Digital Signal Transmission/Redundancy-Free Coding - Revision history

Hwang at 14:01, 23 January 2023

2023-01-23T14:01:24Z

Hwang at 14:01, 23 January 2023

2023-01-23T14:01:00Z

Guenter at 15:02, 3 June 2022

2022-06-03T15:02:57Z

Hwang at 15:56, 23 May 2022

2022-05-23T15:56:31Z

Hwang at 15:55, 23 May 2022

2022-05-23T15:55:20Z

Hwang at 15:53, 23 May 2022

2022-05-23T15:53:09Z

Hwang at 15:47, 23 May 2022

2022-05-23T15:47:54Z

Hwang at 15:47, 23 May 2022

2022-05-23T15:47:13Z

Hwang at 15:46, 23 May 2022

2022-05-23T15:46:41Z

Hwang at 15:43, 23 May 2022

2022-05-23T15:43:11Z

← Older revision		Revision as of 14:01, 23 January 2023
Line 156:		Line 156:
	\frac{ 4}{3} \cdot {\rm Q} \left( \frac{s_0/2}{\sigma_d}\right)\hspace{0.05cm}.$$		\frac{ 4}{3} \cdot {\rm Q} \left( \frac{s_0/2}{\sigma_d}\right)\hspace{0.05cm}.$$

−	*Please note that this equation no longer specifies the bit error probability  $p_{\rm B}$, but the  "symbol error probability"  $p_{\rm S}$.  The corresponding a posteriori parameters are  "bit error rate"  $\rm (BER)$  and  "symbol error rate"  $\rm (SER)$.  More details are given in the  [[Digital_Signal_Transmission/Redundancy-Free_Coding#Symbol_and_bit_error_probability\|"last section"]]  of this chapter.<br>	+	*Please note that this equation no longer specifies the bit error probability  $p_{\rm B}$, but the  "symbol error probability"  $p_{\rm S}$.  The corresponding a-posteriori parameters are  "bit error rate"  $\rm (BER)$  and  "symbol error rate"  $\rm (SER)$.  More details are given in the  [[Digital_Signal_Transmission/Redundancy-Free_Coding#Symbol_and_bit_error_probability\|"last section"]]  of this chapter.<br>

@@ Line 276: / Line 276: @@
 [[File:EN_Dig_T_2_2_S6b.png|right|frame|Comparison of  dual code and Gray code|class=fit]]
 To calculate the bit error probability,&nbsp; one must also consider the mapping between the binary and the quaternary symbols:
-*In &nbsp;"dual coding"&nbsp; according to the table with yellow background,&nbsp; one symbol error &nbsp;$(w_\nu \ne c_\nu)$&nbsp; can result in one or two bit errors &nbsp;$(v_\nu \ne q_\nu)$.&nbsp; Of the six corruption possibilities at the quaternary symbol level,&nbsp; four result in one bit error each and only the two inner ones result in two bit errors.&nbsp; It follows:
+*In &nbsp;"dual coding"&nbsp; according to the table with yellow background,&nbsp; one symbol error &nbsp;$(w_\nu \ne c_\nu)$&nbsp; can result in one or two bit errors &nbsp;$(v_\nu \ne q_\nu)$.&nbsp; Of the six falsification possibilities at the quaternary symbol level,&nbsp; four result in one bit error each and only the two inner ones result in two bit errors.&nbsp; It follows:
 :$$p_{\rm B}  = {1}/{4}\cdot (4 \cdot 1 \cdot p + 2 \cdot 2 \cdot p ) \cdot {1}/{2} = p\hspace{0.05cm}.$$

@@ Line 12: / Line 12: @@
 '''Symbolwise coding'''
-*Here,&nbsp; a code '''KORREKTUR: encoder''' symbol &nbsp;$c_\nu$&nbsp; is generated with each incoming source symbol &nbsp;$q_\nu$,&nbsp; which can depend not only on the current symbol but also on previous symbols &nbsp;$q_{\nu -1}$, &nbsp;$q_{\nu -2}$, ... <br>
+*Here,&nbsp; an encoder symbol &nbsp;$c_\nu$&nbsp; is generated with each incoming source symbol &nbsp;$q_\nu$,&nbsp; which can depend not only on the current symbol but also on previous symbols &nbsp;$q_{\nu -1}$, &nbsp;$q_{\nu -2}$, ... <br>
-*It is typical for all transmission codes for symbolwise coding that the symbol duration &nbsp;$T_c$&nbsp; of the usually multilevel and redundant encoder '''KORREKTUR: encoded''' signal &nbsp;$c(t)$&nbsp; corresponds to the bit duration &nbsp;$T_q$&nbsp; of the source signal,&nbsp; which is assumed to be binary and redundancy-free.<br>
+*It is typical for all transmission codes for symbolwise coding that the symbol duration &nbsp;$T_c$&nbsp; of the usually multilevel and redundant encoded  signal &nbsp;$c(t)$&nbsp; corresponds to the bit duration &nbsp;$T_q$&nbsp; of the source signal,&nbsp; which is assumed to be binary and redundancy-free.<br>
@@ Line 21: / Line 21: @@
 '''Blockwise coding'''
-*Here,&nbsp; a block of &nbsp;$m_q$&nbsp; binary source symbols &nbsp;$(M_q = 2)$&nbsp; of bit duration &nbsp;$T_q$&nbsp; is assigned a one-to-one sequence of &nbsp;$m_c$&nbsp; code '''KORREKTUR: encoder''' symbols from an alphabet with code '''KORREKTUR: encoder''' symbol set size &nbsp;$M_c  \ge 2$.&nbsp;
+*Here,&nbsp; a block of &nbsp;$m_q$&nbsp; binary source symbols &nbsp;$(M_q = 2)$&nbsp; of bit duration &nbsp;$T_q$&nbsp; is assigned a one-to-one sequence of &nbsp;$m_c$&nbsp;  encoder symbols from an alphabet with  encoder symbol set size &nbsp;$M_c  \ge 2$.&nbsp;
-*For the&nbsp; '''symbol duration of a code '''KORREKTUR: encoder''' symbol'''&nbsp; then holds:
+*For the&nbsp; '''symbol duration of an encoder symbol'''&nbsp; then holds:
 :$$T_c = \frac{m_q}{m_c} \cdot T_q \hspace{0.05cm},$$
 *The&nbsp; '''relative redundancy of a block code'''&nbsp; is in general
@@ Line 41: / Line 41: @@
 *Starting from the redundancy-free binary source signal &nbsp;$q(t)$&nbsp; with bit duration &nbsp;$T_q$,&nbsp;
-*a &nbsp;$M_c$&ndash;level code '''KORREKTUR: encoded''' signal &nbsp;$c(t)$&nbsp; with symbol duration &nbsp;$T_c = T_q \cdot \log_2 \hspace{0.05cm} (M_c)$&nbsp; is generated.
+*a &nbsp;$M_c$&ndash;level encoded signal &nbsp;$c(t)$&nbsp; with symbol duration &nbsp;$T_c = T_q \cdot \log_2 \hspace{0.05cm} (M_c)$&nbsp; is generated.
@@ Line 48: / Line 48: @@
 Thereby holds:
-#If &nbsp;$M_c$&nbsp; is a power to the base &nbsp;$2$,&nbsp; then &nbsp;$m_q = \log_2 \hspace{0.05cm} (M_c)$&nbsp; are combined into a single code '''KORREKTUR: encoder''' symbol &nbsp;$(m_c = 1)$.&nbsp; In this case, the relative redundancy is actually &nbsp;$r_c = 0$.<br>
+#If &nbsp;$M_c$&nbsp; is a power to the base &nbsp;$2$,&nbsp; then &nbsp;$m_q = \log_2 \hspace{0.05cm} (M_c)$&nbsp; are combined into a single  encoder symbol &nbsp;$(m_c = 1)$.&nbsp; In this case, the relative redundancy is actually &nbsp;$r_c = 0$.<br>
 #If &nbsp;$M_c$&nbsp; is not a power of two,&nbsp; a hundred percent redundancy-free block coding is not possible.&nbsp; For example, if &nbsp;$m_q = 3$&nbsp; binary symbols are encoded by &nbsp;$m_c = 2$&nbsp; ternary symbols and &nbsp;$T_c = 1.5 \cdot T_q$&nbsp; is set,&nbsp; a relative redundancy of &nbsp;$r_c = 1-1.5/ \log_2 \hspace{0.05cm} (3) \approx 5\%$&nbsp; remains.<br>
 #Encoding a block of &nbsp;$128$&nbsp; binary symbols with &nbsp;$81$&nbsp; ternary symbols results in a relative code redundancy of less than &nbsp;$r_c = 0.3\%$.<br><br>
@@ Line 55: / Line 55: @@
 To simplify the notation and to align the nomenclature with the [[Digital_Signal_Transmission| "first main chapter"]],&nbsp; we use in the following
 *the bit duration &nbsp;$T_{\rm B} = T_q$&nbsp; of the redundancy-free binary source signal,
-*the symbol duration &nbsp;$T = T_c$&nbsp; of the encoder '''KORREKTUR: encoded''' signal and the transmitted signal, and
+*the symbol duration &nbsp;$T = T_c$&nbsp; of the encoded  signal and the transmitted signal, and
 *the number &nbsp;$M = M_c$&nbsp; of levels.<br>}}
@@ Line 66: / Line 66: @@
 *Thus,&nbsp; for bipolar signaling &nbsp;$(-1 \le a_\nu \le +1)$,&nbsp; the following applies to the possible amplitude coefficients with index &nbsp;$\mu = 1$, ... , $M$:
 :$$a_\mu = \frac{2\mu - M - 1}{M-1} \hspace{0.05cm}.$$
-*Independently of the '''KORREKTUR: level''' number &nbsp;$M$&nbsp; one obtains from this for the outer amplitude coefficients &nbsp;$a_1 = -1$&nbsp; and &nbsp;$a_M = +1$.
+*Independently of the level number &nbsp;$M$&nbsp; one obtains from this for the outer amplitude coefficients &nbsp;$a_1 = -1$&nbsp; and &nbsp;$a_M = +1$.
 *For a ternary signal &nbsp;$(M = 3)$,&nbsp; the possible amplitude coefficients are &nbsp;$-1$, &nbsp;$0$&nbsp; and &nbsp;$+1$.
@@ Line 84: / Line 84: @@
 The lower plot shows the $($almost$)$ redundancy-free ternary signal &nbsp;$s_3(t)$&nbsp; and the mapping of three binary symbols each to two ternary symbols.
-*The possible amplitude coefficients are &nbsp;$-1$, &nbsp;$0$&nbsp; and &nbsp;$+1$&nbsp; and the symbol duration of the coded '''KORREKTUR: encoded''' signal &nbsp;$T = 3/2 \cdot T_{\rm B}$.
+*The possible amplitude coefficients are &nbsp;$-1$, &nbsp;$0$&nbsp; and &nbsp;$+1$&nbsp; and the symbol duration of the encoded  signal &nbsp;$T = 3/2 \cdot T_{\rm B}$.
 *It can be seen from the green mapping table that the coefficients &nbsp;$+1$&nbsp; and &nbsp;$-1$&nbsp; occur somewhat more frequently than the coefficient &nbsp;$a_\nu = 0$.&nbsp; This results in the above mentioned relative redundancy of&nbsp; $5\%$.
@@ Line 173: / Line 173: @@
   \frac{ 2 + 2 \cdot (M-2)}{M} \cdot {\rm Q} \left( \frac{s_0/(M-1)}{\sigma_d(M)}\right) = \frac{ 2  \cdot (M-1)}{M} \cdot {\rm Q} \left( \frac{s_0}{\sigma_d (M)\cdot (M-1)}\right)\hspace{0.05cm}.$$
-*The notation &nbsp;$\sigma_d(M)$&nbsp; is intended to make clear that the rms value of the noise component &nbsp;$d_{\rm N}(t)$&nbsp; depends significantly on the '''KORREKTUR: level''' number &nbsp;$M$.&nbsp;}}<br>
+*The notation &nbsp;$\sigma_d(M)$&nbsp; is intended to make clear that the rms value of the noise component &nbsp;$d_{\rm N}(t)$&nbsp; depends significantly on the  level number &nbsp;$M$.&nbsp;}}<br>
@@ Line 179: / Line 179: @@
 <br>
 For this system comparison under fair conditions,&nbsp; the following are assumed:
-*Let the equivalent bit rate &nbsp;$R_{\rm B} = 1/T_{\rm B}$&nbsp; be constant.&nbsp; Depending on the '''KORREKTUR: level''' number &nbsp;$M$,&nbsp; the symbol duration of the encoder '''KORREKTUR: encoded''' signal and the transmitted signal is thus:
+*Let the equivalent bit rate &nbsp;$R_{\rm B} = 1/T_{\rm B}$&nbsp; be constant.&nbsp; Depending on the  level number &nbsp;$M$,&nbsp; the symbol duration of the encoded  signal and the transmitted signal is thus:
 :$$T = T_{\rm B} \cdot {\rm log_2} (M)   \hspace{0.05cm}.$$
 *The Nyquist condition is satisfied by a &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems|"root&ndash;root characteristic"]]&nbsp; with rolloff factor &nbsp;$r$.&nbsp; Furthermore,&nbsp; no intersymbol interference occurs.&nbsp; The detection noise power is:
@@ Line 192: / Line 192: @@
   \frac{ 2  \cdot (M-1)}{M} \cdot {\rm Q} \left( \sqrt{\frac{3 \cdot {\rm log_2}\hspace{0.05cm} (M)}{M^2 -1}\cdot
   \frac{2 \cdot E_{\rm B}}{N_0}}\right)$$
-[[File:EN_Dig_T_2_3_S3b_v2.png|right|frame|Symbol error probability curves for different step '''KORREKTUR: level''' numbers &nbsp;$M$]]
+[[File:EN_Dig_T_2_3_S3b_v2.png|right|frame|Symbol error probability curves for different level numbers &nbsp;$M$]]
 :$$\Rightarrow \hspace{0.3cm} p_{\rm S}  =
    K_1 \cdot {\rm Q} \left( \sqrt{K_2\cdot
   \frac{2 \cdot E_{\rm B}}{N_0}}\right)\hspace{0.05cm}.$$
-For &nbsp;$M = 2$,&nbsp; set &nbsp;$K_1 = K_2 = 1$.&nbsp; For larger step '''KORREKTUR: level''' numbers,&nbsp; one obtains  for the symbol error probability that can be achieved with &nbsp;$M$&ndash;level redundancy-free coding:
+For &nbsp;$M = 2$,&nbsp; set &nbsp;$K_1 = K_2 = 1$.&nbsp; For larger level numbers,&nbsp; one obtains  for the symbol error probability that can be achieved with &nbsp;$M$&ndash;level redundancy-free coding:
 :$$M = 3\text{:} \ \ K_1 = 1.333, \ K_2 = 0.594;\hspace{0.5cm}M = 4\text{:} \ \ K_1 = 1.500, \ K_2 = 0.400;$$
 :$$M = 5\text{:} \ \ K_1 = 1.600, \ K_2 = 0.290;\hspace{0.5cm}M = 6\text{:} \ \ K_1 = 1.666, \ K_2 = 0.221;$$
@@ Line 221: / Line 221: @@
 {{BlaueBox|TEXT=
 $\text{System comparison under the peak limitation constraint:}$&nbsp;
-*With the constraint&nbsp; "peak limitation",&nbsp; the combination of rectangular &nbsp;$g_s(t)$&nbsp; and rectangular &nbsp;$h_{\rm E}(t)$&nbsp; leads to the optimum regardless of the step '''KORREKTUR: level''' number  &nbsp;$M$.&nbsp; <br>
+*With the constraint&nbsp; "peak limitation",&nbsp; the combination of rectangular &nbsp;$g_s(t)$&nbsp; and rectangular &nbsp;$h_{\rm E}(t)$&nbsp; leads to the optimum regardless of the level number  &nbsp;$M$.&nbsp; <br>
 *The loss of the multilevel system compared to the binary system is here even greater than with power limitation.
@@ Line 260: / Line 260: @@
 *In general,&nbsp; the symbol error probability &nbsp;$p_{\rm S}$&nbsp; can be calculated somewhat more easily than the bit error probability &nbsp;$p_{\rm B}$&nbsp; for redundancy-containing multilevel systems.
-* However,&nbsp; a comparison of systems with different step '''KORREKTUR: level''' numbers&nbsp; $M$&nbsp; or different types of coding should always be based on the bit error probability &nbsp;$p_{\rm B}$&nbsp; for reasons of fairness.&nbsp; The mapping between the source and coded '''KORREKTUR: encoder''' symbols must also be taken into account,&nbsp; as shown in the following example.}}<br>
+* However,&nbsp; a comparison of systems with different level numbers&nbsp; $M$&nbsp; or different types of coding should always be based on the bit error probability &nbsp;$p_{\rm B}$&nbsp; for reasons of fairness.&nbsp; The mapping between the source and encoder symbols must also be taken into account,&nbsp; as shown in the following example.}}<br>
 {{GraueBox|TEXT=
@@ Line 271: / Line 271: @@
 <br>
-For equally probable binary source symbols &nbsp;$q_\nu$&nbsp; the quaternary code '''KORREKTUR: encoder''' symbols &nbsp;$c_\nu$&nbsp; also occur with equal probability.&nbsp; Thus,&nbsp; we obtain for the symbol error probability:
+For equally probable binary source symbols &nbsp;$q_\nu$&nbsp; the quaternary encoder symbols &nbsp;$c_\nu$&nbsp; also occur with equal probability.&nbsp; Thus,&nbsp; we obtain for the symbol error probability:
 :$$p_{\rm S}  ={1}/{4}\cdot (2 \cdot p + 2 \cdot 2 \cdot  p) =  {3}/{2} \cdot p\hspace{0.05cm}.$$

@@ Line 260: / Line 260: @@
 *In general,&nbsp; the symbol error probability &nbsp;$p_{\rm S}$&nbsp; can be calculated somewhat more easily than the bit error probability &nbsp;$p_{\rm B}$&nbsp; for redundancy-containing multilevel systems.
-* However,&nbsp; a comparison of systems with different step '''KORREKTUR: level''' numbers&nbsp; $M$&nbsp; or different types of coding should always be based on the bit error probability &nbsp;$p_{\rm B}$&nbsp; for reasons of fairness.&nbsp; The mapping between the source and coded symbols must also be taken into account,&nbsp; as shown in the following example.}}<br>
+* However,&nbsp; a comparison of systems with different step '''KORREKTUR: level''' numbers&nbsp; $M$&nbsp; or different types of coding should always be based on the bit error probability &nbsp;$p_{\rm B}$&nbsp; for reasons of fairness.&nbsp; The mapping between the source and coded '''KORREKTUR: encoder''' symbols must also be taken into account,&nbsp; as shown in the following example.}}<br>
 {{GraueBox|TEXT=

@@ Line 12: / Line 12: @@
 '''Symbolwise coding'''
-*Here,&nbsp; a code symbol &nbsp;$c_\nu$&nbsp; is generated with each incoming source symbol &nbsp;$q_\nu$,&nbsp; which can depend not only on the current symbol but also on previous symbols &nbsp;$q_{\nu -1}$, &nbsp;$q_{\nu -2}$, ... <br>
+*Here,&nbsp; a code '''KORREKTUR: encoder''' symbol &nbsp;$c_\nu$&nbsp; is generated with each incoming source symbol &nbsp;$q_\nu$,&nbsp; which can depend not only on the current symbol but also on previous symbols &nbsp;$q_{\nu -1}$, &nbsp;$q_{\nu -2}$, ... <br>
 *It is typical for all transmission codes for symbolwise coding that the symbol duration &nbsp;$T_c$&nbsp; of the usually multilevel and redundant encoder '''KORREKTUR: encoded''' signal &nbsp;$c(t)$&nbsp; corresponds to the bit duration &nbsp;$T_q$&nbsp; of the source signal,&nbsp; which is assumed to be binary and redundancy-free.<br>
@@ Line 21: / Line 21: @@
 '''Blockwise coding'''
-*Here,&nbsp; a block of &nbsp;$m_q$&nbsp; binary source symbols &nbsp;$(M_q = 2)$&nbsp; of bit duration &nbsp;$T_q$&nbsp; is assigned a one-to-one sequence of &nbsp;$m_c$&nbsp; code symbols from an alphabet with code symbol set size &nbsp;$M_c  \ge 2$.&nbsp;
+*Here,&nbsp; a block of &nbsp;$m_q$&nbsp; binary source symbols &nbsp;$(M_q = 2)$&nbsp; of bit duration &nbsp;$T_q$&nbsp; is assigned a one-to-one sequence of &nbsp;$m_c$&nbsp; code '''KORREKTUR: encoder''' symbols from an alphabet with code '''KORREKTUR: encoder''' symbol set size &nbsp;$M_c  \ge 2$.&nbsp;
-*For the&nbsp; '''symbol duration of a code symbol'''&nbsp; then holds:
+*For the&nbsp; '''symbol duration of a code '''KORREKTUR: encoder''' symbol'''&nbsp; then holds:
 :$$T_c = \frac{m_q}{m_c} \cdot T_q \hspace{0.05cm},$$
 *The&nbsp; '''relative redundancy of a block code'''&nbsp; is in general
@@ Line 48: / Line 48: @@
 Thereby holds:
-#If &nbsp;$M_c$&nbsp; is a power to the base &nbsp;$2$,&nbsp; then &nbsp;$m_q = \log_2 \hspace{0.05cm} (M_c)$&nbsp; are combined into a single code symbol &nbsp;$(m_c = 1)$.&nbsp; In this case, the relative redundancy is actually &nbsp;$r_c = 0$.<br>
+#If &nbsp;$M_c$&nbsp; is a power to the base &nbsp;$2$,&nbsp; then &nbsp;$m_q = \log_2 \hspace{0.05cm} (M_c)$&nbsp; are combined into a single code '''KORREKTUR: encoder''' symbol &nbsp;$(m_c = 1)$.&nbsp; In this case, the relative redundancy is actually &nbsp;$r_c = 0$.<br>
 #If &nbsp;$M_c$&nbsp; is not a power of two,&nbsp; a hundred percent redundancy-free block coding is not possible.&nbsp; For example, if &nbsp;$m_q = 3$&nbsp; binary symbols are encoded by &nbsp;$m_c = 2$&nbsp; ternary symbols and &nbsp;$T_c = 1.5 \cdot T_q$&nbsp; is set,&nbsp; a relative redundancy of &nbsp;$r_c = 1-1.5/ \log_2 \hspace{0.05cm} (3) \approx 5\%$&nbsp; remains.<br>
 #Encoding a block of &nbsp;$128$&nbsp; binary symbols with &nbsp;$81$&nbsp; ternary symbols results in a relative code redundancy of less than &nbsp;$r_c = 0.3\%$.<br><br>
@@ Line 271: / Line 271: @@
 <br>
-For equally probable binary source symbols &nbsp;$q_\nu$&nbsp; the quaternary code symbols &nbsp;$c_\nu$&nbsp; also occur with equal probability.&nbsp; Thus,&nbsp; we obtain for the symbol error probability:
+For equally probable binary source symbols &nbsp;$q_\nu$&nbsp; the quaternary code '''KORREKTUR: encoder''' symbols &nbsp;$c_\nu$&nbsp; also occur with equal probability.&nbsp; Thus,&nbsp; we obtain for the symbol error probability:
 :$$p_{\rm S}  ={1}/{4}\cdot (2 \cdot p + 2 \cdot 2 \cdot  p) =  {3}/{2} \cdot p\hspace{0.05cm}.$$

← Older revision		Revision as of 15:47, 23 May 2022
Line 173:		Line 173:
	\frac{ 2 + 2 \cdot (M-2)}{M} \cdot {\rm Q} \left( \frac{s_0/(M-1)}{\sigma_d(M)}\right) = \frac{ 2 \cdot (M-1)}{M} \cdot {\rm Q} \left( \frac{s_0}{\sigma_d (M)\cdot (M-1)}\right)\hspace{0.05cm}.$$		\frac{ 2 + 2 \cdot (M-2)}{M} \cdot {\rm Q} \left( \frac{s_0/(M-1)}{\sigma_d(M)}\right) = \frac{ 2 \cdot (M-1)}{M} \cdot {\rm Q} \left( \frac{s_0}{\sigma_d (M)\cdot (M-1)}\right)\hspace{0.05cm}.$$

−	*The notation  $\sigma_d(M)$  is intended to make clear that the rms value of the noise component  $d_{\rm N}(t)$  depends significantly on the number  $M$. }}<br>	+	*The notation  $\sigma_d(M)$  is intended to make clear that the rms value of the noise component  $d_{\rm N}(t)$  depends significantly on the '''KORREKTUR: level''' number  $M$. }}<br>