$. 
*For example:  ${\rm Q}(x=4)=3.1671 \cdot 10^{-5}$   ⇒   ${\rm LB}(x=4)=3.1366 \cdot 10^{-5}$,   ${\rm UB}(x=4)=3.3458 \cdot 10^{-5}$.
*The upper bound has greater significance for assessing a communications system than "LB",  since this corresponds to a "worst case" consideration.

{{BlueBox|TEXT=
'''(3)'''   Try to use the app to determine  ${\rm Q}(x=2 \cdot \sqrt{2} \approx 2.828)$  as accurately as possible despite the quantization of the input parameter. }}
*The program returns for  $x=2.8$  the too large result  $2.5551 \cdot 10^{-3}$  and for  $x=2.85$  the result  $2.186 \cdot 10^{-3}$.  The exact value lies in between.
*But it also holds:  ${\rm Q}(x=2 \cdot \sqrt{2})=0.5 \cdot {\rm erfc}(x=2)$.  This gives the exact value  ${\rm Q}(x=2 \cdot \sqrt{2})=2.3389 \cdot 10^{-3}$.

{{BlueBox|TEXT=
'''(4)'''   Find the values of the function  $0.5 \cdot {\rm erfc}(x)$  for  $x=1$,  $x=2$,  $x=3$  and  $x=4$.  Interpret the exact results and the bounds.}}

*The applet returns:  $0.5 \cdot {\rm erfc}(1)=7.865 \cdot 10^{-2}$,  $0.5 \cdot {\rm erfc}(2)=2. 3389 \cdot 10^{-3}$,  $0.5 \cdot {\rm erfc}(3)=1.1045 \cdot 10^{-5}$  and  $0.5 \cdot {\rm erfc}(4)=7.7086 \cdot 10^{-9}$.
*All the above statements about  ${\rm Q}(x)$  with respect to suitable representation type and upper and lower bounds also apply to the function  $0.5 \cdot {\rm erfc}(x)$.

{{BlueBox|TEXT=
'''(5)'''   The results of  '''(4)'''  are now to be converted for the case of a logarithmic abscissa.  The conversion is done according to  $\rho\big[{\rm dB}\big ] = 20 \cdot \lg(x)$. }}

* The linear abscissa value  $x=1$  leads to the logarithmic abscissa value  $\rho=0\ \rm dB$   ⇒   $0. 5 \cdot {\rm erfc}(\rho=0\ {\rm dB})={0.5 \cdot \rm erfc}(x=1)=7.865 \cdot 10^{-2}$.
*Similarly  $0.5 \cdot {\rm erfc}(\rho=6.021\ {\rm dB}) =0.5 \cdot {\rm erfc}(x=2)=2. 3389 \cdot 10^{-3}$,     $0.5 \cdot {\rm erfc}(\rho=9.542\ {\rm dB})=0.5 \cdot {\rm erfc}(3)=1.1045 \cdot 10^{-5}$,   
*$0.5 \cdot {\rm erfc}(\rho=12.041\ {\rm dB})= 0.5 \cdot {\rm erfc}(4)=7.7086 \cdot 10^{-9}$.
*As per right diagram:  $0.5 \cdot {\rm erfc}(\rho=6\ {\rm dB}) =2.3883 \cdot 10^{-3}$,     $0.5 \cdot {\rm erfc}(\rho=9. 5\ {\rm dB}) =1.2109 \cdot 10^{-5}$,     $0.5 \cdot {\rm erfc}(\rho=12\ {\rm dB}) =9.006 \cdot 10^{-9}$.

{{BlueBox|TEXT=
'''(6)'''   Find  ${\rm Q}(\rho=0\ {\rm dB})$,  ${\rm Q}(\rho=5\ {\rm dB})$  and  ${\rm Q}(\rho=10\ {\rm dB})$,  and establish the relationship between linear and logarithmic abscissa.}}
*The program returns for logarithmic abscissa  ${\rm Q}(\rho=0\ {\rm dB})=1. 5866 \cdot 10^{-1}$,  ${\rm Q}(\rho=5\ {\rm dB})=3.7679 \cdot 10^{-2}$,  ${\rm Q}(\rho=10\ {\rm dB})=7.827 \cdot 10^{-4}$.
*The conversion is done according to the equation  $x=10^{\hspace{0.05cm}0.05\hspace{0.05cm} \cdot\hspace{0.05cm} \rho[{\rm dB}]}$.  For  $\rho=0\ {\rm dB}$  we get  $x=1$   ⇒   ${\rm Q}(\rho=0\ {\rm dB})={\rm Q}(x=1) =1.5866 \cdot 10^{-1}$.
*For  $\rho=5\ {\rm dB}$  we get  $x=1.1778$   ⇒   ${\rm Q}(\rho=5\ {\rm dB})={\rm Q}(x=1. 778) =3.7679 \cdot 10^{-2}$.  From the left diagram:  ${\rm Q}(x=1.8) =3.593 \cdot 10^{-2}$.
*For  $\rho=10\ {\rm dB}$  we get  $x=3.162$   ⇒   ${\rm Q}(\rho=10\ {\rm dB})={\rm Q}(x=3. 162) =7.827 \cdot 10^{-4}$.  After "quantization":  ${\rm Q}(x=3.15) =8.1635 \cdot 10^{-4}$.

==Applet Manual==
 

[[File:Qfunction bedienung.png|right|550px]]
    '''(A)'''     Verwendete Gleichungen am Beispiel  ${\rm Q}(x)$

    '''(B)'''     Auswahloption für  ${\rm Q}(x)$  oder  ${\rm 0.5 \cdot erfc}(x)$

    '''(C)'''     Schranken  ${\rm LB}$  und  ${\rm UB}$  werden gezeichnet

    '''(D)'''     Auswahl, ob Abszisse linear  $\rm (lin)$  oder logarithmisch  $\rm (log)$ 

    '''(E)'''     Auswahl, ob Ordinate linear  $\rm (lin)$  oder logarithmisch  $\rm (log)$ 

    '''(F)'''     Numerikausgabe am Beispiel  ${\rm Q}(x)$  bei linearer Abszisse

    '''(G)'''     Slidereingabe des Abszissenwertes  $x$  für lineare Abszisse

    '''(H)'''     Slidereingabe des Abszissenwertes  $\rho \ \rm [dB]$  für logarithmische Abszisse

    '''(I)'''     Grafikausgabe der Funktion  ${\rm Q}(x)$  – hier:  lineare Abszisse

    '''(J)'''     Grafikausgabe der Funktion  ${\rm 0.5 \cdot erfc}(x)$  – hier:  lineare Abszisse

    '''(K)'''     Variationsmöglichkeit für die graphischen Darstellungen

$\hspace{1.5cm}$"$+$" (Vergrößern),

$\hspace{1.5cm}$ "$-$" (Verkleinern)

$\hspace{1.5cm}$ "$\rm o$" (Zurücksetzen)

$\hspace{1.5cm}$ "$\leftarrow$" (Verschieben nach links), usw.
 
==About the Authors==
 
This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
*The first version was created in 2007 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|Thomas Großer]]  as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 the program was redesigned by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]]  as part of her bachelor thesis  (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) via "HTML5".
*Last revision and English version 2021 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  in the context of a working student activity. 

The conversion of this applet to HTML 5 was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ "Studienzuschüsse"]  (Faculty EI of the TU Munich).  We thank.

==Once again: Open Applet in new Tab==
 
{{LntAppletLinkEnDe|qfunction_en|qfunction}}

Applets:Generation of Walsh functions

2023-03-21T18:05:15Z

Hwang:

{{LntAppletLinkEnDe|walsh_en|walsh}}

== Program description==
 
This applet allows to display the Hadamard matrices  $\mathbf{H}_J$  for the construction of the Walsh functions  $w_j$.  The factor  $J$  of the band spreading as well as the selection of the individual Walsh functions  (by means of a blue border around rows of the matrix)  can be changed.

==Theoretical background==
 
===Application===
 
The  '''Walsh functions'''  are a group of periodic orthogonal functions.  Their application in digital signal processing mainly lies in the use for band spreading in CDMA systems, for example the mobile radio standard UMTS.
*Due to their orthogonal properties and the favourable periodic cross-correlation function  $\rm (PCCF)$, the Walsh functions represent optimal spreading sequences for a distortion-free channel and a synchronous CDMA system.  If you take any two lines and form the correlation (averaging over the products), the PCCF value is always zero.
*In asynchronous operation  (example:   uplink of a mobile radio system)  or de-orthogonalization due to multipath propagation, Walsh functions alone are not necessarily suitable for band spreading.
*In terms of  $\rm (PACF)$  (''periodic autocorrelation function'') these sequences are not as good:  Each individual Walsh function has a different PACF and each individual PACF is less good than a comparable pseudo noise  $\rm (PN)$  sequence. That means:   The synchronization is more difficult with Walsh functions than with PN sequences.
 

=== Construction===
 
The construction of Walsh functions can be done recursively using the  '''Hadamard matrices'''.
*A Hadamard matrix  $\mathbf{H}_J$  of order  $J$  is a  $J\times J$  matrix, which contains line by line the  $\pm 1$  weights of the Walsh sequences.
*The orders of the Hadamard matrices are fixed to powers of two, i.e.  $J = 2^G$  applies to a natural number  $G$. Starting from $\mathbf{H}_1 = [+1]$ and

:$$
\mathbf{H}_2 =
\left[ \begin{array}{rr}
+1 & +1\\
+1 & -1 \\
\end{array}\right]
$$
the following relationship applies to the generation of further Hadamard matrices:
:$$
\mathbf{H}_{2N} =
\left[ \begin{array}{rr}
+\mathbf{H}_N & +\mathbf{H}_N\\
+\mathbf{H}_N & -\mathbf{H}_N \\
\end{array}\right]
$$
 

{{GraueBox|TEXT=
$\text{Example:}$  The graphic shows the Hadamard matrix  $\mathbf H_8$  (right) and the  $J\hspace{-0.09cm} -\hspace{-0.09cm}1$  spreading sequences which can be constructed with it.
[[File:P_ID1882__Mod_T_5_3_S7_neu.png|right|frame| Walsh spreading sequences  $(J = 8)$  and Hadamard matrix  $\mathbf H_8$ ]]
*Only  $J\hspace{-0.09cm} -\hspace{-0.09cm}1$, because the unspreaded sequence  $w_0(t)$  is usually not used.
*Please note the color assignment between the lines of the Hadamard matrix and the spreading sequences  $w_j(t)$.
*The submatrix  $\mathbf H_4$  is highlighted in yellow.}}
 

==How to use the applet==

 
[[File:Bildschirm_Walsh_EN_3.png|right|600px]]

    '''(A)'''     Selection of  $G$   ⇒   Band spread factor:  $J= 2^G$

    '''(B)'''     Selection of the Walsh function  $w_j$  to be marked 
 

== About the authors==

This interactive calculation tool was designed and realized at the  [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik]  $\rm (LNT)$  of the  [https://www.tum.de/ Technical University of Munich]  $\rm (TUM)$.

*The first German version was created in 2007 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|Thomas Großer]]   in the context of his diploma thesis with "FlashMX–Actionscript"  (Supervisor:  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*2018/2019 the applet was converted on "HTML5" and redesigned by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  (Engineering practice, supervisor:  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ).
*2020 this English version was made by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]  (working student) and  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]. 
==Call the applet again==
 
{{LntAppletLinkEnDe|walsh_en|walsh}}

Applets:Period Duration of Periodic Signals

2023-03-21T18:05:06Z

Hwang:

{{LntAppletLinkEnDe|signalPeriod_en|signalPeriod}}

==Applet Descripition==
 
This applet draws the course and calculates the period duration  $T_0$  of the periodic function
:$$x(t) = A_1\cdot \cos\left(2\pi f_1\cdot t- \varphi_1\right)+A_2\cdot \cos\left(2\pi f_2\cdot t- \varphi_2\right).$$

Please note:
*The phases  $\varphi_i$  must be entered here in radians.  Conversion from the input value:  
:$$\varphi_i \text{[in radians]} =\varphi_i \text{[in degrees]}/360 \cdot 2\pi.$$
*The maximum value  $x_{\rm max}$  and a signal value  $x(t_*)$  at a given time  $t_*$ are also output.

==Theoretical background==
 
A periodic signal  $x(t)$  is present exactly when it is not constant and if for all arbitrary values of  $t$  and all integer values of  $i$  with an appropriate  $T_{0}$  applies:  
:$$x(t+i\cdot T_{0}) = x(t).$$
*$T_0$  is called the  '''period duration'''   and  $f_0 = 1/T_0$  the  '''basic frequency'''.

*For a harmonic oscillation  $x_1(t) = A_1\cdot \cos\left(2\pi f_1\cdot t- \varphi_1\right)$  applies  $f_0 = f_1$  and  $T_0 = 1/f_1$,  independent of the phase  $\varphi_1$  and the amplitude  $A_1 \ne 0$.

{{BlueBox|TEXT=
$\text{Calculation Rule: }$  If the periodic signal  $x(t)$  consists of two parts  $x_1(t)$  and  $x_2(t)$  like in this applet, then applies for the basic frequency and the period duration with  $A_1 \ne 0$,  $f_1 \ne 0$,  $A_2 \ne 0$,  $f_2 \ne 0$:

:$$f_0 = {\rm gcd}(f_1, \ f_2) \hspace{0.3cm} \Rightarrow \hspace{0.3cm}T_0 = 1/f_0.$$

Here  $\rm gcd$  denotes the '''greatest common divisor'''.}}

{{GraueBox|TEXT=
$\text{Examples:}$   In the following  $f_0'$,  $f_1'$  and $f_2'$  denote signal frequencies normalized to $1\ \rm kHz$:

'''(a)'''   $f_1' = 1.0$,   $f_2' = 3.0$   ⇒   $f_0' = {\rm gcd}(1.0, \ 3.0) = 1.0$   ⇒   $T_0 = 1.0\ \rm ms$;

'''(b)'''   $f_1' = 1.0$,   $f_2' = 3.5$   ⇒   $f_0' = {\rm gcd}(1.0, \ 3.5)= 0.5$   ⇒   $T_0 = 2.0\ \rm ms$;

'''(c)'''   $f_1' = 1.0$,   $f_2' = 2.5$   ⇒   $f_0' = {\rm gcd}(1.0, \ 2.5) = 0.5$   ⇒   $T_0 = 2.0\ \rm ms$;

'''(d)'''   $f_1' = 0.9$,   $f_2' = 2.5$   ⇒   $f_0' = {\rm gcd}(0.9, \ 2.5) = 0.1$   ⇒   $T_0 = 10.0 \ \rm ms$;

'''(e)'''   $f_2' = \sqrt{2} \cdot f_1' $   ⇒   $f_0' = {\rm gcd}(f_1', \ f_2') \to 0$   ⇒   $T_0 \to \infty$  ⇒   the signal  $x(t)$  is not periodic.}}

$\text{Note:}$  The period duration could also be determined as  '''least common multiple'''  $\rm (lcm)$  according to  $T_0 = {\rm lcm}(T_1, \ T_2)$:

:'''(c)'''   $T_1 = 1.0\ \rm ms$,   $T_2 = 0.4\ \rm kHz$   ⇒   $T_0 = {\rm lcm}(1.0, \ 0.4) \ \rm ms = 2.0\ \rm ms$

With all other parameter values, however, there would be numerical problems, for example

:'''(a)'''   $T_1 = 1.0\ \rm ms$  and  $T_2 = 0.333\text{...} \ \rm ms$  have no  "least common multiple"  due to the limited representation of real numbers.

==Exercises==

* First select the number  (1, 2, ... )  of the exercise.  The number  '''0'''  corresponds to a  "Reset":  Same setting as at the program start.
* An exercise description is displayed.  Parameter values are adjusted.  Solution after pressing "Show solution". 
* $A_1'$  and  $A_2'$  denote the signal amplitudes normalized to  $1\ \rm V$.  $f_0'$,  $f_1'$  and  $f_2'$  are the frequencies normalized to  $1\ \rm kHz$.
 
{{BlaueBox|TEXT=
'''(1)'''   Consider  $A_1' = 1.0, \ A_2' = 0.5, \ f_1' = 2.0, \ f_2' = 2.5, \ \varphi_1 = 0^\circ \ \varphi_2 = 90^\circ$.  How large is the period  $T_0$?}}

$\hspace{1.0cm}\Rightarrow\hspace{0.3cm}$The period is  $T_0 = 2.0 \ \rm ms$   due to   $\rm{gcd}(2.0, 2.5) = 0.5$.

{{BlaueBox|TEXT=
'''(2)'''   Vary  $\varphi_1$  and  $\varphi_2$  in the whole possible range 
$\pm 180^\circ$.  How does this affect the period  $T_0$?
}}

$\hspace{1.0cm}\Rightarrow\hspace{0.3cm}$The period  $T_0 = 2.0 \ \rm ms$  remains the same for all  $\varphi_1$  and  $\varphi_2$.

{{BlaueBox|TEXT=
'''(3)'''   Select the default setting   ⇒   "Recall Parameters".  Vary  $A_1'$  in the entire possible range  $0 \le A_1' \le 1$.}}

$\hspace{1.0cm}\Rightarrow\hspace{0.3cm}$The period  $T_0 = 2.0 \ \rm ms$  remains the same with the exception of  $A_1' =0$.  In the latter case:  $T_0 = 0.4 \ \rm ms$.

{{BlaueBox|TEXT=
'''(4)'''   Choose the default setting   ⇒   "Recall Parameters"  and vary  $f_2'$.  Does this affect  $T_0$?  Which value is the result for  $f_2' = 0.2$?}}

$\hspace{1.0cm}\Rightarrow\hspace{0.3cm}$The period jumps back and forth.  For  $f_2' = 0.2$  the result is  $T_0 = 5.0 \ \rm ms$  because of $\ \rm{gcd} (2.0,0.2)=0.2$.

{{BlaueBox|TEXT=
'''(5)'''   Consider  $A_1' = 1.0, \ A_2' = 0.5, \ f_1' = 0.2, \ f_2' = 2.5, \ \varphi_1 = 0^\circ \ \varphi_2 = 90^\circ$.  How large is the period  $T_0$?  Save this setting with 
"Store Parameters".
}}

$\hspace{1.0cm}\Rightarrow\hspace{0.3cm}$The period is  $T_0 = 10.0 \ \rm ms$    due to  $\rm{gcd}(0.2, 2.5) = 0.1$.

{{BlaueBox|TEXT=
'''(6)'''   Select the last setting   ⇒  "Recall Parameters"  and change  $f_2' = 0.6$.  Save this setting with  "Store Parameters".
}}

$\hspace{1.0cm}\Rightarrow\hspace{0.3cm}$The period is  $T_0 = 5.0 \ \rm ms$  due to  $\rm{gcd}(0.2,0.6) = 0.2$.

{{BlaueBox|TEXT=
'''(7)'''   How large is the maximum signal value  $x_{\rm max}$  with the same settings?`
}}

$\hspace{1.0cm}\Rightarrow\hspace{0.3cm}$ $x_{\rm max} =x(t_* + i \cdot T_0) = 1.38 \ {\rm V} < A_1 + A_2$  with  $t_* = 0.3 \ \rm ms$  and  $T_0 = 5.0 \ \rm ms$.

{{BlaueBox|TEXT=
'''(8)'''   What changes with  $\varphi_2 = 0^\circ$   ⇒   Sum of two cosine waves?}}

$\hspace{1.0cm}\Rightarrow\hspace{0.3cm}$ $t_* = 0$,  $T_0 = 5.0 \ \rm ms$  ⇒   $x_{\rm max} =x(t_* + i \cdot T_0) = 1.5 \ {\rm V}=A_1 + A_2$.

{{BlaueBox|TEXT=
'''(9)'''   Now consider  $\varphi_1 = \varphi_2 = 90^\circ$   ⇒   Sum of two sine waves?}}

$\hspace{1.0cm}\Rightarrow\hspace{0.3cm}$The maximum signal value is now  $x_{\rm{max}} = 1.07 \ \rm V < A_1 + A_2$. 
This value results from  $T_0 = 5.0 \ \rm ms$  and  $t_* = 0.6 \ \rm ms$  or  $t_* = 1.9 \ \rm ms$.

==Applet Manual==
[[File:Anleitung_Periodendauer.png|right|frame|Screenshot]]

    '''(A)'''     Parameter input for harmonic oscillation 1

    '''(B)'''     Parameter input for harmonic oscillation 2 and time  $t_*$.

    '''(C)'''     Numerical output of the main result  $T_0$; graphical illustration by red line

    '''(D)'''     Save parameter sets

    '''(E)'''     Retrieve parameter sets

    '''(F)'''     Output of  $x_{\rm max}$  and the signal values  $x(t_*) = x(t_* + T_0)= x(t_* + 2T_0)$

    '''(G)'''     Graphic field for displaying the signals

                  The signal values  $x(t_*) = x(t_* + T_0)= x(t_* + 2T_0)$  are marked by green dots

                  At the bottom of the graphic field you will find the following buttons:

                  '''(1)'''     Zoom funktions:    "$+$" (Zoom In),    "$-$" (Zoom Out),    $\rm o$ (Reset)

                  '''(2)'''     Move with    "←"    (Section to the left, ordinate to the right),    "$\uparrow$"    "$\downarrow$",     "$\rightarrow$"

    '''(H)'''     Task selection according to the task number
 
In all applets top right:    Changeable graphical interface design   ⇒   '''Theme''':
* Dark:   black background  (recommended by the authors).
* Bright:   white background  (recommended for beamers and printouts)
* Deuteranopia:   for users with pronounced green–visual impairment
* Protanopia:   for users with pronounced red–visual impairment

==About the Authors==

This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
*The first German version was created in 2004 by  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]]  as part of her diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2017 the program was redesigned by [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#David_Jobst_.28Ingenieurspraxis_Math_2017.29|David Jobst]] (Bachelor thesis LB, Supervisor:  [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) via "HTML5".
* The English version was done in 2020 by  [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]. 

==Once again: Open Applet in new Tab==
{{LntAppletLinkEnDe|signalPeriod_en|signalPeriod}}

Examples of Communication Systems/Further Developments of UMTS

2023-03-08T10:49:40Z

Hwang:

{{LastPage}}
{{Header
|Untermenü=UMTS – Universal Mobile Telecommunications System
|Vorherige Seite=Telecommunications Aspects of UMTS
|Nächste Seite=
}}

==High–Speed Downlink Packet Access==
 
In order to meet the increasing demand for higher data rates in mobile communications and to ensure ever better quality of service, the UMTS Release 99 standard has been further developed in five phases up to the present day  $(2011)$.  The diagram shows the individual development phases in chronological order.

The most important further developments were
[[File:EN_Bei_T_4_4_S1_v2.png|right|frame|Further development of UMTS between 2000 and 2011 ]]
[[File:EN_Bei_T_4_4_S1.png|right|frame|Characteristics of HSDPA]]

*the UMTS Release 5 with  »'''HSDPA'''«  and

*the UMTS Release 6 with  »'''HSUPA'''«.

For these two standards, the main focus was on increasing the data rates provided for downlink and uplink as well as greater bandwidth efficiency and cell capacity.

Together,  HSDPA and HSUPA make up the  »'''HSPA standard'''«.
*In 2002,  "High-Speed Downlink Packet Access"  – abbreviated  $\rm HSDPA$  – was specified with UMTS Release 5 and introduced in 2006 to increase data rate and throughput compared to the original UMTS standard as well as to reduce response times for packet-switched transmission.

*In HSPDA, the data rates provided are between  $\text{500 kbit/s}$  and  $\text{3.6 Mbit/s}$  – theoretically even up to  $\text{14.4 Mbit/s}$.

*Compared to the data rate of UMTS R'99  $\text{(14.4 kbit/s}$  to  $\text{2 Mbit/s)}$  these values represent a doubling to quadrupling.

The following technical procedures contribute to the increase in performance of HSDPA compared to UMTS. The diagram compiles these features:
#Introduction of an additional shared channel:  »'''HS-PDSCH'''«,
#use of the  »'''Hybrid ARQ'''«  method,
#'''delay'''  minimization,
#introduction of  »'''Node B scheduling'''«,
#use of  '''adaptive'''  modulation,  coding and transmission rate.

==Additional channels in HSDPA==
 
[[File:EN_Bei_T_4_4_S3neu.png|right|frame|Transport channels,  logical channels,  and physical channels in HSDPA]]
The  "High-Speed Downlink Physical High Speed Channel"  – abbreviated  '''HS-PDCH'''  – is a high-speed transport channel used for the transmission of subscriber data.  It combines the characteristics of a shared and a dedicated channel:

*In the downlink,  one or more channels can be used by multiple subscribers simultaneously.  This allows simultaneous transmission of the same data to different subscribers as well as a significant increase in transmission speed by bundling several channels of this type.

*In any HS-PDCH,  the spreading factor  $J = 16$.  This means that theoretically up to fifteen such channels can be used simultaneously in one cell.  In practice,  however,  only between five and ten channels are ever used,  since the remaining channels are required for the operation of other services.

Resource allocation for the  "High-Speed Shared Data Channel"  $($'''HS-DSCH'''$)$  is done via so-called  "High-Speed Shared Control Channels"  $($'''HS-SCCH'''$)$.  A receiver must therefore be able to receive and decode up to four such channels simultaneously.  In addition to the channels presented above

#The  "Dedicated Physical Control Channel"  $($'''DPCCH'''$)$  is used for the transmission of control data in the uplink.
# The  "Dedicated Control Channel"  $($'''DCCH'''$)$  is used for the localization procedure in downlink and uplink.
#The  "Dedicated Traffic Channel"  $($'''DTCH'''$)$  is responsible for the transmission of Internet Protocol payload data in the uplink direction.

==HARQ procedure and Node B scheduling ==
 
Another feature of HSDPA is the reduction of packet round-trip delay  $\rm (RTD)$  and the use of  "HARQ":
*The  »'''round-trip delay'''»  has been reduced by HSDPA to  $\text{70 ms}$  $($compared  $\text{160 ... 200 ms}$  with UMTS R'99$)$, which is of great importance for some applications,  e.g. web browsing.  This reduction was achieved by decreasing the transport block length to about  $2$  milliseconds  $($previously  $\text{10 ms}$  or  $\text{20 ms)}$.

*In each Node B,  a  »'''HARQ'''«  $($"Hybrid Automatic Repeat Request"$)$  has been implemented to minimize transmission delays.  This mechanism prevents significant delays from occurring due to retransmission of erroneous blocks.  This is because such delays can be interpreted by TCP protocol as blocking,  which then leads to further delays.

*Using the HARQ mechanism and with transport block lengths of  $\text{2 ms}$  the transmission delays in HSPDA are less than  $\text{10 ms}$.  This is a crucial improvement over UMTS,  where error detection  $($associated with retransmission$)$  takes approximately  $\text{90 ms}$.

*In the HARQ procedure,  the detection of  "error"  or  "no error"  is acknowledged for each individual transport frame.  This procedure is referred to as  "Stop and Wait"  $\rm (SAW)$.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The graph shows the achievable data rate as a function of the quotient  $E_{\rm B}/N_0$  $($in dB$)$.
[[File:EN_Bei_T_4_4_S4.png|right|frame|Increasing the data rate by HARQ]]

*You can see decisive improvements by the HARQ mechanism,  especially for small  $E_{\rm B}/N_0$  values.

*In contrast,  HARQ does not further increase the data rate if  $10 \cdot \lg \E_{\rm B}/N_0 > 2 \ \rm dB$.}}
 
{{GraueBox|TEXT=
$\text{Example 2:}$ 
This graphic is intended to illustrate  how the HARQ process works.  The following steps are to be distinguished:
[[File:EN_Bei_T_4_4_S4b.png|right|frame|On the HARQ procedure '''KORREKTUR: ""''']]
*Before transmission,  the base station informs the receiver of an upcoming transmission using the  '''HS-SCCH'''  channel,  where the  '''HS-SCCH''' frame has three time slots.

*Control data arrives at the receiver and is evaluated immediately after the arrival of the first  '''SCCH''' time slot.

*Data transmission on the  '''HS-PDSCH'''  starts as soon as the user has received the first two time slots of the control data block.

*Within  $\text{5 ms}$  of receiving a frame of data,  the receiver must have decoded the entire frame and checked for errors.

*If the transmission is error-free, 
:*a positive acknowledgement  $($'''ACK'''$)$  is sent upstream, 
:*otherwise a  "non acknowledgement"  $($'''NACK'''$)$  is sent.}}

Since the HARQ does not send a new frame until the acknowledgement of the already transmitted frames is received,  the receiver must be able to manage up to eight HARQs. This guarantees the correct sequence and thereby the correct processing of the data in the higher levels.

{{BlaueBox|TEXT=
$\text{It is also worth mentioning:}$  In addition to HARQ,  in  UMTS Release 5  a  »'''Node B Scheduling'''«  was introduced to be able to react quickly to changes in the transmission conditions of individual nodes  $($for example,  due to fading$)$.
#This scheduling is used to decide which frames are assigned to which transmission channel.
#Priorities are assigned during scheduling.  A frame is only sent when it has the highest priority,  which means that it is most likely to be received correctly.
#This scheduling makes better use of the available bandwidth and significantly increases cell capacity.}}

==Adaptive modulation, adaptive coding and adaptive transmission rate==
 
In HSDPA,  the signals are  "adaptively modulated".  This means:
[[File:EN_Bei_T_4_4_S5.png|right|frame|Adaptive modulation and coding in HSDPA]]
*Under good transmission conditions,  a higher-level modulation is used:  $\text{16-QAM}$  or  $\text{64-QAM}$.

*In poorer conditions,  it is switched to  "Quaternary Phase Shift Keying"  $\text{QPSK}$  or  $\text{4-QAM}$.

In addition to modulation,  the coding as well as the number of  '''HS-DSCH'''  channels used simultaneously by a subscriber can be flexibly and quickly  $($all  $\text{2 ms)}$  changed depending on the channel quality.  Despite the simultaneous use of adaptive modulation and adaptive coding,  the power is always kept constant.

⇒   Power control runs differently in  "HSDPA"  than in  "UMTS R'99":
*The transmit power is always adapted to the signal quality, while the bandwidth should be kept as constant as possible.
*Only if the power can no longer be increased, the spreading factor is increased and thus the data rate is lowered.

⇒   The maximum achievable data rate depends mainly on the  receiver performance  and on the  "transport format and resource combinations"  $\rm (TFRC)$.

In the table,  various parameter combinations for modulation and code rate are given and the resulting bit rates can be seen.  Not considered in this table is the  "overhead".
 
==High–Speed Uplink Packet Access==
 
Since  "UMTS R'99",  the specifications for the uplink have not been further developed,  although bidirectional symmetrical applications have become increasingly important and ever greater demands have been placed on transmission speeds.  Until the introduction of  "Release 6",  data rates were between   $\text{64 kbit/s}$   and   $\text{128 kbit/s}$,  and up to   $\text{384 kbit/s}$   under ideal conditions.

With  "UMTS Release 6",  "High-Speed Uplink Packet Access"  $\rm (HSUPA)$  was defined in 2004 and introduced in 2007. 

This significantly increased the data rates on the uplink:
*These are theoretically up to  $\text{5.8 Mbit/s}$.

*In practice – taking into account simultaneous transmission for multiple users and receiver capacity – transmission rates of up to  $\approx\text{800 kbit/s}$  are still achieved.

[[File:P_ID3116__Bei_T_4_4_S6_v1.png|right|frame|Chart of HSUPA]]

The essential improvement by HSUPA is due to the introduction of an additional uplink channel,  the  "Enhanced Dedicated Channel"  $($'''E-DCH'''$)$.  This minimizes,  among other things in the dedicated uplink channels,  the impact of applications with highly varying and sometimes very intensive data volumes  $($"Bursty Traffic"$)$.

However,  although the  '''E-DCH'''  is a dedicated transport channel,  it does not guarantee the subscriber a fixed bandwidth in the uplink direction,  as is the case with  "UMTS R'99".  This flexible and efficient allocation of bandwidth depending on channel conditions allows a substantial increase in cell capacity.

In addition to the new transport channel  '''E-DCH''',  the following procedures were additional introduced in the uplink  $($'''HSUPA'''$)$  analogous to the downlink  $($'''HSDPA'''$)$:
*"Node B Scheduling",

*"Hybrid Automatic Repeat Request"  $\rm (HARQ)$.

The use of HSUPA in the uplink only makes sense if it is combined with HSDPA in the downlink.  Their interaction significantly increases the performance of the overall system.

==UTRAN Long Term Evolution==
 
"Long Term Evolution"  $\rm (LTE)$  represents a fourth-generation mobile communications system designed and standardized by the  [http://www.3gpp.org/ $\rm 3gpp$]  in parallel with the various further development phases of UMTS,  in order to meet the ever-increasing demands on future mobile communications systems.  This system is also referred to as  "High Speed OFDM Packet Access"  $\rm (HSOPA)$.  The chart summarizes the development of mobile communications systems from the perspective of 2011.

LTE was developed as a forward-looking alternative to third-generation mobile communications systems.  The basic LTE features of were defined in 2004,  but concrete requirements were not drawn up until 2006.  The first systems began operating in 2011.

[[File:EN_Bei_T_4_4_S7.png|right|frame|From  '''UMTS'''  to  '''LTE'''   Attention please:   The data given here is from  $2011$,  shortly after the LTE launch.   You can find the currently valid LTE features on the Internet. ]]

Some features of UTRAN-LTE are listed below in bullet points without comment:
#The frequency ranges allocated for GSM and UMTS  continue to be used.  An extension into the range around  $\text{2600 MHz}$  is planned.
#Between  $200$  and  $400$  active subscribers can be served simultaneously,  which means an increase in  ''cell capacity''  compared to UMTS by a factor of $2$  to $3$.
#The range is  $\text{5 km}$  $($optimal quality$)$ up to  $\text{100 km}$  $($reduced quality$)$.  Maximum data rates are given as  $\text{100 Mbit/s}$  in the downlink and  $\text{50 Mbit/s}$  in the uplink.
#The delay times are reduced to less than  $\text{5 ms}$  for larger bandwidth allocations and to  $\text{10 ms}$  for smaller bandwidth allocations.
#Bandwidths can be flexibly allocated over a very wide range with  $\text{1.25 MHz}$,  $\text{2.5 MHz}$,  $\text{5 MHz}$,  $\text{10 MHz}$,  $\text{15 MHz}$  and  $\text{20 MHz}$  .
#The multiple access techniques used are  "Orthogonal Frequency Division Multiple Access"  $\rm (OFDMA)$  in the downlink and  "Single Carrier Frequency Division Multiple Multiplexing"  $\text{(SC-FDMA)}$  in the uplink.
#Despite these innovations,  there is compatibility with the mobile communications systems of previous generations and a seamless transition to them is possible.

A detailed description of LTE can be found in the fourth main chapter of the book  "[[Mobile Communications]]". However, this was also written in 2011, shortly after its introduction.

== Exercises for the chapter==
 
[[Aufgaben:Exercise_4.8:_HSDPA_and_HSUPA|Exercise 4.8: HSDPA and HSUPA]]

{{Display}}

Examples of Communication Systems/Further Developments of UMTS

2023-03-08T10:45:25Z

Hwang:

{{LastPage}}
{{Header
|Untermenü=UMTS – Universal Mobile Telecommunications System
|Vorherige Seite=Telecommunications Aspects of UMTS
|Nächste Seite=
}}

==High–Speed Downlink Packet Access==
 
In order to meet the increasing demand for higher data rates in mobile communications and to ensure ever better quality of service, the UMTS Release 99 standard has been further developed in five phases up to the present day  $(2011)$.  The diagram shows the individual development phases in chronological order.

The most important further developments were
[[File:EN_Bei_T_4_4_S1_v2.png|right|frame|Further development of UMTS between 2000 and 2011 ]]
[[File:EN_Bei_T_4_4_S1.png|right|frame|Characteristics of HSDPA]]

*the UMTS Release 5 with  »'''HSDPA'''«  and

*the UMTS Release 6 with  »'''HSUPA'''«.

For these two standards, the main focus was on increasing the data rates provided for downlink and uplink as well as greater bandwidth efficiency and cell capacity.

Together,  HSDPA and HSUPA make up the  »'''HSPA standard'''«.
*In 2002,  "High-Speed Downlink Packet Access"  – abbreviated  $\rm HSDPA$  – was specified with UMTS Release 5 and introduced in 2006 to increase data rate and throughput compared to the original UMTS standard as well as to reduce response times for packet-switched transmission.

*In HSPDA, the data rates provided are between  $\text{500 kbit/s}$  and  $\text{3.6 Mbit/s}$  – theoretically even up to  $\text{14.4 Mbit/s}$.

*Compared to the data rate of UMTS R'99  $\text{(14.4 kbit/s}$  to  $\text{2 Mbit/s)}$  these values represent a doubling to quadrupling.

The following technical procedures contribute to the increase in performance of HSDPA compared to UMTS. The diagram compiles these features:
#Introduction of an additional shared channel:  »'''HS-PDSCH'''«,
#use of the  »'''Hybrid ARQ'''«  method,
#'''delay'''  minimization,
#introduction of  »'''Node B scheduling'''«,
#use of  '''adaptive'''  modulation,  coding and transmission rate.

==Additional channels in HSDPA==
 
[[File:EN_Bei_T_4_4_S3neu.png|right|frame|Transport channels,  logical channels,  and physical channels in HSDPA]]
The  "High-Speed Downlink Physical High Speed Channel"  – abbreviated  '''HS-PDCH'''  – is a high-speed transport channel used for the transmission of subscriber data.  It combines the characteristics of a shared and a dedicated channel:

*In the downlink,  one or more channels can be used by multiple subscribers simultaneously.  This allows simultaneous transmission of the same data to different subscribers as well as a significant increase in transmission speed by bundling several channels of this type.

*In any HS-PDCH,  the spreading factor  $J = 16$.  This means that theoretically up to fifteen such channels can be used simultaneously in one cell.  In practice,  however,  only between five and ten channels are ever used,  since the remaining channels are required for the operation of other services.

Resource allocation for the  "High-Speed Shared Data Channel"  $($'''HS-DSCH'''$)$  is done via so-called  "High-Speed Shared Control Channels"  $($'''HS-SCCH'''$)$.  A receiver must therefore be able to receive and decode up to four such channels simultaneously.  In addition to the channels presented above

#The  "Dedicated Physical Control Channel"  $($'''DPCCH'''$)$  is used for the transmission of control data in the uplink.
# The  "Dedicated Control Channel"  $($'''DCCH'''$)$  is used for the localization procedure in downlink and uplink.
#The  "Dedicated Traffic Channel"  $($'''DTCH'''$)$  is responsible for the transmission of Internet Protocol payload data in the uplink direction.

==HARQ procedure and Node B scheduling ==
 
Another feature of HSDPA is the reduction of packet round-trip delay  $\rm (RTD)$  and the use of  "HARQ":
*The  »'''round-trip delay'''»  has been reduced by HSDPA to  $\text{70 ms}$  $($compared  $\text{160 ... 200 ms}$  with UMTS R'99$)$, which is of great importance for some applications,  e.g. web browsing.  This reduction was achieved by decreasing the transport block length to about  $2$  milliseconds  $($previously  $\text{10 ms}$  or  $\text{20 ms)}$.

*In each Node B,  a  »'''HARQ'''«  $($"Hybrid Automatic Repeat Request"$)$  has been implemented to minimize transmission delays.  This mechanism prevents significant delays from occurring due to retransmission of erroneous blocks.  This is because such delays can be interpreted by TCP protocol as blocking,  which then leads to further delays.

*Using the HARQ mechanism and with transport block lengths of  $\text{2 ms}$  the transmission delays in HSPDA are less than  $\text{10 ms}$.  This is a crucial improvement over UMTS,  where error detection  $($associated with retransmission$)$  takes approximately  $\text{90 ms}$.

*In the HARQ procedure,  the detection of  "error"  or  "no error"  is acknowledged for each individual transport frame.  This procedure is referred to as  "Stop and Wait"  $\rm (SAW)$.

{{GraueBox|TEXT=
$\text{Example 1:}$ 
The graph shows the achievable data rate as a function of the quotient  $E_{\rm B}/N_0$  $($in dB$)$.
[[File:EN_Bei_T_4_4_S4.png|right|frame|Increasing the data rate by HARQ]]

*You can see decisive improvements by the HARQ mechanism,  especially for small  $E_{\rm B}/N_0$  values.

*In contrast,  HARQ does not further increase the data rate if  $10 \cdot \lg \E_{\rm B}/N_0 > 2 \ \rm dB$.}}
 
{{GraueBox|TEXT=
$\text{Example 2:}$ 
This graphic is intended to illustrate  how the HARQ process works.  The following steps are to be distinguished:
[[File:EN_Bei_T_4_4_S4b.png|right|frame|On the HARQ procedure]]
*Before transmission,  the base station informs the receiver of an upcoming transmission using the  '''HS-SCCH'''  channel,  where the  '''HS-SCCH''' frame has three time slots.

*Control data arrives at the receiver and is evaluated immediately after the arrival of the first  '''SCCH''' time slot.

*Data transmission on the  '''HS-PDSCH'''  starts as soon as the user has received the first two time slots of the control data block.

*Within  $\text{5 ms}$  of receiving a frame of data,  the receiver must have decoded the entire frame and checked for errors.

*If the transmission is error-free, 
:*a positive acknowledgement  $($'''ACK'''$)$  is sent upstream, 
:*otherwise a  "non acknowledgement"  $($'''NACK'''$)$  is sent.}}

Since the HARQ does not send a new frame until the acknowledgement of the already transmitted frames is received,  the receiver must be able to manage up to eight HARQs. This guarantees the correct sequence and thereby the correct processing of the data in the higher levels.

{{BlaueBox|TEXT=
$\text{It is also worth mentioning:}$  In addition to HARQ,  in  UMTS Release 5  a  »'''Node B Scheduling'''«  was introduced to be able to react quickly to changes in the transmission conditions of individual nodes  $($for example,  due to fading$)$.
#This scheduling is used to decide which frames are assigned to which transmission channel.
#Priorities are assigned during scheduling.  A frame is only sent when it has the highest priority,  which means that it is most likely to be received correctly.
#This scheduling makes better use of the available bandwidth and significantly increases cell capacity.}}

==Adaptive modulation, adaptive coding and adaptive transmission rate==
 
In HSDPA,  the signals are  "adaptively modulated".  This means:
[[File:EN_Bei_T_4_4_S5.png|right|frame|Adaptive modulation and coding in HSDPA]]
*Under good transmission conditions,  a higher-level modulation is used:  $\text{16-QAM}$  or  $\text{64-QAM}$.

*In poorer conditions,  it is switched to  "Quaternary Phase Shift Keying"  $\text{QPSK}$  or  $\text{4-QAM}$.

In addition to modulation,  the coding as well as the number of  '''HS-DSCH'''  channels used simultaneously by a subscriber can be flexibly and quickly  $($all  $\text{2 ms)}$  changed depending on the channel quality.  Despite the simultaneous use of adaptive modulation and adaptive coding,  the power is always kept constant.

⇒   Power control runs differently in  "HSDPA"  than in  "UMTS R'99":
*The transmit power is always adapted to the signal quality, while the bandwidth should be kept as constant as possible.
*Only if the power can no longer be increased, the spreading factor is increased and thus the data rate is lowered.

⇒   The maximum achievable data rate depends mainly on the  receiver performance  and on the  "transport format and resource combinations"  $\rm (TFRC)$.

In the table,  various parameter combinations for modulation and code rate are given and the resulting bit rates can be seen.  Not considered in this table is the  "overhead".
 
==High–Speed Uplink Packet Access==
 
Since  "UMTS R'99",  the specifications for the uplink have not been further developed,  although bidirectional symmetrical applications have become increasingly important and ever greater demands have been placed on transmission speeds.  Until the introduction of  "Release 6",  data rates were between   $\text{64 kbit/s}$   and   $\text{128 kbit/s}$,  and up to   $\text{384 kbit/s}$   under ideal conditions.

With  "UMTS Release 6",  "High-Speed Uplink Packet Access"  $\rm (HSUPA)$  was defined in 2004 and introduced in 2007. 

This significantly increased the data rates on the uplink:
*These are theoretically up to  $\text{5.8 Mbit/s}$.

*In practice – taking into account simultaneous transmission for multiple users and receiver capacity – transmission rates of up to  $\approx\text{800 kbit/s}$  are still achieved.

[[File:P_ID3116__Bei_T_4_4_S6_v1.png|right|frame|Chart of HSUPA]]

The essential improvement by HSUPA is due to the introduction of an additional uplink channel,  the  "Enhanced Dedicated Channel"  $($'''E-DCH'''$)$.  This minimizes,  among other things in the dedicated uplink channels,  the impact of applications with highly varying and sometimes very intensive data volumes  $($"Bursty Traffic"$)$.

However,  although the  '''E-DCH'''  is a dedicated transport channel,  it does not guarantee the subscriber a fixed bandwidth in the uplink direction,  as is the case with  "UMTS R'99".  This flexible and efficient allocation of bandwidth depending on channel conditions allows a substantial increase in cell capacity.

In addition to the new transport channel  '''E-DCH''',  the following procedures were additional introduced in the uplink  $($'''HSUPA'''$)$  analogous to the downlink  $($'''HSDPA'''$)$:
*"Node B Scheduling",

*"Hybrid Automatic Repeat Request"  $\rm (HARQ)$.

The use of HSUPA in the uplink only makes sense if it is combined with HSDPA in the downlink.  Their interaction significantly increases the performance of the overall system.

==UTRAN Long Term Evolution==
 
"Long Term Evolution"  $\rm (LTE)$  represents a fourth-generation mobile communications system designed and standardized by the  [http://www.3gpp.org/ $\rm 3gpp$]  in parallel with the various further development phases of UMTS,  in order to meet the ever-increasing demands on future mobile communications systems.  This system is also referred to as  "High Speed OFDM Packet Access"  $\rm (HSOPA)$.  The chart summarizes the development of mobile communications systems from the perspective of 2011.

LTE was developed as a forward-looking alternative to third-generation mobile communications systems.  The basic LTE features of were defined in 2004,  but concrete requirements were not drawn up until 2006.  The first systems began operating in 2011.

[[File:EN_Bei_T_4_4_S7.png|right|frame|From  '''UMTS'''  to  '''LTE'''   Attention please:   The data given here is from  $2011$,  shortly after the LTE launch.   You can find the currently valid LTE features on the Internet. ]]

Some features of UTRAN-LTE are listed below in bullet points without comment:
#The frequency ranges allocated for GSM and UMTS  continue to be used.  An extension into the range around  $\text{2600 MHz}$  is planned.
#Between  $200$  and  $400$  active subscribers can be served simultaneously,  which means an increase in  ''cell capacity''  compared to UMTS by a factor of $2$  to $3$.
#The range is  $\text{5 km}$  $($optimal quality$)$ up to  $\text{100 km}$  $($reduced quality$)$.  Maximum data rates are given as  $\text{100 Mbit/s}$  in the downlink and  $\text{50 Mbit/s}$  in the uplink.
#The delay times are reduced to less than  $\text{5 ms}$  for larger bandwidth allocations and to  $\text{10 ms}$  for smaller bandwidth allocations.
#Bandwidths can be flexibly allocated over a very wide range with  $\text{1.25 MHz}$,  $\text{2.5 MHz}$,  $\text{5 MHz}$,  $\text{10 MHz}$,  $\text{15 MHz}$  and  $\text{20 MHz}$  .
#The multiple access techniques used are  "Orthogonal Frequency Division Multiple Access"  $\rm (OFDMA)$  in the downlink and  "Single Carrier Frequency Division Multiple Multiplexing"  $\text{(SC-FDMA)}$  in the uplink.
#Despite these innovations,  there is compatibility with the mobile communications systems of previous generations and a seamless transition to them is possible.

A detailed description of LTE can be found in the fourth main chapter of the book  "[[Mobile Communications]]". However, this was also written in 2011, shortly after its introduction.

== Exercises for the chapter==
 
[[Aufgaben:Exercise_4.8:_HSDPA_and_HSUPA|Exercise 4.8: HSDPA and HSUPA]]

{{Display}}

Examples of Communication Systems/Telecommunications Aspects of UMTS

2023-03-08T10:43:22Z

Hwang:

{{Header
|Untermenü=UMTS – Universal Mobile Telecommunications System
|Vorherige Seite=UMTS Network Architecture
|Nächste Seite=Further Developments of UMTS
}}

==Improvements regarding speech coding ==
 
In the chapter  [[Examples_of_Communication_Systems/General_Description_of_GSM|"Global System for Mobile Communications"]]  $\rm (GSM)$  of this book,  several speech codecs have already been described in detail.

{{BlaueBox|TEXT=
$\text{Reminder:}$ 
A speech codec is used to reduce the data rate of a digitized speech or music signal.
#In the process,  redundancy and irrelevance are removed from the original signal.
#The artificial word  "codec"  indicates that the same functional unit is used for both,  encoding and decoding.}}

Among others,  the  [[Examples_of_Communication_Systems/Speech_Coding#Adaptive_Multi_Rate_Codec|"Adaptive Multi-Rate Codec"]]  $\rm (AMR)$  based on  [[Examples_of_Communication_Systems/Speech_Coding#Algebraic_Code_Excited_Linear_Prediction|$\rm ACELP$]]  $($"Algebraic Code Excited Linear Prediction"$)$  was introduced, 
*which in the frequency range from  $\text{300 Hz}$  to  $\text{3400 Hz}$ 

*dynamically switches between eight different modes  $($single codecs$)$ 

*of different data rate in the range of  $\text{4. 75 kbit/s}$  to  $\text{12.2 kbit/s}$.

These codecs are also supported in UMTS Release 99 and Release 4.  Compared to the earlier speech codecs  $($Full Rate,  Half Rate, Enhanced Full Rate Vocoder$)$,  they allow
#independence from channel conditions and network load,
#the ability to adapt data rates to conditions,
#improved flexible error protection in the event of more severe radio interference, and
#thereby providing better overall voice quality.

[[File:EN_Bei_T_4_3_S1_v2.png|right|frame|Composition of wideband AMR modes]]

In 2001,  the  "3rd Generation Partnership Project"  $\text{(3gpp)}$  and the  "International Telecommuncation Union"    $\text{(ITU)}$  specified the new voice codec  »'''Wideband AMR'''«  for UMTS Release 5.  This is a further development of AMR and offers

*an extended bandwidth from  $\text{50 Hz}$  to  $\text{7 kHz}$  $($sampling frequency  $\text{16 kHz})$,

*a total of nine modes between  $\text{6.6 kbit/s}$  and  $\text{23.85 kbit/s}$  $($of which only five modes are used$)$,  and

*improved voice quality and better (more natural) sound.

{{BlaueBox|TEXT=
$\text{Some features of wideband AMR}$ 

#Speech data is delivered to the codec as PCM encoded speech with  $16\hspace{0.05cm}000$  samples per second. 
#The speech coding is done in blocks of  $\text{20 ms}$  and the data rate is adjusted every  $\text{20 ms}$.
#The band  $\text{(50 Hz}$  to  $\text{7000 Hz})$  is divided into two sub-bands,  which are encoded differently to allocate more bits to the subjectively important frequencies.
#The upper band  $\text{(6400 Hz}$  to  $\text{7000 Hz})$  is transmitted only in the highest mode $($with  $\text{23.85 kbit/s)}$ .
#In all other modes,  only frequencies  $\text{50 Hz}$  to  $\text{6400 Hz}$  are considered in encoding.
#Wideband AMR supports  "discontinuous transmission"'  $\rm (DTX)$.  This feature means that transmission is paused during voice pauses,  reducing both mobile station power consumption and overall interference at the air interface.  This process is also known as  "Source-Controlled Rate"  $\rm (SCR)$.
#The  "Voice Activity Detection"  $\rm (VAD)$  determines whether speech is in progress or not and inserts a  "silence descriptor frame"  during speech pauses.
#The subscriber is suggested the feeling of a continuous connection by the decoder inserting synthetically generated  "comfort noise"  during speech pauses.}}

==Application of the CDMA method to UMTS==
 
UMTS uses the multiple access method  "Direct Sequence Code Division Multiple Access"  $\rm (DS-CDMA)$,  which has already been discussed in the  [[Modulation_Methods/Direct-Sequence_Spread_Spectrum_Modulation#Block_diagram_and_equivalent_low-pass_model|"PN modulation"]]  chapter  of the book "Modulation Methods".

Here follows a brief summary of this method according to the diagram describing such a system in the equivalent low-pass range and highly simplified:
[[File:EN_Bei_T_4_3_S2c.png|right|frame|CDMA transmission system for two subscribers]]

*The two data signals  $q_1(t)$  and  $q_2(t)$  are supposed to use the same channel without interfering with each other. The bit duration of each is  $T_{\rm B}$.

*Each of the data signals is multiplied by an associated spreading code   $c_1(t)$  resp.  $c_2(t)$.

*The sum signal  $s(t) = q_1(t) · c_1(t) + q_2(t) · c_2(t)$  is formed and transmitted.

*At the receiver,  the same spreading codes  $c_1(t)$  resp.  $c_2(t)$  are added,  thus separating the signals again.

*Assuming orthogonal spreading codes and a small AWGN noise,  the two reconstructed signals at the receiver output are:
:$$v_1(t) = q_1(t) \ \text{and} \ v_2(t) = q_2(t).$$
*For AWGN noise signal  $n(t)$  and orthogonal spreading codes,  this does not change the error probability due to other participants.

{{GraueBox|TEXT=
$\text{Example1:}$ 
The upper graph shows three data bits   $(+1, -1, +1)$   of the rectangular signal  $q_1(t)$  from subscriber '''1''',  each with symbol duration  $T_{\rm B}$.

[[File:EN Mob T 3 4 S4.png|right|frame|Signals at  "Direct–Sequence Spread Spectrum"]]
*Here,  the symbol duration  $T_{\rm C}$  of the spreading code  $c_1(t)$   ⇒   also called  "chip duration"  is smaller by a factor  $4$.

*The multiplication  $s_1(t) = q_1(t) · c_1(t)$  results in a chip sequence of length  $12 · T_{\rm C}$.

One recognizes from this sketch that the signal  $s_1(t)$  is of higher frequency than  $q_1(t)$.

*This is why this modulation method is often also called  "spread spectrum".

*The CDMA receiver reverses this  "spreading".  We refer to this  "receiver-side spreading"  as  "despreading". }}

{{BlaueBox|TEXT=
$\text{Summarizing:}$  
By applying  "Direct Sequence Code Division Multiple Access"  $\rm (DS-CDMA)$  to a data bit sequence  $q(t)$ 
*increases the bandwidth of  $s(t) = q(t) \cdot c(t)$  by the  »'''spreading factor'''«   $J = T_{\rm B}/T_{\rm C}$  –  this is equal to the number of  "chips per bit";

*the chip rate  $R_{\rm C}$  is greater than the bit rate  $R_{\rm B}$ by a factor  $J$;

*the bandwidth of the entire CDMA signal is greater than the bandwidth of each user by a factor  $J$.

That is:    $\text{In UMTS, the entire bandwidth is available to each subscriber for the entire transmission duration}$.

Recall:  In GSM,  both  "Frequency Division Multiple Access"  and  "Time Division Multiple Access"  are used as multiple access methods.
*Here,  each subscriber has only a limited frequency band  $\rm (FDMA)$,  and

*he only has access to the channel within time slots  $\rm (TDMA)$.}}

==Spreading codes and scrambling with UMTS==
 
The spreading codes for UMTS should
*be orthogonal to each other to avoid mutual interference between subscribers,

*allow a flexible realization of different spreading factors  $J$.

⇒   The issue presented here is also illustrated by the German-language SWF applet  [[Applets:OVSF_codes_(Applet)|"OVSF codes"]].

{{GraueBox|TEXT=
$\text{Example 2:}$ 
An example of this is the   »'''orthogonal variable spreading factor'''«   $\rm (OVSF)$,  which provide codes of lengths from  $J = 4$  to  $J = 512$.

[[File:P_ID1535__Bei_T_4_3_S3c_v1.png|right|frame|Chart on the OVSF code family]]

These can be created using a code tree, as shown in the diagram.  Here,  at each branch,  two new codes are created from one code  $C$:
:#  $(+C \ +\hspace{-0.1cm}C)$,  and
:#  $(+C \ -\hspace{-0.1cm}C)$.

Note that no predecessor and successor of a code may be used.
*So in the drawn example,  eight spreading codes with spreading factor  $J = 8$  could be used.

*Also possible are the four codes with yellow background
:*once with  $J = 2$,
:*once with  $J = 4$,  and
:*twice with  $J = 8$.

But the lower four codes with spreading factor  $J = 8$  cannot be used,  because they all start with  "$+1 \ -\hspace{-0.1cm}1$",  which is already occupied by the OVSF code with spreading factor  $J = 2$.}}

[[File:EN_Mob_T_3_4_S5.png|right|frame|Additional scrambling after spreading]]
To obtain more spreading codes and thus be able to supply more participants,  after the band spreading with  $c(t)$ 
*the sequence is chip-wise scrambled again with  $w(t)$, 

*without any further spreading.

The  »'''scrambling code'''« $w(t)$  has same length and rate as the spreading code  $c(t)$.
 
[[File:EN_Mob_T_3_4_S5b_v3.png|right|frame|Typical spreading and scrambling codes for UMTS ]]

⇒   Scrambling causes the codes to lose their complete orthogonality; they are called  "quasi-orthogonal".

*For these codes,  although the  [[Modulation_Methods/Spreading_Sequences_for_CDMA#Properties_of_the_correlation_functions|"cross-correlation function"]]  $\rm (CCF)$  between different spreading codes is non-zero.

*But they are characterized by a pronounced  [[Modulation_Methods/Spreading_Sequences_for_CDMA#Properties_of_the_correlation_functions|"auto-correlation function"]]  $\rm (ACF)$  around zero,  which facilitates detection at the receiver.

*Using quasi-orthogonal codes makes sense because the set of orthogonal codes is limited and scrambling allows also different users to use the same spreading codes.

The table summarizes some data of spreading and scrambling codes.
 
{{GraueBox|TEXT=
[[File:P_ID1537__Bei_T_4_3_S3b_v2.png|right|frame|Generator for creating Gold codes]]
$\text{Example 3:}$ 
In UMTS, so-called  »'''Gold codes'''«  are used for scrambling. The graphic from  [3gpp]<ref>3gpp Group:  UMTS Release 6 - Technical Specification 25.213 V6.4.0,  Sept. 2005.</ref>  shows the block diagram for the circuitry generation of such codes.
*Two different pseudonoise sequences of equal length  $($here:  $N = 18)$  are first generated in parallel using shift registers and added bitwise using  "exclusive-or"  gates''.

*In the uplink,  each mobile station has its own scrambling code and the separation of each channel is done using the same code.

*In contrast,  in the downlink, each coverage area of a  "Node B"  has a common scrambling code.}}

==Channel coding for UMTS==
 
With UMTS,  the EFR- and AMR-encoded voice data pass through a two-stage error protection  $($similar to GSM$)$,  consisting of
[[File:EN_Bei_T_4_3_S4a_v1.png|right|frame|Insertion of CRC bits and tailbits in UMTS ]]
#formation of  "cyclic redundancy check bits"  $\rm (CRC)$,
#subsequent convolutional encoding.

However,  these methods differ from those used for GSM in that they are more flexible,  since for UMTS they have to take different data rates into account.

⇒   For  »'''error detection'''«,  eight, twelve, sixteen or  $24$  CRC bits are formed depending on the size of the transport block  $\text{(10 ms}$  or  $\text{20 ms})$,  and appended to it.

*Eight tail bits are also inserted at the end of each frame for synchronization purposes.

*The diagram shows a transport block of the '''DCH''' channel with  $164$  user data bits,  to which  $16$  CRC bits and eight tail bits are appended.
 

⇒   For  »'''error correction'''«,  UMTS uses two different methods,  depending on the data rate:
*For low data rates,  [[Channel_Coding/Basics_of_Convolutional_Coding|"convolutional codes"]]  with code rates  $R = 1/2$   or   $R = 1/3$  are used as with GSM.  These are generated with eight memory elements of a feedback shift register  $(256$  states$)$.  The coding gain is approximately  $4.5$  to  $6$  dB with code rate  $R = 1/3$  and at low error rates.

*For higher data rates,  one uses  [[Channel_Coding/The_Basics_of_Turbo_Codes|"turbo codes"]]  of rate   $R = 1/3$.  The shift register consists here of three memory cells,  which can assume a total of eight states.  The gain of turbo codes is larger by  $2$  to  $3$  dB than by convolutional codes and depends on the number of iterations.  You need a processor with high processing power for this and there may be relatively large delays.

After channel coding,  the data is fed to an  [[Examples_of_Communication_Systems/Entire_GSM_Transmission_System#Interleaving_for_speech_signals|"interleaver"]]  as in GSM,  in order to be able to resolve bundle errors caused by fading on the receiving side.  Finally, for  "rate matching"  of the resulting data to the physical channel,  individual bits are removed  $($"puncturing"$)$  or repeated  $($"repetition"$)$  according to a predetermined algorithm.

{{GraueBox|TEXT=
[[File:EN_Bei_T_4_3_S4b.png|right|frame|Error correction mechanisms in UMTS]]
$\text{Example 4:}$ 
The graph first shows the increase in bits due to a convolutional or turbo code of rate  $R =1/3$, where  the  $188$  bit time frame  $($after the CRC checksum and tail bits$)$  becomes a  $564$  bit frame.

*Followed by a first  $($external$)$  nesting and then a second  $($internal$)$  nesting.

*After this,  the time frame is divided into four subframes of  $141$  bits each,  and these are then matched to the physical channel by rate matching.}}

==Frequency responses and pulse shaping for UMTS==
 
[[File:EN_Bei_T_4_3_UMTS1_v2.png|right|frame|Block diagram of the optimal Nyquist equalizer at ideal channel|class=fit]]
In this section,  we assume the following block diagram of a binary system with ideal channel   ⇒   $H_{\rm K}(f) = 1$.

In particular,  let hold:

*The  "transmitter pulse filter"  converts the binary  $\{0, \ 1\}$ data into physical signals.  The filter is described by the frequency response  $H_{\rm S}(f)$,  which is identical in shape to the spectrum of a single transmitted pulse.

*In UMTS,  the receive '''KORREKTUR: receiver''' filter  $H_{\rm E}f) = H_{\rm S}(f)$  is matched to the transmitter  $($"matched filter"$)$  and the overall frequency response  $H(f) = H_{\rm S}(f) \cdot H_{\rm E}(f)$  satisfies the  [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_frequency_domain|"first Nyquist criterion"]]:
:$$ H(f) = H_{\rm CRO}(f) = \left\{ \begin{array}{c} 1 \\ 0 \\ \cos^2 \left( \frac {\pi \cdot (|f| - f_1)}{2 \cdot (f_2 - f_1)} \right)\end{array} \right.\quad
\begin{array}{*{1}c} {\rm{for}} \\ {\rm{for}}\\ {\rm else }\hspace{0.05cm}. \end{array}
\begin{array}{*{20}c} |f| \le f_1, \\ |f| \ge f_2.\\ \\\end{array}$$

This means:   Consecutive pulses in time do not interfere with each other   ⇒   no  [[Digital_Signal_Transmission/Causes_and_Effects_of_Intersymbol_Interference|"intersymbol interference"]]  $\rm (ISI)$  occur.  The associated time function is:

:$$h(t) = h_{\rm CRO}(t) ={\rm sinc}(t/ T_{\rm C}) \cdot \frac{\cos(r \cdot \pi t/T_{\rm C})}{1- (2r \cdot t/T_{\rm C})^2}\hspace{0.4cm} \text{with } \hspace{0.4cm} r = \frac{f_2 - f_1}{f_2 + f_1}. $$

*"CRO" here stands for  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Raised-cosine_low-pass_filter|"raised cosine low-pass"]].

*The sum  $f_1 + f_2$  is equal to the inverse of the chip duration  $T_{\rm C} = 260 \ \rm ns$,  so it is equal to  $3.84 \ \rm MHz$.

*The  "rolloff factor"  has been determined to  $r = 0.22$  for UMTS.  The two  "corner frequencies"  are thus

:$$f_1 = {1}/(2 T_{\rm C}) \cdot (1-r) \approx 1.5\,{\rm MHz},$$
:$$f_2 ={1}/(2 T_{\rm C}) \cdot (1+r) \approx 2.35\,{\rm MHz}.$$

*The required bandwidth is  $B = 2 \cdot f_2 = 4.7 \ \rm MHz$.  Thus,  there is sufficient bandwidth available for each UMTS channel with  $5 \ \rm MHz$.

{{BlaueBox|TEXT=
$\text{Conclusion:}$  The graph shows.
[[File:P_ID1547__Bei_T_4_3_S5b_v1.png|right|frame|Raised cosine spectrum and impulse response]]
*on the left,  the  $($normalized$)$  Nyquist spectrum  $H(f)$,

*on the right,  the corresponding Nyquist pulse  $h(t)$,  whose zero crossings are equidistant with distance  $T_{\rm C}$.

$\text{It should be noted:}$
# The transmission filter  $H_{\rm S}(f)$  and the matched filter  $H_{\rm E}(f)$  are each  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems|"root raised cosine"]].
#Only the product  $H(f) = H_{\rm S}(f) \cdot H_{\rm E}(f)$  leads to the raised cosine.  This also means:
# The impulse responses  $h_{\rm S}(t)$  and  $h_{\rm E}(t)$  by themselves do not satisfy the first Nyquist condition.
#Only the combination of the two  $($in the time domain the convolution$)$  leads to the desired equidistant zeros.}}

==Modulation methods for UMTS==
 
The modulation techniques used in UMTS can be summarized as follows:
#In the downlink:  "Quaternary Phase Shift Keying"  is used for modulation  both in  "frequency division duplex"  $\rm (FDD)$  and in  "time division duplex""  $\rm (TDD)$.
#Here,  user data  $($DPDCH channel$)$  and control data  $($DPCCH channel$)$  are multiplexed in time.
#With TDD,  the signal is modulated in the uplink also by means of QPSK,  but not with  FDD. 
#Here,  a  "dual channel binary phase shift keying"  is used   ⇒   different channels are transmitted in  "in-phase"  and  "quadrature components".
#Thus,  two chips are transmitted per modulation step.  The gross chip rate is therefore twice the modulation rate of  $3.84$ Mchip per second.

{{GraueBox|TEXT=
$\text{Example 5:}$ 
The graph shows in the equivalent low-pass domain this  "I/Q multiplexing method",  as it is also called:
[[File:EN_Mob_T_3_4_S6.png|right|frame|Modulation and pulse shaping for UMTS|class=fit]]

#The spread useful data of the DPDCH channel is modulated onto the inphase component.
#The spread control data of the DPCCH channel is modulated onto the quadrature component.
#After modulation,  the quadrature component is weighted by the root of the power ratio  $G$  between the two channels to minimize the influence of power differences between  $I$  and  $Q$.
#Finally,  the complex sum signal  $(I +{\rm j} \cdot Q)$  is multiplied by a scrambling code that is also complex.}}

{{BlaueBox|TEXT=
$\text{Conclusion:}$  An advantage of dual channel BPSK modulation is the  ''' possibility of usinglow-power amplifiers'''.
*But time division multiplexing of user and control data as in the uplink  '''is not possible in the downlink'''.

*One reason for this is the use of  "Discontinuous Transmission"  $\rm (DTX)$  and the associated time constraints.}}

==Single-user receiver==
 
The task of a CDMA receiver is to separate and reconstruct the transmitted data of the individual subscribers from the sum of the spread data streams.  A distinction is made between  "single-user receivers"  and  "multi-user receivers".

In the UMTS downlink,  it is always used a  »'''single-user receiver'''«,  since in the mobile station a joint detection of all subscribers would be too costly
*due to the large number of active subscribers

*as well as the length of the scrambling codes and the asynchronous operation.

Such a receiver consists of a bank of independent correlators.
*Each one of the total  $J$  correlators belongs to a specific spreading sequence.

*The correlation is usually formed in a so-called  "correlator database"  by software.

Thereby one receives at the correlator output the sum of
[[File:EN_Bei_T_4_3_S6a_v2.png|right|frame|Single-user receiver with matched filter]]
*the  "autocorrelation '''KORREKTUR: auto-correlation''' function"  of the spreading code and

*the  "cross-correlation function"  of all other users with their own spreading code.

The graphic shows the simplest realization of such a receiver with matched filter.

#The received signal  $r(t)$  is first multiplied by the spreading code  $c(t)$  of the considered subscriber,  which is called  "despreading"  $($yellow background$)$.
#Followed by convolution with the matched filter impulse response  $($"Root Raised Cosine"$)$  to maximize SNR,  and sampling in bit clock  $(T_{\rm B})$.
#Finally, the threshold decision is made, which provides the sink signal  $v(t)$ and  thus the data bits of the considered subscriber.

{{BlaueBox|TEXT=
$\text{Please note:}$ 

*For the AWGN channel,  spreading at the transmitter and the matched despreading at the receiver have no effect on the bit error probability because of  $c(t)^2 = 1$.  As shown in  [[Aufgaben:Exercise_4.5:_Pseudo_Noise_Modulation|$\text{Exercise 4.5}$]],  even with spreading/despreading at the optimal receiver, regardless of spreading factor  $J$:

:$$p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt { {2 \cdot E_{\rm B} }/{N_{\rm 0} } } \hspace{0.05cm} \right )\hspace{0.05cm}. $$

*This result can be justified as follows:  The statistical properties of white noise  $n(t)$  are not changed by multiplication with the  $±1$  signal  $c(t)$.}}

==Rake receiver==
 
Another receiver for single-user detection is the  »'''rake receiver'''«,  which leads to significant improvements for a multipath channel.
[[File:EN_Bei_T_4_3_S6b_v2.png|right|frame|Structure of the rake receiver  $($shown in the equivalent low-pass domain$)$]]

The diagram shows its setup for a two-way channel with 
*a direct path with coefficient  $h_0$  and delay time  $τ_0$,

*an echo with coefficient  $h_1$  and delay time  $τ_1$.

For simplicity,  the coefficients  $h_0$  and  $h_1$  are assumed to be real.  Due to the representation in the equivalent low-pass domain,  these could also be complex.

⇒   The task of the rake receiver is to concentrate the signal energies of all paths  $($in this example only two$)$  to a single instant.  It works accordingly like a  "rake"  for the garden.
 
If one applies a Dirac delta impulse at time  $t = 0$  to the channel input,  there will be three Dirac delta impulses at the output of the rake receiver:
:$$ s(t) = \delta(t) \hspace{0.3cm}\Rightarrow\hspace{0.3cm}
y(t) = h_0 \cdot h_1 \cdot \delta(t - 2\tau_0) + (h_0^2 + h_1^2) \cdot \delta(t - \tau_0 - \tau_1)+
h_0 \cdot h_1 \cdot \delta(t - 2\tau_1) .$$

*The signal energy is concentrated at the time  $τ_0 + τ_1$.  Of the total four paths,  two contribute  $($middle term$)$.

*The Dirac delta functions at  $2τ_0$  and  $2τ_1$  do cause momentum interference.  However, their  weights are much smaller than those of the main path.

{{GraueBox|TEXT=
$\text{Example 6:}$ 
With channel parameters  $h_0 = 0.8$  and  $h_1 = 0.6$  the main path  $($with weight  $h_0)$  contains  $0.82/(0.82 + 0.62) = 64\%$  of the total signal energy.
*With rake receiver and the same weights,  the above equation is:

:$$ y(t) = 0.48 \cdot \delta(t - 2\tau_0) + 1.0 \cdot \delta(t - \tau_0 - \tau_1)+
0.48 \cdot \delta(t - 2\tau_1) .$$

*The share of the main path in the total energy amounts in this simple example to  ${1^2}/{(1^2 + 0.48^2 + 0.48^2)} ≈ 68\%.$}}

Rake receivers are preferred for implementation in mobile devices,  but have a limited performance when there are many active participants.
#In a multipath channel with many  $(M)$  paths,  the Rake has also  $M$  fingers.
#The main finger  – also called  "searcher"  – is responsible for identifying and ranking the individual paths of multiple propagation.
#It searches for the strongest paths and assigns them to other fingers along with their control information.
#In the process,  the time and frequency synchronization of all fingers is continuously compared with the control data of the received signal.

==Multi-user receiver ==
 
In a single-user receiver,  only the data signal of one subscriber is decided,  while all other subscriber signals are considered as additional noise.  However,  the bit error rate of such a detector will be very large
*if there is large  "intracell interference"  $($many active subscribers in the considered radio cell$)$

*or large  "intercell interference"  $($highly interfering subscribers in neighboring cells$)$.

In contrast,  »'''multi-user receivers'''«  make a joint decision for all active subscribers.» Their characteristics can be summarized as follows:
#Such a multi-user receiver does not consider the interference from other participants as noise,  but also uses the information contained in the interference signals for detection.
#The receiver is expensive to implement and the algorithms are extremely computationally intensive.  It contains an extremely large correlator database followed by a common detector.
#The multi-user receiver must know the spreading codes of all active users.  This requirement precludes use in the UMTS downlink  $($i.e.,  at the mobile station$)$.  In contrast,  all subscriber-specific spreading codes are known a-priori to the base stations,  so that multi-user detection is only used in the uplink.
#Some detection algorithms additionally require knowledge of other signal parameters such as energies and delay times.  The common detector  – the heart of the receiver –  is responsible for applying the appropriate detection algorithm in each case.
#Examples of multi-user detection are  "decorrelating detection"  and  "Interference Cancellation".

==Near–far problem==
 
The  "near-far problem"  is exclusively an uplink problem,  i.e.,  the transmission from mobile subscribers to a base station.  We consider a scenario with two users at different distances from the base station according to the following graph.  This can be interpreted as follows:

[[File:EN_Mob_T_3_2_S3.png|right|frame|Scenarios for the near-far problem|class=fit]]

#If both mobile stations transmit with the same power,  the received power of the red user  $\rm A$  at the base station is significantly smaller than that of the blue user  $\rm B$  $($left scenario$)$ due to path loss.
#In large macrocells,  the difference can be as much as  $100$  dB.  As a result,  the red signal is largely obscured by the blue.
#You can largely avoid the near-far problem if user  $\rm A$  transmits with higher power than user  $\rm B$,  as indicated in the right scenario.
#Then,  at the base station,  the received power of both mobile stations is then  $($almost$)$  equal.

Note:   In an idealized system  $($one-way channel,  ideal A/D converters,  fully linear amplifiers$)$  the transmitted data of the users are orthogonal to each other and one could detect the users individually even with very different received powers.  This statement is true
*for UMTS  $($multiple access:  CDMA$)$  as well as

*for the 2G system GSM  $($FDMA/TDMA$)$,  and

*for the 4G system LTE  $($TDMA/OFDMA$)$.

In reality,  however,  orthogonality is not always given due to the following reasons:
#Different receive paths   ⇒   multipath channel,
#non-ideal characteristics of the spreading and scrambling codes in CDMA,
#asynchrony of users in the time domain  $($basic propagation delay of paths$)$,
#asynchrony of users in the frequency domain  $($non-ideal oscillators and Doppler shift due to mobility of users$)$.

Consequently,  the users are no longer orthogonal to each other and the signal-to-noise ratio of the user to be detected with respect to the other users is not arbitrarily high:
*For  [[Examples_of_Communication_Systems/General_Description_of_GSM|"GSM"]]  and  [[Mobile_Communications/General_Information_on_the_LTE_Mobile_Communications_Standard|"LTE"]]  one can assume signal-to-noise ratios of  $25 $  dB  and more,

*but for UMTS  $($CDMA$)$  only approx.  $15$  dB,  with high-rate data transmission rather less.

==Carrier-to-interference power ratio==

The term  "capacity"  is generally understood to mean the number of available transmission channels per cell.  However,  since the number of subscribers is not strictly limited in UMTS unlike in GSM,  no fixed capacity can be specified here.
*In perfect codes,  the subscribers do not interfere with each other.  Thus,  the maximum number of users is determined solely by the spreading factor  $J$  and the available number of mutually orthogonal codes,  which,  however,  is also limited.

*More practical are non-perfect,  only quasi-orthogonal codes.  Here,  the  "capacity"  of a radio cell is predominantly determined by the resulting interference or the  "carrier-to-interference power ratio"  $\rm (CIR)$.

[[File:EN_Bei_T_4_3_S8_v2.png|right|frame|Carrier-to-interference ratio depending on the number of subscribers]]

As can be seen from this graph,  CIR depends directly on the number of active participants.  The more active subscribers there are,  the more interference power is generated and the smaller the CIR becomes.

Furthermore,  this decisive criterion for UMTS also depends on the following variables:
#The topology and user behavior  $($number of services called up$)$,
#the spreading factor  $J$  and the orthogonality of the used spreading code.

In order to limit the disturbing influence of the interference power on the transmission quality,  there are two possible criteria:

»'''Cell breathing'''«:   If the number of active subscribers increases significantly with UMTS,  the cell radius is reduced and  $($because of the now fewer subscribers in the cell$)$  also the current interference power is lower.  A less loaded neighboring cell then steps in to supply the subscribers at the edge of the reduced cell.

»'''Power control'''«:   If the total interference power within a radio cell exceeds a specified limit,  the transmission power of all subscribers is reduced accordingly and/or the data rate is reduced,  resulting in poorer transmission quality for all.  More about this in the next section.

==Power and power control in UMTS==
 
The ratio between the signal power and the interference power is used as the controlled variable for power control in UMTS.  There are differences between the  "frequency division duplex"  $\rm (FDD)$  and the  "time division duplex"  $\rm (TDD)$  modes.

[[File:EN_Bei_T_4_3_S10a.png|right|frame|Power control in the FDD mode '''KORREKTUR: transmit power''']]
We take a closer look at the FDD power control.  In the diagram you can see two different control loops:
*The  »'''inner control loop'''«  controls the transmitter power based on time slots,  where one power command is transmitted in each time slot. 

::The power of the transmitter is determined and changed using the CIR estimates in the receiver and the specifications of the  "radio network controller"  $\rm (RNC)$  from the outer control loop.

*The  '''outer loop'''  controls based on $10$ millisecond duration frames. It is implemented in the RNC and is responsible for determining the set point for the inner loop.

The FDD power control sequence is as follows:
#The RNC provides a carrier-to-interference ratio  $\rm (CIR)$  setpoint.
#The receiver estimates the actual CIR value and generates control commands for the transmitter.
#The transmitter changes the transmitted power according to these control commands.

The principle of  "power control in TDD mode"  is similar to the control presented here for the FDD mode.  In fact in the downlink direction they are practically identical.

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
The  »'''TDD power control'''«  is much slower and thus less precise than in  FDD.

*However,  fast power control is not even possible in this case,  since each participant has only a fraction of the time frame available for him.}}

==Link budget ==
 
When planning UMTS networks,  calculating the link budget is an important step.  Knowledge of the link budget is required both for dimensioning the coverage areas and for determining the capacity and quality of service requirements.

{{BlaueBox|TEXT=
The  »'''objective of the link budget'''«  is to calculate the  '''maximum cell size'''  considering the following criteria:
#Type and data rate of the services,
#topology of the environment,
#system configuration  $($location and power of base stations,  handover gain$)$,
#service requirements  $($availability$)$,
#type of mobile station  $($speed,  power$)$,
#financial and economic aspects}}

{{GraueBox|TEXT=
[[File:EN_Bei_T_4_3_S9_v2.png|right|frame|Budget for a voice transmission channel '''KORREKTUR: fault''']]
$\text{Example 7:}$ 
The calculation of the link budget is illustrated using the example of a voice transmission channel in the UMTS downlink.  Regarding the exemplary numerical values,  it should be noted:
#The transmitted power is  $P_{\rm S} =19$  dBm,  which corresponds to approx.  $79$  mW.  Here,  the antenna loss is considered to be  $2$  dB.
#The noise power  $P_{\rm R} = 5 \cdot 10^{-11}$  mW   ⇒   $P_{\rm R} = -103$  dBm   product of UMTS bandwidth and noise power density.
#The interference power is  $P_{\rm I} = -99$  dBm  corresponding to  $1.25 \cdot 10^{-10}$  mW.
#This gives the total interference power  $P_{\rm R+I} = P_{\rm R} + P_{\rm I} = 1.25 \cdot 10^{-10}$  mW   ⇒   $P_{\rm R+I} =- 97.5$  dBm.
#The antenna sensitivity results in $-97.5 - 27 + 5 - 17 + 3.5 = - 133$  dBm.  A large negative value here is  "good".
#Maximum allowable path loss should be as large as possible.  Here  $19 - (-133) = 152$  dB.
#The  '''link budget'''  includes the margin for fading and the handover gain,  in the example:  $140$  dB.
#The  '''maximum cell radius'''  can be determined from the link budget using an [https://en.wikipedia.org/wiki/Path_loss "empirical formula"] of Okumura-Hata.  It holds:  $ {r}\ [{\rm km}] = 10^{({\rm LinkBudget}- 137)/35}= 10^{0.0857}\approx 1.22 . $
 

Notes:  
*"$\rm dB$"  denotes a logarithmic power specification,  referenced to  $1 \rm W$.

*In contrast  "$\rm dBm$"  refers to the power  $1 \rm mW$.}}

==UMTS radio resource management ==
 
The central task of  »'''radio resource management'''«  $\rm (RRM)$  is the dynamic adaptation of radio transmission parameters to the current situation  $($fading,  mobile station movement,  load,  etc.$)$  with the aim to
[[File:EN_Bei_T_4_3_S11.png|right|frame|Radio Resource Management in UMTS]]
#increase the transmission and subscriber capacity,
#improve the individual transmission quality,  and
#use existing radio resources economically.

The main RRM mechanisms summarized in the diagram are explained below.

'''Transmit power control''' The  radio resource management  attempts to keep the received power and thus the carrier-to-interference ratio  $\rm (CIR)$  at the receiver constant,  or at least to prevent it from falling below a specified limit.

An example of the need for power control is the  [[Examples_of_Communication_Systems/Telecommunications_Aspects_of_UMTS#Near.E2.80.93far_problem|"near-far problem"]],  which is known to cause a disconnect.

The step size of the power control is  $1 \ \rm dB$  or  $2 \ \rm dB$,  and the frequency of the control commands is  $1500$  commands per second.

'''Regulation of data rate'''
 UMTS allows an exchange between data rate and transmission quality,  which can be realized by selecting the spreading factor.  Doubling the spreading factor corresponds to halving the data rate and increases the quality by  $3\ \rm dB$  $($spreading gain$)$.

'''Access control'''
 To avoid overload situations in the overall network,  a check is made before a new connection is established to see whether the necessary resources are available.  If not,  the new connection is rejected.  This check is realized by estimating the transmission power distribution after the new connection is established.

'''Load control'''
 This becomes active if an overload occurs despite access control.  In this case,  a handover to another  base station is initiated and  – if this is not possible –  the data rates of certain nodes are lowered.

'''Handover'''
 Finally,  radio resource management is also responsible for handover to ensure uninterrupted connections.  Mobile stations are assigned to the individual radio cells on the basis of CIR measurements.

==Exercises for the chapter ==
 
[[Aufgaben:Exercise_4.5:_Pseudo_Noise_Modulation|Exercise 4.5: Pseudo Noise Modulation]]

[[Aufgaben:Exercise_4.5Z:_About_Band_Spreading_with_UMTS|Exercise 4.5Z: About Spread Spectrum with UMTS]]

[[Aufgaben:Exercise_4.6:_OVSF_Codes|Exercise 4.6: OVSF Codes]]

[[Aufgaben:Exercise_4.7:_To_the_Rake_Receiver|Exercise 4.7: About the Rake Receiver]]

==Sources==
<references />

{{Display}}

Examples of Communication Systems/Telecommunications Aspects of UMTS

2023-03-08T10:42:03Z

Hwang:

{{Header
|Untermenü=UMTS – Universal Mobile Telecommunications System
|Vorherige Seite=UMTS Network Architecture
|Nächste Seite=Further Developments of UMTS
}}

==Improvements regarding speech coding ==
 
In the chapter  [[Examples_of_Communication_Systems/General_Description_of_GSM|"Global System for Mobile Communications"]]  $\rm (GSM)$  of this book,  several speech codecs have already been described in detail.

{{BlaueBox|TEXT=
$\text{Reminder:}$ 
A speech codec is used to reduce the data rate of a digitized speech or music signal.
#In the process,  redundancy and irrelevance are removed from the original signal.
#The artificial word  "codec"  indicates that the same functional unit is used for both,  encoding and decoding.}}

Among others,  the  [[Examples_of_Communication_Systems/Speech_Coding#Adaptive_Multi_Rate_Codec|"Adaptive Multi-Rate Codec"]]  $\rm (AMR)$  based on  [[Examples_of_Communication_Systems/Speech_Coding#Algebraic_Code_Excited_Linear_Prediction|$\rm ACELP$]]  $($"Algebraic Code Excited Linear Prediction"$)$  was introduced, 
*which in the frequency range from  $\text{300 Hz}$  to  $\text{3400 Hz}$ 

*dynamically switches between eight different modes  $($single codecs$)$ 

*of different data rate in the range of  $\text{4. 75 kbit/s}$  to  $\text{12.2 kbit/s}$.

These codecs are also supported in UMTS Release 99 and Release 4.  Compared to the earlier speech codecs  $($Full Rate,  Half Rate, Enhanced Full Rate Vocoder$)$,  they allow
#independence from channel conditions and network load,
#the ability to adapt data rates to conditions,
#improved flexible error protection in the event of more severe radio interference, and
#thereby providing better overall voice quality.

[[File:EN_Bei_T_4_3_S1_v2.png|right|frame|Composition of wideband AMR modes]]

In 2001,  the  "3rd Generation Partnership Project"  $\text{(3gpp)}$  and the  "International Telecommuncation Union"    $\text{(ITU)}$  specified the new voice codec  »'''Wideband AMR'''«  for UMTS Release 5.  This is a further development of AMR and offers

*an extended bandwidth from  $\text{50 Hz}$  to  $\text{7 kHz}$  $($sampling frequency  $\text{16 kHz})$,

*a total of nine modes between  $\text{6.6 kbit/s}$  and  $\text{23.85 kbit/s}$  $($of which only five modes are used$)$,  and

*improved voice quality and better (more natural) sound.

{{BlaueBox|TEXT=
$\text{Some features of wideband AMR}$ 

#Speech data is delivered to the codec as PCM encoded speech with  $16\hspace{0.05cm}000$  samples per second. 
#The speech coding is done in blocks of  $\text{20 ms}$  and the data rate is adjusted every  $\text{20 ms}$.
#The band  $\text{(50 Hz}$  to  $\text{7000 Hz})$  is divided into two sub-bands,  which are encoded differently to allocate more bits to the subjectively important frequencies.
#The upper band  $\text{(6400 Hz}$  to  $\text{7000 Hz})$  is transmitted only in the highest mode $($with  $\text{23.85 kbit/s)}$ .
#In all other modes,  only frequencies  $\text{50 Hz}$  to  $\text{6400 Hz}$  are considered in encoding.
#Wideband AMR supports  "discontinuous transmission"'  $\rm (DTX)$.  This feature means that transmission is paused during voice pauses,  reducing both mobile station power consumption and overall interference at the air interface.  This process is also known as  "Source-Controlled Rate"  $\rm (SCR)$.
#The  "Voice Activity Detection"  $\rm (VAD)$  determines whether speech is in progress or not and inserts a  "silence descriptor frame"  during speech pauses.
#The subscriber is suggested the feeling of a continuous connection by the decoder inserting synthetically generated  "comfort noise"  during speech pauses.}}

==Application of the CDMA method to UMTS==
 
UMTS uses the multiple access method  "Direct Sequence Code Division Multiple Access"  $\rm (DS-CDMA)$,  which has already been discussed in the  [[Modulation_Methods/Direct-Sequence_Spread_Spectrum_Modulation#Block_diagram_and_equivalent_low-pass_model|"PN modulation"]]  chapter  of the book "Modulation Methods".

Here follows a brief summary of this method according to the diagram describing such a system in the equivalent low-pass range and highly simplified:
[[File:EN_Bei_T_4_3_S2c.png|right|frame|CDMA transmission system for two subscribers]]

*The two data signals  $q_1(t)$  and  $q_2(t)$  are supposed to use the same channel without interfering with each other. The bit duration of each is  $T_{\rm B}$.

*Each of the data signals is multiplied by an associated spreading code   $c_1(t)$  resp.  $c_2(t)$.

*The sum signal  $s(t) = q_1(t) · c_1(t) + q_2(t) · c_2(t)$  is formed and transmitted.

*At the receiver,  the same spreading codes  $c_1(t)$  resp.  $c_2(t)$  are added,  thus separating the signals again.

*Assuming orthogonal spreading codes and a small AWGN noise,  the two reconstructed signals at the receiver output are:
:$$v_1(t) = q_1(t) \ \text{and} \ v_2(t) = q_2(t).$$
*For AWGN noise signal  $n(t)$  and orthogonal spreading codes,  this does not change the error probability due to other participants.

{{GraueBox|TEXT=
$\text{Example1:}$ 
The upper graph shows three data bits   $(+1, -1, +1)$   of the rectangular signal  $q_1(t)$  from subscriber '''1''',  each with symbol duration  $T_{\rm B}$.

[[File:EN Mob T 3 4 S4.png|right|frame|Signals at  "Direct–Sequence Spread Spectrum"]]
*Here,  the symbol duration  $T_{\rm C}$  of the spreading code  $c_1(t)$   ⇒   also called  "chip duration"  is smaller by a factor  $4$.

*The multiplication  $s_1(t) = q_1(t) · c_1(t)$  results in a chip sequence of length  $12 · T_{\rm C}$.

One recognizes from this sketch that the signal  $s_1(t)$  is of higher frequency than  $q_1(t)$.

*This is why this modulation method is often also called  "spread spectrum".

*The CDMA receiver reverses this  "spreading".  We refer to this  "receiver-side spreading"  as  "despreading". }}

{{BlaueBox|TEXT=
$\text{Summarizing:}$  
By applying  "Direct Sequence Code Division Multiple Access"  $\rm (DS-CDMA)$  to a data bit sequence  $q(t)$ 
*increases the bandwidth of  $s(t) = q(t) \cdot c(t)$  by the  »'''spreading factor'''«   $J = T_{\rm B}/T_{\rm C}$  –  this is equal to the number of  "chips per bit";

*the chip rate  $R_{\rm C}$  is greater than the bit rate  $R_{\rm B}$ by a factor  $J$;

*the bandwidth of the entire CDMA signal is greater than the bandwidth of each user by a factor  $J$.

That is:    $\text{In UMTS, the entire bandwidth is available to each subscriber for the entire transmission duration}$.

Recall:  In GSM,  both  "Frequency Division Multiple Access"  and  "Time Division Multiple Access"  are used as multiple access methods.
*Here,  each subscriber has only a limited frequency band  $\rm (FDMA)$,  and

*he only has access to the channel within time slots  $\rm (TDMA)$.}}

==Spreading codes and scrambling with UMTS==
 
The spreading codes for UMTS should
*be orthogonal to each other to avoid mutual interference between subscribers,

*allow a flexible realization of different spreading factors  $J$.

⇒   The issue presented here is also illustrated by the German-language SWF applet  [[Applets:OVSF_codes_(Applet)|"OVSF codes"]].

{{GraueBox|TEXT=
$\text{Example 2:}$ 
An example of this is the   »'''orthogonal variable spreading factor'''«   $\rm (OVSF)$,  which provide codes of lengths from  $J = 4$  to  $J = 512$.

[[File:P_ID1535__Bei_T_4_3_S3c_v1.png|right|frame|Chart on the OVSF code family]]

These can be created using a code tree, as shown in the diagram.  Here,  at each branch,  two new codes are created from one code  $C$:
:#  $(+C \ +\hspace{-0.1cm}C)$,  and
:#  $(+C \ -\hspace{-0.1cm}C)$.

Note that no predecessor and successor of a code may be used.
*So in the drawn example,  eight spreading codes with spreading factor  $J = 8$  could be used.

*Also possible are the four codes with yellow background
:*once with  $J = 2$,
:*once with  $J = 4$,  and
:*twice with  $J = 8$.

But the lower four codes with spreading factor  $J = 8$  cannot be used,  because they all start with  "$+1 \ -\hspace{-0.1cm}1$",  which is already occupied by the OVSF code with spreading factor  $J = 2$.}}

[[File:EN_Mob_T_3_4_S5.png|right|frame|Additional scrambling after spreading]]
To obtain more spreading codes and thus be able to supply more participants,  after the band spreading with  $c(t)$ 
*the sequence is chip-wise scrambled again with  $w(t)$, 

*without any further spreading.

The  »'''scrambling code'''« $w(t)$  has same length and rate as the spreading code  $c(t)$.
 
[[File:EN_Mob_T_3_4_S5b_v3.png|right|frame|Typical spreading and scrambling codes for UMTS ]]

⇒   Scrambling causes the codes to lose their complete orthogonality; they are called  "quasi-orthogonal".

*For these codes,  although the  [[Modulation_Methods/Spreading_Sequences_for_CDMA#Properties_of_the_correlation_functions|"cross-correlation function"]]  $\rm (CCF)$  between different spreading codes is non-zero.

*But they are characterized by a pronounced  [[Modulation_Methods/Spreading_Sequences_for_CDMA#Properties_of_the_correlation_functions|"auto-correlation function"]]  $\rm (ACF)$  around zero,  which facilitates detection at the receiver.

*Using quasi-orthogonal codes makes sense because the set of orthogonal codes is limited and scrambling allows also different users to use the same spreading codes.

The table summarizes some data of spreading and scrambling codes.
 
{{GraueBox|TEXT=
[[File:P_ID1537__Bei_T_4_3_S3b_v2.png|right|frame|Generator for creating Gold codes]]
$\text{Example 3:}$ 
In UMTS, so-called  »'''Gold codes'''«  are used for scrambling. The graphic from  [3gpp]<ref>3gpp Group:  UMTS Release 6 - Technical Specification 25.213 V6.4.0,  Sept. 2005.</ref>  shows the block diagram for the circuitry generation of such codes.
*Two different pseudonoise sequences of equal length  $($here:  $N = 18)$  are first generated in parallel using shift registers and added bitwise using  "exclusive-or"  gates''.

*In the uplink,  each mobile station has its own scrambling code and the separation of each channel is done using the same code.

*In contrast,  in the downlink, each coverage area of a  "Node B"  has a common scrambling code.}}

==Channel coding for UMTS==
 
With UMTS,  the EFR- and AMR-encoded voice data pass through a two-stage error protection  $($similar to GSM$)$,  consisting of
[[File:EN_Bei_T_4_3_S4a_v1.png|right|frame|Insertion of CRC bits and tailbits in UMTS ]]
#formation of  "cyclic redundancy check bits"  $\rm (CRC)$,
#subsequent convolutional encoding.

However,  these methods differ from those used for GSM in that they are more flexible,  since for UMTS they have to take different data rates into account.

⇒   For  »'''error detection'''«,  eight, twelve, sixteen or  $24$  CRC bits are formed depending on the size of the transport block  $\text{(10 ms}$  or  $\text{20 ms})$,  and appended to it.

*Eight tail bits are also inserted at the end of each frame for synchronization purposes.

*The diagram shows a transport block of the '''DCH''' channel with  $164$  user data bits,  to which  $16$  CRC bits and eight tail bits are appended.
 

⇒   For  »'''error correction'''«,  UMTS uses two different methods,  depending on the data rate:
*For low data rates,  [[Channel_Coding/Basics_of_Convolutional_Coding|"convolutional codes"]]  with code rates  $R = 1/2$   or   $R = 1/3$  are used as with GSM.  These are generated with eight memory elements of a feedback shift register  $(256$  states$)$.  The coding gain is approximately  $4.5$  to  $6$  dB with code rate  $R = 1/3$  and at low error rates.

*For higher data rates,  one uses  [[Channel_Coding/The_Basics_of_Turbo_Codes|"turbo codes"]]  of rate   $R = 1/3$.  The shift register consists here of three memory cells,  which can assume a total of eight states.  The gain of turbo codes is larger by  $2$  to  $3$  dB than by convolutional codes and depends on the number of iterations.  You need a processor with high processing power for this and there may be relatively large delays.

After channel coding,  the data is fed to an  [[Examples_of_Communication_Systems/Entire_GSM_Transmission_System#Interleaving_for_speech_signals|"interleaver"]]  as in GSM,  in order to be able to resolve bundle errors caused by fading on the receiving side.  Finally, for  "rate matching"  of the resulting data to the physical channel,  individual bits are removed  $($"puncturing"$)$  or repeated  $($"repetition"$)$  according to a predetermined algorithm.

{{GraueBox|TEXT=
[[File:EN_Bei_T_4_3_S4b.png|right|frame|Error correction mechanisms in UMTS]]
$\text{Example 4:}$ 
The graph first shows the increase in bits due to a convolutional or turbo code of rate  $R =1/3$, where  the  $188$  bit time frame  $($after the CRC checksum and tail bits$)$  becomes a  $564$  bit frame.

*Followed by a first  $($external$)$  nesting and then a second  $($internal$)$  nesting.

*After this,  the time frame is divided into four subframes of  $141$  bits each,  and these are then matched to the physical channel by rate matching.}}

==Frequency responses and pulse shaping for UMTS==
 
[[File:EN_Bei_T_4_3_UMTS1_v2.png|right|frame|Block diagram of the optimal Nyquist equalizer at ideal channel|class=fit]]
In this section,  we assume the following block diagram of a binary system with ideal channel   ⇒   $H_{\rm K}(f) = 1$.

In particular,  let hold:

*The  "transmitter pulse filter"  converts the binary  $\{0, \ 1\}$ data into physical signals.  The filter is described by the frequency response  $H_{\rm S}(f)$,  which is identical in shape to the spectrum of a single transmitted pulse.

*In UMTS,  the receive '''KORREKTUR: receiver''' filter  $H_{\rm E}f) = H_{\rm S}(f)$  is matched to the transmitter  $($"matched filter"$)$  and the overall frequency response  $H(f) = H_{\rm S}(f) \cdot H_{\rm E}(f)$  satisfies the  [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_frequency_domain|"first Nyquist criterion"]]:
:$$ H(f) = H_{\rm CRO}(f) = \left\{ \begin{array}{c} 1 \\ 0 \\ \cos^2 \left( \frac {\pi \cdot (|f| - f_1)}{2 \cdot (f_2 - f_1)} \right)\end{array} \right.\quad
\begin{array}{*{1}c} {\rm{for}} \\ {\rm{for}}\\ {\rm else }\hspace{0.05cm}. \end{array}
\begin{array}{*{20}c} |f| \le f_1, \\ |f| \ge f_2.\\ \\\end{array}$$

This means:   Consecutive pulses in time do not interfere with each other   ⇒   no  [[Digital_Signal_Transmission/Causes_and_Effects_of_Intersymbol_Interference|"intersymbol interference"]]  $\rm (ISI)$  occur.  The associated time function is:

:$$h(t) = h_{\rm CRO}(t) ={\rm sinc}(t/ T_{\rm C}) \cdot \frac{\cos(r \cdot \pi t/T_{\rm C})}{1- (2r \cdot t/T_{\rm C})^2}\hspace{0.4cm} \text{with } \hspace{0.4cm} r = \frac{f_2 - f_1}{f_2 + f_1}. $$

*"CRO" here stands for  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Raised-cosine_low-pass_filter|"raised cosine low-pass"]].

*The sum  $f_1 + f_2$  is equal to the inverse of the chip duration  $T_{\rm C} = 260 \ \rm ns$,  so it is equal to  $3.84 \ \rm MHz$.

*The  "rolloff factor"  has been determined to  $r = 0.22$  for UMTS.  The two  "corner frequencies"  are thus

:$$f_1 = {1}/(2 T_{\rm C}) \cdot (1-r) \approx 1.5\,{\rm MHz},$$
:$$f_2 ={1}/(2 T_{\rm C}) \cdot (1+r) \approx 2.35\,{\rm MHz}.$$

*The required bandwidth is  $B = 2 \cdot f_2 = 4.7 \ \rm MHz$.  Thus,  there is sufficient bandwidth available for each UMTS channel with  $5 \ \rm MHz$.

{{BlaueBox|TEXT=
$\text{Conclusion:}$  The graph shows.
[[File:P_ID1547__Bei_T_4_3_S5b_v1.png|right|frame|Raised cosine spectrum and impulse response]]
*on the left,  the  $($normalized$)$  Nyquist spectrum  $H(f)$,

*on the right,  the corresponding Nyquist pulse  $h(t)$,  whose zero crossings are equidistant with distance  $T_{\rm C}$.

$\text{It should be noted:}$
# The transmission filter  $H_{\rm S}(f)$  and the matched filter  $H_{\rm E}(f)$  are each  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems|"root raised cosine"]].
#Only the product  $H(f) = H_{\rm S}(f) \cdot H_{\rm E}(f)$  leads to the raised cosine.  This also means:
# The impulse responses  $h_{\rm S}(t)$  and  $h_{\rm E}(t)$  by themselves do not satisfy the first Nyquist condition.
#Only the combination of the two  $($in the time domain the convolution$)$  leads to the desired equidistant zeros.}}

==Modulation methods for UMTS==
 
The modulation techniques used in UMTS can be summarized as follows:
#In the downlink:  "Quaternary Phase Shift Keying"  is used for modulation  both in  "frequency division duplex"  $\rm (FDD)$  and in  "time division duplex""  $\rm (TDD)$.
#Here,  user data  $($DPDCH channel$)$  and control data  $($DPCCH channel$)$  are multiplexed in time.
#With TDD,  the signal is modulated in the uplink also by means of QPSK,  but not with  FDD. 
#Here,  a  "dual channel binary phase shift keying"  is used   ⇒   different channels are transmitted in  "in-phase"  and  "quadrature components".
#Thus,  two chips are transmitted per modulation step.  The gross chip rate is therefore twice the modulation rate of  $3.84$ Mchip per second.

{{GraueBox|TEXT=
$\text{Example 5:}$ 
The graph shows in the equivalent low-pass domain this  "I/Q multiplexing method",  as it is also called:
[[File:EN_Mob_T_3_4_S6.png|right|frame|Modulation and pulse shaping for UMTS|class=fit]]

#The spread useful data of the DPDCH channel is modulated onto the inphase component.
#The spread control data of the DPCCH channel is modulated onto the quadrature component.
#After modulation,  the quadrature component is weighted by the root of the power ratio  $G$  between the two channels to minimize the influence of power differences between  $I$  and  $Q$.
#Finally,  the complex sum signal  $(I +{\rm j} \cdot Q)$  is multiplied by a scrambling code that is also complex.}}

{{BlaueBox|TEXT=
$\text{Conclusion:}$  An advantage of dual channel BPSK modulation is the  ''' possibility of usinglow-power amplifiers'''.
*But time division multiplexing of user and control data as in the uplink  '''is not possible in the downlink'''.

*One reason for this is the use of  "Discontinuous Transmission"  $\rm (DTX)$  and the associated time constraints.}}

==Single-user receiver==
 
The task of a CDMA receiver is to separate and reconstruct the transmitted data of the individual subscribers from the sum of the spread data streams.  A distinction is made between  "single-user receivers"  and  "multi-user receivers".

In the UMTS downlink,  it is always used a  »'''single-user receiver'''«,  since in the mobile station a joint detection of all subscribers would be too costly
*due to the large number of active subscribers

*as well as the length of the scrambling codes and the asynchronous operation.

Such a receiver consists of a bank of independent correlators.
*Each one of the total  $J$  correlators belongs to a specific spreading sequence.

*The correlation is usually formed in a so-called  "correlator database"  by software.

Thereby one receives at the correlator output the sum of
[[File:EN_Bei_T_4_3_S6a_v2.png|right|frame|Single-user receiver with matched filter]]
*the  "autocorrelation '''KORREKTUR: auto-correlation''' function"  of the spreading code and

*the  "cross-correlation function"  of all other users with their own spreading code.

The graphic shows the simplest realization of such a receiver with matched filter.

#The received signal  $r(t)$  is first multiplied by the spreading code  $c(t)$  of the considered subscriber,  which is called  "despreading"  $($yellow background$)$.
#Followed by convolution with the matched filter impulse response  $($"Root Raised Cosine"$)$  to maximize SNR,  and sampling in bit clock  $(T_{\rm B})$.
#Finally, the threshold decision is made, which provides the sink signal  $v(t)$ and  thus the data bits of the considered subscriber.

{{BlaueBox|TEXT=
$\text{Please note:}$ 

*For the AWGN channel,  spreading at the transmitter and the matched despreading at the receiver have no effect on the bit error probability because of  $c(t)^2 = 1$.  As shown in  [[Aufgaben:Exercise_4.5:_Pseudo_Noise_Modulation|$\text{Exercise 4.5}$]],  even with spreading/despreading at the optimal receiver, regardless of spreading factor  $J$:

:$$p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt { {2 \cdot E_{\rm B} }/{N_{\rm 0} } } \hspace{0.05cm} \right )\hspace{0.05cm}. $$

*This result can be justified as follows:  The statistical properties of white noise  $n(t)$  are not changed by multiplication with the  $±1$  signal  $c(t)$.}}

==Rake receiver==
 
Another receiver for single-user detection is the  »'''rake receiver'''«,  which leads to significant improvements for a multipath channel.
[[File:EN_Bei_T_4_3_S6b_v2.png|right|frame|Structure of the rake receiver  $($shown in the equivalent low-pass domain$)$]]

The diagram shows its setup for a two-way channel with 
*a direct path with coefficient  $h_0$  and delay time  $τ_0$,

*an echo with coefficient  $h_1$  and delay time  $τ_1$.

For simplicity,  the coefficients  $h_0$  and  $h_1$  are assumed to be real.  Due to the representation in the equivalent low-pass domain,  these could also be complex.

⇒   The task of the rake receiver is to concentrate the signal energies of all paths  $($in this example only two$)$  to a single instant.  It works accordingly like a  "rake"  for the garden.
 
If one applies a Dirac delta impulse at time  $t = 0$  to the channel input,  there will be three Dirac delta impulses at the output of the rake receiver:
:$$ s(t) = \delta(t) \hspace{0.3cm}\Rightarrow\hspace{0.3cm}
y(t) = h_0 \cdot h_1 \cdot \delta(t - 2\tau_0) + (h_0^2 + h_1^2) \cdot \delta(t - \tau_0 - \tau_1)+
h_0 \cdot h_1 \cdot \delta(t - 2\tau_1) .$$

*The signal energy is concentrated at the time  $τ_0 + τ_1$.  Of the total four paths,  two contribute  $($middle term$)$.

*The Dirac delta functions at  $2τ_0$  and  $2τ_1$  do cause momentum interference.  However, their  weights are much smaller than those of the main path.

{{GraueBox|TEXT=
$\text{Example 6:}$ 
With channel parameters  $h_0 = 0.8$  and  $h_1 = 0.6$  the main path  $($with weight  $h_0)$  contains  $0.82/(0.82 + 0.62) = 64\%$  of the total signal energy.
*With rake receiver and the same weights,  the above equation is:

:$$ y(t) = 0.48 \cdot \delta(t - 2\tau_0) + 1.0 \cdot \delta(t - \tau_0 - \tau_1)+
0.48 \cdot \delta(t - 2\tau_1) .$$

*The share of the main path in the total energy amounts in this simple example to  ${1^2}/{(1^2 + 0.48^2 + 0.48^2)} ≈ 68\%.$}}

Rake receivers are preferred for implementation in mobile devices,  but have a limited performance when there are many active participants.
#In a multipath channel with many  $(M)$  paths,  the Rake has also  $M$  fingers.
#The main finger  – also called  "searcher"  – is responsible for identifying and ranking the individual paths of multiple propagation.
#It searches for the strongest paths and assigns them to other fingers along with their control information.
#In the process,  the time and frequency synchronization of all fingers is continuously compared with the control data of the received signal.

==Multi-user receiver ==
 
In a single-user receiver,  only the data signal of one subscriber is decided,  while all other subscriber signals are considered as additional noise.  However,  the bit error rate of such a detector will be very large
*if there is large  "intracell interference"  $($many active subscribers in the considered radio cell$)$

*or large  "intercell interference"  $($highly interfering subscribers in neighboring cells$)$.

In contrast,  »'''multi-user receivers'''«  make a joint decision for all active subscribers.» Their characteristics can be summarized as follows:
#Such a multi-user receiver does not consider the interference from other participants as noise,  but also uses the information contained in the interference signals for detection.
#The receiver is expensive to implement and the algorithms are extremely computationally intensive.  It contains an extremely large correlator database followed by a common detector.
#The multi-user receiver must know the spreading codes of all active users.  This requirement precludes use in the UMTS downlink  $($i.e.,  at the mobile station$)$.  In contrast,  all subscriber-specific spreading codes are known a-priori to the base stations,  so that multi-user detection is only used in the uplink.
#Some detection algorithms additionally require knowledge of other signal parameters such as energies and delay times.  The common detector  – the heart of the receiver –  is responsible for applying the appropriate detection algorithm in each case.
#Examples of multi-user detection are  "decorrelating detection"  and  "Interference Cancellation".

==Near–far problem==
 
The  "near-far problem"  is exclusively an uplink problem,  i.e.,  the transmission from mobile subscribers to a base station.  We consider a scenario with two users at different distances from the base station according to the following graph.  This can be interpreted as follows:

[[File:EN_Mob_T_3_2_S3.png|right|frame|Scenarios for the near-far problem|class=fit]]

#If both mobile stations transmit with the same power,  the received power of the red user  $\rm A$  at the base station is significantly smaller than that of the blue user  $\rm B$  $($left scenario$)$ due to path loss.
#In large macrocells,  the difference can be as much as  $100$  dB.  As a result,  the red signal is largely obscured by the blue.
#You can largely avoid the near-far problem if user  $\rm A$  transmits with higher power than user  $\rm B$,  as indicated in the right scenario.
#Then,  at the base station,  the received power of both mobile stations is then  $($almost$)$  equal.

Note:   In an idealized system  $($one-way channel,  ideal A/D converters,  fully linear amplifiers$)$  the transmitted data of the users are orthogonal to each other and one could detect the users individually even with very different received powers.  This statement is true
*for UMTS  $($multiple access:  CDMA$)$  as well as

*for the 2G system GSM  $($FDMA/TDMA$)$,  and

*for the 4G system LTE  $($TDMA/OFDMA$)$.

In reality,  however,  orthogonality is not always given due to the following reasons:
#Different receive paths   ⇒   multipath channel,
#non-ideal characteristics of the spreading and scrambling codes in CDMA,
#asynchrony of users in the time domain  $($basic propagation delay of paths$)$,
#asynchrony of users in the frequency domain  $($non-ideal oscillators and Doppler shift due to mobility of users$)$.

Consequently,  the users are no longer orthogonal to each other and the signal-to-noise ratio of the user to be detected with respect to the other users is not arbitrarily high:
*For  [[Examples_of_Communication_Systems/General_Description_of_GSM|"GSM"]]  and  [[Mobile_Communications/General_Information_on_the_LTE_Mobile_Communications_Standard|"LTE"]]  one can assume signal-to-noise ratios of  $25 $  dB  and more,

*but for UMTS  $($CDMA$)$  only approx.  $15$  dB,  with high-rate data transmission rather less.

==Carrier-to-interference power ratio==

The term  "capacity"  is generally understood to mean the number of available transmission channels per cell.  However,  since the number of subscribers is not strictly limited in UMTS unlike in GSM,  no fixed capacity can be specified here.
*In perfect codes,  the subscribers do not interfere with each other.  Thus,  the maximum number of users is determined solely by the spreading factor  $J$  and the available number of mutually orthogonal codes,  which,  however,  is also limited.

*More practical are non-perfect,  only quasi-orthogonal codes.  Here,  the  "capacity"  of a radio cell is predominantly determined by the resulting interference or the  "carrier-to-interference power ratio"  $\rm (CIR)$.

[[File:EN_Bei_T_4_3_S8_v2.png|right|frame|Carrier-to-interference ratio depending on the number of subscribers]]

As can be seen from this graph,  CIR depends directly on the number of active participants.  The more active subscribers there are,  the more interference power is generated and the smaller the CIR becomes.

Furthermore,  this decisive criterion for UMTS also depends on the following variables:
#The topology and user behavior  $($number of services called up$)$,
#the spreading factor  $J$  and the orthogonality of the used spreading code.

In order to limit the disturbing influence of the interference power on the transmission quality,  there are two possible criteria:

»'''Cell breathing'''«:   If the number of active subscribers increases significantly with UMTS,  the cell radius is reduced and  $($because of the now fewer subscribers in the cell$)$  also the current interference power is lower.  A less loaded neighboring cell then steps in to supply the subscribers at the edge of the reduced cell.

»'''Power control'''«:   If the total interference power within a radio cell exceeds a specified limit,  the transmission power of all subscribers is reduced accordingly and/or the data rate is reduced,  resulting in poorer transmission quality for all.  More about this in the next section.

==Power and power control in UMTS==
 
The ratio between the signal power and the interference power is used as the controlled variable for power control in UMTS.  There are differences between the  "frequency division duplex"  $\rm (FDD)$  and the  "time division duplex"  $\rm (TDD)$  modes.

[[File:EN_Bei_T_4_3_S10a.png|right|frame|Power control in the FDD mode '''KORREKTUR: transmit power''']]
We take a closer look at the FDD power control.  In the diagram you can see two different control loops:
*The  »'''inner control loop'''«  controls the transmitter power based on time slots,  where one power command is transmitted in each time slot. 

::The power of the transmitter is determined and changed using the CIR estimates in the receiver and the specifications of the  "radio network controller"  $\rm (RNC)$  from the outer control loop.

*The  '''outer loop'''  controls based on $10$ millisecond duration frames. It is implemented in the RNC and is responsible for determining the set point for the inner loop.

The FDD power control sequence is as follows:
#The RNC provides a carrier-to-interference ratio  $\rm (CIR)$  setpoint.
#The receiver estimates the actual CIR value and generates control commands for the transmitter.
#The transmitter changes the transmitted power according to these control commands.

The principle of  "power control in TDD mode"  is similar to the control presented here for the FDD mode.  In fact in the downlink direction they are practically identical.

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
The  »'''TDD power control'''«  is much slower and thus less precise than in  FDD.

*However,  fast power control is not even possible in this case,  since each participant has only a fraction of the time frame available for him.}}

==Link budget ==
 
When planning UMTS networks,  calculating the link budget is an important step.  Knowledge of the link budget is required both for dimensioning the coverage areas and for determining the capacity and quality of service requirements.

{{BlaueBox|TEXT=
The  »'''objective of the link budget'''«  is to calculate the  '''maximum cell size'''  considering the following criteria:
#Type and data rate of the services,
#topology of the environment,
#system configuration  $($location and power of base stations,  handover gain$)$,
#service requirements  $($availability$)$,
#type of mobile station  $($speed,  power$)$,
#financial and economic aspects}}

{{GraueBox|TEXT=
[[File:EN_Bei_T_4_3_S9_v2.png|right|frame|Budget for a voice transmission channel '''KORREKTUR: fault''']]
$\text{Example 7:}$ 
The calculation of the link budget is illustrated using the example of a voice transmission channel in the UMTS downlink.  Regarding the exemplary numerical values,  it should be noted:
#The transmitted power is  $P_{\rm S} =19$  dBm,  which corresponds to approx.  $79$  mW.  Here,  the antenna loss is considered to be  $2$  dB.
#The noise power  $P_{\rm R} = 5 \cdot 10^{-11}$  mW   ⇒   $P_{\rm R} = -103$  dBm   product of UMTS bandwidth and noise power density.
#The interference power is  $P_{\rm I} = -99$  dBm  corresponding to  $1.25 \cdot 10^{-10}$  mW.
#This gives the total interference power  $P_{\rm R+I} = P_{\rm R} + P_{\rm I} = 1.25 \cdot 10^{-10}$  mW   ⇒   $P_{\rm R+I} =- 97.5$  dBm.
#The antenna sensitivity results in $-97.5 - 27 + 5 - 17 + 3.5 = - 133$  dBm.  A large negative value here is  "good".
#Maximum allowable path loss should be as large as possible.  Here  $19 - (-133) = 152$  dB.
#The  '''link budget'''  includes the margin for fading and the handover gain,  in the example:  $140$  dB.
#The  '''maximum cell radius'''  can be determined from the link budget using an [https://en.wikipedia.org/wiki/Path_loss "empirical formula"] of Okumura-Hata.  It holds:  $ {r}\ [{\rm km}] = 10^{({\rm LinkBudget}- 137)/35}= 10^{0.0857}\approx 1.22 . $
 

Notes:  
*"$\rm dB$"  denotes a logarithmic power specification,  referenced to  $1 \rm W$.

*In contrast  "$\rm dBm$"  refers to the power  $1 \rm mW$.}}

==UMTS radio resource management ==
 
The central task of  »'''radio resource management'''«  $\rm (RRM)$  is the dynamic adaptation of radio transmission parameters to the current situation  $($fading,  mobile station movement,  load,  etc.$)$  with the aim to
[[File:EN_Bei_T_4_3_S11.png|right|frame|Radio Resource Management in UMTS]]
#increase the transmission and subscriber capacity,
#improve the individual transmission quality,  and
#use existing radio resources economically.

The main RRM mechanisms summarized in the diagram are explained below.

'''Transmit power control''' The  radio resource management  attempts to keep the received power and thus the carrier-to-interference ratio  $\rm (CIR)$  at the receiver constant,  or at least to prevent it from falling below a specified limit.

An example of the need for power control is the  [[Examples_of_Communication_Systems/Telecommunications_Aspects_of_UMTS#Near.E2.80.93Far.E2.80.93Effect|"near-far problem"]],  which is known to cause a disconnect.

The step size of the power control is  $1 \ \rm dB$  or  $2 \ \rm dB$,  and the frequency of the control commands is  $1500$  commands per second.

'''Regulation of data rate'''
 UMTS allows an exchange between data rate and transmission quality,  which can be realized by selecting the spreading factor.  Doubling the spreading factor corresponds to halving the data rate and increases the quality by  $3\ \rm dB$  $($spreading gain$)$.

'''Access control'''
 To avoid overload situations in the overall network,  a check is made before a new connection is established to see whether the necessary resources are available.  If not,  the new connection is rejected.  This check is realized by estimating the transmission power distribution after the new connection is established.

'''Load control'''
 This becomes active if an overload occurs despite access control.  In this case,  a handover to another  base station is initiated and  – if this is not possible –  the data rates of certain nodes are lowered.

'''Handover'''
 Finally,  radio resource management is also responsible for handover to ensure uninterrupted connections.  Mobile stations are assigned to the individual radio cells on the basis of CIR measurements.

==Exercises for the chapter ==
 
[[Aufgaben:Exercise_4.5:_Pseudo_Noise_Modulation|Exercise 4.5: Pseudo Noise Modulation]]

[[Aufgaben:Exercise_4.5Z:_About_Band_Spreading_with_UMTS|Exercise 4.5Z: About Spread Spectrum with UMTS]]

[[Aufgaben:Exercise_4.6:_OVSF_Codes|Exercise 4.6: OVSF Codes]]

[[Aufgaben:Exercise_4.7:_To_the_Rake_Receiver|Exercise 4.7: About the Rake Receiver]]

==Sources==
<references />

{{Display}}

Examples of Communication Systems/Telecommunications Aspects of UMTS

2023-03-08T10:41:37Z

Hwang:

{{Header
|Untermenü=UMTS – Universal Mobile Telecommunications System
|Vorherige Seite=UMTS Network Architecture
|Nächste Seite=Further Developments of UMTS
}}

==Improvements regarding speech coding ==
 
In the chapter  [[Examples_of_Communication_Systems/General_Description_of_GSM|"Global System for Mobile Communications"]]  $\rm (GSM)$  of this book,  several speech codecs have already been described in detail.

{{BlaueBox|TEXT=
$\text{Reminder:}$ 
A speech codec is used to reduce the data rate of a digitized speech or music signal.
#In the process,  redundancy and irrelevance are removed from the original signal.
#The artificial word  "codec"  indicates that the same functional unit is used for both,  encoding and decoding.}}

Among others,  the  [[Examples_of_Communication_Systems/Speech_Coding#Adaptive_Multi_Rate_Codec|"Adaptive Multi-Rate Codec"]]  $\rm (AMR)$  based on  [[Examples_of_Communication_Systems/Speech_Coding#Algebraic_Code_Excited_Linear_Prediction|$\rm ACELP$]]  $($"Algebraic Code Excited Linear Prediction"$)$  was introduced, 
*which in the frequency range from  $\text{300 Hz}$  to  $\text{3400 Hz}$ 

*dynamically switches between eight different modes  $($single codecs$)$ 

*of different data rate in the range of  $\text{4. 75 kbit/s}$  to  $\text{12.2 kbit/s}$.

These codecs are also supported in UMTS Release 99 and Release 4.  Compared to the earlier speech codecs  $($Full Rate,  Half Rate, Enhanced Full Rate Vocoder$)$,  they allow
#independence from channel conditions and network load,
#the ability to adapt data rates to conditions,
#improved flexible error protection in the event of more severe radio interference, and
#thereby providing better overall voice quality.

[[File:EN_Bei_T_4_3_S1_v2.png|right|frame|Composition of wideband AMR modes]]

In 2001,  the  "3rd Generation Partnership Project"  $\text{(3gpp)}$  and the  "International Telecommuncation Union"    $\text{(ITU)}$  specified the new voice codec  »'''Wideband AMR'''«  for UMTS Release 5.  This is a further development of AMR and offers

*an extended bandwidth from  $\text{50 Hz}$  to  $\text{7 kHz}$  $($sampling frequency  $\text{16 kHz})$,

*a total of nine modes between  $\text{6.6 kbit/s}$  and  $\text{23.85 kbit/s}$  $($of which only five modes are used$)$,  and

*improved voice quality and better (more natural) sound.

{{BlaueBox|TEXT=
$\text{Some features of wideband AMR}$ 

#Speech data is delivered to the codec as PCM encoded speech with  $16\hspace{0.05cm}000$  samples per second. 
#The speech coding is done in blocks of  $\text{20 ms}$  and the data rate is adjusted every  $\text{20 ms}$.
#The band  $\text{(50 Hz}$  to  $\text{7000 Hz})$  is divided into two sub-bands,  which are encoded differently to allocate more bits to the subjectively important frequencies.
#The upper band  $\text{(6400 Hz}$  to  $\text{7000 Hz})$  is transmitted only in the highest mode $($with  $\text{23.85 kbit/s)}$ .
#In all other modes,  only frequencies  $\text{50 Hz}$  to  $\text{6400 Hz}$  are considered in encoding.
#Wideband AMR supports  "discontinuous transmission"'  $\rm (DTX)$.  This feature means that transmission is paused during voice pauses,  reducing both mobile station power consumption and overall interference at the air interface.  This process is also known as  "Source-Controlled Rate"  $\rm (SCR)$.
#The  "Voice Activity Detection"  $\rm (VAD)$  determines whether speech is in progress or not and inserts a  "silence descriptor frame"  during speech pauses.
#The subscriber is suggested the feeling of a continuous connection by the decoder inserting synthetically generated  "comfort noise"  during speech pauses.}}

==Application of the CDMA method to UMTS==
 
UMTS uses the multiple access method  "Direct Sequence Code Division Multiple Access"  $\rm (DS-CDMA)$,  which has already been discussed in the  [[Modulation_Methods/Direct-Sequence_Spread_Spectrum_Modulation#Block_diagram_and_equivalent_low-pass_model|"PN modulation"]]  chapter  of the book "Modulation Methods".

Here follows a brief summary of this method according to the diagram describing such a system in the equivalent low-pass range and highly simplified:
[[File:EN_Bei_T_4_3_S2c.png|right|frame|CDMA transmission system for two subscribers]]

*The two data signals  $q_1(t)$  and  $q_2(t)$  are supposed to use the same channel without interfering with each other. The bit duration of each is  $T_{\rm B}$.

*Each of the data signals is multiplied by an associated spreading code   $c_1(t)$  resp.  $c_2(t)$.

*The sum signal  $s(t) = q_1(t) · c_1(t) + q_2(t) · c_2(t)$  is formed and transmitted.

*At the receiver,  the same spreading codes  $c_1(t)$  resp.  $c_2(t)$  are added,  thus separating the signals again.

*Assuming orthogonal spreading codes and a small AWGN noise,  the two reconstructed signals at the receiver output are:
:$$v_1(t) = q_1(t) \ \text{and} \ v_2(t) = q_2(t).$$
*For AWGN noise signal  $n(t)$  and orthogonal spreading codes,  this does not change the error probability due to other participants.

{{GraueBox|TEXT=
$\text{Example1:}$ 
The upper graph shows three data bits   $(+1, -1, +1)$   of the rectangular signal  $q_1(t)$  from subscriber '''1''',  each with symbol duration  $T_{\rm B}$.

[[File:EN Mob T 3 4 S4.png|right|frame|Signals at  "Direct–Sequence Spread Spectrum"]]
*Here,  the symbol duration  $T_{\rm C}$  of the spreading code  $c_1(t)$   ⇒   also called  "chip duration"  is smaller by a factor  $4$.

*The multiplication  $s_1(t) = q_1(t) · c_1(t)$  results in a chip sequence of length  $12 · T_{\rm C}$.

One recognizes from this sketch that the signal  $s_1(t)$  is of higher frequency than  $q_1(t)$.

*This is why this modulation method is often also called  "spread spectrum".

*The CDMA receiver reverses this  "spreading".  We refer to this  "receiver-side spreading"  as  "despreading". }}

{{BlaueBox|TEXT=
$\text{Summarizing:}$  
By applying  "Direct Sequence Code Division Multiple Access"  $\rm (DS-CDMA)$  to a data bit sequence  $q(t)$ 
*increases the bandwidth of  $s(t) = q(t) \cdot c(t)$  by the  »'''spreading factor'''«   $J = T_{\rm B}/T_{\rm C}$  –  this is equal to the number of  "chips per bit";

*the chip rate  $R_{\rm C}$  is greater than the bit rate  $R_{\rm B}$ by a factor  $J$;

*the bandwidth of the entire CDMA signal is greater than the bandwidth of each user by a factor  $J$.

That is:    $\text{In UMTS, the entire bandwidth is available to each subscriber for the entire transmission duration}$.

Recall:  In GSM,  both  "Frequency Division Multiple Access"  and  "Time Division Multiple Access"  are used as multiple access methods.
*Here,  each subscriber has only a limited frequency band  $\rm (FDMA)$,  and

*he only has access to the channel within time slots  $\rm (TDMA)$.}}

==Spreading codes and scrambling with UMTS==
 
The spreading codes for UMTS should
*be orthogonal to each other to avoid mutual interference between subscribers,

*allow a flexible realization of different spreading factors  $J$.

⇒   The issue presented here is also illustrated by the German-language SWF applet  [[Applets:OVSF_codes_(Applet)|"OVSF codes"]].

{{GraueBox|TEXT=
$\text{Example 2:}$ 
An example of this is the   »'''orthogonal variable spreading factor'''«   $\rm (OVSF)$,  which provide codes of lengths from  $J = 4$  to  $J = 512$.

[[File:P_ID1535__Bei_T_4_3_S3c_v1.png|right|frame|Chart on the OVSF code family]]

These can be created using a code tree, as shown in the diagram.  Here,  at each branch,  two new codes are created from one code  $C$:
:#  $(+C \ +\hspace{-0.1cm}C)$,  and
:#  $(+C \ -\hspace{-0.1cm}C)$.

Note that no predecessor and successor of a code may be used.
*So in the drawn example,  eight spreading codes with spreading factor  $J = 8$  could be used.

*Also possible are the four codes with yellow background
:*once with  $J = 2$,
:*once with  $J = 4$,  and
:*twice with  $J = 8$.

But the lower four codes with spreading factor  $J = 8$  cannot be used,  because they all start with  "$+1 \ -\hspace{-0.1cm}1$",  which is already occupied by the OVSF code with spreading factor  $J = 2$.}}

[[File:EN_Mob_T_3_4_S5.png|right|frame|Additional scrambling after spreading]]
To obtain more spreading codes and thus be able to supply more participants,  after the band spreading with  $c(t)$ 
*the sequence is chip-wise scrambled again with  $w(t)$, 

*without any further spreading.

The  »'''scrambling code'''« $w(t)$  has same length and rate as the spreading code  $c(t)$.
 
[[File:EN_Mob_T_3_4_S5b_v3.png|right|frame|Typical spreading and scrambling codes for UMTS ]]

⇒   Scrambling causes the codes to lose their complete orthogonality; they are called  "quasi-orthogonal".

*For these codes,  although the  [[Modulation_Methods/Spreading_Sequences_for_CDMA#Properties_of_the_correlation_functions|"cross-correlation function"]]  $\rm (CCF)$  between different spreading codes is non-zero.

*But they are characterized by a pronounced  [[Modulation_Methods/Spreading_Sequences_for_CDMA#Properties_of_the_correlation_functions|"auto-correlation function"]]  $\rm (ACF)$  around zero,  which facilitates detection at the receiver.

*Using quasi-orthogonal codes makes sense because the set of orthogonal codes is limited and scrambling allows also different users to use the same spreading codes.

The table summarizes some data of spreading and scrambling codes.
 
{{GraueBox|TEXT=
[[File:P_ID1537__Bei_T_4_3_S3b_v2.png|right|frame|Generator for creating Gold codes]]
$\text{Example 3:}$ 
In UMTS, so-called  »'''Gold codes'''«  are used for scrambling. The graphic from  [3gpp]<ref>3gpp Group:  UMTS Release 6 - Technical Specification 25.213 V6.4.0,  Sept. 2005.</ref>  shows the block diagram for the circuitry generation of such codes.
*Two different pseudonoise sequences of equal length  $($here:  $N = 18)$  are first generated in parallel using shift registers and added bitwise using  "exclusive-or"  gates''.

*In the uplink,  each mobile station has its own scrambling code and the separation of each channel is done using the same code.

*In contrast,  in the downlink, each coverage area of a  "Node B"  has a common scrambling code.}}

==Channel coding for UMTS==
 
With UMTS,  the EFR- and AMR-encoded voice data pass through a two-stage error protection  $($similar to GSM$)$,  consisting of
[[File:EN_Bei_T_4_3_S4a_v1.png|right|frame|Insertion of CRC bits and tailbits in UMTS ]]
#formation of  "cyclic redundancy check bits"  $\rm (CRC)$,
#subsequent convolutional encoding.

However,  these methods differ from those used for GSM in that they are more flexible,  since for UMTS they have to take different data rates into account.

⇒   For  »'''error detection'''«,  eight, twelve, sixteen or  $24$  CRC bits are formed depending on the size of the transport block  $\text{(10 ms}$  or  $\text{20 ms})$,  and appended to it.

*Eight tail bits are also inserted at the end of each frame for synchronization purposes.

*The diagram shows a transport block of the '''DCH''' channel with  $164$  user data bits,  to which  $16$  CRC bits and eight tail bits are appended.
 

⇒   For  »'''error correction'''«,  UMTS uses two different methods,  depending on the data rate:
*For low data rates,  [[Channel_Coding/Basics_of_Convolutional_Coding|"convolutional codes"]]  with code rates  $R = 1/2$   or   $R = 1/3$  are used as with GSM.  These are generated with eight memory elements of a feedback shift register  $(256$  states$)$.  The coding gain is approximately  $4.5$  to  $6$  dB with code rate  $R = 1/3$  and at low error rates.

*For higher data rates,  one uses  [[Channel_Coding/The_Basics_of_Turbo_Codes|"turbo codes"]]  of rate   $R = 1/3$.  The shift register consists here of three memory cells,  which can assume a total of eight states.  The gain of turbo codes is larger by  $2$  to  $3$  dB than by convolutional codes and depends on the number of iterations.  You need a processor with high processing power for this and there may be relatively large delays.

After channel coding,  the data is fed to an  [[Examples_of_Communication_Systems/Entire_GSM_Transmission_System#Interleaving_for_speech_signals|"interleaver"]]  as in GSM,  in order to be able to resolve bundle errors caused by fading on the receiving side.  Finally, for  "rate matching"  of the resulting data to the physical channel,  individual bits are removed  $($"puncturing"$)$  or repeated  $($"repetition"$)$  according to a predetermined algorithm.

{{GraueBox|TEXT=
[[File:EN_Bei_T_4_3_S4b.png|right|frame|Error correction mechanisms in UMTS]]
$\text{Example 4:}$ 
The graph first shows the increase in bits due to a convolutional or turbo code of rate  $R =1/3$, where  the  $188$  bit time frame  $($after the CRC checksum and tail bits$)$  becomes a  $564$  bit frame.

*Followed by a first  $($external$)$  nesting and then a second  $($internal$)$  nesting.

*After this,  the time frame is divided into four subframes of  $141$  bits each,  and these are then matched to the physical channel by rate matching.}}

==Frequency responses and pulse shaping for UMTS==
 
[[File:EN_Bei_T_4_3_UMTS1_v2.png|right|frame|Block diagram of the optimal Nyquist equalizer at ideal channel|class=fit]]
In this section,  we assume the following block diagram of a binary system with ideal channel   ⇒   $H_{\rm K}(f) = 1$.

In particular,  let hold:

*The  "transmitter pulse filter"  converts the binary  $\{0, \ 1\}$ data into physical signals.  The filter is described by the frequency response  $H_{\rm S}(f)$,  which is identical in shape to the spectrum of a single transmitted pulse.

*In UMTS,  the receive '''KORREKTUR: receiver''' filter  $H_{\rm E}f) = H_{\rm S}(f)$  is matched to the transmitter  $($"matched filter"$)$  and the overall frequency response  $H(f) = H_{\rm S}(f) \cdot H_{\rm E}(f)$  satisfies the  [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_frequency_domain|"first Nyquist criterion"]]:
:$$ H(f) = H_{\rm CRO}(f) = \left\{ \begin{array}{c} 1 \\ 0 \\ \cos^2 \left( \frac {\pi \cdot (|f| - f_1)}{2 \cdot (f_2 - f_1)} \right)\end{array} \right.\quad
\begin{array}{*{1}c} {\rm{for}} \\ {\rm{for}}\\ {\rm else }\hspace{0.05cm}. \end{array}
\begin{array}{*{20}c} |f| \le f_1, \\ |f| \ge f_2.\\ \\\end{array}$$

This means:   Consecutive pulses in time do not interfere with each other   ⇒   no  [[Digital_Signal_Transmission/Causes_and_Effects_of_Intersymbol_Interference|"intersymbol interference"]]  $\rm (ISI)$  occur.  The associated time function is:

:$$h(t) = h_{\rm CRO}(t) ={\rm sinc}(t/ T_{\rm C}) \cdot \frac{\cos(r \cdot \pi t/T_{\rm C})}{1- (2r \cdot t/T_{\rm C})^2}\hspace{0.4cm} \text{with } \hspace{0.4cm} r = \frac{f_2 - f_1}{f_2 + f_1}. $$

*"CRO" here stands for  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Raised-cosine_low-pass_filter|"raised cosine low-pass"]].

*The sum  $f_1 + f_2$  is equal to the inverse of the chip duration  $T_{\rm C} = 260 \ \rm ns$,  so it is equal to  $3.84 \ \rm MHz$.

*The  "rolloff factor"  has been determined to  $r = 0.22$  for UMTS.  The two  "corner frequencies"  are thus

:$$f_1 = {1}/(2 T_{\rm C}) \cdot (1-r) \approx 1.5\,{\rm MHz},$$
:$$f_2 ={1}/(2 T_{\rm C}) \cdot (1+r) \approx 2.35\,{\rm MHz}.$$

*The required bandwidth is  $B = 2 \cdot f_2 = 4.7 \ \rm MHz$.  Thus,  there is sufficient bandwidth available for each UMTS channel with  $5 \ \rm MHz$.

{{BlaueBox|TEXT=
$\text{Conclusion:}$  The graph shows.
[[File:P_ID1547__Bei_T_4_3_S5b_v1.png|right|frame|Raised cosine spectrum and impulse response]]
*on the left,  the  $($normalized$)$  Nyquist spectrum  $H(f)$,

*on the right,  the corresponding Nyquist pulse  $h(t)$,  whose zero crossings are equidistant with distance  $T_{\rm C}$.

$\text{It should be noted:}$
# The transmission filter  $H_{\rm S}(f)$  and the matched filter  $H_{\rm E}(f)$  are each  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems|"root raised cosine"]].
#Only the product  $H(f) = H_{\rm S}(f) \cdot H_{\rm E}(f)$  leads to the raised cosine.  This also means:
# The impulse responses  $h_{\rm S}(t)$  and  $h_{\rm E}(t)$  by themselves do not satisfy the first Nyquist condition.
#Only the combination of the two  $($in the time domain the convolution$)$  leads to the desired equidistant zeros.}}

==Modulation methods for UMTS==
 
The modulation techniques used in UMTS can be summarized as follows:
#In the downlink:  "Quaternary Phase Shift Keying"  is used for modulation  both in  "frequency division duplex"  $\rm (FDD)$  and in  "time division duplex""  $\rm (TDD)$.
#Here,  user data  $($DPDCH channel$)$  and control data  $($DPCCH channel$)$  are multiplexed in time.
#With TDD,  the signal is modulated in the uplink also by means of QPSK,  but not with  FDD. 
#Here,  a  "dual channel binary phase shift keying"  is used   ⇒   different channels are transmitted in  "in-phase"  and  "quadrature components".
#Thus,  two chips are transmitted per modulation step.  The gross chip rate is therefore twice the modulation rate of  $3.84$ Mchip per second.

{{GraueBox|TEXT=
$\text{Example 5:}$ 
The graph shows in the equivalent low-pass domain this  "I/Q multiplexing method",  as it is also called:
[[File:EN_Mob_T_3_4_S6.png|right|frame|Modulation and pulse shaping for UMTS|class=fit]]

#The spread useful data of the DPDCH channel is modulated onto the inphase component.
#The spread control data of the DPCCH channel is modulated onto the quadrature component.
#After modulation,  the quadrature component is weighted by the root of the power ratio  $G$  between the two channels to minimize the influence of power differences between  $I$  and  $Q$.
#Finally,  the complex sum signal  $(I +{\rm j} \cdot Q)$  is multiplied by a scrambling code that is also complex.}}

{{BlaueBox|TEXT=
$\text{Conclusion:}$  An advantage of dual channel BPSK modulation is the  ''' possibility of usinglow-power amplifiers'''.
*But time division multiplexing of user and control data as in the uplink  '''is not possible in the downlink'''.

*One reason for this is the use of  "Discontinuous Transmission"  $\rm (DTX)$  and the associated time constraints.}}

==Single-user receiver==
 
The task of a CDMA receiver is to separate and reconstruct the transmitted data of the individual subscribers from the sum of the spread data streams.  A distinction is made between  "single-user receivers"  and  "multi-user receivers".

In the UMTS downlink,  it is always used a  »'''single-user receiver'''«,  since in the mobile station a joint detection of all subscribers would be too costly
*due to the large number of active subscribers

*as well as the length of the scrambling codes and the asynchronous operation.

Such a receiver consists of a bank of independent correlators.
*Each one of the total  $J$  correlators belongs to a specific spreading sequence.

*The correlation is usually formed in a so-called  "correlator database"  by software.

Thereby one receives at the correlator output the sum of
[[File:EN_Bei_T_4_3_S6a_v2.png|right|frame|Single-user receiver with matched filter]]
*the  "autocorrelation '''KORREKTUR: auto-correlation''' function"  of the spreading code and

*the  "cross-correlation function"  of all other users with their own spreading code.

The graphic shows the simplest realization of such a receiver with matched filter.

#The received signal  $r(t)$  is first multiplied by the spreading code  $c(t)$  of the considered subscriber,  which is called  "despreading"  $($yellow background$)$.
#Followed by convolution with the matched filter impulse response  $($"Root Raised Cosine"$)$  to maximize SNR,  and sampling in bit clock  $(T_{\rm B})$.
#Finally, the threshold decision is made, which provides the sink signal  $v(t)$ and  thus the data bits of the considered subscriber.

{{BlaueBox|TEXT=
$\text{Please note:}$ 

*For the AWGN channel,  spreading at the transmitter and the matched despreading at the receiver have no effect on the bit error probability because of  $c(t)^2 = 1$.  As shown in  [[Aufgaben:Exercise_4.5:_Pseudo_Noise_Modulation|$\text{Exercise 4.5}$]],  even with spreading/despreading at the optimal receiver, regardless of spreading factor  $J$:

:$$p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt { {2 \cdot E_{\rm B} }/{N_{\rm 0} } } \hspace{0.05cm} \right )\hspace{0.05cm}. $$

*This result can be justified as follows:  The statistical properties of white noise  $n(t)$  are not changed by multiplication with the  $±1$  signal  $c(t)$.}}

==Rake receiver==
 
Another receiver for single-user detection is the  »'''rake receiver'''«,  which leads to significant improvements for a multipath channel.
[[File:EN_Bei_T_4_3_S6b_v2.png|right|frame|Structure of the rake receiver  $($shown in the equivalent low-pass domain$)$]]

The diagram shows its setup for a two-way channel with 
*a direct path with coefficient  $h_0$  and delay time  $τ_0$,

*an echo with coefficient  $h_1$  and delay time  $τ_1$.

For simplicity,  the coefficients  $h_0$  and  $h_1$  are assumed to be real.  Due to the representation in the equivalent low-pass domain,  these could also be complex.

⇒   The task of the rake receiver is to concentrate the signal energies of all paths  $($in this example only two$)$  to a single instant.  It works accordingly like a  "rake"  for the garden.
 
If one applies a Dirac delta impulse at time  $t = 0$  to the channel input,  there will be three Dirac delta impulses at the output of the rake receiver:
:$$ s(t) = \delta(t) \hspace{0.3cm}\Rightarrow\hspace{0.3cm}
y(t) = h_0 \cdot h_1 \cdot \delta(t - 2\tau_0) + (h_0^2 + h_1^2) \cdot \delta(t - \tau_0 - \tau_1)+
h_0 \cdot h_1 \cdot \delta(t - 2\tau_1) .$$

*The signal energy is concentrated at the time  $τ_0 + τ_1$.  Of the total four paths,  two contribute  $($middle term$)$.

*The Dirac delta functions at  $2τ_0$  and  $2τ_1$  do cause momentum interference.  However, their  weights are much smaller than those of the main path.

{{GraueBox|TEXT=
$\text{Example 6:}$ 
With channel parameters  $h_0 = 0.8$  and  $h_1 = 0.6$  the main path  $($with weight  $h_0)$  contains  $0.82/(0.82 + 0.62) = 64\%$  of the total signal energy.
*With rake receiver and the same weights,  the above equation is:

:$$ y(t) = 0.48 \cdot \delta(t - 2\tau_0) + 1.0 \cdot \delta(t - \tau_0 - \tau_1)+
0.48 \cdot \delta(t - 2\tau_1) .$$

*The share of the main path in the total energy amounts in this simple example to  ${1^2}/{(1^2 + 0.48^2 + 0.48^2)} ≈ 68\%.$}}

Rake receivers are preferred for implementation in mobile devices,  but have a limited performance when there are many active participants.
#In a multipath channel with many  $(M)$  paths,  the Rake has also  $M$  fingers.
#The main finger  – also called  "searcher"  – is responsible for identifying and ranking the individual paths of multiple propagation.
#It searches for the strongest paths and assigns them to other fingers along with their control information.
#In the process,  the time and frequency synchronization of all fingers is continuously compared with the control data of the received signal.

==Multi-user receiver ==
 
In a single-user receiver,  only the data signal of one subscriber is decided,  while all other subscriber signals are considered as additional noise.  However,  the bit error rate of such a detector will be very large
*if there is large  "intracell interference"  $($many active subscribers in the considered radio cell$)$

*or large  "intercell interference"  $($highly interfering subscribers in neighboring cells$)$.

In contrast,  »'''multi-user receivers'''«  make a joint decision for all active subscribers.» Their characteristics can be summarized as follows:
#Such a multi-user receiver does not consider the interference from other participants as noise,  but also uses the information contained in the interference signals for detection.
#The receiver is expensive to implement and the algorithms are extremely computationally intensive.  It contains an extremely large correlator database followed by a common detector.
#The multi-user receiver must know the spreading codes of all active users.  This requirement precludes use in the UMTS downlink  $($i.e.,  at the mobile station$)$.  In contrast,  all subscriber-specific spreading codes are known a-priori to the base stations,  so that multi-user detection is only used in the uplink.
#Some detection algorithms additionally require knowledge of other signal parameters such as energies and delay times.  The common detector  – the heart of the receiver –  is responsible for applying the appropriate detection algorithm in each case.
#Examples of multi-user detection are  "decorrelating detection"  and  "Interference Cancellation".

==Near–far problem==
 
The  "near-far problem"  is exclusively an uplink problem,  i.e.,  the transmission from mobile subscribers to a base station.  We consider a scenario with two users at different distances from the base station according to the following graph.  This can be interpreted as follows:

[[File:EN_Mob_T_3_2_S3.png|right|frame|Scenarios for the near-far problem|class=fit]]

#If both mobile stations transmit with the same power,  the received power of the red user  $\rm A$  at the base station is significantly smaller than that of the blue user  $\rm B$  $($left scenario$)$ due to path loss.
#In large macrocells,  the difference can be as much as  $100$  dB.  As a result,  the red signal is largely obscured by the blue.
#You can largely avoid the near-far problem if user  $\rm A$  transmits with higher power than user  $\rm B$,  as indicated in the right scenario.
#Then,  at the base station,  the received power of both mobile stations is then  $($almost$)$  equal.

Note:   In an idealized system  $($one-way channel,  ideal A/D converters,  fully linear amplifiers$)$  the transmitted data of the users are orthogonal to each other and one could detect the users individually even with very different received powers.  This statement is true
*for UMTS  $($multiple access:  CDMA$)$  as well as

*for the 2G system GSM  $($FDMA/TDMA$)$,  and

*for the 4G system LTE  $($TDMA/OFDMA$)$.

In reality,  however,  orthogonality is not always given due to the following reasons:
#Different receive paths   ⇒   multipath channel,
#non-ideal characteristics of the spreading and scrambling codes in CDMA,
#asynchrony of users in the time domain  $($basic propagation delay of paths$)$,
#asynchrony of users in the frequency domain  $($non-ideal oscillators and Doppler shift due to mobility of users$)$.

Consequently,  the users are no longer orthogonal to each other and the signal-to-noise ratio of the user to be detected with respect to the other users is not arbitrarily high:
*For  [[Examples_of_Communication_Systems/General_Description_of_GSM|"GSM"]]  and  [[Mobile_Communications/General_Information_on_the_LTE_Mobile_Communications_Standard|"LTE"]]  one can assume signal-to-noise ratios of  $25 $  dB  and more,

*but for UMTS  $($CDMA$)$  only approx.  $15$  dB,  with high-rate data transmission rather less.

==Carrier-to-interference power ratio==

The term  "capacity"  is generally understood to mean the number of available transmission channels per cell.  However,  since the number of subscribers is not strictly limited in UMTS unlike in GSM,  no fixed capacity can be specified here.
*In perfect codes,  the subscribers do not interfere with each other.  Thus,  the maximum number of users is determined solely by the spreading factor  $J$  and the available number of mutually orthogonal codes,  which,  however,  is also limited.

*More practical are non-perfect,  only quasi-orthogonal codes.  Here,  the  "capacity"  of a radio cell is predominantly determined by the resulting interference or the  "carrier-to-interference power ratio"  $\rm (CIR)$.

[[File:EN_Bei_T_4_3_S8_v2.png|right|frame|Carrier-to-interference ratio depending on the number of subscribers]]

As can be seen from this graph,  CIR depends directly on the number of active participants.  The more active subscribers there are,  the more interference power is generated and the smaller the CIR becomes.

Furthermore,  this decisive criterion for UMTS also depends on the following variables:
#The topology and user behavior  $($number of services called up$)$,
#the spreading factor  $J$  and the orthogonality of the used spreading code.

In order to limit the disturbing influence of the interference power on the transmission quality,  there are two possible criteria:

»'''Cell breathing'''«:   If the number of active subscribers increases significantly with UMTS,  the cell radius is reduced and  $($because of the now fewer subscribers in the cell$)$  also the current interference power is lower.  A less loaded neighboring cell then steps in to supply the subscribers at the edge of the reduced cell.

»'''Power control'''«:   If the total interference power within a radio cell exceeds a specified limit,  the transmission power of all subscribers is reduced accordingly and/or the data rate is reduced,  resulting in poorer transmission quality for all.  More about this in the next section.

==Power and power control in UMTS==
 
The ratio between the signal power and the interference power is used as the controlled variable for power control in UMTS.  There are differences between the  "frequency division duplex"  $\rm (FDD)$  and the  "time division duplex"  $\rm (TDD)$  modes.

[[File:EN_Bei_T_4_3_S10a.png|right|frame|Power control in the FDD mode]]
We take a closer look at the FDD power control.  In the diagram you can see two different control loops:
*The  »'''inner control loop'''«  controls the transmitter power based on time slots,  where one power command is transmitted in each time slot. 

::The power of the transmitter is determined and changed using the CIR estimates in the receiver and the specifications of the  "radio network controller"  $\rm (RNC)$  from the outer control loop.

*The  '''outer loop'''  controls based on $10$ millisecond duration frames. It is implemented in the RNC and is responsible for determining the set point for the inner loop.

The FDD power control sequence is as follows:
#The RNC provides a carrier-to-interference ratio  $\rm (CIR)$  setpoint.
#The receiver estimates the actual CIR value and generates control commands for the transmitter.
#The transmitter changes the transmitted power according to these control commands.

The principle of  "power control in TDD mode"  is similar to the control presented here for the FDD mode.  In fact in the downlink direction they are practically identical.

{{BlaueBox|TEXT=
$\text{Conclusion:}$ 
The  »'''TDD power control'''«  is much slower and thus less precise than in  FDD.

*However,  fast power control is not even possible in this case,  since each participant has only a fraction of the time frame available for him.}}

==Link budget ==
 
When planning UMTS networks,  calculating the link budget is an important step.  Knowledge of the link budget is required both for dimensioning the coverage areas and for determining the capacity and quality of service requirements.

{{BlaueBox|TEXT=
The  »'''objective of the link budget'''«  is to calculate the  '''maximum cell size'''  considering the following criteria:
#Type and data rate of the services,
#topology of the environment,
#system configuration  $($location and power of base stations,  handover gain$)$,
#service requirements  $($availability$)$,
#type of mobile station  $($speed,  power$)$,
#financial and economic aspects}}

{{GraueBox|TEXT=
[[File:EN_Bei_T_4_3_S9_v2.png|right|frame|Budget for a voice transmission channel '''KORREKTUR: fault''']]
$\text{Example 7:}$ 
The calculation of the link budget is illustrated using the example of a voice transmission channel in the UMTS downlink.  Regarding the exemplary numerical values,  it should be noted:
#The transmitted power is  $P_{\rm S} =19$  dBm,  which corresponds to approx.  $79$  mW.  Here,  the antenna loss is considered to be  $2$  dB.
#The noise power  $P_{\rm R} = 5 \cdot 10^{-11}$  mW   ⇒   $P_{\rm R} = -103$  dBm   product of UMTS bandwidth and noise power density.
#The interference power is  $P_{\rm I} = -99$  dBm  corresponding to  $1.25 \cdot 10^{-10}$  mW.
#This gives the total interference power  $P_{\rm R+I} = P_{\rm R} + P_{\rm I} = 1.25 \cdot 10^{-10}$  mW   ⇒   $P_{\rm R+I} =- 97.5$  dBm.
#The antenna sensitivity results in $-97.5 - 27 + 5 - 17 + 3.5 = - 133$  dBm.  A large negative value here is  "good".
#Maximum allowable path loss should be as large as possible.  Here  $19 - (-133) = 152$  dB.
#The  '''link budget'''  includes the margin for fading and the handover gain,  in the example:  $140$  dB.
#The  '''maximum cell radius'''  can be determined from the link budget using an [https://en.wikipedia.org/wiki/Path_loss "empirical formula"] of Okumura-Hata.  It holds:  $ {r}\ [{\rm km}] = 10^{({\rm LinkBudget}- 137)/35}= 10^{0.0857}\approx 1.22 . $
 

Notes:  
*"$\rm dB$"  denotes a logarithmic power specification,  referenced to  $1 \rm W$.

*In contrast  "$\rm dBm$"  refers to the power  $1 \rm mW$.}}

==UMTS radio resource management ==
 
The central task of  »'''radio resource management'''«  $\rm (RRM)$  is the dynamic adaptation of radio transmission parameters to the current situation  $($fading,  mobile station movement,  load,  etc.$)$  with the aim to
[[File:EN_Bei_T_4_3_S11.png|right|frame|Radio Resource Management in UMTS]]
#increase the transmission and subscriber capacity,
#improve the individual transmission quality,  and
#use existing radio resources economically.

The main RRM mechanisms summarized in the diagram are explained below.

'''Transmit power control''' The  radio resource management  attempts to keep the received power and thus the carrier-to-interference ratio  $\rm (CIR)$  at the receiver constant,  or at least to prevent it from falling below a specified limit.

An example of the need for power control is the  [[Examples_of_Communication_Systems/Telecommunications_Aspects_of_UMTS#Near.E2.80.93Far.E2.80.93Effect|"near-far problem"]],  which is known to cause a disconnect.

The step size of the power control is  $1 \ \rm dB$  or  $2 \ \rm dB$,  and the frequency of the control commands is  $1500$  commands per second.

'''Regulation of data rate'''
 UMTS allows an exchange between data rate and transmission quality,  which can be realized by selecting the spreading factor.  Doubling the spreading factor corresponds to halving the data rate and increases the quality by  $3\ \rm dB$  $($spreading gain$)$.

'''Access control'''
 To avoid overload situations in the overall network,  a check is made before a new connection is established to see whether the necessary resources are available.  If not,  the new connection is rejected.  This check is realized by estimating the transmission power distribution after the new connection is established.

'''Load control'''
 This becomes active if an overload occurs despite access control.  In this case,  a handover to another  base station is initiated and  – if this is not possible –  the data rates of certain nodes are lowered.

'''Handover'''
 Finally,  radio resource management is also responsible for handover to ensure uninterrupted connections.  Mobile stations are assigned to the individual radio cells on the basis of CIR measurements.

==Exercises for the chapter ==
 
[[Aufgaben:Exercise_4.5:_Pseudo_Noise_Modulation|Exercise 4.5: Pseudo Noise Modulation]]

[[Aufgaben:Exercise_4.5Z:_About_Band_Spreading_with_UMTS|Exercise 4.5Z: About Spread Spectrum with UMTS]]

[[Aufgaben:Exercise_4.6:_OVSF_Codes|Exercise 4.6: OVSF Codes]]

[[Aufgaben:Exercise_4.7:_To_the_Rake_Receiver|Exercise 4.7: About the Rake Receiver]]

==Sources==
<references />

{{Display}}