Digital Signal Transmission/Redundancy-Free Coding - Revision history

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	<br>		<br>
	For a redundancy-free coded  $M$–level bipolar digital signal  $s(t)$,  the following holds for the  [[Digital_Signal_Transmission/Basics_of_Coded_Transmission#ACF_calculation_of_a_digital_signal\|"discrete auto-correlation function"]]  $\rm (ACF)$  of the amplitude coefficients and for the corresponding  [[Digital_Signal_Transmission/Basics_of_Coded_Transmission#PSD_calculation_of_a_digital_signal\|"power-spectral density"]]  $\rm (PSD)$:		For a redundancy-free coded  $M$–level bipolar digital signal  $s(t)$,  the following holds for the  [[Digital_Signal_Transmission/Basics_of_Coded_Transmission#ACF_calculation_of_a_digital_signal\|"discrete auto-correlation function"]]  $\rm (ACF)$  of the amplitude coefficients and for the corresponding  [[Digital_Signal_Transmission/Basics_of_Coded_Transmission#PSD_calculation_of_a_digital_signal\|"power-spectral density"]]  $\rm (PSD)$:
	:$$\varphi_a(\lambda) = \left\{ \begin{array}{c} \frac{M+ 1}{3 \cdot (M-1)} \\		:$$\varphi_a(\lambda) = \left\{ \begin{array}{c} \frac{M+ 1}{3 \cdot (M-1)} \\\\ 0 \\ \end{array} \right.\quad\begin{array}{{1}c} {\rm{for}}\\ \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c}\lambda = 0, \\ \\ \lambda \ne 0 \\\end{array}\hspace{0.9cm}\Rightarrow \hspace{0.9cm}{\it \Phi_a(f)} =\frac{M+ 1}{3 \cdot (M-1)}= {\rm const.}$$Considering the spectral shaping by the basic transmission pulse  $g_s(t)$  with spectrum  $G_s(f)$,  we obtain::$$\varphi_{s}(\tau) = \frac{M+ 1}{3 \cdot (M-1)} \cdot \varphi^{^{\bullet}}_{gs}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.4cm}{\it \Phi}_{s}(f) = \frac{M+ 1}{3 \cdot (M-1)}\cdot \|G_s(f)\|^2\hspace{0.05cm}.$$
	\\ 0 \\ \end{array} \right.\quad
	\begin{array}{*{1}c} {\rm{for}}\\ \\ {\rm{for}} \\ \end{array}
	\begin{array}{*{20}c}\lambda = 0, \\ \\ \lambda \ne 0 \\
	\end{array}
	\hspace{0.9cm}\Rightarrow \hspace{0.9cm}{\it \Phi_a(f)} =
	\frac{M+ 1}{3 \cdot (M-1)}= {\rm const.}$$Considering the spectral shaping by the basic transmission pulse  $g_s(t)$  with spectrum  $G_s(f)$,  we obtain::$$\varphi_{s}(\tau) = \frac{M+ 1}{3 \cdot (M-1)} \cdot \varphi^{^{\bullet}}_{gs}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet
	\hspace{0.4cm}
	{\it \Phi}_{s}(f) = \frac{M+ 1}{3 \cdot (M-1)}\cdot \|G_s(f)\|^2
	\hspace{0.05cm}.$$

	One can see from these equations:		One can see from these equations:

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	\end{array}		\end{array}
	\hspace{0.9cm}\Rightarrow \hspace{0.9cm}{\it \Phi_a(f)} =		\hspace{0.9cm}\Rightarrow \hspace{0.9cm}{\it \Phi_a(f)} =
	\frac{M+ 1}{3 \cdot (M-1)}= {\rm const.}$$		\frac{M+ 1}{3 \cdot (M-1)}= {\rm const.}$$Considering the spectral shaping by the basic transmission pulse  $g_s(t)$  with spectrum  $G_s(f)$,  we obtain::$$\varphi_{s}(\tau) = \frac{M+ 1}{3 \cdot (M-1)} \cdot \varphi^{^{\bullet}}_{gs}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet

	Considering the spectral shaping by the basic transmission pulse  $g_s(t)$  with spectrum  $G_s(f)$,  we obtain:
	:$$\varphi_{s}(\tau) = \frac{M+ 1}{3 \cdot (M-1)} \cdot \varphi^{^{\bullet}}_{gs}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet
	\hspace{0.4cm}		\hspace{0.4cm}
	{\it \Phi}_{s}(f) = \frac{M+ 1}{3 \cdot (M-1)}\cdot \|G_s(f)\|^2		{\it \Phi}_{s}(f) = \frac{M+ 1}{3 \cdot (M-1)}\cdot \|G_s(f)\|^2
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	*Accordingly,  an amplitude coefficient  $a_\nu = 0$  is falsified twice as often compared to  $a_\nu = +1$  or  $a_\nu = -1$.  For AWGN noise with rms value  $\sigma_d$  as well as equal probability amplitude coefficients,  according to the section  [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Definition_of_the_bit_error_probability\|"Definition of the bit error probability"]]  for the  "symbol error probability":		*Accordingly,  an amplitude coefficient  $a_\nu = 0$  is falsified twice as often compared to  $a_\nu = +1$  or  $a_\nu = -1$.  For AWGN noise with rms value  $\sigma_d$  as well as equal probability amplitude coefficients,  according to the section  [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Definition_of_the_bit_error_probability\|"Definition of the bit error probability"]]  for the  "symbol error probability":

	:$$p_{\rm S} = { 1}/{3} \cdot \left[{\rm Q} \left( \frac{s_0/2}{\sigma_d}\right)+		:$$p_{\rm S} = { 1}/{3} \cdot \left[{\rm Q} \left( \frac{s_0/2}{\sigma_d}\right)+2 \cdot {\rm Q} \left( \frac{s_0/2}{\sigma_d}\right)+ {\rm Q} \left(\frac{s_0/2}{\sigma_d}\right)\right]=\frac{ 4}{3} \cdot {\rm Q} \left( \frac{s_0/2}{\sigma_d}\right)\hspace{0.05cm}.$$
	2 \cdot {\rm Q} \left( \frac{s_0/2}{\sigma_d}\right)+ {\rm Q} \left(
	\frac{s_0/2}{\sigma_d}\right)\right]=
	\frac{ 4}{3} \cdot {\rm Q} \left( \frac{s_0/2}{\sigma_d}\right)\hspace{0.05cm}.$$

	*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>
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	Taking into account the occurrence probabilities  $(1/4$  for equally probable symbols$)$  and the six possibilities of falsification (see arrows in the graph),  we obtain:		Taking into account the occurrence probabilities  $(1/4$  for equally probable symbols$)$  and the six possibilities of falsification (see arrows in the graph),  we obtain:
	:$$p_{\rm S} =		:$$p_{\rm S} ={ 6}/{4} \cdot {\rm Q} \left( \frac{s_0/3}{\sigma_d}\right)\hspace{0.05cm}.$$
	{ 6}/{4} \cdot {\rm Q} \left( \frac{s_0/3}{\sigma_d}\right)\hspace{0.05cm}.$$

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Conclusion:}$  In general, the  '''symbol error probability'''  for  $M$–level digital signal transmission is:		$\text{Conclusion:}$  In general, the  '''symbol error probability'''  for  $M$–level digital signal transmission is:
	:$$p_{\rm S} =		:$$p_{\rm S} =\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 level 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 level number  $M$. }}<br>
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	::$$\sigma_d^2 = \frac{N_0}{2T} \hspace{0.05cm}.$$		::$$\sigma_d^2 = \frac{N_0}{2T} \hspace{0.05cm}.$$
	*The comparison of the symbol error probabilities  $p_{\rm S}$  is performed for  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#System_optimization_with_power_limitation\|"power limitation"]]. The energy per bit for  $M$–level transmission is:		*The comparison of the symbol error probabilities  $p_{\rm S}$  is performed for  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#System_optimization_with_power_limitation\|"power limitation"]]. The energy per bit for  $M$–level transmission is:
	:$$E_{\rm B} = \frac{M+ 1}{3 \cdot (M-1)} \cdot s_0^2 \cdot T_{\rm B}		:$$E_{\rm B} = \frac{M+ 1}{3 \cdot (M-1)} \cdot s_0^2 \cdot T_{\rm B}\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	Substituting these equations into the general result on the  [[Digital_Signal_Transmission/Redundancy-Free_Coding#Error_probability_of_a_multilevel_system\|"last section"]],  we obtain for the symbol error probability:		Substituting these equations into the general result on the  [[Digital_Signal_Transmission/Redundancy-Free_Coding#Error_probability_of_a_multilevel_system\|"last section"]],  we obtain for the symbol error probability:
	:$$p_{\rm S} =		:$$p_{\rm S} =\frac{ 2 \cdot (M-1)}{M} \cdot {\rm Q} \left( \sqrt{\frac{s_0^2 /(M-1)^2}{\sigma_d^2}}\right) =\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)$$
	\frac{ 2 \cdot (M-1)}{M} \cdot {\rm Q} \left( \sqrt{\frac{s_0^2 /(M-1)^2}{\sigma_d^2}}\right) =
	\frac{ 2 \cdot (M-1)}{M} \cdot {\rm Q} \left( \sqrt{\frac{3 \cdot {\rm log_2}\hspace{0.05cm} (M)}{M^2 -1}\cdot
	\frac{2 \cdot E_{\rm B}}{N_0}}\right)$$
	[[File:EN_Dig_T_2_3_S3b_v2.png\|right\|frame\|Symbol error probability curves for different level numbers  $M$]]		[[File:EN_Dig_T_2_3_S3b_v2.png\|right\|frame\|Symbol error probability curves for different level numbers  $M$]]
	:$$\Rightarrow \hspace{0.3cm} p_{\rm S} =		:$$\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}.$$
	K_1 \cdot {\rm Q} \left( \sqrt{K_2\cdot
	\frac{2 \cdot E_{\rm B}}{N_0}}\right)\hspace{0.05cm}.$$

	For  $M = 2$,  set  $K_1 = K_2 = 1$.  For larger level numbers,  one obtains for the symbol error probability that can be achieved with  $M$–level redundancy-free coding:		For  $M = 2$,  set  $K_1 = K_2 = 1$.  For larger level numbers,  one obtains for the symbol error probability that can be achieved with  $M$–level redundancy-free coding:
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	*This can be seen from the factor  $K_2$  decreasing with  $M$,  for which then applies:		*This can be seen from the factor  $K_2$  decreasing with  $M$,  for which then applies:
	:$$p_{\rm S} = K_1 \cdot {\rm Q} \left( \sqrt{K_2\cdot		:$$p_{\rm S} = K_1 \cdot {\rm Q} \left( \sqrt{K_2\cdot\frac{2 \cdot s_{\rm 0}^2 \cdot T}{N_0} }\right)\hspace{0.3cm}{\rm with}\hspace{0.3cm}K_2 = \frac{ {\rm log_2}\,(M)}{(M-1)^2}\hspace{0.05cm}.$$
	\frac{2 \cdot s_{\rm 0}^2 \cdot T}{N_0} }\right)\hspace{0.3cm}{\rm with}\hspace{0.3cm}
	K_2 = \frac{ {\rm log_2}\,(M)}{(M-1)^2}
	\hspace{0.05cm}.$$
	*The constant  $K_1$  is unchanged from the above specification for power limitation,  while  $K_2$  is smaller by a factor of  $3$:		*The constant  $K_1$  is unchanged from the above specification for power limitation,  while  $K_2$  is smaller by a factor of  $3$:
	:$$M = 3\text{:} \ \ K_1 = 1.333, \ K_2 = 0.198;\hspace{1cm}M = 4\text{:} \ \ K_1 = 1.500, \ K_2 = 0.133;$$		:$$M = 3\text{:} \ \ K_1 = 1.333, \ K_2 = 0.198;\hspace{1cm}M = 4\text{:} \ \ K_1 = 1.500, \ K_2 = 0.133;$$
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	*The  '''symbol error probability'''  refers to the  $M$–level and possibly redundant sequences  $\langle c_\nu \rangle$  and  $\langle w_\nu \rangle$:		*The  '''symbol error probability'''  refers to the  $M$–level and possibly redundant sequences  $\langle c_\nu \rangle$  and  $\langle w_\nu \rangle$:
	:$$p_{\rm S} = \overline{{\rm Pr} (w_\nu \ne c_\nu)} =		:$$p_{\rm S} = \overline{{\rm Pr} (w_\nu \ne c_\nu)} =\lim_{N \to \infty} \frac{1}{N} \cdot \sum \limits^{N} _{\nu = 1} {\rm Pr} (w_\nu \ne c_\nu) \hspace{0.05cm}.$$
	\lim_{N \to \infty} \frac{1}{N} \cdot \sum \limits^{N} _{\nu = 1} {\rm Pr} (w_\nu \ne c_\nu) \hspace{0.05cm}.$$
	*The  '''bit error probability'''  describes the falsifications with respect to the binary sequences  $\langle q_\nu \rangle$  and  $\langle v_\nu \rangle$  of source and sink:		*The  '''bit error probability'''  describes the falsifications with respect to the binary sequences  $\langle q_\nu \rangle$  and  $\langle v_\nu \rangle$  of source and sink:
	:$$p_{\rm B} = \overline{{\rm Pr} (v_\nu \ne q_\nu)} =		:$$p_{\rm B} = \overline{{\rm Pr} (v_\nu \ne q_\nu)} =\lim_{N \to \infty} \frac{1}{N} \cdot \sum \limits^{N} _{\nu = 1} {\rm Pr} (v_\nu \ne q_\nu) \hspace{0.05cm}.$$
	\lim_{N \to \infty} \frac{1}{N} \cdot \sum \limits^{N} _{\nu = 1} {\rm Pr} (v_\nu \ne q_\nu) \hspace{0.05cm}.$$

	The diagram illustrates these two definitions and is also valid for the next chapters.  The block  "encoder"  causes		The diagram illustrates these two definitions and is also valid for the next chapters.  The block  "encoder"  causes

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Add German interlanguage link

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	{{Display}}		{{Display}}
			[[de:Digitalsignalübertragung/Redundanzfreie_Codierung]]

Hwang at 14:01, 23 January 2023

2023-01-23T14:01:24Z

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	\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>

Hwang at 14:01, 23 January 2023

2023-01-23T14:01:00Z

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

Guenter at 15:02, 3 June 2022

2022-06-03T15:02:57Z

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Hwang at 15:56, 23 May 2022

2022-05-23T15:56:31Z

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	*In general,  the symbol error probability  $p_{\rm S}$  can be calculated somewhat more easily than the bit error probability  $p_{\rm B}$  for redundancy-containing multilevel systems.		*In general,  the symbol error probability  $p_{\rm S}$  can be calculated somewhat more easily than the bit error probability  $p_{\rm B}$  for redundancy-containing multilevel systems.

	* However,  a comparison of systems with different step '''KORREKTUR: level''' numbers  $M$  or different types of coding should always be based on the bit error probability  $p_{\rm B}$  for reasons of fairness.  The mapping between the source and coded symbols must also be taken into account,  as shown in the following example.}}<br>		* However,  a comparison of systems with different step '''KORREKTUR: level''' numbers  $M$  or different types of coding should always be based on the bit error probability  $p_{\rm B}$  for reasons of fairness.  The mapping between the source and coded '''KORREKTUR: encoder''' symbols must also be taken into account,  as shown in the following example.}}<br>

	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=

Hwang at 15:55, 23 May 2022

2022-05-23T15:55:20Z

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	'''Symbolwise coding'''		'''Symbolwise coding'''
	*Here,  a code symbol  $c_\nu$  is generated with each incoming source symbol  $q_\nu$,  which can depend not only on the current symbol but also on previous symbols  $q_{\nu -1}$,  $q_{\nu -2}$, ... <br>		*Here,  a code '''KORREKTUR: encoder''' symbol  $c_\nu$  is generated with each incoming source symbol  $q_\nu$,  which can depend not only on the current symbol but also on previous symbols  $q_{\nu -1}$,  $q_{\nu -2}$, ... <br>

	*It is typical for all transmission codes for symbolwise coding that the symbol duration  $T_c$  of the usually multilevel and redundant encoder '''KORREKTUR: encoded''' signal  $c(t)$  corresponds to the bit duration  $T_q$  of the source signal,  which is assumed to be binary and redundancy-free.<br>		*It is typical for all transmission codes for symbolwise coding that the symbol duration  $T_c$  of the usually multilevel and redundant encoder '''KORREKTUR: encoded''' signal  $c(t)$  corresponds to the bit duration  $T_q$  of the source signal,  which is assumed to be binary and redundancy-free.<br>
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	'''Blockwise coding'''		'''Blockwise coding'''
	*Here,  a block of  $m_q$  binary source symbols  $(M_q = 2)$  of bit duration  $T_q$  is assigned a one-to-one sequence of  $m_c$  code symbols from an alphabet with code symbol set size  $M_c \ge 2$.		*Here,  a block of  $m_q$  binary source symbols  $(M_q = 2)$  of bit duration  $T_q$  is assigned a one-to-one sequence of  $m_c$  code '''KORREKTUR: encoder''' symbols from an alphabet with code '''KORREKTUR: encoder''' symbol set size  $M_c \ge 2$.

	*For the  '''symbol duration of a code symbol'''  then holds:		*For the  '''symbol duration of a code '''KORREKTUR: encoder''' symbol'''  then holds:
	:$$T_c = \frac{m_q}{m_c} \cdot T_q \hspace{0.05cm},$$		:$$T_c = \frac{m_q}{m_c} \cdot T_q \hspace{0.05cm},$$
	*The  '''relative redundancy of a block code'''  is in general		*The  '''relative redundancy of a block code'''  is in general
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	Thereby holds:		Thereby holds:
	#If  $M_c$  is a power to the base  $2$,  then  $m_q = \log_2 \hspace{0.05cm} (M_c)$  are combined into a single code symbol  $(m_c = 1)$.  In this case, the relative redundancy is actually  $r_c = 0$.<br>		#If  $M_c$  is a power to the base  $2$,  then  $m_q = \log_2 \hspace{0.05cm} (M_c)$  are combined into a single code '''KORREKTUR: encoder''' symbol  $(m_c = 1)$.  In this case, the relative redundancy is actually  $r_c = 0$.<br>
	#If  $M_c$  is not a power of two,  a hundred percent redundancy-free block coding is not possible.  For example, if  $m_q = 3$  binary symbols are encoded by  $m_c = 2$  ternary symbols and  $T_c = 1.5 \cdot T_q$  is set,  a relative redundancy of  $r_c = 1-1.5/ \log_2 \hspace{0.05cm} (3) \approx 5\%$  remains.<br>		#If  $M_c$  is not a power of two,  a hundred percent redundancy-free block coding is not possible.  For example, if  $m_q = 3$  binary symbols are encoded by  $m_c = 2$  ternary symbols and  $T_c = 1.5 \cdot T_q$  is set,  a relative redundancy of  $r_c = 1-1.5/ \log_2 \hspace{0.05cm} (3) \approx 5\%$  remains.<br>
	#Encoding a block of  $128$  binary symbols with  $81$  ternary symbols results in a relative code redundancy of less than  $r_c = 0.3\%$.<br><br>		#Encoding a block of  $128$  binary symbols with  $81$  ternary symbols results in a relative code redundancy of less than  $r_c = 0.3\%$.<br><br>
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	<br>		<br>
	For equally probable binary source symbols  $q_\nu$  the quaternary code symbols  $c_\nu$  also occur with equal probability.  Thus,  we obtain for the symbol error probability:		For equally probable binary source symbols  $q_\nu$  the quaternary code '''KORREKTUR: encoder''' symbols  $c_\nu$  also occur with equal probability.  Thus,  we obtain for the symbol error probability:
	:$$p_{\rm S} ={1}/{4}\cdot (2 \cdot p + 2 \cdot 2 \cdot p) = {3}/{2} \cdot p\hspace{0.05cm}.$$		:$$p_{\rm S} ={1}/{4}\cdot (2 \cdot p + 2 \cdot 2 \cdot p) = {3}/{2} \cdot p\hspace{0.05cm}.$$

Hwang at 15:53, 23 May 2022

2022-05-23T15:53:09Z

Hwang at 15:47, 23 May 2022

2022-05-23T15:47:54Z

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	<br>		<br>
	For this system comparison under fair conditions,  the following are assumed:		For this system comparison under fair conditions,  the following are assumed:
	*Let the equivalent bit rate  $R_{\rm B} = 1/T_{\rm B}$  be constant.  Depending on the number  $M$,  the symbol duration of the encoder signal and the transmitted signal is thus:		*Let the equivalent bit rate  $R_{\rm B} = 1/T_{\rm B}$  be constant.  Depending on the '''KORREKTUR: level''' number  $M$,  the symbol duration of the encoder signal and the transmitted signal is thus:
	:$$T = T_{\rm B} \cdot {\rm log_2} (M) \hspace{0.05cm}.$$		:$$T = T_{\rm B} \cdot {\rm log_2} (M) \hspace{0.05cm}.$$
	*The Nyquist condition is satisfied by a  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems\|"root–root characteristic"]]  with rolloff factor  $r$.  Furthermore,  no intersymbol interference occurs.  The detection noise power is:		*The Nyquist condition is satisfied by a  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems\|"root–root characteristic"]]  with rolloff factor  $r$.  Furthermore,  no intersymbol interference occurs.  The detection noise power is:

← Older revision		Revision as of 17:53, 23 May 2022
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	*Here,  a code symbol  $c_\nu$  is generated with each incoming source symbol  $q_\nu$,  which can depend not only on the current symbol but also on previous symbols  $q_{\nu -1}$,  $q_{\nu -2}$, ... <br>		*Here,  a code symbol  $c_\nu$  is generated with each incoming source symbol  $q_\nu$,  which can depend not only on the current symbol but also on previous symbols  $q_{\nu -1}$,  $q_{\nu -2}$, ... <br>

	*It is typical for all transmission codes for symbolwise coding that the symbol duration  $T_c$  of the usually multilevel and redundant encoder signal  $c(t)$  corresponds to the bit duration  $T_q$  of the source signal,  which is assumed to be binary and redundancy-free.<br>		*It is typical for all transmission codes for symbolwise coding that the symbol duration  $T_c$  of the usually multilevel and redundant encoder '''KORREKTUR: encoded''' signal  $c(t)$  corresponds to the bit duration  $T_q$  of the source signal,  which is assumed to be binary and redundancy-free.<br>


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	*Starting from the redundancy-free binary source signal  $q(t)$  with bit duration  $T_q$,		*Starting from the redundancy-free binary source signal  $q(t)$  with bit duration  $T_q$,
	*a  $M_c$–level code signal  $c(t)$  with symbol duration  $T_c = T_q \cdot \log_2 \hspace{0.05cm} (M_c)$  is generated.		*a  $M_c$–level code '''KORREKTUR: encoded''' signal  $c(t)$  with symbol duration  $T_c = T_q \cdot \log_2 \hspace{0.05cm} (M_c)$  is generated.


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	To simplify the notation and to align the nomenclature with the [[Digital_Signal_Transmission\| "first main chapter"]],  we use in the following		To simplify the notation and to align the nomenclature with the [[Digital_Signal_Transmission\| "first main chapter"]],  we use in the following
	*the bit duration  $T_{\rm B} = T_q$  of the redundancy-free binary source signal,		*the bit duration  $T_{\rm B} = T_q$  of the redundancy-free binary source signal,
	*the symbol duration  $T = T_c$  of the encoder signal and the transmitted signal, and		*the symbol duration  $T = T_c$  of the encoder '''KORREKTUR: encoded''' signal and the transmitted signal, and
	*the number  $M = M_c$  of levels.<br>}}		*the number  $M = M_c$  of levels.<br>}}

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	The lower plot shows the $($almost$)$ redundancy-free ternary signal  $s_3(t)$  and the mapping of three binary symbols each to two ternary symbols.		The lower plot shows the $($almost$)$ redundancy-free ternary signal  $s_3(t)$  and the mapping of three binary symbols each to two ternary symbols.
	*The possible amplitude coefficients are  $-1$,  $0$  and  $+1$  and the symbol duration of the coded signal  $T = 3/2 \cdot T_{\rm B}$.		*The possible amplitude coefficients are  $-1$,  $0$  and  $+1$  and the symbol duration of the coded '''KORREKTUR: encoded''' signal  $T = 3/2 \cdot T_{\rm B}$.

	*It can be seen from the green mapping table that the coefficients  $+1$  and  $-1$  occur somewhat more frequently than the coefficient  $a_\nu = 0$.  This results in the above mentioned relative redundancy of  $5\%$.		*It can be seen from the green mapping table that the coefficients  $+1$  and  $-1$  occur somewhat more frequently than the coefficient  $a_\nu = 0$.  This results in the above mentioned relative redundancy of  $5\%$.
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	<br>		<br>
	For this system comparison under fair conditions,  the following are assumed:		For this system comparison under fair conditions,  the following are assumed:
	*Let the equivalent bit rate  $R_{\rm B} = 1/T_{\rm B}$  be constant.  Depending on the '''KORREKTUR: level''' number  $M$,  the symbol duration of the encoder signal and the transmitted signal is thus:		*Let the equivalent bit rate  $R_{\rm B} = 1/T_{\rm B}$  be constant.  Depending on the '''KORREKTUR: level''' number  $M$,  the symbol duration of the encoder '''KORREKTUR: encoded''' signal and the transmitted signal is thus:
	:$$T = T_{\rm B} \cdot {\rm log_2} (M) \hspace{0.05cm}.$$		:$$T = T_{\rm B} \cdot {\rm log_2} (M) \hspace{0.05cm}.$$
	*The Nyquist condition is satisfied by a  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems\|"root–root characteristic"]]  with rolloff factor  $r$.  Furthermore,  no intersymbol interference occurs.  The detection noise power is:		*The Nyquist condition is satisfied by a  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems\|"root–root characteristic"]]  with rolloff factor  $r$.  Furthermore,  no intersymbol interference occurs.  The detection noise power is:
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	\lim_{N \to \infty} \frac{1}{N} \cdot \sum \limits^{N} _{\nu = 1} {\rm Pr} (v_\nu \ne q_\nu) \hspace{0.05cm}.$$		\lim_{N \to \infty} \frac{1}{N} \cdot \sum \limits^{N} _{\nu = 1} {\rm Pr} (v_\nu \ne q_\nu) \hspace{0.05cm}.$$

	The diagram illustrates these two definitions and is also valid for the next chapters.  The block  "~~coder~~"  causes		The diagram illustrates these two definitions and is also valid for the next chapters.  The block  "encoder"  causes
	*in the present chapter a redundancy-free coding,		*in the present chapter a redundancy-free coding,
	*in the  [[Digital_Signal_Transmission/Block_Coding_with_4B3T_Codes\|"following chapter"]]  a blockwise transmission coding,  and finally		*in the  [[Digital_Signal_Transmission/Block_Coding_with_4B3T_Codes\|"following chapter"]]  a blockwise transmission coding,  and finally