Digital Signal Transmission/Basics of Coded Transmission - Revision history

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	:$$N = T_{\rm M}/(2T), \hspace{0.5cm}\lambda = \kappa- \nu,\hspace{0.5cm}t - \nu \cdot T \to T$$		:$$N = T_{\rm M}/(2T), \hspace{0.5cm}\lambda = \kappa- \nu,\hspace{0.5cm}t - \nu \cdot T \to T$$
	:for this can also be written:		:for this can also be written:
	:$$\varphi_s(\tau) = \lim_{T_{\rm M} \to \infty}\frac{1}{T_{\rm M}}		:$$\varphi_s(\tau) = \lim_{T_{\rm M} \to \infty}\frac{1}{T_{\rm M}}\cdot\int_{-T_{\rm M}/2}^{+T_{\rm M}/2}\sum_{\nu = -\infty}^{+\infty} \sum_{\kappa = -\infty}^{+\infty}a_\nu \cdot g_s ( t - \nu \cdot T ) \cdota_\kappa \cdot g_s ( t + \tau - \kappa \cdot T )\,{\rm d} t \hspace{0.05cm}.$$
	\cdot
	\int_{-T_{\rm M}/2}^{+T_{\rm M}/2}
	\sum_{\nu = -\infty}^{+\infty} \sum_{\kappa = -\infty}^{+\infty}
	a_\nu \cdot g_s ( t - \nu \cdot T ) \~~cdot~~
	a_\kappa \cdot g_s ( t + \tau - \kappa \cdot T )
	\,{\rm d} t \hspace{0.05cm}.$$
	Now the following quantities are introduced for abbreviation:		Now the following quantities are introduced for abbreviation:

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	*Considering the relation between energy ACF and energy spectrum,		*Considering the relation between energy ACF and energy spectrum,
	:$$\varphi^{^{\hspace{0.05cm}\bullet}}_{gs}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{gs}(f) = \|G_s(f)\|^2\hspace{0.05cm},$$		:$$\varphi^{^{\hspace{0.05cm}\bullet}}_{gs}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{gs}(f) = \|G_s(f)\|^2\hspace{0.05cm},$$
	:and the   [[Signal_Representation/~~Fourier_Transform_Theorems~~#Shifting_Theorem\|"shifting theorem"]],  the  '''power-spectral density of the digital signal'''  $s(t)$  can be represented in the following way:		:and the   [[Signal_Representation/The_Fourier_Transform_Theorems#Shifting_Theorem\|"shifting theorem"]],  the  '''power-spectral density of the digital signal'''  $s(t)$  can be represented in the following way:
	:$${\it \Phi}_s(f) = \sum_{\lambda =-\infty}^{+\infty}{1}/{T} \cdot \varphi_a(\lambda)\cdot {\it\Phi}^{^{\hspace{0.05cm}\bullet}}_{gs}(f) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}2 \pi f \hspace{0.02cm} \lambda T} = {1}/{T} \cdot \|G_s(f)\|^2 \cdot \sum_{\lambda =-\infty}^{+\infty}\varphi_a(\lambda)\cdot \cos (2 \pi f \lambda T)\hspace{0.05cm}.$$		:$${\it \Phi}_s(f) = \sum_{\lambda =-\infty}^{+\infty}{1}/{T} \cdot \varphi_a(\lambda)\cdot {\it\Phi}^{^{\hspace{0.05cm}\bullet}}_{gs}(f) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}2 \pi f \hspace{0.02cm} \lambda T} = {1}/{T} \cdot \|G_s(f)\|^2 \cdot \sum_{\lambda =-\infty}^{+\infty}\varphi_a(\lambda)\cdot \cos (2 \pi f \lambda T)\hspace{0.05cm}.$$
	:Here it is considered that  ${\it \Phi}_s(f)$  and  $\|G_s(f)\|^2$  are real-valued and at the same time  $\varphi_a(-\lambda) =\varphi_a(+\lambda)$  holds.<br><br>		:Here it is considered that  ${\it \Phi}_s(f)$  and  $\|G_s(f)\|^2$  are real-valued and at the same time  $\varphi_a(-\lambda) =\varphi_a(+\lambda)$  holds.<br><br>
	*If we now define the  '''spectral power density of the amplitude coefficients'''  to be		*If we now define the  '''spectral power density of the amplitude coefficients'''  to be
	:$${\it \Phi}_a(f) = \sum_{\lambda =-\infty}^{+\infty}\varphi_a(\lambda)\cdot {\rm e}^{- {\~~rmj~~}\hspace{0.05cm} 2 \pi f \hspace{0.02cm} \lambda \hspace{0.02cm}T} =\varphi_a(0) + 2 \cdot \sum_{\lambda =1}^{\infty}\varphi_a(\lambda)\cdot\cos ( 2 \pi f\lambda T) \hspace{0.05cm},$$		:$${\it \Phi}_a(f) = \sum_{\lambda =-\infty}^{+\infty}\varphi_a(\lambda)\cdot {\rm e}^{- {\rm j}\hspace{0.05cm} 2 \pi f \hspace{0.02cm} \lambda \hspace{0.02cm}T} =\varphi_a(0) + 2 \cdot \sum_{\lambda =1}^{\infty}\varphi_a(\lambda)\cdot\cos ( 2 \pi f\lambda T) \hspace{0.05cm},$$
	:then the following expression is obtained:		:then the following expression is obtained:
	:$${\it \Phi}_s(f) = {\it \Phi}_a(f) \cdot {1}/{T} \cdot\|G_s(f)\|^2 \hspace{0.05cm}.$$		:$${\it \Phi}_s(f) = {\it \Phi}_a(f) \cdot {1}/{T} \cdot\|G_s(f)\|^2 \hspace{0.05cm}.$$
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	The previous results are now illustrated by examples.  Starting from binary bipolar amplitude coefficients  $a_\nu \in \{-1, +1\}$,  if there are no bonds between the individual amplitude coefficients  $a_\nu$,  we obtain:<br>		The previous results are now illustrated by examples.  Starting from binary bipolar amplitude coefficients  $a_\nu \in \{-1, +1\}$,  if there are no bonds between the individual amplitude coefficients  $a_\nu$,  we obtain:<br>
	[[File:P_ID1306__Dig_T_2_1_S6_v2.png\|right\|frame\|Signal section  ACF and PSD for binary bipolar signaling\|class=fit]]		[[File:P_ID1306__Dig_T_2_1_S6_v2.png\|right\|frame\|Signal section  ACF and PSD for binary bipolar signaling\|class=fit]]
	:$$\varphi_a(\lambda) = \left\{ \begin{array}{c} 1		:$$\varphi_a(\lambda) = \left\{ \begin{array}{c} 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{0.5cm}\Rightarrow \hspace{0.5cm}\varphi_s(\tau)={1}/{T} \cdot \varphi^{^{\bullet}}_{gs}(\tau)\hspace{0.05cm}.$$
	\\ 0 \\ \end{array} \right.\quad
	\begin{array}{*{1}c} {\rm{for}}\\ {\rm{for}} \\ \end{array}
	\begin{array}{*{20}c}\lambda = 0, \\ \lambda \ne 0 \\
	\end{array}
	\hspace{0.5cm}\Rightarrow \hspace{0.5cm}\varphi_s(\tau)=
	{1}/{T} \cdot \varphi^{^{\bullet}}_{gs}(\tau)\hspace{0.05cm}.$$

	The graph shows two signal sections each with rectangular pulses  $g_s(t)$,  which accordingly lead to a triangular auto-correlation function  $\rm (ACF)$  and to a  $\rm sinc^2$–shaped power-spectral density  $\rm (PSD)$.		The graph shows two signal sections each with rectangular pulses  $g_s(t)$,  which accordingly lead to a triangular auto-correlation function  $\rm (ACF)$  and to a  $\rm sinc^2$–shaped power-spectral density  $\rm (PSD)$.

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

Guenter at 14:53, 23 March 2023

2023-03-23T14:53:25Z

← Older revision		Revision as of 16:53, 23 March 2023
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	== Source coding – Channel coding – ~~Transmission~~ coding ==		== Source coding – Channel coding – Line coding ==
	<br>		<br>
	"Coding"  is the conversion of the source symbol sequence  $\langle q_\nu \rangle$  with symbol set size  $M_q$  into an encoder symbol sequence  $\langle c_\nu \rangle$  with symbol set size  $M_c$.  Usually,  coding manipulates the redundancy contained in a digital signal.  Often – but not always –  $M_q$  and  $M_c$  are different.<br>		"Coding"  is the conversion of the source symbol sequence  $\langle q_\nu \rangle$  with symbol set size  $M_q$  into an encoder symbol sequence  $\langle c_\nu \rangle$  with symbol set size  $M_c$.  Usually,  coding manipulates the redundancy contained in a digital signal.  Often – but not always –  $M_q$  and  $M_c$  are different.<br>
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	*On the other hand,  with  '''channel coding'''  a noticeable improvement in the transmission behavior is achieved by using a redundancy specifically added at the transmitter to detect and correct transmission errors at the receiver end.  Such codes,  the most important of which are block codes,  convolutional codes and turbo codes,  are particularly important in the case of heavily disturbed channels.  The greater the relative redundancy of the encoded signal,  the better the correction properties of the code,  albeit at a reduced user data rate.		*On the other hand,  with  '''channel coding'''  a noticeable improvement in the transmission behavior is achieved by using a redundancy specifically added at the transmitter to detect and correct transmission errors at the receiver end.  Such codes,  the most important of which are block codes,  convolutional codes and turbo codes,  are particularly important in the case of heavily disturbed channels.  The greater the relative redundancy of the encoded signal,  the better the correction properties of the code,  albeit at a reduced user data rate.

	*'''~~Transmission~~ coding''' ~~– often referred to as~~&nbsp~~; "line coding"  &ndash~~; is used to adapt the transmitted signal to the spectral characteristics of the transmission channel and the receiving equipment by recoding the source symbols.  For example,  in the case of a channel with the frequency response characteristic  $H_{\rm K}(f=0) = 0$,  over which consequently no DC signal can be transmitted,  transmission coding must ensure that the encoder symbol sequence contains neither a long  $\rm L$ sequence nor a long  $\rm H$ sequence.<br>		*'''Line coding'''  is used to adapt the transmitted signal to the spectral characteristics of the transmission channel and the receiving equipment by recoding the source symbols.  For example,  in the case of a channel with the frequency response characteristic  $H_{\rm K}(f=0) = 0$,  over which consequently no DC signal can be transmitted,  transmission coding must ensure that the encoder symbol sequence contains neither a long  $\rm L$ sequence nor a long  $\rm H$ sequence.<br>

Hwang at 13:59, 23 January 2023

2023-01-23T13:59:55Z

← Older revision		Revision as of 15:59, 23 January 2023
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	*[[Channel_Coding\|"Channel Coding"]]  has its own book dedicated to it in our learning tutorial.		*[[Channel_Coding\|"Channel Coding"]]  has its own book dedicated to it in our learning tutorial.
	*Source coding is covered in detail in the book  [[Information_Theory\|"Information Theory"]]  (main chapter 2).		*Source coding is covered in detail in the book  [[Information_Theory\|"Information Theory"]]  (main chapter 2).
	*[[Examples_of_Communication_Systems/~~Sprachcodierung~~\|"Speech coding"]]  – described in the book "Examples of Communication Systems" –  is a special form of source coding.<br>		*[[Examples_of_Communication_Systems/Speech_Coding\|"Speech coding"]]  – described in the book "Examples of Communication Systems" –  is a special form of source coding.<br>

Hwang at 13:58, 23 January 2023

2023-01-23T13:58:32Z

← Older revision		Revision as of 15:58, 23 January 2023
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	*[[Channel_Coding\|"Channel Coding"]]  has its own book dedicated to it in our learning tutorial.		*[[Channel_Coding\|"Channel Coding"]]  has its own book dedicated to it in our learning tutorial.
	*Source coding is covered in detail in the book  [[Information_Theory\|"Information Theory"]]  (main chapter 2).		*Source coding is covered in detail in the book  [[Information_Theory\|"Information Theory"]]  (main chapter 2).
	*[[Examples_of_Communication_Systems/Sprachcodierung\|"~~Voice~~ coding"]]  – described in the book "Examples of Communication Systems" –  is a special form of source coding.<br>		*[[Examples_of_Communication_Systems/Sprachcodierung\|"Speech coding"]]  – described in the book "Examples of Communication Systems" –  is a special form of source coding.<br>

Hwang at 13:57, 23 January 2023

2023-01-23T13:57:38Z

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	*The task of  '''source coding'''  is redundancy reduction for data compression,  as applied for example in image coding.  By exploiting statistical ties between the individual points of an image or between the brightness values of a point at different times  (in the case of moving image sequences),  methods can be developed that lead to a noticeable reduction in the amount of data  (measured in  "bit"  or "byte"),  while maintaining virtually the same  (subjective)  image quality.  A simple example of this is "differential pulse code modulation"  $\rm (DPCM)$.<br>		*The task of  '''source coding'''  is redundancy reduction for data compression,  as applied for example in image coding.  By exploiting statistical ties between the individual points of an image or between the brightness values of a point at different times  (in the case of moving image sequences),  methods can be developed that lead to a noticeable reduction in the amount of data  (measured in  "bit"  or "byte"),  while maintaining virtually the same  (subjective)  image quality.  A simple example of this is "differential pulse code modulation"  $\rm (DPCM)$.<br>

	*On the other hand,  with  '''channel coding'''  a noticeable improvement in the transmission behavior is achieved by using a redundancy specifically added at the transmitter to detect and correct transmission errors at the receiver end.  Such codes,  the most important of which are block codes,  convolutional codes and turbo codes,  are particularly important in the case of heavily disturbed channels.  The greater the relative redundancy of the ~~coded~~ signal,  the better the correction properties of the code,  albeit at a reduced user data rate.		*On the other hand,  with  '''channel coding'''  a noticeable improvement in the transmission behavior is achieved by using a redundancy specifically added at the transmitter to detect and correct transmission errors at the receiver end.  Such codes,  the most important of which are block codes,  convolutional codes and turbo codes,  are particularly important in the case of heavily disturbed channels.  The greater the relative redundancy of the encoded signal,  the better the correction properties of the code,  albeit at a reduced user data rate.

	*'''Transmission coding''' – often referred to as  "line coding"  – is used to adapt the transmitted signal to the spectral characteristics of the transmission channel and the receiving equipment by recoding the source symbols.  For example,  in the case of a channel with the frequency response characteristic  $H_{\rm K}(f=0) = 0$,  over which consequently no DC signal can be transmitted,  transmission coding must ensure that the encoder symbol sequence contains neither a long  $\rm L$ sequence nor a long  $\rm H$ sequence.<br>		*'''Transmission coding''' – often referred to as  "line coding"  – is used to adapt the transmitted signal to the spectral characteristics of the transmission channel and the receiving equipment by recoding the source symbols.  For example,  in the case of a channel with the frequency response characteristic  $H_{\rm K}(f=0) = 0$,  over which consequently no DC signal can be transmitted,  transmission coding must ensure that the encoder symbol sequence contains neither a long  $\rm L$ sequence nor a long  $\rm H$ sequence.<br>

Hwang at 13:56, 23 January 2023

2023-01-23T13:56:30Z

← Older revision		Revision as of 15:56, 23 January 2023
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	== # OVERVIEW OF THE SECOND MAIN CHAPTER # ==		== # OVERVIEW OF THE SECOND MAIN CHAPTER # ==
	<br>		<br>
	The second main chapter deals with so-called  '''transmission coding''',  which is sometimes also referred to as  "line coding"  in literature.  In this process,  an adaptation of the digital ~~transmission~~ signal to the characteristics of the transmission channel is achieved through the targeted addition of redundancy.  In detail,  the following are dealt with:		The second main chapter deals with so-called  '''transmission coding''',  which is sometimes also referred to as  "line coding"  in literature.  In this process,  an adaptation of the digital transmitted signal to the characteristics of the transmission channel is achieved through the targeted addition of redundancy.  In detail,  the following are dealt with:

	#  Some basic concepts of information theory such as  »information content«  and  »entropy«,		#  Some basic concepts of information theory such as  »information content«  and  »entropy«,

Guenter at 12:53, 31 August 2022

2022-08-31T12:53:03Z

← Older revision		Revision as of 14:53, 31 August 2022
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	The second main chapter deals with so-called  '''transmission coding''',  which is sometimes also referred to as  "line coding"  in literature.  In this process,  an adaptation of the digital transmission signal to the characteristics of the transmission channel is achieved through the targeted addition of redundancy.  In detail,  the following are dealt with:		The second main chapter deals with so-called  '''transmission coding''',  which is sometimes also referred to as  "line coding"  in literature.  In this process,  an adaptation of the digital transmission signal to the characteristics of the transmission channel is achieved through the targeted addition of redundancy.  In detail,  the following are dealt with:

	*some basic concepts of information theory such as  »information content«  and  »entropy«,		#  Some basic concepts of information theory such as  »information content«  and  »entropy«,
	*the  »auto-correlation function«  and the  »power-spectral densities«  of digital signals,		#  the  »auto-correlation function«  and the  »power-spectral densities«  of digital signals,
	*the  »redundancy-free coding«  which leads to a non-binary transmitted signal,		#  the  »redundancy-free coding«  which leads to a non-binary transmitted signal,
	*the calculation of  »symbol and bit error probability«  for  »multilevel systems« ,		#  the calculation of  »symbol and bit error probability«  for  »multilevel systems« ,
	*the so-called  »4B3T codes«  as an important example of  »block-wise coding«,  and		#  the so-called  »4B3T codes«  as an important example of  »block-wise coding«,  and
	*the  »pseudo-ternary codes«,  each of which realizes symbol-wise coding.		#  the  »pseudo-ternary codes«,  each of which realizes symbol-wise coding.

Guenter at 15:15, 24 August 2022

2022-08-24T15:15:04Z

← Older revision		Revision as of 17:15, 24 August 2022
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	<br>		<br>
	In the following we always assume the block diagram sketched on the right and the following agreements:		In the following we always assume the block diagram sketched on the right and the following agreements:
	[[File:~~EN_Dig_T_2_1_S3_v2~~.png\|right\|frame\|Block diagram for the description of multilevel and coded transmission systems ~~'''Korrektur''' bzw.~~\|class=fit]]		[[File:EN_Dig_T_2_1_S3_v23.png\|right\|frame\|Block diagram for the description of multilevel and coded transmission systems\|class=fit]]
	*Let the digital source signal  $q(t)$  be binary  $(M_q = 2)$  and redundancy-free  $(H_q = 1 \ \rm bit/symbol)$.		*Let the digital source signal  $q(t)$  be binary  $(M_q = 2)$  and redundancy-free  $(H_q = 1 \ \rm bit/symbol)$.

← Older revision		Revision as of 15:59, 23 January 2023
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	*[[Channel_Coding\|"Channel Coding"]]  has its own book dedicated to it in our learning tutorial.		*[[Channel_Coding\|"Channel Coding"]]  has its own book dedicated to it in our learning tutorial.
	*Source coding is covered in detail in the book  [[Information_Theory\|"Information Theory"]]  (main chapter 2).		*Source coding is covered in detail in the book  [[Information_Theory\|"Information Theory"]]  (main chapter 2).
	*[[Examples_of_Communication_Systems/~~Sprachcodierung~~\|"Speech coding"]]  – described in the book "Examples of Communication Systems" –  is a special form of source coding.<br>		*[[Examples_of_Communication_Systems/Speech_Coding\|"Speech coding"]]  – described in the book "Examples of Communication Systems" –  is a special form of source coding.<br>

← Older revision		Revision as of 15:58, 23 January 2023
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	*[[Channel_Coding\|"Channel Coding"]]  has its own book dedicated to it in our learning tutorial.		*[[Channel_Coding\|"Channel Coding"]]  has its own book dedicated to it in our learning tutorial.
	*Source coding is covered in detail in the book  [[Information_Theory\|"Information Theory"]]  (main chapter 2).		*Source coding is covered in detail in the book  [[Information_Theory\|"Information Theory"]]  (main chapter 2).
	*[[Examples_of_Communication_Systems/Sprachcodierung\|"~~Voice~~ coding"]]  – described in the book "Examples of Communication Systems" –  is a special form of source coding.<br>		*[[Examples_of_Communication_Systems/Sprachcodierung\|"Speech coding"]]  – described in the book "Examples of Communication Systems" –  is a special form of source coding.<br>