View source for Digital Signal Transmission/Causes and Effects of Intersymbol Interference


{{Header
|Untermenü=Intersymbol Interfering and Equalization Methods
|Vorherige Seite=Symbolwise_Coding_with_Pseudo_Ternary_Codes
|Nächste Seite=Fehlerwahrscheinlichkeit unter Berücksichtigung von Impulsinterferenzen
}}

== # OVERVIEW OF THE THIRD MAIN CHAPTER # ==
<br>
The third main chapter focuses on&nbsp; '''intersymbol interference'''&nbsp; $\rm (ISI)$,&nbsp; which arises,&nbsp; for example,&nbsp; from distortions of the transmission channel or is related to a realization of the receiver filter that deviates from the Nyquist condition.&nbsp;  Subsequently,&nbsp; some equalization methods are described which can be used to mitigate the system degradation due to intersymbol interference.

The description is given throughout in the baseband.&nbsp; However,&nbsp; the results can easily be applied to the carrier frequency systems discussed in the chapter &nbsp;[[Digital_Signal_Transmission/Lineare_digitale_Modulation_–_Kohärente_Demodulation|"Linear Digital Modulation - Coherent Demodulation"]].&nbsp;

In detail,&nbsp; this chapter deals with:
#&nbsp; the&nbsp; &raquo;causes and effects&laquo;&nbsp; of intersymbol interference,
#&nbsp; the&nbsp; &raquo;eye diagram&laquo;&nbsp; as a suitable tool for the description of intersymbol interferences,
#&nbsp; the&nbsp; &raquo;error probability calculation&laquo;&nbsp; considering channel distortions,
#&nbsp; the&nbsp; &raquo;influence of intersymbol interference in multilevel and/or coded transmission&laquo;,
#&nbsp; the&nbsp; &raquo;optimal Nyquist equalizer as an example of linear channel equalization,
#&nbsp; the&nbsp; &raquo;decision feedback equalization&laquo;&nbsp; $\rm (DFE)$ – an effective nonlinear decision realization,
#&nbsp; the&nbsp; &raquo;correlation receiver&laquo;&nbsp; as an example of&nbsp; &raquo;maximum likelihood or maximum a-posteriori&laquo;&nbsp; $\rm (MAP)$&nbsp; decision strategy,
#&nbsp; the&nbsp; &raquo;Viterbi receiver&laquo;,&nbsp; a reduced-effort MAP decision algorithm.



== Definition of the term "Intersymbol Interference" ==
<br>
For the first two main chapters of this book,&nbsp; it was assumed that the basic detection pulse''&nbsp; $g_d(t)$
*either is limited to the time domain &nbsp;$|t| \le T$,&nbsp; or<br>
*has equidistant zero crossings in the symbol spacing $T$.<br><br>

If we denote the samples of &nbsp;$g_d(t)$&nbsp; at multiples of the symbol duration &nbsp;$T$&nbsp; (spacing of the pulses)&nbsp; as the&nbsp; "basic detection pulse values",&nbsp; it has been tacitly assumed so far:
[[File:EN_Dig_T_3_1_S1.png|right|frame|Detection signals with and without intersymbol interferences|class=fit]]
:$$g_\nu = g_d(\nu T)   =   \left\{ \begin{array}{c} g_0    
 \\ 0  \\  \end{array} \right.\quad
\begin{array}{*{1}c} {\rm{for}}\\  {\rm{for}} \\ \end{array}
\begin{array}{*{20}c}\nu = 0, \\   \nu \ne 0. \\
\end{array}$$

As a consequence of this assumption it has resulted that in the binary case the signal component&nbsp; (index "S")&nbsp; 
:$$d_{\rm S}(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g_d ( t - \nu \cdot T)
 \hspace{0.3cm}{\rm with}\hspace{0.3cm}a_\nu \in \{ -1, +1\}$$

of the detection signal at the instants &nbsp;$\nu \cdot T$&nbsp; can take only two different values,&nbsp; namely &nbsp;$\pm g_0$.<br>

*The upper of the following two time plots shows &nbsp;$d_{\rm S}(t)$&nbsp; for this ISI-free case with &nbsp;$g_{\nu \ne 0} = 0$&nbsp; and &nbsp;$g_0 = s_0$ &nbsp; &rArr; &nbsp; the value &nbsp;$g_0$&nbsp; is equal to the maximum value &nbsp;$s_0$&nbsp; of the transmitted signal.

*Drawn below is the signal waveform for a set of basic detection pulse values that cause intersymbol interference:
:$$g_0 = 0.6 \cdot s_0,\hspace{0.15cm} g_{-1} = g_{+1} =0.2 \cdot s_0,\hspace{0.15cm}g_\nu =0\hspace{0.25cm}{\rm for}\hspace{0.25cm} |\nu| \ge 2 .$$

In both plots,&nbsp; the&nbsp; (triangular)&nbsp; basic detection pulse&nbsp; $g_d(t)$&nbsp; is drawn in red.&nbsp; The detection time points &nbsp;$\nu \cdot T$&nbsp; are marked by blue circles. 

One can see from the lower signal plot:
*The basic detection pulse &nbsp;$g_d(t)$&nbsp; is now different from zero in the range &nbsp;$|t| \le 1.5 \cdot T$&nbsp;and thus no longer fulfills the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain|"Nyquist condition in the time domain"]]&nbsp; for intersymbol interference freedom.<br>

*As a consequence,&nbsp; at the detection times&nbsp; (marked with circles)&nbsp; not only two values &nbsp;$(\pm s_0)$&nbsp; are possible as in the upper figure.&nbsp; Rather,&nbsp; the following applies here for the detection sampling values:
:$$d_{\rm S}(\nu \cdot T)  \in \{ \pm s_0, \ \pm 0.6 s_0, \ \pm 0.2  s_0\}\hspace{0.05cm}.$$
*The samples that are close to the threshold due to unfavorable neighboring pulses are more often falsified by the AWGN noise $($with noise rms value &nbsp;$\sigma_d)$&nbsp; than the samples further out.&nbsp; 

*Exemplarily,&nbsp; with &nbsp;$\sigma_d = 0.2 \cdot s_0$&nbsp; the blue filled points close to the threshold are falsified with probability &nbsp;$p_{\rm S} ={\rm Q} (1) \approx 16 \%$&nbsp; and the outer points&nbsp; (with white core)&nbsp; are falsified only with &nbsp;$p_{\rm S} ={\rm Q} (5) \approx 3 \cdot 10^{-7}$.&nbsp; The error probability of the red filled points&nbsp; (distance &nbsp;$0.6 \cdot s_0$&nbsp;  from zero line)&nbsp; is in between: &nbsp; $p_{\rm S} ={\rm Q} (3) \approx 0.13 \%$.


So far,&nbsp; the effects of intersymbol interference have been presented as vividly as possible.&nbsp; An exact definition is still missing.

{{BlaueBox|TEXT=  
$\text{Definition:}$&nbsp; '''Intersymbol interference'''&nbsp; $\rm (ISI)$&nbsp; is 
* the impairment of a symbol decision due to pulse broadening&nbsp; ("time dispersion"),&nbsp; and 
*the associated dependence of the error probability on the neighboring symbols.<br>


In other words:
#Falling edges of preceding pulses&nbsp; ("trailers")&nbsp; and rising edges of following pulses&nbsp; ("precursors")&nbsp; change the currently applied detection sample value.<br>
#This can increase or decrease the probability of a wrong decision for the current symbol,&nbsp; depending on whether the distance to the threshold becomes smaller or larger.<br>
#On statistical average &ndash; i.e. when considering an&nbsp; (infinitely)&nbsp; long symbol sequence &ndash; '''this always leads to a&nbsp; (considerable)&nbsp; increase of the&nbsp; (mean)&nbsp; symbol error probability''' &nbsp;$p_{\rm S} $.}}


== Possible causes for intersymbol interference ==
<br>
The graphic shows the &nbsp;[[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference#Definition_and_statements_of_the_eye_diagram| "eye diagram"]]&nbsp; for a
[[File:EN_Dig_T_3_1_S2.png|right|frame|Eye diagrams with and without intersymbol interference|class=fit]] 
#intersymbol interference&nbsp; $\rm (ISI)$&nbsp; system without noise&nbsp; (left),<br>
#an ISI-free system without noise&nbsp; (middle),<br>
#the same ISI-free system with noise&nbsp; (right).<br><br>

The definition,&nbsp; meaning and calculation of the eye diagram will be discussed in detail in the chapter &nbsp;[[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference|"Error Probability with Intersymbol Interference"]].&nbsp; These screenshots can be interpreted as follows:
*The middle diagram is from a Nyquist system with cosine rolloff characteristic $($rolloff factor &nbsp;$r = 0.5)$.&nbsp; Thus,&nbsp; no intersymbol interference occurs.<br>

*The right diagram is from the same ISI-free system,&nbsp; although here &nbsp;$d(t) = \pm s_0$&nbsp; does not apply.&nbsp; The deviations from the nominal values &nbsp;$\pm s_0$&nbsp; are here due to the AWGN noise.<br>

*From this follows the important insight: &nbsp; The question whether there is an ISI-free or  ISI-affected system can only be decided on the basis of the detection signal&nbsp; (or eye diagram)&nbsp;  '''without noise'''.<br>


The left diagram indicates intersymbol interference,&nbsp; since no noise is taken into account here. 
*The reason for this intersymbol interference could be that the overall frequency response of transmitter and receiver does not exactly fulfill the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_frequency_domain|"first Nyquist criterion"]]&nbsp; due to tolerances.<br>

*However,&nbsp; intersymbol interference also occurs with a channel with frequency-dependent frequency response &nbsp;$H_{\rm K}(f)$,&nbsp; if the receiver does not succeed in compensating the attenuation and phase distortions of the channel completely&nbsp; (i.e. one hundred percent).<br>

*Finally,&nbsp; even with the middle system,&nbsp; intersymbol interference occurs if the decision is not made exactly in the center of the eye,&nbsp; but at a detection time &nbsp;$T_{\rm D} \ne 0$.&nbsp; Then the basic detection pulse values must be defined to &nbsp;$g_\nu = g_d(T_{\rm D} + \nu \cdot T)$.<br>


== Some remarks on the channel frequency response ==
<br>
For the further sections in this third main chapter the following block diagram is&nbsp;  (mostly)&nbsp; assumed.&nbsp; The main difference to the&nbsp; [[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Block_diagram_and_prerequisites_for_the_first_main_chapter|block diagram of the first main chapter]]&nbsp; is the channel frequency response&nbsp; $($German:&nbsp; "Kanalfrequenzgang" &nbsp; &rArr; &nbsp; subscript&nbsp; "K"$)$,&nbsp; which is always assumed to be ideal &nbsp; &rArr; &nbsp; &nbsp;$H_{\rm K}(f) = 1$.&nbsp; <br>

The following applies  to the&nbsp; "frequency response"&nbsp; and the&nbsp; &nbsp;"impulse response"&nbsp; of the channel &nbsp;$(\rm exp[\hspace{0.05cm} . ]$&nbsp; denotes the&nbsp; "exponential function"$)$:
:$$H_{\rm K}(f)  =  {\rm exp} \left[ - a_{{\star} \hspace{0.01cm}({\rm Np})} \cdot \sqrt{\frac{f}{R_{\rm B}/2}}\hspace{0.1cm}\right] \cdot
  {\rm exp} \left[ - {\rm j} \cdot a_{{\star} \hspace{0.01cm}({\rm Np})} \cdot \sqrt{\frac{f}{R_{\rm B}/2}}\hspace{0.1cm}\right]
  \hspace{0.05cm}, $$
[[File:EN_Dig_T_3_1_S5.png|right|frame|Block diagram of a system with distorting channel|class=fit]]
:$$h_{\rm K}(t)  =    \frac{ a_{{\star}\hspace{0.01cm}({\rm Np})}}{  \sqrt{2  \pi^2 \cdot R_{\rm B} \cdot t^3}}\hspace{0.1cm} \cdot
  {\rm exp} \left[ - \frac{a_{{\star} \hspace{0.01cm}({\rm Np})}^2}{2  \pi \cdot R_{\rm B} \cdot t}\hspace{0.1cm}\right]
  \hspace{0.05cm}.$$

Here &nbsp;$a_{{\star} \hspace{0.01cm}({\rm Np})}$&nbsp; indicates the cable attenuation at half the bit rate.&nbsp; We call this quantity the &nbsp;'''characteristic cable attenuation'''&nbsp; in Neper&nbsp; $\rm (Np)$:
:$$a_{{\star} \hspace{0.01cm}({\rm Np})} = a_{\rm K}(f = {R_{\rm B}}/{2})= 0.1151 \cdot a_{{\star} \hspace{0.01cm}({\rm dB})}
  \hspace{0.05cm}.$$

#The corresponding dB value &nbsp; &rArr; &nbsp; $a_{{\star} \hspace{0.01cm}({\rm dB})}$&nbsp; is larger by a factor of &nbsp;$1/0.1151 = 8.686$.&nbsp; 
#In realized systems, the characteristic cable attenuation &nbsp;$a_{{\star} \hspace{0.01cm}({\rm dB})}$&nbsp; is in the range between &nbsp;$40 \ \rm dB$&nbsp; and &nbsp;$100 \ \rm dB$. 
#The addition&nbsp; "(Np)"&nbsp; or&nbsp; "(dB)"&nbsp; is omitted in the following.<br>


In the main chapter 4: &nbsp; "Properties of electrical lines" of the book &nbsp;[[Linear_and_Time_Invariant_Systems|"Linear and Time Invariant Systems"]]&nbsp; it is shown that these equations reproduce the conditions with good approximation for wireline communication systems via coaxial cable.&nbsp; For a two-wire line,&nbsp; the deviation between this very simple&nbsp; (analytically manageable)&nbsp; formula and the actual conditions is somewhat larger.<br>

A short summary of these derivations follows in the next two sections. 
*For simplicity,&nbsp; we will use a redundancy-free binary system. 
*Thus,&nbsp; the bit rate &nbsp;$R_{\rm B}$&nbsp; is equal to the reciprocal of the symbol duration &nbsp;$T$.<br>


== Frequency response of a coaxial cable==
<br>
A &nbsp;'''coaxial cable'''&nbsp; with core diameter 2.6 mm,&nbsp; outer diameter 9.5 mm and length &nbsp;$l$&nbsp; has the following frequency response:
:$$H_{\rm K}(f) = {\rm e}^{-\left[ a_{\rm K}(f) + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} b_{\rm K}(f)\right] } = {\rm e}^{- \alpha_0 \hspace{0.05cm} \cdot \hspace{0.05cm} l}  \cdot
  {\rm e}^{- (\alpha_1 + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_1) \hspace{0.05cm}\cdot f \hspace{0.05cm}\cdot \hspace{0.05cm}l}  \cdot
  {\rm e}^{- (\alpha_2 + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_2) \hspace{0.05cm}\cdot \sqrt{f} \hspace{0.05cm}\cdot \hspace{0.05cm}l}
    \hspace{0.05cm},$$

where the following parameters apply for these dimensions &ndash; referred to as&nbsp; "normal coaxial cable":&nbsp;
:$$\alpha_0 = 0.00162 \hspace{0.15cm}\frac {\rm Np}{\rm km} \hspace{0.05cm},\hspace{0.2cm}
  \alpha_1 = 0.000435 \hspace{0.15cm}\frac {\rm Np}{{\rm km} \cdot {\rm MHz}} \hspace{0.05cm},
  \hspace{0.2cm}
  \alpha_2 = 0.2722 \hspace{0.15cm}\frac {\rm Np}{{\rm km} \cdot \sqrt{\rm MHz}} \hspace{0.05cm},\hspace{0.2cm}
\beta_1 = 21.78 \hspace{0.15cm}\frac {\rm rad}{{\rm km} \cdot {\rm MHz}} \hspace{0.05cm},
  \hspace{0.2cm}
  \beta_2 = 0.2722 \hspace{0.15cm}\frac {\rm rad}{{\rm km} \cdot \sqrt{\rm MHz}}
  \hspace{0.05cm}.$$

In the above equation the attenuation parameters&nbsp; ("alpha")&nbsp; are to be inserted in&nbsp; "Np"&nbsp; and the phase parameters&nbsp; ("beta")&nbsp;  in&nbsp; "rad".<br>

[[File:EN_Dig_T_3_1_S3a.png|right|frame|Attenuation curve of a coaxial cable and approximation&nbsp; (skin effect only) |class=fit]]<br>
The graph shows the exact attenuation curve for a standard coaxial cable of one kilometer length and an approximation for frequencies up to &nbsp;$f = 1000\ \rm  MHz$:
:$$a_{\rm K}(f)  = \alpha_0  \cdot l \hspace{0.05cm} + \hspace{0.05cm} \alpha_1  \cdot f \cdot l
 + \hspace{0.05cm} \alpha_2  \cdot \sqrt{f} \cdot l \hspace{0.05cm},$$
:$$a_{\rm K}(f)  \approx \alpha_2  \cdot \sqrt{f} \cdot l.$$
 
*The axis is labeled in&nbsp; "$\rm dB$"&nbsp; on the left and&nbsp; "$\rm Np$"&nbsp; on the right.
*One&nbsp; $\rm Np$&nbsp; ("Neper")&nbsp; corresponds to&nbsp; $8.686 \ \rm dB$.


We refer to the interactive HTML5/JavaScript applet&nbsp; [[Applets:Attenuation_of_Copper_Cables|"Attenuation of copper cables".]]<br>

You can see from the diagram and the above numerical values:
*The first term &nbsp;$(\alpha_0 \cdot l)$&nbsp; originating from the ohmic losses is negligible.&nbsp; Moreover,&nbsp; this term causes only frequency-independent attenuation and no signal distortion.<br>

*The second term &nbsp;$(\alpha_1 \cdot f \cdot l)$,&nbsp; due to the transverse losses,&nbsp; is proportional to frequency and therefore becomes noticeable only at very high frequencies;&nbsp; it will be neglected in the following.<br>

*The frequency-proportional phase &nbsp;$(\beta_1 \cdot f \cdot l)$&nbsp; only results in a signal delay by the transit time &nbsp;$\beta_1/(2\pi)  \cdot l$,&nbsp; but no distortion.&nbsp; This transit time will also be disregarded in the following.<br>

*With these simplifications,&nbsp; the frequency response is thus determined by the skin effect alone.&nbsp; Since the numerical values for &nbsp;$\alpha_2$&nbsp; $($in Np$)$&nbsp; and &nbsp;$\beta_2$&nbsp; $($in rad$)$&nbsp; are the same,&nbsp; it follows:
:$$H_{\rm K}(f)  ={\rm e}^{- (\alpha_2 + {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \beta_2) \hspace{0.05cm}\cdot \sqrt{f} \hspace{0.05cm}\cdot \hspace{0.05cm}l}
  \cdot {\rm e}^{- \alpha_2  \hspace{0.05cm}\cdot \hspace{0.05cm}l \hspace{0.05cm}\cdot \sqrt{{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}
  2f}} \hspace{0.05cm}.$$
*Often in the literature &ndash; and also in this tutorial &ndash; the attenuation measure at half the bit rate is used,&nbsp; which we call the &nbsp;"characteristic cable attenuation"&nbsp; $($in Neper$)$:
:$$a_{\star} = a_{\rm K}(f ={R_{\rm B}}/{2})= a_{\rm K}(f = \frac{1}{2 \cdot T})\approx \frac{\alpha_2 \cdot
  l }{ \sqrt {2\cdot T}}
  \hspace{0.05cm}.$$

{{GraueBox|TEXT=  
$\text{Example 1:}$&nbsp; For a binary system with half the bit rate &nbsp;$R_{\rm B}/2 = 280 \ \rm Mbit/s$&nbsp; and &nbsp;$l = 1\ \rm  km$ &nbsp;  &nbsp;$\Rightarrow \ a_{\star}  \approx 4.55 \ \rm Np$&nbsp; or &nbsp;$a_{\star}  \approx 40 \ \rm dB$&nbsp; (green markings in the diagram). 

*But if the half bit rate is only &nbsp;$70 \ \rm Mbit/s$, &nbsp;$a_{\star} = 40 \ \rm dB$&nbsp; characterizes a transmission system with cable length &nbsp;$l = 2\ \rm  km$.

*Note: &nbsp; The above approximation &nbsp;$a_{\rm K}(f)   \approx \alpha_2  \cdot \sqrt{f} \cdot l$&nbsp; is only admissible for coaxial cables,&nbsp; since for these the coefficients &nbsp;$\alpha_0$&nbsp; and  &nbsp;$\alpha_1$&nbsp; can be neglected.

*However,&nbsp; for a &nbsp;[[Linear_and_Time_Invariant_Systems/Properties_of_Balanced_Copper_Pairs|"balanced two-wire cable"]],&nbsp; the coefficients &nbsp;$\alpha_0$&nbsp; and  &nbsp;$\alpha_1$&nbsp; are much larger and the above approximation is invalid.<br>}}



== Impulse response of a coaxial cable ==
<br>
We now consider the coaxial cable impulse response,&nbsp; which is for a binary system &nbsp;$(R_{\rm B} = 1/T)$&nbsp; as follows:
:$$h_{\rm K}(t) =   \frac{ a_{\rm \star \hspace{0.01cm}(Np)}/T}{  \sqrt{2  \pi^2 \cdot (t/T)^3}}\hspace{0.1cm} \cdot
  {\rm exp} \left[ - \frac{a_{\rm \star \hspace{0.01cm}(Np)}^2}{2  \pi  \cdot t/T}\hspace{0.1cm}\right]
  \hspace{0.05cm}.$$

This time course is shown here for characteristic cable attenuations &nbsp;$(a_{\rm \star})$&nbsp; between &nbsp;$40 \ \rm dB$&nbsp; and &nbsp;$100 \ \rm dB$.&nbsp; Note the conversion &nbsp; $\rm  1 \ Np = 8.686 \ dB.$<br>
[[File:EN_Dig_T_3_1_S3b.png|right|frame|Impulse and rectangular pulse response of the coaxial cable&nbsp; (skin effect only); &nbsp; <u>Note:</u> <br>Here,&nbsp; the basic receiver pulse &nbsp;$g_r(t)$&nbsp; is identical with the&nbsp; "rectangular pulse response"]]

One can see from this time domain plot:
*Even with the relatively small characteristic cable attenuation value  &nbsp;$a_{\rm \star} = 40 \ \rm dB$,&nbsp; the impulse response extends over more than &nbsp;$100$&nbsp; symbol durations &nbsp;$(T)$.<br>

*The larger &nbsp;$a_{\rm \star}$&nbsp; is chosen,&nbsp; the broader and lower the impulse response becomes.&nbsp; The integral over &nbsp;$h_{\rm K}(t)$&nbsp; from zero to infinity is the same for all curves,&nbsp; since &nbsp;$H_{\rm K}(f=0) = 1$&nbsp; always holds&nbsp; $(\alpha_0 = 0$,&nbsp; $\alpha_1 = 0).$<br>

*The basic receiver pulse &nbsp;$g_r(t) = g_s(t) \star h_{\rm K}(t)$&nbsp; is nearly equal in shape to &nbsp;$h_{\rm K}(t)$.&nbsp; The right ordinate axis shows &nbsp;$g_r(t)/s_0$&nbsp; when &nbsp;$g_s(t)$&nbsp; is an NRZ rectangular pulse with height &nbsp;$s_0$&nbsp; and duration &nbsp;$T$.&nbsp; <br>

*For &nbsp;$a_{\rm \star}  \ge 60 \ \rm dB$,&nbsp; &nbsp;$h_{\rm K}(t)$&nbsp; and &nbsp;$g_r(t)$&nbsp; are indistinguishable within the character accuracy when suitably normalized.&nbsp; For &nbsp;$a_{\rm \star} = 40 \ \rm dB$,&nbsp; one can see a small difference at the peak&nbsp; (yellow background); &nbsp; $g_r(t)/s_0$&nbsp; is here minimally smaller than &nbsp;$T \cdot h_{\rm K}(t)$.<br>

*However,&nbsp; with the characteristic attenuation &nbsp;$a_{\rm \star} = 40 \ \rm dB$,&nbsp; the pulse amplitude at the cable end is only less than &nbsp;$7\%$&nbsp; of the input amplitude.&nbsp; At &nbsp;$60 \ \rm dB$&nbsp; and &nbsp;$100 \ \rm dB$,&nbsp; this value drops to &nbsp;$3\%$&nbsp; and &nbsp;$2\%$.<br><br>

In &nbsp;[[Aufgaben:Exercise_3.1:_Impulse_response_of_the_Coaxial_Cable|Exercise 3.1]],&nbsp; the coaxial cable impulse response is analyzed in detail.&nbsp; We also refer to the interactive&nbsp; (German language)&nbsp; SWF applet &nbsp;[[Applets:Zeitverhalten_von_Kupferkabeln|"Time behavior of copper cables"]].<br>

== Prerequisites for the entire third main chapter ==
<br>
Consider again the block diagram of a transmission system, assuming a highly distorting channel, such as is present in wireline transmission.<br>

Due to the channel frequency response &nbsp;$H_{\rm K}(f)$,&nbsp; which is highlighted in red in the diagram,&nbsp; there are also certain limitations for the other system components:

[[File:EN_Dig_T_3_1_S5.png|right|frame|Block diagram of a system with&nbsp; (highly)&nbsp; distorting channel]]

*The basic receiver pulse &nbsp;$g_r(t)  = g_s(t) \star h_{\rm K}(t)$&nbsp; extends over hundreds of bits.&nbsp; Therefore,&nbsp; the receiver filter &nbsp;$H_{\rm E}(f)$&nbsp; cannot be applied as a &nbsp;[[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Optimal_binary_receiver_.E2.80.93_.22Matched_Filter.22_realization|"matched filter"]];&nbsp; this would approximately double the duration of the basic detection pulse &nbsp;$g_d(t)$&nbsp; compared to &nbsp;$g_r(t)$.&nbsp; <br>

*Rather,&nbsp; the receiver filter &nbsp;$H_{\rm E}(f)$&nbsp; must compensate for the enormous attenuation distortions &nbsp;$(\alpha_2$&ndash;term$)$ and the enormous phase distortions &nbsp;$(\beta_2$&ndash;term$)$ of the coaxial channel &nbsp;$H_{\rm K}(f)$,&nbsp; especially if a simple threshold decision is assumed.<br>

*Although this linear form of signal equalization can be supported by more elaborate decision strategies &ndash; e.g.  &nbsp; [[Digital_Signal_Transmission/Entscheidungsrückkopplung|"decision feedback"]], &nbsp;[[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Matched_filter_receiver_vs._correlation_receiver|"correlation receivers"]],&nbsp; [[Digital_Signal_Transmission/Viterbi–Empfänger|"Viterbi receivers"]].&nbsp; For wired transmission,&nbsp; a linear signal equalization &nbsp; &rArr; &nbsp; receiver filter &nbsp;$H_{\rm E}(f)$&nbsp; cannot be completely dispensed with due to the very strong distortions.<br>

*The noise &nbsp;$n(t)$&nbsp; is further assumed to be additive,&nbsp; white,&nbsp; and Gaussian distributed&nbsp; $\rm (AWGN)$,&nbsp; which is justified for a coaxial cable.&nbsp; For a two-wire line,&nbsp; crosstalk from adjacent copper wires is the dominant noise,&nbsp; as explained in detail in the chapter &nbsp;[[Examples_of_Communication_Systems| "ISDN"]]&nbsp; ("Integrated Services Digital Network").<br><br>

{{BlaueBox|TEXT=  
$\text{Conclusion:}$&nbsp; In the following chapters  we consider only the binary bipolar redundancy-free transmission &nbsp; &rArr; &nbsp; bit rate &nbsp;$R_{\rm B} = 1/T$.&nbsp; It is always assumed that:

*The basic transmission pulse &nbsp;$g_s(t)$&nbsp; is NRZ rectangular with amplitude &nbsp;$s_0$&nbsp; and duration &nbsp;$T$. 

*Thus,&nbsp;  the transmitted signal &nbsp;$s(t)$&nbsp; is equal to &nbsp;$\pm s_0$&nbsp; at all times and the spectral function is: &nbsp; $G_s(f) = s_0 \cdot T \cdot {\rm sinc}(f T)$.<br>

*Splitting the equalization between transmitter and receiver according to the root&ndash;root characteristic does not make sense for conducted transmission. 

*There would be too much intersymbol interference already at the transmitter.<br><br>}}


== Exercises for the chapter==
<br>
[[Aufgaben:Exercise_3.1:_Impulse_response_of_the_Coaxial_Cable|Exercise 3.1: Impulse response of the Coaxial Cable]]

[[Aufgaben:Exercise_3.1Z:_Frequency_Response_of_the_Coaxial_Cable|Exercise 3.1Z: Frequency Response of the Coaxial Cable]]


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