Digital Signal Transmission/Block Coding with 4B3T Codes - Revision history

Guenter at 15:19, 24 August 2022

2022-08-24T15:19:23Z

Guenter at 15:22, 3 June 2022

2022-06-03T15:22:06Z

Hwang at 08:37, 24 May 2022

2022-05-24T08:37:42Z

Hwang at 08:36, 24 May 2022

2022-05-24T08:36:11Z

Hwang at 08:34, 24 May 2022

2022-05-24T08:34:56Z

Hwang at 08:31, 24 May 2022

2022-05-24T08:31:09Z

Hwang at 08:27, 24 May 2022

2022-05-24T08:27:06Z

Guenter at 14:13, 18 May 2022

2022-05-18T14:13:29Z

Guenter at 12:47, 18 May 2022

2022-05-18T12:47:56Z

Guenter at 12:46, 18 May 2022

2022-05-18T12:46:48Z

@@ Line 76: / Line 76: @@
 The statements of this graph can be summarized as follows:
-[[File:P_ID1336__Dig_T_2_3_S3b_v1.png|right|frame|Power-spectral density (of amplitude coefficients) of 4B3T compared to redundancy-free and AMI coding '''Korrektur'''|class=fit]]
+[[File:EN_Dig_T_2_4_S3b_v23.png|right|frame|Power-spectral density (of amplitude coefficients) of 4B3T compared to redundancy-free and AMI coding|class=fit]]
 *The graph shows the power-spectral density &nbsp;${\it \Phi}_a(f)$&nbsp; of the amplitude coefficients &nbsp;$a_\nu$&nbsp; of the 4B3T code &nbsp; &rArr; &nbsp; red curve.

@@ Line 9: / Line 9: @@
 == General description of block codes ==
 <br>
-In &nbsp;'''block coding'''&nbsp; each sequence of &nbsp;$m_q$&nbsp; binary source symbols &nbsp;$(M_q = 2)$&nbsp; is represented by a block of &nbsp;$m_c$&nbsp; coded '''KORREKTUR: encoder''' symbols with symbol set size &nbsp;$M_c$.&nbsp; In order to convert each source symbol sequence &nbsp;$\langle q_\nu \rangle$&nbsp; into another encoder symbol sequence &nbsp;$\langle c_\nu \rangle$,&nbsp; the following condition must be satisfied:
+In &nbsp;'''block coding'''&nbsp; each sequence of &nbsp;$m_q$&nbsp; binary source symbols &nbsp;$(M_q = 2)$&nbsp; is represented by a block of &nbsp;$m_c$&nbsp; encoder symbols with symbol set size &nbsp;$M_c$.&nbsp; In order to convert each source symbol sequence &nbsp;$\langle q_\nu \rangle$&nbsp; into another encoder symbol sequence &nbsp;$\langle c_\nu \rangle$,&nbsp; the following condition must be satisfied:
 :$$M_c^{\hspace{0.1cm}m_c} \ge
 M_q^{\hspace{0.1cm}m_q}\hspace{0.05cm}.$$
@@ Line 25: / Line 25: @@
 Such a 4B3T code has the following properties:
-*Because of &nbsp;$m_q \cdot T_{\rm B} =  m_c \cdot T$,&nbsp; the symbol duration &nbsp;$T$&nbsp; of the ternary encoder '''KORREKTUR: encoded''' signal is larger than the bit duration &nbsp;$T_{\rm B}$&nbsp; of the binary source signal by a factor of &nbsp;$4/3$.&nbsp; This results in the favorable property that the bandwidth requirement is a quarter less than for redundancy-free binary transmission.
+*Because of &nbsp;$m_q \cdot T_{\rm B} =  m_c \cdot T$,&nbsp; the symbol duration &nbsp;$T$&nbsp; of the ternary encoded  signal is larger than the bit duration &nbsp;$T_{\rm B}$&nbsp; of the binary source signal by a factor of &nbsp;$4/3$.&nbsp; 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 &nbsp;$r_c \approx 16\%$.&nbsp; This redundancy is used in the 4B3T code to achieve DC freedom.
@@ Line 119: / Line 119: @@
 The two other eye diagrams can be interpreted as follows:
-*For the redundancy-free ternary system,&nbsp; the eye opening is only half as large as in the binary case,&nbsp; and the noise power &nbsp;$\sigma_2^2$&nbsp; is smaller than &nbsp;$\sigma_1^2$ by a factor of &nbsp;$\log_2 \hspace{0.05cm}(3)$.&nbsp; The factor &nbsp;$4/3$&nbsp; in front of the Q&ndash;function takes into account that the ternary&nbsp; "0"&nbsp; can be distorted '''KORREKTUR: falsified''' in both directions.&nbsp; This results in the following numerical values:
+*For the redundancy-free ternary system,&nbsp; the eye opening is only half as large as in the binary case,&nbsp; and the noise power &nbsp;$\sigma_2^2$&nbsp; is smaller than &nbsp;$\sigma_1^2$ by a factor of &nbsp;$\log_2 \hspace{0.05cm}(3)$.&nbsp; The factor &nbsp;$4/3$&nbsp; in front of the Q&ndash;function takes into account that the ternary&nbsp; "0"&nbsp; can be  falsified in both directions.&nbsp; This results in the following numerical values:
 :$$\frac{ \sigma_2}{s_0}\hspace{-0.05cm} =\hspace{-0.05cm} \frac{ \sigma_1/s_0}{\sqrt{{\rm log_2} (3)}}\hspace{-0.05cm} =\hspace{-0.05cm}\frac{ 0.167}{1.259}
   \approx 0.133 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}p_{\rm S}\hspace{-0.05cm}=\hspace{-0.05cm} {4}/{3} \cdot {\rm Q}\left (\frac{ 0.5}{0.133}\right)\approx

@@ Line 9: / Line 9: @@
 == General description of block codes ==
 <br>
-In &nbsp;'''block coding'''&nbsp; each sequence of &nbsp;$m_q$&nbsp; binary source symbols &nbsp;$(M_q = 2)$&nbsp; is represented by a block of &nbsp;$m_c$&nbsp; coded symbols with symbol set size &nbsp;$M_c$.&nbsp; In order to convert each source symbol sequence &nbsp;$\langle q_\nu \rangle$&nbsp; into another encoder symbol sequence &nbsp;$\langle c_\nu \rangle$,&nbsp; the following condition must be satisfied:
+In &nbsp;'''block coding'''&nbsp; each sequence of &nbsp;$m_q$&nbsp; binary source symbols &nbsp;$(M_q = 2)$&nbsp; is represented by a block of &nbsp;$m_c$&nbsp; coded '''KORREKTUR: encoder''' symbols with symbol set size &nbsp;$M_c$.&nbsp; In order to convert each source symbol sequence &nbsp;$\langle q_\nu \rangle$&nbsp; into another encoder symbol sequence &nbsp;$\langle c_\nu \rangle$,&nbsp; the following condition must be satisfied:
 :$$M_c^{\hspace{0.1cm}m_c} \ge
 M_q^{\hspace{0.1cm}m_q}\hspace{0.05cm}.$$

@@ Line 25: / Line 25: @@
 Such a 4B3T code has the following properties:
-*Because of &nbsp;$m_q \cdot T_{\rm B} =  m_c \cdot T$,&nbsp; the symbol duration &nbsp;$T$&nbsp; of the ternary encoder signal is larger than the bit duration &nbsp;$T_{\rm B}$&nbsp; of the binary source signal by a factor of &nbsp;$4/3$.&nbsp; This results in the favorable property that the bandwidth requirement is a quarter less than for redundancy-free binary transmission.
+*Because of &nbsp;$m_q \cdot T_{\rm B} =  m_c \cdot T$,&nbsp; the symbol duration &nbsp;$T$&nbsp; of the ternary encoder '''KORREKTUR: encoded''' signal is larger than the bit duration &nbsp;$T_{\rm B}$&nbsp; of the binary source signal by a factor of &nbsp;$4/3$.&nbsp; 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 &nbsp;$r_c \approx 16\%$.&nbsp; This redundancy is used in the 4B3T code to achieve DC freedom.

@@ Line 119: / Line 119: @@
 The two other eye diagrams can be interpreted as follows:
-*For the redundancy-free ternary system,&nbsp; the eye opening is only half as large as in the binary case,&nbsp; and the noise power &nbsp;$\sigma_2^2$&nbsp; is smaller than &nbsp;$\sigma_1^2$ by a factor of &nbsp;$\log_2 \hspace{0.05cm}(3)$.&nbsp; The factor &nbsp;$4/3$&nbsp; in front of the Q&ndash;function takes into account that the ternary&nbsp; "0"&nbsp; can be distorted in both directions.&nbsp; This results in the following numerical values:
+*For the redundancy-free ternary system,&nbsp; the eye opening is only half as large as in the binary case,&nbsp; and the noise power &nbsp;$\sigma_2^2$&nbsp; is smaller than &nbsp;$\sigma_1^2$ by a factor of &nbsp;$\log_2 \hspace{0.05cm}(3)$.&nbsp; The factor &nbsp;$4/3$&nbsp; in front of the Q&ndash;function takes into account that the ternary&nbsp; "0"&nbsp; can be distorted '''KORREKTUR: falsified''' in both directions.&nbsp; This results in the following numerical values:
 :$$\frac{ \sigma_2}{s_0}\hspace{-0.05cm} =\hspace{-0.05cm} \frac{ \sigma_1/s_0}{\sqrt{{\rm log_2} (3)}}\hspace{-0.05cm} =\hspace{-0.05cm}\frac{ 0.167}{1.259}
   \approx 0.133 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}p_{\rm S}\hspace{-0.05cm}=\hspace{-0.05cm} {4}/{3} \cdot {\rm Q}\left (\frac{ 0.5}{0.133}\right)\approx

@@ Line 88: / Line 88: @@
 *The zero point at &nbsp;$f = 0$&nbsp; has the advantage that the 4B3T signal can also be transmitted without major losses via a so-called&nbsp; "telephone channel",&nbsp; which is not suitable for a DC signal due to transformers.<br>
-*The zero point at &nbsp;$f = 1/T$&nbsp; has the disadvantage that this makes clock recovery at the receiver more difficult.&nbsp; Outside of these zeros,&nbsp; the 4B3T codes have a flatter &nbsp;${\it \Phi}_a(f)$&nbsp; than the  &nbsp;[[Digital_Signal_Transmission/Symbol-Wise_Coding_with_Pseudo_Ternary_Codes#Properties_of_the_AMI_code|"AMI code"]]&nbsp; discussed in the next chapter&nbsp; (blue curve), which is advantageous.<br>
+*The zero point at &nbsp;$f = 1/T$&nbsp; has the disadvantage that this makes clock recovery at the receiver more difficult.&nbsp; Outside of these zeros,&nbsp; the 4B3T codes have a flatter &nbsp;${\it \Phi}_a(f)$&nbsp; than the  &nbsp;[[Digital_Signal_Transmission/Symbolwise_Coding_with_Pseudo-Ternary_Codes#Properties_of_the_AMI_code|"AMI code"]]&nbsp; discussed in the next chapter&nbsp; (blue curve), which is advantageous.<br>
 *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 &nbsp;$+1$&nbsp; coefficients&nbsp; (resp. &nbsp;$-1$ coefficients)&nbsp; can follow each other.&nbsp; With the AMI code,&nbsp; these symbols occur only in isolation.<br>

@@ Line 13: / Line 13: @@
 M_q^{\hspace{0.1cm}m_q}\hspace{0.05cm}.$$
-*For the &nbsp;[[Digital_Signal_Transmission/Redundancy-Free_Coding|redundancy-free codes]]&nbsp; discussed in the last chapter,&nbsp; the equal sign applies in this equation if &nbsp;$M_q$&nbsp; is a power of two.
+*For the &nbsp;[[Digital_Signal_Transmission/Redundancy-Free_Coding|"redundancy-free codes"]]&nbsp; discussed in the last chapter,&nbsp; the equal sign applies in this equation if &nbsp;$M_q$&nbsp; is a power of two.
 *Using the greater sign results in a redundant digital signal,&nbsp; and the &nbsp;"relative encoder redundancy"&nbsp; can be calculated as follows:
@@ Line 22: / Line 22: @@
 ,\hspace{0.2cm}M_c = 3\hspace{0.05cm},$$
-which was developed in the 1970s and is used,&nbsp; for example,&nbsp; in &nbsp;[[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_ISDN| ISDN]]&nbsp; ("Integrated Services Digital Networks").
+which was developed in the 1970s and is used,&nbsp; for example,&nbsp; in &nbsp;[[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_ISDN|"ISDN"]]&nbsp; ("Integrated Services Digital Networks").
 Such a 4B3T code has the following properties:
@@ Line 88: / Line 88: @@
 *The zero point at &nbsp;$f = 0$&nbsp; has the advantage that the 4B3T signal can also be transmitted without major losses via a so-called&nbsp; "telephone channel",&nbsp; which is not suitable for a DC signal due to transformers.<br>
-*The zero point at &nbsp;$f = 1/T$&nbsp; has the disadvantage that this makes clock recovery at the receiver more difficult.&nbsp; Outside of these zeros,&nbsp; the 4B3T codes have a flatter &nbsp;${\it \Phi}_a(f)$&nbsp; than the  &nbsp;[[Digital_Signal_Transmission/Symbol-Wise_Coding_with_Pseudo_Ternary_Codes#Properties_of_the_AMI_code|AMI code]]&nbsp; discussed in the next chapter&nbsp; (blue curve), which is advantageous.<br>
+*The zero point at &nbsp;$f = 1/T$&nbsp; has the disadvantage that this makes clock recovery at the receiver more difficult.&nbsp; Outside of these zeros,&nbsp; the 4B3T codes have a flatter &nbsp;${\it \Phi}_a(f)$&nbsp; than the  &nbsp;[[Digital_Signal_Transmission/Symbol-Wise_Coding_with_Pseudo_Ternary_Codes#Properties_of_the_AMI_code|"AMI code"]]&nbsp; discussed in the next chapter&nbsp; (blue curve), which is advantageous.<br>
 *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 &nbsp;$+1$&nbsp; coefficients&nbsp; (resp. &nbsp;$-1$ coefficients)&nbsp; can follow each other.&nbsp; With the AMI code,&nbsp; these symbols occur only in isolation.<br>
@@ Line 104: / Line 104: @@
-The graph shows the &nbsp;[[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference#Definition_and_statements_of_the_eye_diagram|eye diagrams]]&nbsp; (with noise) of the three systems to be compared.&nbsp; It also contains&nbsp; $($in row &nbsp;$\rm A)$&nbsp; the equations for calculating the symbol error probability.&nbsp; Approximately &nbsp;2000&nbsp; eye lines are drawn for each diagram.
+The graph shows the &nbsp;[[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference#Definition_and_statements_of_the_eye_diagram|"eye diagrams"]]&nbsp; (with noise) of the three systems to be compared.&nbsp; It also contains&nbsp; $($in row &nbsp;$\rm A)$&nbsp; the equations for calculating the symbol error probability.&nbsp; Approximately &nbsp;2000&nbsp; eye lines are drawn for each diagram.
 The first two rows of the table describe the system comparison at peak limitation.&nbsp; For the binary system&nbsp; (first column),&nbsp; the noise power&nbsp; (taking into account the &nbsp;$12\%$ increase)&nbsp; is given by

@@ Line 56: / Line 56: @@
-You can seee from the graph:
+You can see from the graph:
 #The two code tables of the Jessop&ndash;Waters code are selected in such a way that the running digital sum &nbsp;${\it \Sigma}_l$&nbsp; always lies between&nbsp; $0$&nbsp; and&nbsp; $5$.<br><br>
 #For the other two codes&nbsp; (MS43,&nbsp; FoMoT),&nbsp; the restriction of the running digital sum to the range &nbsp;$0 \le {\it \Sigma}_l \le 3$&nbsp; is achieved by three resp. four alternative tables.
@@ Line 104: / Line 104: @@
-The following graph shows the &nbsp;[[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference#Definition_and_statements_of_the_eye_diagram|eye diagrams]]&nbsp; (with noise) of the three systems to be compared.&nbsp; It also contains&nbsp; $($in row &nbsp;$\rm A)$&nbsp; the equations for calculating the symbol error probability.&nbsp; Approximately &nbsp;2000&nbsp; eye lines are drawn for each diagram.
+The graph shows the &nbsp;[[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference#Definition_and_statements_of_the_eye_diagram|eye diagrams]]&nbsp; (with noise) of the three systems to be compared.&nbsp; It also contains&nbsp; $($in row &nbsp;$\rm A)$&nbsp; the equations for calculating the symbol error probability.&nbsp; Approximately &nbsp;2000&nbsp; eye lines are drawn for each diagram.
 The first two rows of the table describe the system comparison at peak limitation.&nbsp; For the binary system&nbsp; (first column),&nbsp; the noise power&nbsp; (taking into account the &nbsp;$12\%$ increase)&nbsp; is given by

@@ Line 131: / Line 131: @@
 $\text{Row C: &nbsp;   Symbol error probabilities under power limitation}$&nbsp;
-&rArr; &nbsp; Thereby is given the rectangular basic transmission pulse&nbsp; $g_{s}(t)$&nbsp; and  &nbsp;$10 \cdot \lg \hspace{0.05cm}(E_{\rm B}/N_0) = 13 \ \rm dB$:
+&nbsp; &nbsp; &rArr; &nbsp; Thereby is given the rectangular basic transmission pulse&nbsp; $g_{s}(t)$&nbsp; and  &nbsp;$10 \cdot \lg \hspace{0.05cm}(E_{\rm B}/N_0) = 13 \ \rm dB$:
 *For the&nbsp; '''redundancy-free binary system'''&nbsp; with NRZ rectangular pulses, &nbsp;$p_{\rm S}$&nbsp; is not changed with respect to row &nbsp;$\rm B$&nbsp; because of &nbsp;$E_{\rm B} = s_0^2 \cdot T$:&nbsp; <br>
 :$$p_\text{S, power limitation} = p_\text{S, peak limitation} \approx 10^{-9}.$$