Applets:Principle of 4B3T Coding

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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:

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

The best known block code for transmission coding is the   4B3T code   with the code parameters

$$m_q = 4,\hspace{0.2cm}M_q = 2,\hspace{0.2cm}m_c = 3,\hspace{0.2cm}M_c = 3\hspace{0.05cm},$$

which was developed in the 1970s and is used,  for example,  in  "ISDN"  ("Integrated Services Digital Networks").

Such a 4B3T code has the following properties:

• Because of  $m_q \cdot T_{\rm B} = m_c \cdot T$,  the symbol duration  $T$  of the ternary encoded signal is larger than the bit duration  $T_{\rm B}$  of the binary source signal by a factor of  $4/3$.  This results in the favorable property that the bandwidth requirement is a quarter less than for redundancy-free binary transmission.

• The relative redundancy can be calculated with the above equation and results in  $r_c \approx 16\%$.  This redundancy is used in the 4B3T code to achieve DC freedom.

• The 4B3T encoder signal can thus also be transmitted over a channel  (German:  "Kanal"   ⇒  subscript:  "K")  with the property  $H_{\rm K}(f= 0) = 0$  without noticeable degradation.

The encoding of the sixteen possible binary blocks into the corresponding ternary blocks could in principle be performed according to a fixed code table. To further improve the spectral properties of these codes, the common 4B3T codes, viz.

1. the 4B3T code according to Jessop and Waters,
   
2. the MS43 code (from:  $\rm M$onitored $\rm S$um $\rm 4$B$\rm 3$T Code),
   
3. the FoMoT code (from:  $\rm Fo$ur $\rm Mo$de $\rm T$ernary),
   
   

two or more code tables are used,  the selection of which is controlled by the  "running digital sum"  of the amplitude coefficients.  The principle is explained in the next section.

Running digital sum

After the transmission of  $l$  coded blocks,  the  "running digital sum"  with ternary amplitude coefficients  $a_\nu \in \{ -1, \ 0, +1\}$:

Code tables for three 4B3T codes

$${\it \Sigma}_l = \sum_{\nu = 1}^{3 \hspace{0.02cm}\cdot \hspace{0.05cm} l}\hspace{0.02cm} a_\nu \hspace{0.05cm}.$$

The selection of the table for encoding the  $(l + 1)$–th block is done depending on the current   ${\it \Sigma}_l$  value.

The table shows the coding rules for the three 4B3T codes mentioned above.  To simplify the notation,

•   "+" stands for the amplitude coefficient "+1" and
•   "–" for the coefficient "–1".
  

You can see from the graph:

1. The two code tables of the Jessop–Waters code are selected in such a way that the running digital sum  ${\it \Sigma}_l$  always lies between  $0$  and  $5$.
   
   
2. For the other two codes  (MS43,  FoMoT),  the restriction of the running digital sum to the range  $0 \le {\it \Sigma}_l \le 3$  is achieved by three resp. four alternative tables.

ACF and PSD of the 4B3T codes

The procedure for calculating the auto-correlation function  $\rm (ACF)$  and the power-spectral density  $\rm (PSD)$  is only outlined here in bullet points:

Markov diagram for the analysis of the 4B3T FoMoT code

(1)   The transition of the running digital sum from  ${\it \Sigma}_l$  to  ${\it \Sigma}_{l+1}$  is described by a homogeneous stationary first-order Markov chain with six  $($Jessop–Waters$)$  or four states  $($MS43, FoMoT$)$.  For the FoMoT code, the Markov diagram sketched on the right applies.

(2)   The values at the arrows denote the transition probabilities  ${\rm Pr}({\it \Sigma}_{l+1}|{\it \Sigma}_{l})$,  resulting from the respective code tables.  The colors correspond to the backgrounds of the table on the last section.  Due to the symmetry of the FoMoT Markov diagram,  the four probabilities are all the same:

$${\rm Pr}({\it \Sigma}_{l} = 0) = \text{...} = {\rm Pr}({\it \Sigma}_{l} = 3) = 1/4.$$

(3)   The auto-correlation function  $\varphi_a(\lambda) = {\rm E}\big [a_\nu \cdot a_{\nu+\lambda}\big ]$  of the amplitude coefficients can be determined from this diagram.  Simpler than the analytical calculation,  which requires a very large computational effort,  is the simulative determination of the ACF values by computer.

Fourier transforming the ACF yields the power-spectral density  ${\it \Phi}_a(f)$  of the amplitude coefficients corresponding to the following graph from  [TS87].  The outlined PSD was determined for the FoMoT code,  whose Markov diagram is shown above.  The differences between the individual 4B3T codes are not particularly pronounced.  Thus,  for the MS43 code  ${\rm E}\big [a_\nu^2 \big ] \approx 0.65$  and for the other two 4B3T codes  (Jessop/Waters, MS43)   ${\rm E}\big [a_\nu^2 \big ] \approx 0.69$.
The statements of this graph can be summarized as follows:

Power-spectral density (of amplitude coefficients) of 4B3T compared to redundancy-free and AMI coding

• The graph shows the power-spectral density  ${\it \Phi}_a(f)$  of the amplitude coefficients  $a_\nu$  of the 4B3T code   ⇒   red curve.

• The PSD  ${\it \Phi}_s(f)$  including the transmission pulse is obtained by multiplying by  $1/T \cdot |G_s(f)|^2$   ⇒   ${\it \Phi}_a(f)$  must be multiplied by a  $\rm sinc^2$ function, if  $g_s(t)$  describes a rectangular pulse.
  

• Redundancy-free binary or ternary coding results in a constant  ${\it \Phi}_a(f)$  in each case,  the magnitude of which depends on the number  $M$  of levels  (different signal power).

• In contrast,  the 4B3T power-spectral density has zeros at  $f = 0$  and multiples of  $f = 1/T$. 
  

• The zero point at  $f = 0$  has the advantage that the 4B3T signal can also be transmitted without major losses via a so-called  "telephone channel",  which is not suitable for a DC signal due to transformers.
  

• The zero point at  $f = 1/T$  has the disadvantage that this makes clock recovery at the receiver more difficult.  Outside of these zeros,  the 4B3T codes have a flatter  ${\it \Phi}_a(f)$  than the  "AMI code"  discussed in the next chapter  (blue curve), which is advantageous.
  

• The reason for the flatter PSD curve at medium frequencies as well as the steeper drop towards the zeros is that for the 4B3T codes up to five  $+1$  coefficients  (resp.  $-1$ coefficients)  can follow each other.  With the AMI code,  these symbols occur only in isolation.
  

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.

(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) $   ⇒   ${\it \Sigma}_3= 2.$

(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) $   ⇒   ${\it \Sigma}_3= 2.$

(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.

(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.

(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.

(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.

(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

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    (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  Institute for Communications Engineering  at the  Technical University of Munich.

• The first version was created in 2010 by Stefan Müller  as part of his bachelor thesis with “FlashMX – Actionscript” (Supervisor: Günter Söder).

• Last revision and English version 2020/2021 by  Carolin Mirschina  in the context of a working student activity. 

The conversion of this applet to HTML 5 was financially supported by  Studienzuschüsse  ("study grants")  of the TUM Faculty EI.  We thank.

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