View source for Applets:Principle of Pseudo-Ternary Coding

{{LntAppletLinkEnDe|pseudoternarycodes_en|pseudoternarycodes}}

 
==Applet Description==
<br>
The applet covers the properties of the best known pseudo-ternary codes, namely:
#&nbsp; First-order bipolar code,&nbsp; $\rm AMI$ code&nbsp; (from: ''Alternate Mark Inversion''),&nbsp; characterized by the parameters&nbsp; $N_{\rm C} = 1, \ K_{\rm C} = +1$,
#&nbsp; Duobinary code,&nbsp; $(\rm DUOB)$,&nbsp; code parameters:&nbsp; $N_{\rm C} = 1, \ K_{\rm C} = -1$,
#&nbsp; Second order bipolar code&nbsp; $(\rm BIP2)$,&nbsp; code parameters:&nbsp; $N_{\rm C} = 2, \ K_{\rm C} = +1$.


At the input is the redundancy-free binary bipolar source symbol sequence&nbsp; $\langle \hspace{0.05cm}q_\nu \hspace{0.05cm}\rangle \ \in \{+1, -1\}$ &nbsp; &rArr; &nbsp; rectangular signal&nbsp; $q(t)$&nbsp; an.&nbsp; Illustrating the generation. 
*of the binary&ndash;precoded sequence&nbsp; $\langle \hspace{0.05cm}b_\nu \hspace{0.05cm}\rangle \ \in \{+1, -1\}$,&nbsp; represented by the also redundancy-free binary bipolar rectangular signal&nbsp; $b(t)$,
*the pseudo-ternary code sequence&nbsp; $\langle \hspace{0.05cm}c_\nu \hspace{0.05cm}\rangle \ \in \{+1,\ 0, -1\}$,&nbsp; represented by the redundant ternary bipolar rectangular signal&nbsp; $c(t)$,
*the equally redundant ternary transmitted signal&nbsp; $s(t)$, characterized by the amplitude coefficients&nbsp; $a_\nu $,&nbsp; and the (transmitted&ndash;) base impulse&nbsp; $g(t)$:
:$$s(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g( t - \nu \cdot T)\hspace{0.05cm}.$$
         
The base impulse&nbsp; $g(t)$&nbsp; &ndash;&nbsp; in the applet&nbsp; "Rectangle",&nbsp; "Nyquist" and&nbsp; "Root&ndash;Nyquist"&nbsp; &ndash;&nbsp; determines not only the shape of the transmitted signal, but also the course of
* of the auto-correlation function&nbsp; $\rm (ACF)$&nbsp; $\varphi_s (\tau)$&nbsp; and
* of the associated power spectral density&nbsp; $\rm (PSD)$&nbsp; ${\it \Phi}_s (f)$.


The applet also shows that the total power spectral density&nbsp; ${\it \Phi}_s (f)$ can be split into the part&nbsp; ${\it \Phi}_a (f)$ that takes into account the statistical relations of the amplitude coefficients&nbsp; $a_\nu$ &nbsp; and the energy spectral density $ {\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = |G(f)|^2 $, characterized by the shape&nbsp; $g(t)$.

''Note'' &nbsp; In the applet, no distinction is made between the encoder symbols&nbsp; $c_\nu \in \{+1,\ 0, -1\}$&nbsp; and the amplitude coefficients&nbsp; $a_\nu \in \{+1,\ 0, -1\}$&nbsp; . &nbsp; It should be remembered that the&nbsp; $a_\nu$&nbsp; are always numerical values, while for the encoder symbols also the notation&nbsp; $c_\nu \in \{\text{plus},\ \text{zero},\  \text{minus}\}$&nbsp; would be admissible.

==Theoretical Background==

== General description of the pseudo-multilevel codes ==
<br>
In symbolwise coding,&nbsp; each incoming source symbol &nbsp;$q_\nu$&nbsp; generates an encoder symbol &nbsp;$c_\nu$,&nbsp; which depends not only on the current input symbol &nbsp;$q_\nu$&nbsp; but also on the &nbsp;$N_{\rm C}$&nbsp; preceding symbols &nbsp;$q_{\nu-1}$, ... , $q_{\nu-N_{\rm C}} $.&nbsp; &nbsp;$N_{\rm C}$&nbsp; is referred to as the&nbsp; "order"&nbsp; of the code.<br>

Typical for symbolwise coding is that
*the symbol duration &nbsp;$T$&nbsp; of the encoded signal&nbsp; (and of the transmitted signal)&nbsp; matches the bit duration &nbsp;$T_{\rm B}$&nbsp; of the binary source signal,&nbsp; and

*encoding and decoding do not lead to major time delays,&nbsp; which are unavoidable when block codes are used.<br><br>

The&nbsp; "pseudo-multilevel codes"&nbsp; &ndash; better known as&nbsp; "partial response codes"&nbsp; &ndash;&nbsp; are of special importance.&nbsp; In the following,&nbsp; only&nbsp; "pseudo-ternary codes" &nbsp; &rArr; &nbsp;  level number &nbsp;$M = 3$&nbsp; are considered.&nbsp; 
*These can be described by the block diagram corresponding to the left graph.&nbsp; 
*In the right graph an equivalent circuit is given,&nbsp; which is very suitable for an analysis of these codes.

[[File:EN_Dig_T_2_4_S1_v23.png|right|frame|Block diagram&nbsp; (above)&nbsp; and equivalent circuit&nbsp; (below)&nbsp; of a pseudo-ternary encoder|class=fit]]


One can see from the two representations:
*The pseudo-ternary encoder can be split into the&nbsp; "non-linear pre-encoder"&nbsp; and the&nbsp; "linear coding network",&nbsp; if  the delay by &nbsp;$N_{\rm C} \cdot T$&nbsp; and the weighting by &nbsp;$K_{\rm C}$&nbsp; are drawn twice for clarity&nbsp; &ndash; as shown in the right equivalent figure.

*The&nbsp; "non-linear pre-encoder"&nbsp; obtains the precoded symbols &nbsp;$b_\nu$,&nbsp; which are also binary,&nbsp; by a modulo&ndash;2 addition&nbsp; ("antivalence")&nbsp; between the symbols &nbsp;$q_\nu$&nbsp; and &nbsp;$K_{\rm C} \cdot b_{\nu-N_{\rm C}} $:&nbsp; 
:$$q_\nu \in \{-1, +1\},\hspace{0.1cm} K_{\rm C} \in \{-1,
+1\}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}b_\nu \in \{-1,
+1\}\hspace{0.05cm}.$$
*Like the source symbols &nbsp;$q_\nu$,&nbsp; the symbols &nbsp;$b_\nu$&nbsp; are statistically independent of each other.&nbsp; Thus,&nbsp; the pre-encoder does not add any redundancy.&nbsp; However,&nbsp; it allows a simpler realization of the decoder and prevents error propagation after a transmission error.<br>

*The actual encoding from binary &nbsp;$(M_q = 2)$&nbsp; to ternary &nbsp;$(M = M_c = 3)$&nbsp; is done by the&nbsp; "linear coding network"&nbsp; by the conventional subtraction
:$$c(t) ={1}/{2} \cdot \big [b(t) - K_{\rm C} \cdot b(t- N_{\rm
C}\cdot T)\big]  \in \{-1, \ 0, +1\}\hspace{0.05cm},$$

:which can be described by the following &nbsp;[[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Impulse_response|"impulse response"]]&nbsp; resp. &nbsp;[[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function|"transfer function"]]&nbsp; with respect to the input signal &nbsp;$b(t)$&nbsp; and the output signal &nbsp;$c(t)$:&nbsp; 
:$$h_{\rm C}(t) =  {1}/{2} \cdot \big [\delta(t) - K_{\rm C} \cdot \delta(t- N_{\rm
C}\cdot T)\big] \ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ H_{\rm C}(f) ={1}/{2} \cdot \left [1 - K_{\rm C} \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}N_{\rm C}\hspace{0.05cm}\cdot \hspace{0.05cm}T}\right]\hspace{0.05cm}.  $$

*The relative redundancy is the same for all pseudo-ternary codes.&nbsp; Substituting&nbsp; $M_q=2$, &nbsp;$M_c=3$&nbsp; and &nbsp;$T_c =T_q$&nbsp; into the &nbsp;[[Digital_Signal_Transmission/Redundancy-Free_Coding#Symbolwise_coding_vs._blockwise_coding|"general definition equation"]],&nbsp; we obtain
:$$r_c = 1- \frac{R_q}{R_c} = 1- \frac{T_c}{T_q} \cdot \frac{{\rm log_2}\hspace{0.05cm} (M_q)}{{\rm log_2} \hspace{0.05cm}(M_c)} = 1- \frac{T_c}{T_q \cdot {\rm log_2} \hspace{0.05cm}(M_c)}\hspace{0.5cm}\Rightarrow  \hspace{0.5cm} r_c = 1 -1/\log_2\hspace{0.05cm}(3) \approx 36.9 \%\hspace{0.05cm}.$$

{{BlaueBox|TEXT=  
The&nbsp; $\text{transmitted signal of all pseudo-ternary codes}$&nbsp; is always represented as follows:
:$$s(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g( t - \nu \cdot T)\hspace{0.05cm}.$$
*The property of the current pseudo-ternary code is reflected in the statistical bonds between the&nbsp; $a_\nu$&nbsp; In all cases&nbsp; $a_\nu \in \{-1, \ 0, +1\}$.
*The basic transmitting pulse&nbsp; $g(t)$&nbsp; provides on the one hand the required energy, but has also influence on the statistical bonds within the signal.
*In addition to the NRZ rectangular pulse&nbsp; $g_{\rm R}(t)$ can be selected in the program:&nbsp; 
:*the Nyquist impulse&nbsp;&nbsp; &rArr; &nbsp; impulse response of the raised cosine low-pass with rolloff factor $r$:
:$$g_{\rm Nyq}(t)={\rm const.} \cdot \frac{\cos(\pi \cdot r\cdot t/T)}{1-(2\cdot r\cdot  t/T)^2} \cdot {\rm si}(\pi \cdot  t/T) \ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ G_{\rm Nyq}(f),$$
  
:*the root Nyquist impulse&nbsp;&nbsp; &rArr; &nbsp; impulse response of the root raised cosine low-pass with rolloff factor $r$: 
:$$g_{\sqrt{\rm Nyq} }(t)\ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ G_{\sqrt{\rm Nyq} }(f)={\rm const.} \cdot \sqrt{G_{\rm Nyq}(f)} .$$ }}
<br><br>

== Properties of the AMI code==
<br>
The individual pseudo-ternary codes differ in the &nbsp;$N_{\rm C}$&nbsp; and &nbsp;$K_{\rm C}$ parameters.&nbsp; The best-known representative is the&nbsp; '''first-order bipolar code'''&nbsp; with the code parameters 
:*&nbsp;$N_{\rm C} = 1$,&nbsp; 
:*&nbsp;$K_{\rm C} = 1$, 


which is also known as&nbsp;  '''AMI code'''&nbsp;  (from: "Alternate Mark Inversion").&nbsp; This is used e.g. with &nbsp;[[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_ISDN|"ISDN"]]&nbsp; ("Integrated Services Digital Networks")&nbsp; on the so-called&nbsp; $S_0$&nbsp; interface.

[[File:EN_Dig_T_2_4_S2a.png|right|frame|Signals with AMI coding and HDB3 coding|class=fit]]
*The graph above shows the binary source signal &nbsp;$q(t)$.
 
*The second and third diagrams show:
:* the likewise binary signal &nbsp;$b(t)$&nbsp; after the pre-encoder,&nbsp; and

:* the encoded signal &nbsp;$c(t) = s(t)$&nbsp; of the AMI code.


One can see the simple AMI encoding principle:
#Each binary value&nbsp; "&ndash;1"&nbsp; of the source signal&nbsp; $q(t)$  &nbsp; &rArr; &nbsp;  symbol &nbsp;$\rm L$&nbsp; is encoded by the ternary amplitude coefficient &nbsp;$a_\nu = 0$.&nbsp; <br><br>
#The binary value&nbsp; "+1"&nbsp; of the source signal&nbsp; $q(t)$ &nbsp; &rArr; &nbsp;  symbol &nbsp;$\rm H$&nbsp; is alternately represented by &nbsp;$a_\nu = +1$&nbsp; and &nbsp;$a_\nu = -1$.&nbsp; <br><br>

This ensures that the AMI encoded signal does not contain any &nbsp; "long sequences"<br> 
* &nbsp; $ \langle c_\nu \rangle =  \langle \text{...}, +1, +1, +1, +1, +1, \text{...}\rangle$ &nbsp; resp. 
* &nbsp; $ \langle c_\nu \rangle =  \langle \text{...}, -1, -1, -1, -1, -1, \text{...}\rangle$,


which would lead to problems with a DC-free channel. 

On the other hand,&nbsp; the occurrence of long zero sequences is quite possible,&nbsp; where no clock information is transmitted over a longer period of time.
<br clear = all>
To avoid this second problem,&nbsp; some modified AMI codes have been developed, for example the&nbsp; "B6ZS code"&nbsp; and the&nbsp; "HDB3 code":
*In the&nbsp; '''HDB3 code'''&nbsp; (green curve in the graphic),&nbsp; four consecutive zeros in the AMI encoded signal are replaced by a subsequence that violates the AMI encoding rule.<br>

*In the gray shaded area,&nbsp; this is the sequence&nbsp; "$+\ 0\ 0\ +$",&nbsp; since the last symbol before the replacement was a&nbsp; "minus".<br>

*This limits the number of consecutive zeros to &nbsp; $3$ &nbsp; for the HDB3 code and to &nbsp; $5$ &nbsp; for the &nbsp;[https://www.itwissen.info/en/bipolar-with-six-zero-substitution-B6ZS-121675.html#gsc.tab=0 "B6ZS code"].&nbsp;
 
*The decoder detects this code violation and replaces "$+\ 0\ 0\ +$" with "$0\ 0\ 0\ 0$" again.<br> 
<br><br>
=== Calculating the ACF of a digital signal ===
In the execution of the experiment some quantities and correlations are used, which shall be briefly explained here:

*&nbsp; The (time-unlimited) digital signal includes both the source statistics $($amplitude coefficients &nbsp;$a_\nu$)&nbsp; and the transmitted pulse shape &nbsp;$g(t)$:
:$$s(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g ( t - \nu \cdot T)\hspace{0.05cm}.$$

*&nbsp; If&nbsp; $s(t)$&nbsp; is the pattern function of a stationary and ergodic random process, then for the &nbsp;[[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Random_processes|"Auto-Correlation Function"]]&nbsp; $\rm (ACF)$: 
:$$\varphi_s(\tau) = {\rm E}\big [s(t) \cdot s(t + \tau)\big ] =  \sum_{\lambda = -\infty}^{+\infty}{1}/{T}
\cdot \varphi_a(\lambda)\cdot\varphi^{^{\bullet} }_{gs}(\tau -
\lambda \cdot T)\hspace{0.05cm}.$$

*&nbsp; This equation describes the convolution of the discrete ACF&nbsp; $\varphi_a(\lambda) = {\rm E}\big [ a_\nu \cdot a_{\nu + \lambda}\big]$&nbsp; of the amplitude coefficients with the energy&ndash;ACF of the base impulse: 

:$$\varphi^{^{\bullet} }_{g}(\tau) =
\int_{-\infty}^{+\infty} g ( t ) \cdot  g ( t +
\tau)\,{\rm d} t \hspace{0.05cm}.$$

*The point is to indicate that&nbsp; $\varphi^{^{\bullet} }_{g}(\tau)$&nbsp; has the unit of an energy, while&nbsp; $\varphi_s(\tau)$&nbsp; indicates a power and&nbsp; $\varphi_a(\lambda)$&nbsp; is dimensionless.
<br>
=== Calculating the PSD of a digital signal ===
The corresponding quantity to the ACF in the frequency domain is the [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Wiener-Khintchine_Theorem|"Power Spectral Density"]]&nbsp; $\rm (PSD)$&nbsp; ${\it \Phi}_s(f)$, which is fixedly related to&nbsp; $\varphi_s(\tau)$&nbsp; via the Fourier integral:<br>
:$$\varphi_s(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}
  {\it \Phi}_s(f)  =   \int_{-\infty}^{+\infty} \varphi_s(\tau) \cdot
  {\rm e}^{- {\rm j}\hspace{0.05cm} 2 \pi  f \hspace{0.02cm} \tau}
  \,{\rm d} \tau  \hspace{0.05cm}.$$
*The power spectral density &nbsp;${\it \Phi}_s(f)$&nbsp; can be represented as a product of two functions, taking into account the dimensional adjustment&nbsp; $(1/T)$&nbsp;:
:$${\it \Phi}_s(f) =  {\it \Phi}_a(f) \cdot  {1}/{T} \cdot
|G_s(f)|^2 \hspace{0.05cm}.$$
*The first term &nbsp;${\it \Phi}_a(f)$&nbsp; is dimensionless and describes the spectral shaping of the transmitted signal by the statistical relations of the source:<br>
:$$\varphi_a(\lambda) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}_a(f) =  \sum_{\lambda =
-\infty}^{+\infty}\varphi_a(\lambda)\cdot {\rm e}^{- {\rm
j}\hspace{0.05cm} 2 \pi  f \hspace{0.02cm} \lambda \hspace{0.02cm}T} = 
\varphi_a(0) + 2 \cdot \sum_{\lambda =
1}^{\infty}\varphi_a(\lambda)\cdot\cos ( 2 \pi  f 
\lambda T) \hspace{0.05cm}.$$
*${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$&nbsp; takes into account the spectral shaping by &nbsp;$g(t)$. The narrower this is, the wider &nbsp;$\vert G(f) \vert^2$&nbsp; and thus the larger the bandwidth requirement: 
:$$\varphi^{^{\hspace{0.05cm}\bullet}}_{g}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}
  {\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f)  = |G(f)|^2
  \hspace{0.05cm}.$$
*The energy spectral density ${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$&nbsp; has the unit &nbsp;$\rm Ws/Hz$&nbsp; and the power spectral density&nbsp; ${\it \Phi_{s}}(f)$&nbsp; after division by the symbol spacing &nbsp;$T$&nbsp; has the unit &nbsp;$\rm W/Hz$. 





==  Power-spectral density of the AMI code==
<br>
The frequency response of the linear code network of a pseudo-ternary code is generally:
:$$H_{\rm C}(f) = {1}/{2} \cdot \big [1 - K_{\rm C} \cdot {\rm
e}^{-{\rm j}\hspace{0.03cm}\cdot \hspace{0.03cm}
2\pi\hspace{0.03cm}\cdot \hspace{0.03cm}f \hspace{0.03cm}\cdot
\hspace{0.03cm} N_{\rm C}\hspace{0.03cm}\cdot \hspace{0.03cm}T}
\big] ={1}/{2} \cdot \big [1 - K \cdot {\rm
e}^{-{\rm j}\hspace{0.03cm}\cdot \hspace{0.03cm}
\alpha}
\big ]\hspace{0.05cm}.$$

This gives the power-spectral density&nbsp; $\rm (PSD)$&nbsp; of the amplitude coefficients &nbsp;$(K$&nbsp; and &nbsp;$\alpha$&nbsp; are abbreviations according to the above equation$)$:
:$$  {\it \Phi}_a(f)  =   | H_{\rm C}(f)|^2 =  \frac{\big [1 - K \cos
  (\alpha) + {\rm j}\cdot K \sin  (\alpha) \big ] \big [1 - K  \cos
  (\alpha) - {\rm j}\cdot K \sin  (\alpha) \big ] }{4} =  \text{...}  =  {1}/{4} \cdot \big [2 - 2 \cdot K \cdot \cos
  (\alpha)  \big ] $$
[[File:P_ID1347__Dig_T_2_4_S2b_v2.png|right|frame|Power-spectral density of the AMI code|class=fit]]
:$$  \Rightarrow \hspace{0.3cm}{\it \Phi}_a(f)  =  | H_{\rm C}(f)|^2 =  {1}/{2} \cdot \big [1 - K_{\rm C} \cdot \cos
  (2\pi  f    N_{\rm C}  T)\big ]
 \hspace{0.4cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.4cm}
\varphi_a(\lambda \cdot T)\hspace{0.05cm}.$$

In particular,&nbsp; for the power-spectral density of the AMI code&nbsp; $(N_{\rm C} = K_{\rm C} = 1)$,&nbsp; we obtain:
:$${\it \Phi}_a(f)  =   {1}/{2} \cdot \big [1 - \cos
  (2\pi  f    T)\big ] = \sin^2
  (\pi  f    T)\hspace{0.05cm}.$$

The graph shows
*the PSD &nbsp;${\it \Phi}_a(f)$&nbsp; of the amplitude coefficients&nbsp; (red curve),&nbsp; and<br>

*the PSD &nbsp;${\it \Phi}_s(f)$&nbsp; of the total transmitted signal&nbsp; (blue),&nbsp; valid for NRZ rectangular pulses.<br>


One recognizes from this representation
*that the AMI code has no DC component,&nbsp; since &nbsp;${\it \Phi}_a(f = 0) = {\it \Phi}_s(f = 0) = 0$,&nbsp; <br>

*the power &nbsp;$P_{\rm S} = s_0^2/2$&nbsp; of the AMI-coded transmitted signal&nbsp; $($integral over &nbsp;${\it \Phi}_s(f)$&nbsp; from &nbsp;$- \infty$&nbsp; to &nbsp;$+\infty)$.


Notes:
*The PSD of the HDB3 and B6ZS codes differs only slightly from that of the AMI code.<br>

*You can use the&nbsp; (German language)&nbsp; SWF applet &nbsp;[[Applets:Pseudoternaercodierung|"Signals, ACF, and PSD of pseudo-ternary codes"]]&nbsp; to clarify the topic discussed here.


== Properties of the duobinary code ==
<br>
The&nbsp; '''duobinary code'''&nbsp; is defined by the code parameters &nbsp;$N_{\rm C} = 1$&nbsp; and &nbsp;$K_{\rm C} = -1$.&nbsp; This gives the power-spectral density&nbsp; $\rm (PSD)$&nbsp; of the amplitude coefficients and the PSD of the transmitted signal:
[[File:P_ID1348__Dig_T_2_4_S3b_v1.png|right|frame|Power-spectral density of the duobinary code|right|class=fit]]

:$${\it \Phi}_a(f) ={1}/{2} \cdot \big [1 + \cos
  (2\pi  f    T)\big ] = \cos^2
  (\pi  f    T)\hspace{0.05cm},$$
:$$  {\it \Phi}_s(f)  =    s_0^2 \cdot T  \cdot \cos^2
  (\pi  f    T)\cdot {\rm si}^2
  (\pi  f    T)=    s_0^2 \cdot T  \cdot {\rm si}^2
  (2 \pi  f    T)  \hspace{0.05cm}.$$

The graph shows the power-spectral density
*of the amplitude coefficients &nbsp; &#8658; &nbsp; ${\it \Phi}_a(f)$&nbsp;  as a red curve,<br>

*of the total transmitted signal &nbsp; &#8658; &nbsp; ${\it \Phi}_s(f)$&nbsp; as a blue curve.<br><br>

In the second graph,&nbsp; the signals &nbsp;$q(t)$, &nbsp;$b(t)$&nbsp; and &nbsp;$c(t) = s(t)$&nbsp; are sketched. We refer here again to the&nbsp; (German language)&nbsp; SWF applet &nbsp;[[Applets:Pseudoternaercodierung|"Signals, ACF, and PSD of pseudo-ternary codes"]],&nbsp; which also clarifies the duobinary code.

[[File:P_ID1349__Dig_T_2_4_S3a_v2.png|left|frame|Signals in duobinary coding|class=fit]]  
<br><br><br><br><br><br>From these illustrations it is clear:
*In the duobinary code,&nbsp; any number of symbols with same polarity&nbsp; ("+1"&nbsp; or&nbsp; "&ndash;1")&nbsp; can directly succeed each other &nbsp; &rArr;  &nbsp; ${\it \Phi}_a(f = 0)=1$,&nbsp; ${\it \Phi}_s(f = 0) = 1/2 \cdot s_0^2 \cdot T$.<br>

*In contrast,&nbsp; for the duobinary code,&nbsp; the alternating sequence&nbsp;  "... , +1, &ndash;1, +1, &ndash;1, +1, ..."&nbsp;  does not occur,&nbsp; which is particularly disturbing with respect to intersymbol interference.&nbsp; Therefore,&nbsp; in the duobinary code: &nbsp;${\it \Phi}_s(f = 1/(2T) = 0$.<br>

*The power-spectral density &nbsp;${\it \Phi}_s(f)$&nbsp; of the pseudo-ternary duobinary code is identical to the PSD with redundancy-free binary coding at half rate $($symbol duration &nbsp;$2T)$.<br>
<br clear= all>


==Exercises==

<br>
*First, select the number&nbsp; $(1,\ 2,  \text{...} \ )$&nbsp; of the task to be processed.&nbsp; The number&nbsp; "$0$"&nbsp; corresponds to a "Reset":&nbsp; Same setting as at program start.
*A task description is displayed.&nbsp; The parameter values are adjusted.&nbsp; Solution after pressing "Show Solution".


{{BlueBox|TEXT=
'''(1)'''&nbsp; Consider and interpret the binary pre&ndash;coding of the&nbsp; $\text{AMI}$&nbsp; code using the source symbol sequence&nbsp; $\rm C$&nbsp; assuming&nbsp; $b_0 = +1$.  }}
*The modulo&ndash;2 addition can also be taken as "antivalence".&nbsp; It holds&nbsp; $b_{\nu} = +1$,&nbsp; if&nbsp; $q_{\nu}$&nbsp; and&nbsp; $b_{\nu - 1}$&nbsp; differ, otherwise set&nbsp; $b_{\nu} = -1$&nbsp;:
:&nbsp; $b_1 = (q_1 = +1)\ {\rm XOR}\ (b_0= +1) = -1,\ \ b_2 = (q_2 = -1)\ {\rm XOR}\ (b_1= -1) = -1,\ \ b_3 = (q_3 = -1)\ {\rm XOR}\ (b_2= -1) = -1,$ 
:&nbsp; $b_4 = (q_4 = +1)\ {\rm XOR}\ (b_3= -1) = +1,\ \ b_5 = (q_5 = +1)\ {\rm XOR}\ (b_4= +1) = -1,\ \ b_6 = (q_6 = +1)\ {\rm XOR}\ (b_5= -1) = +1,\ \ b_7 = b_8 = \text{...} = -1.$
*With the initial condition&nbsp; $b_0 = -1$&nbsp; we get the negated sequence:&nbsp; $b_4 = b_6 =-1$.&nbsp; All others&nbsp; $b_\nu = +1$. 


{{BlueBox|TEXT=
'''(2)'''&nbsp; Let&nbsp; $b_0 = +1$.&nbsp; Consider the AMI encoded sequence&nbsp; $\langle c_\nu \rangle$&nbsp; of the source symbol sequence&nbsp; $\rm C$&nbsp; and give their amplitude coefficients&nbsp; $a_\nu$.&nbsp; }}

*It holds:&nbsp; $a_1= 0.5 \cdot (b_1-b_0) = -1$,&nbsp; $a_2= 0.5 \cdot (b_2-b_1) =0$,&nbsp; $a_3= 0. 5 \cdot (b_3-b_2) =0$,&nbsp; $a_4= +1$,&nbsp; $a_5= -1$,&nbsp; $a_6= +1$,&nbsp; $a_7= -1$,&nbsp; $a_8= a_9 = \text{...} = 0$.  
*In contrast to the pre&ndash;coding, the conventional addition (subtraction) is to be applied here and not the modulo&ndash;2 addition.


{{BlueBox|TEXT=
'''(3)'''&nbsp; Now consider the AMI coding for several random sequences.&nbsp; What rules can be derived from these experiments for the amplitude coefficients&nbsp; $a_\nu$ ?}}

*Each binary value&nbsp; "&ndash;1"&nbsp; of &nbsp; $q(t)$ &nbsp; &rArr; &nbsp; symbol &nbsp;$\rm L$&nbsp; is encoded by the ternary coefficient &nbsp;$a_\nu = 0$.&nbsp; Any number of &nbsp;$a_\nu = 0$&nbsp; can be consecutive. 
*The binary value&nbsp; "+1"&nbsp; of &nbsp; $q(t)$ &nbsp; &rArr; &nbsp; symbol &nbsp;$\rm H$&nbsp; is  represented alternatively with &nbsp; $a_\nu = +1$&nbsp; and &nbsp;$a_\nu = -1$&nbsp;, starting with&nbsp; $a_\nu = -1$,&nbsp; if&nbsp; $b_0 = +1$.
*From the source symbol sequence&nbsp; $\rm A$ &nbsp; &rArr; &nbsp; $\langle \hspace{0.05cm}q_\nu \equiv +1 \hspace{0.05cm}\rangle$&nbsp; the code symbol sequence&nbsp; $+1, -1, +1, -1, \text{...}$&nbsp;. Long sequences&nbsp; $\langle \hspace{0.05cm}a_\nu \equiv +1 \hspace{0.05cm}\rangle$&nbsp; or &nbsp; $\langle \hspace{0.05cm}a_\nu \equiv -1 \hspace{0.05cm}\rangle$&nbsp; are shot out.


{{BlueBox|TEXT=
'''(4)'''&nbsp; Continue with&nbsp; AMI&nbsp; coding.&nbsp; Interpret the autocorrelation function&nbsp; $\varphi_a(\lambda)$&nbsp; of the amplitude coefficients and the power density spectrum&nbsp; ${\it \Phi}_a(f)$. }}
*The discrete ACF&nbsp; $\varphi_a(\lambda)$&nbsp; of the amplitude coefficients is only defined for integer&nbsp; $\lambda$&nbsp; values. &nbsp; With AMI coding&nbsp; $(N_{\rm C}=1)$&nbsp; holds: &nbsp; For&nbsp; $|\lambda| > 1$ &nbsp; &rArr; &nbsp; all&nbsp; $\varphi_a(\lambda)= 0$.
*$\varphi_a(\lambda = 0)$&nbsp; is equal to the root mean square of the amplitude coefficients &nbsp; &rArr; &nbsp; $\varphi_a(\lambda = 0) = {\rm Pr}(a_\nu = +1) \cdot (+1)^2 + {\rm Pr}(a_\nu = -1) \cdot (-1)^2 = 0.5.$
*Only the combinations&nbsp; $(+1, -1)$&nbsp; and&nbsp; $(-1, +1)$&nbsp; contribute to the expected value&nbsp; ${\rm E}\big [a_\nu \cdot a_{\nu+1}\big]$&nbsp;&nbsp; Result:&nbsp; $\varphi_a(\lambda = \pm 1)={\rm E}\big [a_\nu \cdot a_{\nu+1}\big]=-0.25.$
*The power density spectrum&nbsp; ${\it \Phi}_a(f)$&nbsp; is the Fourier transform of the discrete ACF&nbsp; $\varphi_a(\lambda)$.&nbsp; Result:&nbsp; ${\it \Phi}_a(f) = {1}/{2} \cdot \big [1 - \cos (2\pi f T)\big ] = \sin^2
  (\pi f T)\hspace{0.05cm}.$
*From &nbsp; ${\it \Phi}_a(f = 0) = 0$&nbsp; follows: &nbsp; The AMI code is especially interesting for channels over which no DC component can be transmitted.


{{BlueBox|TEXT=
'''(5)'''&nbsp; We consider further AMI coding and rectangular pulses.&nbsp; Interpret the ACF&nbsp; $\varphi_s(\tau)$&nbsp; of the transmission signal and the PDS&nbsp; ${\it \Phi}_s(f)$. }}
*$\varphi_s(\tau)$&nbsp; results from the convolution of the discrete AKF&nbsp; $\varphi_a(\lambda)$&nbsp; with&nbsp; $\varphi^{^{\hspace{0.05cm}\bullet}}_{g}(\tau)$. &nbsp; For rectangular pulses&nbsp; $($duration $T)$:&nbsp; The energy&ndash;AKF&nbsp; $\varphi^{^{\hspace{0.05cm}\bullet}}_{g}(\tau)$&nbsp; is a triangle of duration&nbsp; $2T$.
*It holds&nbsp; $\varphi_s(\tau = 0)= \varphi_a(\lambda = 0) =0.5, \ \varphi_s(\pm T)= \varphi_a( 1) =-0. 25,\ , \ \varphi_s( \pm 2T)= \varphi_a(2) =0.$&nbsp; Between these discrete values, $\varphi_{s}(\tau)$&nbsp; is always linear.
*The PDS&nbsp; ${\it \Phi}_s(f)$&nbsp; is obtained from&nbsp; ${\it \Phi}_a(f) = \sin^2(\pi f T)$&nbsp; by multiplying with&nbsp; ${\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f)  = {\rm sinc}^2(f T).$&nbsp; This does not change anything at the zeros of&nbsp; ${\it \Phi}_a(f)$&nbsp;. 


{{BlueBox|TEXT=
'''(6)'''&nbsp; What changes with respect to&nbsp; $s(t)$,&nbsp; $\varphi_s(\tau)$&nbsp; and&nbsp; ${\it \Phi}_s(f)$&nbsp; with the&nbsp; $\rm Nyquist$&nbsp; pulse?&nbsp; Vary the roll&ndash;off factor here in the range&nbsp; $0 \le r \le 1$.}}

*A single Nyquist pulse can be represented with the source symbol sequence&nbsp; $\rm B$&nbsp; in the&nbsp; $s(t)$&nbsp; range.&nbsp; You can see equidistant zero crossings in the distance&nbsp; $T$.   
*Also, for any AMI random sequence, the signal values&nbsp; $s(t=\nu \cdot T)$&nbsp; for each&nbsp; $r$&nbsp; correspond exactly to their nominal positions.&nbsp; Outside these points, there are deviations.
*In the special case&nbsp; $r=0$&nbsp; the energy&ndash;LDS&nbsp; ${\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f)$&nbsp; is constant in the range&nbsp; $|f|<1/2T$.&nbsp; Accordingly, the energy&ndash;ACF&nbsp; ${\it \varphi}^{^{\hspace{0.08cm}\bullet}}_{g}(\tau)$&nbsp; has a&nbsp; $\rm sinc$&nbsp; shape.   
*On the other hand, for larger&nbsp; $r$&nbsp; the zeros of&nbsp; ${\it \varphi}^{^{\hspace{0.08cm}\bullet}}_{g}(\tau)$&nbsp; are no longer equidistant, since although&nbsp; $G(f)$&nbsp; satisfies the first Nyquist criterion, it does not&nbsp; ${\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f)= [G(f)]^2$.
*The main advantage of the Nyquist pulse is the much smaller bandwidth.&nbsp; Here only the frequency range&nbsp; $|f| < (1+r)/(2T)$&nbsp; has to be provided.


{{BlueBox|TEXT=
'''(7)'''&nbsp; Repeat the last experiment using the&nbsp; $\text{Root raised cosine}$&nbsp; pulse instead of the Nyquist pulse.&nbsp; Interpret the results. }}

*In the special case&nbsp; $r=0$&nbsp; the results are as in&nbsp; '''(6)'''.&nbsp;${\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f)$&nbsp; is constant in the range&nbsp; $|f|<1/2T$&nbsp; and outside zero;&nbsp; ${\it \varphi}^{^{\hspace{0.08cm}\bullet}}_{g}(\tau)$&nbsp; has a&nbsp; $\rm sinc$&nbsp; shape.
*Also for larger&nbsp; $r$&nbsp; the zeros of&nbsp; ${\it \varphi}^{^{\hspace{0.08cm}\bullet}}_{g}(\tau)$&nbsp; are eqidistant&nbsp; (but not&nbsp; $\rm sinc$&nbsp; shaped) &nbsp; &rArr; &nbsp; ${\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f)= [G(f)]^2$&nbsp; satisfies the first Nyquist criterion. 
*On the other hand, $G(f)$&nbsp; does not satisfy the first Nyquist criterion&nbsp; $($except for&nbsp; $r=0)$.&nbsp; Intersymbol interference occurs already at the transmitter &nbsp; &rArr; &nbsp;  signal&nbsp; $s(t)$. 
*But this is also not a fundamental problem.&nbsp; By using an identically shaped reception filter like&nbsp; $G(f)$&nbsp; intersymbol interference at the decider is avoided.


{{BlueBox|TEXT=
'''(8)'''&nbsp; Consider and check the pre&ndash;coding&nbsp; $(b_\nu)$&nbsp; and the amplitude coefficients&nbsp; $(a_\nu)$&nbsp; with the&nbsp; $\rm duobinary$&nbsp; coding &nbsp; $($source symbol sequence&nbsp; $\rm C$,&nbsp; $b_0 = +1)$.  }}
*$b_1 = (q_1 = +1)\ {\rm XOR}\ (\overline{b_0}= -1) = +1,\ \ b_2 = (q_2 = -1)\ {\rm XOR}\ (\overline{b_1}= -1) = -1,\ \ b_3 = \text{...} =b_7 = +1,$&nbsp; $b_8 = b_{10} = \text{...} =-1$,&nbsp; $b_9 =b_{11} = \text{...}= +1$.
*$a_1= 0.5 \cdot (b_1+b_0) = +1$,&nbsp; $a_2= 0.5 \cdot (b_2+b_1) =0$,&nbsp; $a_3= 0.5 \cdot (b_3+b_2) = 0$,&nbsp; $a_4= \text{...}= a_7=+1$,&nbsp; $a_8=a_9= \text{...}= 0$.
*With the starting condition&nbsp; $b_0 = -1$&nbsp; we get again the negated sequence: &nbsp; &nbsp; $a_1= -1$,&nbsp; $a_2= a_3= 0$,&nbsp; $a_4= \text{...}= a_7=-1$,&nbsp; $a_8=a_9= \text{...}= 0$.


{{BlueBox|TEXT=
'''(9)'''&nbsp; Now consider the Duobinary coding for several random sequences.&nbsp; What rules can be derived from these experiments for the amplitude coefficients&nbsp; $a_\nu$?}}

*The discrete ACF values are&nbsp; $\varphi_a(\lambda = 0) = +0.5$,&nbsp; $\varphi_a(\lambda = 1) = +0.25$,&nbsp; $\varphi_a(\lambda = 2) = 0$ &nbsp; &rArr; &nbsp; ${\it \Phi}_a(f) = {1}/{2} \cdot \big [1 + \cos (2\pi f T)\big ] = \cos^2
  (\pi f T)\hspace{0.05cm}.$
*Unlike the AMI coding, the encoded sequences&nbsp; $\langle \hspace{0.05cm}a_\nu \equiv +1 \hspace{0.05cm}\rangle$&nbsp; and &nbsp; $\langle \hspace{0.05cm}a_\nu \equiv -1 \hspace{0.05cm}\rangle$&nbsp; are  possible here &nbsp; &rArr; &nbsp; For the Duobinary code holds&nbsp; ${\it \Phi}_a(f= 0) = 1 \ (\ne 0).$
*As with the AMI code, the "long zero sequence" &nbsp; &rArr; &nbsp; $\langle \hspace{0.05cm}a_\nu \equiv 0 \hspace{0.05cm}\rangle$&nbsp; is possible, which can again lead to synchronization problems.
*Excluded are the combinations&nbsp; $a_\nu = +1, \ a_{\nu+1} = -1$&nbsp; and &nbsp; $a_\nu = -1, \ a_{\nu+1} = +1$,&nbsp; recognizable by the PDS value&nbsp; ${\it \Phi}_a(f= 1/(2T)) = 0.$    
*Such direct transitions&nbsp; $a_\nu = +1$ &nbsp; &rArr; &nbsp; $a_{\nu+1} = -1$&nbsp; resp. &nbsp; $a_\nu = -1$ &nbsp; &rArr; &nbsp; $a_{\nu+1} = +1$&nbsp; lead to large intersymbol interference and thus to a higher error rate.


{{BlueBox|TEXT=
'''(10)'''&nbsp; Compare the coding results of second order bipolar code&nbsp; $\rm (BIP2)$&nbsp; and first order bipolar code&nbsp; $\rm (AMI)$&nbsp; for different source symbol sequences. }}
*For a single rectangular pulse &nbsp; &rArr; &nbsp; source symbol sequence&nbsp; $\rm B$&nbsp; both codes result in the same encoded sequence and the same encoded signal&nbsp; $c(t)$ &nbsp; &rArr; &nbsp; also an isolated pulse.
*The  "Permanent&ndash;One sequence"&nbsp; $\rm A$&nbsp; now results&nbsp; $\langle c_\nu \rangle = \langle -1, -1, +1, +1, -1, -1, +1, +1, \text{. ..}\rangle $&nbsp; instead of &nbsp; $\langle c_\nu \rangle = \langle -1, +1, -1, +1, -1, +1, \text{...}\rangle $.
*The simple decoding algorithm of the AMI code&nbsp; $($the ternary&nbsp; $0$&nbsp; becomes the binary&nbsp; $-1$,&nbsp; the ternary&nbsp; $\pm 1$&nbsp; becomes the binary&nbsp; $+1)$&nbsp; cannot be applied to&nbsp; $\rm BIP2$&nbsp;.


{{BlueBox|TEXT=
'''(11)'''&nbsp; View and interpret the various ACF and LDS graphs of the&nbsp; $\rm BIP2$&nbsp; compared to the&nbsp; $\rm AMI$&nbsp; code.}}  
*For&nbsp; $\rm AMI$: &nbsp;  $\varphi_a(\lambda = \pm 1) = -0.25, \ \varphi_a(\lambda = \pm 2) = 0$. &nbsp; For&nbsp; $\rm BIP2$: &nbsp;  $\varphi_a(\lambda = \pm 1) = 0, \ \varphi_a(\lambda = \pm 2) = -0.25$.&nbsp; In both cases:&nbsp; $\varphi_a(\lambda = 0) = 0.5$.
*From the&nbsp; $\rm AMI$ power density spectrum&nbsp; ${\it \Phi}_a(f) = \sin^2 (\pi \cdot f T)$&nbsp; follows for&nbsp; $\rm BIP2$:&nbsp; ${\it \Phi}_a(f) = \sin^2 (2\pi \cdot f T)$&nbsp; by compression with respect to the&nbsp; $f$&ndash;axis.
*Zero at&nbsp; $f=0$:&nbsp; At most two&nbsp; $+1$&nbsp; directly follow each other, and also at most only two&nbsp; $-1$.&nbsp; In the AMI code&nbsp; $+1$&nbsp; and&nbsp; $-1$&nbsp; occur only in isolation. 
*Next zero at&nbsp; $f=1/(2T)$:&nbsp; The infinitely long&nbsp; $(+1, -1)$&nbsp; sequence is excluded in&nbsp; $\rm BIP2$&nbsp; as in the&nbsp; $\rm Duobinary$&nbsp; code.
*Consider and interpret also the functions&nbsp; $\varphi_s(\tau)$&nbsp; and&nbsp; ${\it \Phi}_s(f)$&nbsp; for the pulses "rectangle", "Nyquist" and "Root raised cosine".


==Applet Manual==
[[File:BS_Pseudoternär.png|right|600px|frame|Bildschirmabzug (deutsche Version, heller Hintergrund)]]
<br>
&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Theme (changeable graphical user interface design)
:* Dark: &nbsp; dark background&nbsp; (recommended by the authors)
:*  Bright: &nbsp; white background&nbsp; (recommended for beamers and printouts)
:*  Deuteranopia: &nbsp; for users with pronounced green visual impairment
:*  Protanopia: &nbsp; for users with pronounced red visual impairment

&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Zugrundeliegendes Blockschaltbild 

&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Auswahl des Pseudoternörcodes: 
<br>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; AMI&ndash;Code, Duobinärcode, Bipolarcode 2. Ordnung  

&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Auswahl des Grundimpulses&nbsp; $g(t)$:
<br>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Rechteckimpuls, Nyquistimpuls, Wurzel&ndash;Nyquistimpuls  

&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Rolloff&ndash;Faktor (Frequenzbereich) für "Nyquist" und "Wurzel&ndash;Nyquist" 

&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Einstellung von&nbsp; $3 \cdot 4 = 12$&nbsp; Bit der Quellensymbolfolge

&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Auswahl einer drei voreingestellten Quellensymbolfolgen 

&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Zufällige binäre Quellensymbolfolge 

&nbsp; &nbsp; '''( I )''' &nbsp; &nbsp; Schrittweise Verdeutlichung der Pseudoternärcodierung 

&nbsp; &nbsp; '''(J)''' &nbsp; &nbsp; Ergebnis der Pseudoternärcodierung:&nbsp; Signale&nbsp; $q(t)$,&nbsp; $b(t)$,&nbsp; $c(t)$,&nbsp; $s(t)$

&nbsp; &nbsp; '''(K)''' &nbsp; &nbsp; Löschen der Signalverläufe im Grafikbereich&nbsp; $\rm M$

&nbsp; &nbsp; '''(L)''' &nbsp; &nbsp; Skizzen für Autokorrelationsfunktion & Leistungsdichtespektrum

&nbsp; &nbsp; '''(M)''' &nbsp; &nbsp; Grafikbereich:&nbsp; Quellensignal&nbsp; $q(t)$, Signal&nbsp; $b(t)$&nbsp; nach Vorcodierung, <br>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;  Codersignal&nbsp; $c(t)$&nbsp; mit Rechtecken, Sendesignal&nbsp; $s(t)$&nbsp; gemäß&nbsp; $g(t)$ 

&nbsp; &nbsp; '''(N)''' &nbsp; &nbsp; Bereich für Übungen:&nbsp; Aufgabenauswahl, Fragen, Musterlösung
<br clear=all>
==About the Authors==
This interactive calculation tool was designed and implemented at the&nbsp; [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]&nbsp; at the&nbsp; [https://www.tum.de/en Technical University of Munich]. 
*The first version was created in 2010 by&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Stefan_M.C3.BCller_.28Diplomarbeit_LB_2010.29|Stefan Müller]]&nbsp; &nbsp; 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&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; as part of her bachelor thesis&nbsp; (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&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; in the context of a working student activity.&nbsp; 


The conversion of this applet to HTML 5 was financially supported by&nbsp; [https://www.ei.tum.de/studium/studienzuschuesse/ "Studienzuschüsse"]&nbsp; (Faculty EI of the TU Munich).&nbsp; We thank.
<br><br>

==Once again: Open Applet in new Tab==

{{LntAppletLinkEnDe|pseudoternarycodes_en|pseudoternarycodes}}