Channel Coding/Objective of Channel Coding - Revision history

Guenter at 16:26, 18 November 2022

2022-11-18T16:26:53Z

Guenter at 14:22, 17 November 2022

2022-11-17T14:22:34Z

Hwang at 11:22, 17 November 2022

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Hwang at 10:30, 17 November 2022

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Hwang at 10:26, 17 November 2022

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Hwang at 10:23, 17 November 2022

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Hwang at 10:20, 17 November 2022

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Guenter at 16:52, 16 November 2022

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Guenter at 16:50, 16 November 2022

2022-11-16T16:50:35Z

Guenter at 12:55, 14 November 2022

2022-11-14T12:55:32Z

@@ Line 113: / Line 113: @@
 If you book a railway ticket online and print it out,&nbsp; you will find an example of a two-dimensional bar code,&nbsp; namely the&nbsp; [https://en.wikipedia.org/wiki/Aztec_Code $\text{Aztec code}$]&nbsp; developed in 1995 by Andy Longacre at the Welch Allyn company in the USA,&nbsp; with which amounts of data up to&nbsp; $3000$&nbsp; characters can be encoded.&nbsp; Due to the&nbsp; [[Channel_Coding/Definition_und_Eigenschaften_von_Reed–Solomon–Codes|$\text{Reed&ndash;Solomon error correction}$]],&nbsp; reconstruction of the data content is still possible even if up to&nbsp; $40\%$&nbsp; of the code has been destroyed,&nbsp; for example by bending the ticket or by coffee stains.<br>
-On the right you see a $\rm QR\ code$&nbsp; ("Quick Response")&nbsp; with associated content.
+On the right you see a&nbsp; $\rm QR\ code$&nbsp; ("Quick Response")&nbsp; with associated content.
 [[File:EN_KC_T_1_1_S2a_v2.png|right|frame|Two-dimensional bar codes:&nbsp; Aztec and QR code|class=fit]]
@@ Line 120: / Line 120: @@
 *In the meantime,&nbsp; the use of the QR code has become very diverse.&nbsp; In Japan,&nbsp; it can be found on almost every advertising poster and on every business card.&nbsp; It has also been becoming more and more popular in Europe.<br>
-*All two-dimensional bar codes have square markings to calibrate the reader.&nbsp; You can find details on this in&nbsp; [KM+09]<ref>Kötter, R.; Mayer, T.; Tüchler, M.; Schreckenbach, F.; Brauchle, J.:&nbsp; Channel Coding. Lecture Notes, Institute for Communications Engineering, TU München, 2008.</ref>.}}
+*All two-dimensional bar codes have square markings to calibrate the reader.&nbsp; You can find details on this in&nbsp; [KM+09]<ref>Kötter, R.; Mayer, T.; Tüchler, M.; Schreckenbach, F.; Brauchle, J.:&nbsp; Channel Coding. Lecture notes, Institute for Communications Engineering, TU München, 2008.</ref>.}}
@@ Line 153: / Line 153: @@
 *The common code rate of the two RS&ndash;component codes is&nbsp; $R_{\rm RS} = 24/28 &middot; 28/32 = 3/4$.&nbsp; Through the 8&ndash;to&ndash;14 modulation and some control bits,&nbsp; one arrives at the total code rate&nbsp; $R &asymp; 1/3$.<br>
-*In the case of statistically independent errors according to the&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Channel_.E2.80.93_BSC|$\text{BSC model}$]]&nbsp; ("Binary Symmetric Channel"),&nbsp; a complete correction is possible as long as the bit error rate does not exceed the value&nbsp; $10^{-3}$.<br>
+*In the case of statistically independent errors according to the&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Channel_.E2.80.93_BSC|$\text{BSC}$]]&nbsp;  model&nbsp; ("Binary Symmetric Channel"),&nbsp; a complete correction is possible as long as the bit error rate does not exceed the value&nbsp; $10^{-3}$.<br>
 *The CD specific&nbsp; "Cross Interleaver"&nbsp; scrambles&nbsp; $108$&nbsp; blocks together so that the&nbsp; $588$&nbsp; bits of a block &nbsp;$($each bit corresponds to approx. &nbsp; $0.28 \, \rm {&micro; m})$&nbsp; are distributed over approx.&nbsp; $1.75\, \rm cm$.<br>
@@ Line 198: / Line 198: @@
 == Interplay between source and channel coding ==
 <br>
-Transmission of natural sources such as speech,&nbsp; music,&nbsp; images,&nbsp; videos,&nbsp; etc. is usually done according to the discrete-time model outlined below.&nbsp; From&nbsp; [Liv10]<ref name ='Liv10'>Liva, G.:&nbsp; Channel Coding.&nbsp; Lectures manuscript,&nbsp; Lehrstuhl für Nachrichtentechnik,&nbsp; TU München und DLR Oberpfaffenhofen, 2010.</ref>&nbsp; the following should be noted:
+Transmission of natural sources such as speech,&nbsp; music,&nbsp; images,&nbsp; videos,&nbsp; etc. is usually done according to the discrete-time model outlined below.&nbsp; From&nbsp; [Liv10]<ref name ='Liv10'>Liva, G.:&nbsp; Channel Coding.&nbsp; Lectures manuscript,&nbsp; Institute for Communications Engineering, TU München and DLR Oberpfaffenhofen, 2010.</ref>&nbsp; the following should be noted:
 *Source and sink are digitized and represented by&nbsp; (approximately equal numbers of)&nbsp; zeros and ones.<br>
@@ Line 256: / Line 256: @@
 *Each information block&nbsp; $\underline{u}$&nbsp; is represented by a&nbsp; &raquo;'''code word'''&laquo;&nbsp; (or&nbsp; "code block")&nbsp; $\underline{x} = (x_1, x_2, \text{. ..} \hspace{0.05cm}, x_n)$ &nbsp; with &nbsp; $n \ge k$,&nbsp; $x_i \in \{0, \, 1\}.$&nbsp; One then speaks of a binary&nbsp; $(n, k)$&nbsp; block code&nbsp; $C$.&nbsp; We denote the assignment by&nbsp; $\underline{x} = {\rm enc}(\underline{u})$,&nbsp; where&nbsp; "enc"&nbsp; stands for&nbsp; "encoder function".<br>
-*The&nbsp; &raquo;'''received word'''&laquo;&nbsp;  $\underline{y}$&nbsp; results from the code word&nbsp; $\underline{x}$&nbsp; by&nbsp;  [https://en.wikipedia.org/wiki/Modular_arithmetic $\text{as the  modulo&ndash;2 sum}$]&nbsp; with the likewise binary error vector&nbsp; $\underline{e} = (e_1, e_2, \text{. ..} \hspace{0.05cm}, e_n)$,&nbsp; where&nbsp; "$e= 1$"&nbsp; represents a transmission error and&nbsp; "$e= 0$"&nbsp; indicates that the&nbsp; $i$&ndash;th bit of the code word was transmitted correctly.&nbsp; The following therefore applies:
+*The&nbsp; &raquo;'''received word'''&laquo;&nbsp;  $\underline{y}$&nbsp; results from the code word&nbsp; $\underline{x}$&nbsp; by the&nbsp;  [https://en.wikipedia.org/wiki/Modular_arithmetic $\text{modulo&ndash;2}$]&nbsp;  sum&nbsp; with the likewise binary error vector&nbsp; $\underline{e} = (e_1, e_2, \text{. ..} \hspace{0.05cm}, e_n)$,&nbsp; where&nbsp; "$e= 1$"&nbsp; represents a transmission error and&nbsp; "$e= 0$"&nbsp; indicates that the&nbsp; $i$&ndash;th bit of the code word was transmitted correctly.&nbsp; The following therefore applies:
 ::<math>\underline{y} = \underline{x} \oplus \underline{e} \hspace{0.05cm}, \hspace{0.5cm} y_i  =   x_i \oplus e_i \hspace{0.05cm}, \hspace{0.2cm} i = 1, \text{...} \hspace{0.05cm}  , n\hspace{0.05cm}, </math>
 ::<math>x_i \hspace{-0.05cm} \in  \hspace{-0.05cm} \{ 0, 1 \}\hspace{0.05cm}, \hspace{0.5cm}e_i \in  \{ 0, 1 \}\hspace{0.5cm}
 \Rightarrow \hspace{0.5cm}y_i  \in   \{ 0, 1 \}\hspace{0.05cm}.</math>
-*The description by the&nbsp; &raquo;'''digital channel model'''&laquo;&nbsp; &ndash; i.e. with binary input and output &ndash; is,&nbsp; however,&nbsp; only applicable if the transmission system makes hard decisions &ndash; see section&nbsp; [[Channel_Coding/Decoding_of_Linear_Block_Codes#Coding_gain_-_bit_error_rate_with_AWGN|$\text{AWGN channel at binary input}$]].&nbsp; Systems with&nbsp;  [[Channel_Coding/Decodierung_linearer_Blockcodes#Codiergewinn_.E2.80.93_Bitfehlerrate_bei_AWGN|$\text{soft decision}$]]&nbsp; cannot be modelled with this simple model.<br>
+*The description by the&nbsp; &raquo;'''digital channel model'''&laquo;&nbsp; &ndash; i.e. with binary input and output &ndash; is,&nbsp; however,&nbsp; only applicable if the transmission system makes hard decisions &ndash; see section&nbsp; [[Channel_Coding/Decoding_of_Linear_Block_Codes#Coding_gain_-_bit_error_rate_with_AWGN|"AWGN channel at binary input"]].&nbsp; Systems with&nbsp;  [[Channel_Coding/Decodierung_linearer_Blockcodes#Codiergewinn_.E2.80.93_Bitfehlerrate_bei_AWGN|$\text{soft decision}$]]&nbsp; cannot be modelled with this simple model.<br>
 *The vector&nbsp; $\underline{v}$&nbsp; after&nbsp; &raquo;'''channel decoding'''&laquo;&nbsp; has the same length&nbsp; $k$&nbsp; as the information block&nbsp; $\underline{u}$.&nbsp; We describe the decoding process with the&nbsp; "decoder function"&nbsp; as&nbsp; $\underline{v} = {\rm enc}^{-1}(\underline{y}) = {\rm dec}(\underline{y})$.&nbsp; In the error-free case,&nbsp; analogous to&nbsp; $\underline{x} = {\rm enc}(\underline{u})$ &nbsp; &rArr; &nbsp; $\underline{v} = {\rm enc}^{-1}(\underline{y})$.<br>
@@ Line 373: / Line 373: @@
 #The vector $\underline{x} = (x_1, x_2, \hspace{0.05cm}\text{...} \hspace{0.05cm}, x_n)$ &nbsp; now denotes the discrete-time equivalent to the transmitted signal&nbsp; $s(t)$, while the received  signal&nbsp; $r(t)$ is described by the vector&nbsp; $\underline{y} = (y_1, y_2, \hspace{0.05cm}\text{...}\hspace{0.05cm}, y_n)$&nbsp;. The code rate is the quotient </i>&nbsp; $R=k/n$ &nbsp; with &nbsp; $0 \le R \le 1$ and the number of check bits is given by&nbsp; $m = n-k$.<br><br>
 #In the first main chapter,&nbsp; the elements&nbsp; $u_i$&nbsp; and&nbsp; $v_i$&nbsp; &nbsp;$($each with index&nbsp; $i = 1, \hspace{0.05cm}\text{...} \hspace{0.05cm}, k)$ &nbsp; of the vectors &nbsp; $\underline{u}$ &nbsp; and &nbsp; $\underline{v}$ &nbsp; are always binary&nbsp; $(0$&nbsp; or &nbsp;$1)$,&nbsp; as are the&nbsp; $n$&nbsp; elements&nbsp; $x_i$&nbsp; of the code word&nbsp; $\underline{x}$. &nbsp; For digital channel model&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Channel_.E2.80.93_BSC|$\text{(BSC}$]],&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Erasure_Channel_.E2.80.93_BEC|$\text{BEC}$]],&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Error_.26_Erasure_Channel_.E2.80.93_BSEC|$\text{BSEC})$]]&nbsp; this also applies to the&nbsp; $n$&nbsp; received values&nbsp; $y_i \in \{0, 1\}$.<br><br>
-#The&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input|$\text{AWGN channel}$]]&nbsp; is characterised by real-valued output values&nbsp; $y_i$.&nbsp; The&nbsp; "code word estimator"&nbsp; in this case extracts from the vector&nbsp;$\underline{y} = (y_1, y_2, \hspace{0.05cm}\text{...}\hspace{0.05cm}, y_n)$ &nbsp; the binary vector&nbsp; $\underline{z} = (z_1, z_2, \hspace{0.05cm}\text{...} \hspace{0.05cm}, z_n)$&nbsp; to be compared with the code word&nbsp; $\underline{x}$.<br><br>
+#The&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input|$\text{AWGN}$]]&nbsp;  channel&nbsp; is characterised by real-valued output values&nbsp; $y_i$.&nbsp; The&nbsp; "code word estimator"&nbsp; in this case extracts from the vector&nbsp;$\underline{y} = (y_1, y_2, \hspace{0.05cm}\text{...}\hspace{0.05cm}, y_n)$ &nbsp; the binary vector&nbsp; $\underline{z} = (z_1, z_2, \hspace{0.05cm}\text{...} \hspace{0.05cm}, z_n)$&nbsp; to be compared with the code word&nbsp; $\underline{x}$.<br><br>
 #The transition from &nbsp; $\underline{y}$ &nbsp; to &nbsp; $\underline{z}$ &nbsp; is done by threshold decision &nbsp; &#8658; &nbsp; "Hard Decision"&nbsp; or according"&nbsp; to the MAP criterion &nbsp; &#8658; &nbsp; "Soft Decision".&nbsp; For equally likely input symbols,&nbsp; the&nbsp; "maximum Likelihood estimation also leads to the minimum error rate.<br><br>
 #In the context of the AWGN model,&nbsp; it makes sense to represent binary code symbols&nbsp; $x_i$&nbsp; bipolar&nbsp; (i.e. $\pm1$).&nbsp; This does not change the statistical properties.&nbsp; In the following,&nbsp; we mark bipolar signalling with a tilde.&nbsp; Then applies:

@@ Line 355: / Line 355: @@
 *Of the&nbsp; $2^n - 2^k = 28$&nbsp; invalid code words,&nbsp; now&nbsp; $20$&nbsp; can be assigned to a valid code word&nbsp; (fill colour: &nbsp; red, green, blue or ochre),&nbsp; assuming that a single bit error is more likely than their two or more.
-*For each valid code word,&nbsp; there are five invalid code words,&nbsp; each with only one corruption '''KORREKTUR: falsification'''&nbsp; &rArr; &nbsp; Hamming distance&nbsp; $d_{\rm H} =1$.&nbsp; These are indicated in the respective square with red, green, blue or ochre background colour.
+*For each valid code word,&nbsp; there are five invalid code words,&nbsp; each with only one falsification &nbsp; &rArr; &nbsp; Hamming distance&nbsp; $d_{\rm H} =1$.&nbsp; These are indicated in the respective square with red, green, blue or ochre background colour.
 *Error correction is possible for these due to the minimum distance&nbsp; $d_{\rm min} = 3$&nbsp; between the code words.

@@ Line 41: / Line 41: @@
 #&nbsp;&raquo;'''Error Detection&laquo;''': &nbsp; The decoder checks the integrity of the received blocks and marks any errors found.&nbsp;  If necessary,&nbsp;  the receiver informs the transmitter about erroneous blocks via the return channel,&nbsp;  so that the transmitter sends the corresponding block again.<br><br>
 #&nbsp;&raquo;'''Error Correction&laquo;''': &nbsp; The decoder detects one&nbsp;  (or more)&nbsp; bit errors and provides further information for them,&nbsp;  for example their positions in the transmitted block.&nbsp;  In this way,&nbsp;  it may be possible to completely correct the errors that have occurred.<br><br>
-#&nbsp;&raquo;'''Channel Coding&laquo;'''&nbsp; includes both,&nbsp; procedures for&nbsp; '''&raquo;error detection&laquo;'''&nbsp; as well as those for&nbsp; '''&raquo;error correction&laquo;'''.<br>}}
+#&nbsp;&raquo;'''Channel Coding&laquo;'''&nbsp; includes both,&nbsp; procedures for&nbsp; &raquo;'''error detection'''&laquo;&nbsp; as well as those for&nbsp; &raquo;'''error correction'''&laquo;.<br>}}
@@ Line 237: / Line 237: @@
 [[File:EN_KC_T_1_1_S3c_v3.png|left|frame|Image transmission with source coding,&nbsp; but without channel coding|class=fit]]
 <br><br>
-'''&raquo;Source coding but no channel coding&laquo;'''
+&raquo;'''Source coding but no channel coding'''&laquo;
 Now let's consider the case of directly transferring the compressed data (e.g. JPG) without error-proofing measures.&nbsp; Then:
@@ Line 243: / Line 243: @@
 #The compressed source has only little redundancy left.
 #Thus,&nbsp; any single transmission error will cause entire blocks of images to be decoded incorrectly.<br>
-#'''&raquo;This coding scheme should be avoided at all costs&laquo;'''.}}.
+#&raquo;'''This coding scheme should be avoided at all costs'''&laquo;.}}.
@@ Line 252: / Line 252: @@
 [[File:EN_KC_T_1_1_S4_v2.png|right|frame|Block diagram describing channel coding|class=fit]]
-*The vector&nbsp; $\underline{u} = (u_1, u_2, \text{...} \hspace{0.05cm}, u_k)$&nbsp; denotes an&nbsp; '''&raquo;information block&laquo;'''&nbsp; with&nbsp; $k$&nbsp; symbols.&nbsp; We restrict ourselves to binary symbols&nbsp; (bits) &nbsp; &#8658; &nbsp; $u_i \in \{0, \, 1\}$ &nbsp; for $i = 1, 2, \text{...} \hspace{0.05cm}, k$ &nbsp; with equal occurrence probabilities for zeros and ones.<br>
+*The vector&nbsp; $\underline{u} = (u_1, u_2, \text{...} \hspace{0.05cm}, u_k)$&nbsp; denotes an&nbsp; &raquo;'''information block'''&laquo;&nbsp; with&nbsp; $k$&nbsp; symbols.&nbsp; We restrict ourselves to binary symbols&nbsp; (bits) &nbsp; &#8658; &nbsp; $u_i \in \{0, \, 1\}$ &nbsp; for $i = 1, 2, \text{...} \hspace{0.05cm}, k$ &nbsp; with equal occurrence probabilities for zeros and ones.<br>
-*Each information block&nbsp; $\underline{u}$&nbsp; is represented by a&nbsp; '''&raquo;code word&laquo;'''&nbsp; (or&nbsp; "code block")&nbsp; $\underline{x} = (x_1, x_2, \text{. ..} \hspace{0.05cm}, x_n)$ &nbsp; with &nbsp; $n \ge k$,&nbsp; $x_i \in \{0, \, 1\}.$&nbsp; One then speaks of a binary&nbsp; $(n, k)$&nbsp; block code&nbsp; $C$.&nbsp; We denote the assignment by&nbsp; $\underline{x} = {\rm enc}(\underline{u})$,&nbsp; where&nbsp; "enc"&nbsp; stands for&nbsp; "encoder function".<br>
+*Each information block&nbsp; $\underline{u}$&nbsp; is represented by a&nbsp; &raquo;'''code word'''&laquo;&nbsp; (or&nbsp; "code block")&nbsp; $\underline{x} = (x_1, x_2, \text{. ..} \hspace{0.05cm}, x_n)$ &nbsp; with &nbsp; $n \ge k$,&nbsp; $x_i \in \{0, \, 1\}.$&nbsp; One then speaks of a binary&nbsp; $(n, k)$&nbsp; block code&nbsp; $C$.&nbsp; We denote the assignment by&nbsp; $\underline{x} = {\rm enc}(\underline{u})$,&nbsp; where&nbsp; "enc"&nbsp; stands for&nbsp; "encoder function".<br>
-*The&nbsp; '''&raquo;received word&laquo;'''&nbsp;  $\underline{y}$&nbsp; results from the code word&nbsp; $\underline{x}$&nbsp; by&nbsp;  [https://en.wikipedia.org/wiki/Modular_arithmetic $\text{as the  modulo&ndash;2 sum}$]&nbsp; with the likewise binary error vector&nbsp; $\underline{e} = (e_1, e_2, \text{. ..} \hspace{0.05cm}, e_n)$,&nbsp; where&nbsp; "$e= 1$"&nbsp; represents a transmission error and&nbsp; "$e= 0$"&nbsp; indicates that the&nbsp; $i$&ndash;th bit of the code word was transmitted correctly.&nbsp; The following therefore applies:
+*The&nbsp; &raquo;'''received word'''&laquo;&nbsp;  $\underline{y}$&nbsp; results from the code word&nbsp; $\underline{x}$&nbsp; by&nbsp;  [https://en.wikipedia.org/wiki/Modular_arithmetic $\text{as the  modulo&ndash;2 sum}$]&nbsp; with the likewise binary error vector&nbsp; $\underline{e} = (e_1, e_2, \text{. ..} \hspace{0.05cm}, e_n)$,&nbsp; where&nbsp; "$e= 1$"&nbsp; represents a transmission error and&nbsp; "$e= 0$"&nbsp; indicates that the&nbsp; $i$&ndash;th bit of the code word was transmitted correctly.&nbsp; The following therefore applies:
 ::<math>\underline{y} = \underline{x} \oplus \underline{e} \hspace{0.05cm}, \hspace{0.5cm} y_i  =   x_i \oplus e_i \hspace{0.05cm}, \hspace{0.2cm} i = 1, \text{...} \hspace{0.05cm}  , n\hspace{0.05cm}, </math>
 ::<math>x_i \hspace{-0.05cm} \in  \hspace{-0.05cm} \{ 0, 1 \}\hspace{0.05cm}, \hspace{0.5cm}e_i \in  \{ 0, 1 \}\hspace{0.5cm}
 \Rightarrow \hspace{0.5cm}y_i  \in   \{ 0, 1 \}\hspace{0.05cm}.</math>
-*The description by the&nbsp; '''&raquo;digital channel model&laquo;'''&nbsp; &ndash; i.e. with binary input and output &ndash; is,&nbsp; however,&nbsp; only applicable if the transmission system makes hard decisions &ndash; see section&nbsp; [[Channel_Coding/Decoding_of_Linear_Block_Codes#Coding_gain_-_bit_error_rate_with_AWGN|$\text{AWGN channel at binary input}$]].&nbsp; Systems with&nbsp;  [[Channel_Coding/Decodierung_linearer_Blockcodes#Codiergewinn_.E2.80.93_Bitfehlerrate_bei_AWGN|$\text{soft decision}$]]&nbsp; cannot be modelled with this simple model.<br>
+*The description by the&nbsp; &raquo;'''digital channel model'''&laquo;&nbsp; &ndash; i.e. with binary input and output &ndash; is,&nbsp; however,&nbsp; only applicable if the transmission system makes hard decisions &ndash; see section&nbsp; [[Channel_Coding/Decoding_of_Linear_Block_Codes#Coding_gain_-_bit_error_rate_with_AWGN|$\text{AWGN channel at binary input}$]].&nbsp; Systems with&nbsp;  [[Channel_Coding/Decodierung_linearer_Blockcodes#Codiergewinn_.E2.80.93_Bitfehlerrate_bei_AWGN|$\text{soft decision}$]]&nbsp; cannot be modelled with this simple model.<br>
-*The vector&nbsp; $\underline{v}$&nbsp; after&nbsp; '''&raquo;channel decoding&laquo;'''&nbsp; has the same length&nbsp; $k$&nbsp; as the information block&nbsp; $\underline{u}$.&nbsp; We describe the decoding process with the&nbsp; "decoder function"&nbsp; as&nbsp; $\underline{v} = {\rm enc}^{-1}(\underline{y}) = {\rm dec}(\underline{y})$.&nbsp; In the error-free case,&nbsp; analogous to&nbsp; $\underline{x} = {\rm enc}(\underline{u})$ &nbsp; &rArr; &nbsp; $\underline{v} = {\rm enc}^{-1}(\underline{y})$.<br>
+*The vector&nbsp; $\underline{v}$&nbsp; after&nbsp; &raquo;'''channel decoding'''&laquo;&nbsp; has the same length&nbsp; $k$&nbsp; as the information block&nbsp; $\underline{u}$.&nbsp; We describe the decoding process with the&nbsp; "decoder function"&nbsp; as&nbsp; $\underline{v} = {\rm enc}^{-1}(\underline{y}) = {\rm dec}(\underline{y})$.&nbsp; In the error-free case,&nbsp; analogous to&nbsp; $\underline{x} = {\rm enc}(\underline{u})$ &nbsp; &rArr; &nbsp; $\underline{v} = {\rm enc}^{-1}(\underline{y})$.<br>
 *If the error vector&nbsp; $\underline{e} \ne \underline{0}$,&nbsp; then&nbsp; $\underline{y}$&nbsp; is usually not a valid element of the block code used,&nbsp; and the decoding is then not a pure mapping&nbsp; $\underline{y} \rightarrow \underline{v}$,&nbsp; but an estimate of&nbsp; $\underline{v}$&nbsp; based on maximum match &nbsp; ("mimimum error probability").<br>
@@ Line 274: / Line 274: @@
 This code would be unsuitable for the purpose of error detection or error correction.&nbsp; But it is constructed in such a way that it clearly illustrates the calculation of important descriptive variables:
-*Here,&nbsp; each individual code word&nbsp; $\underline{u}$&nbsp; is described by five bits.&nbsp; Throughout the book,&nbsp; we express this fact by the&nbsp; '''&raquo;code word length&laquo;'''&nbsp;  $n = 5$.
+*Here,&nbsp; each individual code word&nbsp; $\underline{u}$&nbsp; is described by five bits.&nbsp; Throughout the book,&nbsp; we express this fact by the&nbsp; &raquo;'''code word length'''&laquo;&nbsp;  $n = 5$.
-*The above code contains four elements.&nbsp; Thus the&nbsp; '''&raquo;code size&laquo;'''&nbsp;  $|C| = 4$.&nbsp; Accordingly,&nbsp; there are also four unique mappings between&nbsp; $\underline{u}$&nbsp; and&nbsp; $\underline{x}$.
+*The above code contains four elements.&nbsp; Thus the&nbsp; &raquo;'''code size'''&laquo;&nbsp;  $|C| = 4$.&nbsp; Accordingly,&nbsp; there are also four unique mappings between&nbsp; $\underline{u}$&nbsp; and&nbsp; $\underline{x}$.
-*The length of an information block&nbsp; $\underline{u}$ &nbsp; &rArr; &nbsp; '''&raquo;information block length&laquo;'''&nbsp; is denoted by&nbsp; $k$.&nbsp; Since for all binary codes&nbsp; $|C| = 2^k$&nbsp; holds,&nbsp; it follows from&nbsp; $|C| = 4$&nbsp; that&nbsp; $k = 2$.&nbsp; The assignments between&nbsp; $\underline{u}$&nbsp; and&nbsp; $\underline{x}$&nbsp; in the above code&nbsp; $C$&nbsp; are:
+*The length of an information block&nbsp; $\underline{u}$ &nbsp; &rArr; &nbsp; &raquo;'''information block length'''&laquo;&nbsp; is denoted by&nbsp; $k$.&nbsp; Since for all binary codes&nbsp; $|C| = 2^k$&nbsp; holds,&nbsp; it follows from&nbsp; $|C| = 4$&nbsp; that&nbsp; $k = 2$.&nbsp; The assignments between&nbsp; $\underline{u}$&nbsp; and&nbsp; $\underline{x}$&nbsp; in the above code&nbsp; $C$&nbsp; are:
 ::<math>\underline{u_0} = (0, 0) \hspace{0.2cm}\leftrightarrow \hspace{0.2cm}(0, 0, 0, 0, 0) = \underline{x_0}\hspace{0.05cm}, \hspace{0.8cm}
@@ Line 286: / Line 286: @@
 \underline{u_3} = (1, 1) \hspace{0.2cm} \leftrightarrow \hspace{0.2cm}(1, 1, 1, 1, 1) = \underline{x_3}\hspace{0.05cm}.</math>
-*The code has the&nbsp; '''&raquo;code rate&laquo;'''&nbsp; $R = k/n = 2/5$.&nbsp; Accordingly,&nbsp; its redundancy is&nbsp; $1-R$,&nbsp; that is&nbsp; $60\%$.&nbsp; Without error protection&nbsp; $(n = k)$&nbsp; the code rate&nbsp; $R = 1$.<br>
+*The code has the&nbsp; &raquo;'''code rate'''&laquo;&nbsp; $R = k/n = 2/5$.&nbsp; Accordingly,&nbsp; its redundancy is&nbsp; $1-R$,&nbsp; that is&nbsp; $60\%$.&nbsp; Without error protection&nbsp; $(n = k)$&nbsp; the code rate&nbsp; $R = 1$.<br>
 *A small code rate indicates that of the&nbsp; $n$&nbsp; bits of a code word, very few actually carry information.&nbsp; A repetition code&nbsp; $(k = 1,\ n = 10)$&nbsp; has the code rate&nbsp; $R = 0.1$.<br>
-*The&nbsp; '''&raquo;Hamming weight&laquo;'''&nbsp; $w_{\rm H}(\underline{x})$&nbsp; of the code word&nbsp; $\underline{x}$&nbsp; indicates the number of code word elements&nbsp; $x_i \in  \{0, \, 1\}$.&nbsp; For a binary code &nbsp; &rArr; &nbsp; $w_{\rm H}(\underline{x})$&nbsp; is equal to the sum&nbsp; $x_1 + x_2 + \hspace{0.05cm}\text{...} \hspace{0.05cm}+ x_n$.&nbsp; In the example:
+*The&nbsp; &raquo;'''Hamming weight'''&laquo;&nbsp; $w_{\rm H}(\underline{x})$&nbsp; of the code word&nbsp; $\underline{x}$&nbsp; indicates the number of code word elements&nbsp; $x_i \in  \{0, \, 1\}$.&nbsp; For a binary code &nbsp; &rArr; &nbsp; $w_{\rm H}(\underline{x})$&nbsp; is equal to the sum&nbsp; $x_1 + x_2 + \hspace{0.05cm}\text{...} \hspace{0.05cm}+ x_n$.&nbsp; In the example:
 ::<math>w_{\rm H}(\underline{x}_0) = 0\hspace{0.05cm}, \hspace{0.4cm}w_{\rm H}(\underline{x}_1) = 2\hspace{0.05cm}, \hspace{0.4cm} w_{\rm H}(\underline{x}_2) = 3\hspace{0.05cm}, \hspace{0.4cm}w_{\rm H}(\underline{x}_3) = 5\hspace{0.05cm}.
 </math>
-*The&nbsp; '''&raquo;Hamming distance&laquo;'''&nbsp; $d_{\rm H}(\underline{x}, \ \underline{x}\hspace{0.03cm}')$&nbsp; between the code words&nbsp; $\underline{x}$&nbsp; and&nbsp; $\underline{x}\hspace{0.03cm}'$&nbsp; denotes the number of bit positions in which the two code words differ:
+*The&nbsp; &raquo;'''Hamming distance'''&laquo;&nbsp; $d_{\rm H}(\underline{x}, \ \underline{x}\hspace{0.03cm}')$&nbsp; between the code words&nbsp; $\underline{x}$&nbsp; and&nbsp; $\underline{x}\hspace{0.03cm}'$&nbsp; denotes the number of bit positions in which the two code words differ:
 ::<math>d_{\rm H}(\underline{x}_0, \hspace{0.05cm}\underline{x}_1) = 2\hspace{0.05cm}, \hspace{0.4cm}
@@ Line 304: / Line 304: @@
 d_{\rm H}(\underline{x}_2, \hspace{0.05cm}\underline{x}_3) = 2\hspace{0.05cm}.</math>
-*An important property of a code&nbsp; $C$&nbsp; that significantly affects its ability to be corrected is the&nbsp; '''&raquo;minimum distance&laquo;'''&nbsp; between any two code words:
+*An important property of a code&nbsp; $C$&nbsp; that significantly affects its ability to be corrected is the&nbsp; &raquo;'''minimum distance'''&laquo;&nbsp; between any two code words:
 ::<math>d_{\rm min}(\mathcal{C}) =

@@ Line 243: / Line 243: @@
 #The compressed source has only little redundancy left.
 #Thus,&nbsp; any single transmission error will cause entire blocks of images to be decoded incorrectly.<br>
-#'''This coding scheme should be avoided at all costs'''.}}.
+#'''&raquo;This coding scheme should be avoided at all costs&laquo;'''.}}.
@@ Line 252: / Line 252: @@
 [[File:EN_KC_T_1_1_S4_v2.png|right|frame|Block diagram describing channel coding|class=fit]]
-*The vector&nbsp; $\underline{u} = (u_1, u_2, \text{...} \hspace{0.05cm}, u_k)$&nbsp; denotes an&nbsp; '''information block'''&nbsp; with&nbsp; $k$&nbsp; symbols.&nbsp; We restrict ourselves to binary symbols&nbsp; (bits) &nbsp; &#8658; &nbsp; $u_i \in \{0, \, 1\}$ &nbsp; for $i = 1, 2, \text{...} \hspace{0.05cm}, k$ &nbsp; with equal occurrence probabilities for zeros and ones.<br>
+*The vector&nbsp; $\underline{u} = (u_1, u_2, \text{...} \hspace{0.05cm}, u_k)$&nbsp; denotes an&nbsp; '''&raquo;information block&laquo;'''&nbsp; with&nbsp; $k$&nbsp; symbols.&nbsp; We restrict ourselves to binary symbols&nbsp; (bits) &nbsp; &#8658; &nbsp; $u_i \in \{0, \, 1\}$ &nbsp; for $i = 1, 2, \text{...} \hspace{0.05cm}, k$ &nbsp; with equal occurrence probabilities for zeros and ones.<br>
-*Each information block&nbsp; $\underline{u}$&nbsp; is represented by a&nbsp; '''code word'''&nbsp; (or&nbsp; "code block")&nbsp; $\underline{x} = (x_1, x_2, \text{. ..} \hspace{0.05cm}, x_n)$ &nbsp; with &nbsp; $n \ge k$,&nbsp; $x_i \in \{0, \, 1\}.$&nbsp; One then speaks of a binary&nbsp; $(n, k)$&nbsp; block code&nbsp; $C$.&nbsp; We denote the assignment by&nbsp; $\underline{x} = {\rm enc}(\underline{u})$,&nbsp; where&nbsp; "enc"&nbsp; stands for&nbsp; "encoder function".<br>
+*Each information block&nbsp; $\underline{u}$&nbsp; is represented by a&nbsp; '''&raquo;code word&laquo;'''&nbsp; (or&nbsp; "code block")&nbsp; $\underline{x} = (x_1, x_2, \text{. ..} \hspace{0.05cm}, x_n)$ &nbsp; with &nbsp; $n \ge k$,&nbsp; $x_i \in \{0, \, 1\}.$&nbsp; One then speaks of a binary&nbsp; $(n, k)$&nbsp; block code&nbsp; $C$.&nbsp; We denote the assignment by&nbsp; $\underline{x} = {\rm enc}(\underline{u})$,&nbsp; where&nbsp; "enc"&nbsp; stands for&nbsp; "encoder function".<br>
-*The&nbsp; '''received word'''&nbsp;  $\underline{y}$&nbsp; results from the code word&nbsp; $\underline{x}$&nbsp; by&nbsp;  [https://en.wikipedia.org/wiki/Modular_arithmetic $\text{as the  modulo&ndash;2 sum}$]&nbsp; with the likewise binary error vector&nbsp; $\underline{e} = (e_1, e_2, \text{. ..} \hspace{0.05cm}, e_n)$,&nbsp; where&nbsp; "$e= 1$"&nbsp; represents a transmission error and&nbsp; "$e= 0$"&nbsp; indicates that the&nbsp; $i$&ndash;th bit of the code word was transmitted correctly.&nbsp; The following therefore applies:
+*The&nbsp; '''&raquo;received word&laquo;'''&nbsp;  $\underline{y}$&nbsp; results from the code word&nbsp; $\underline{x}$&nbsp; by&nbsp;  [https://en.wikipedia.org/wiki/Modular_arithmetic $\text{as the  modulo&ndash;2 sum}$]&nbsp; with the likewise binary error vector&nbsp; $\underline{e} = (e_1, e_2, \text{. ..} \hspace{0.05cm}, e_n)$,&nbsp; where&nbsp; "$e= 1$"&nbsp; represents a transmission error and&nbsp; "$e= 0$"&nbsp; indicates that the&nbsp; $i$&ndash;th bit of the code word was transmitted correctly.&nbsp; The following therefore applies:
 ::<math>\underline{y} = \underline{x} \oplus \underline{e} \hspace{0.05cm}, \hspace{0.5cm} y_i  =   x_i \oplus e_i \hspace{0.05cm}, \hspace{0.2cm} i = 1, \text{...} \hspace{0.05cm}  , n\hspace{0.05cm}, </math>
 ::<math>x_i \hspace{-0.05cm} \in  \hspace{-0.05cm} \{ 0, 1 \}\hspace{0.05cm}, \hspace{0.5cm}e_i \in  \{ 0, 1 \}\hspace{0.5cm}
 \Rightarrow \hspace{0.5cm}y_i  \in   \{ 0, 1 \}\hspace{0.05cm}.</math>
-*The description by the&nbsp; '''digital channel model'''&nbsp; &ndash; i.e. with binary input and output &ndash; is,&nbsp; however,&nbsp; only applicable if the transmission system makes hard decisions &ndash; see section&nbsp; [[Channel_Coding/Decoding_of_Linear_Block_Codes#Coding_gain_-_bit_error_rate_with_AWGN|$\text{AWGN channel at binary input}$]].&nbsp; Systems with&nbsp;  [[Channel_Coding/Decodierung_linearer_Blockcodes#Codiergewinn_.E2.80.93_Bitfehlerrate_bei_AWGN|$\text{soft decision}$]]&nbsp; cannot be modelled with this simple model.<br>
+*The description by the&nbsp; '''&raquo;digital channel model&laquo;'''&nbsp; &ndash; i.e. with binary input and output &ndash; is,&nbsp; however,&nbsp; only applicable if the transmission system makes hard decisions &ndash; see section&nbsp; [[Channel_Coding/Decoding_of_Linear_Block_Codes#Coding_gain_-_bit_error_rate_with_AWGN|$\text{AWGN channel at binary input}$]].&nbsp; Systems with&nbsp;  [[Channel_Coding/Decodierung_linearer_Blockcodes#Codiergewinn_.E2.80.93_Bitfehlerrate_bei_AWGN|$\text{soft decision}$]]&nbsp; cannot be modelled with this simple model.<br>
-*The vector&nbsp; $\underline{v}$&nbsp; after&nbsp; '''channel decoding'''&nbsp; has the same length&nbsp; $k$&nbsp; as the information block&nbsp; $\underline{u}$.&nbsp; We describe the decoding process with the&nbsp; "decoder function"&nbsp; as&nbsp; $\underline{v} = {\rm enc}^{-1}(\underline{y}) = {\rm dec}(\underline{y})$.&nbsp; In the error-free case,&nbsp; analogous to&nbsp; $\underline{x} = {\rm enc}(\underline{u})$ &nbsp; &rArr; &nbsp; $\underline{v} = {\rm enc}^{-1}(\underline{y})$.<br>
+*The vector&nbsp; $\underline{v}$&nbsp; after&nbsp; '''&raquo;channel decoding&laquo;'''&nbsp; has the same length&nbsp; $k$&nbsp; as the information block&nbsp; $\underline{u}$.&nbsp; We describe the decoding process with the&nbsp; "decoder function"&nbsp; as&nbsp; $\underline{v} = {\rm enc}^{-1}(\underline{y}) = {\rm dec}(\underline{y})$.&nbsp; In the error-free case,&nbsp; analogous to&nbsp; $\underline{x} = {\rm enc}(\underline{u})$ &nbsp; &rArr; &nbsp; $\underline{v} = {\rm enc}^{-1}(\underline{y})$.<br>
 *If the error vector&nbsp; $\underline{e} \ne \underline{0}$,&nbsp; then&nbsp; $\underline{y}$&nbsp; is usually not a valid element of the block code used,&nbsp; and the decoding is then not a pure mapping&nbsp; $\underline{y} \rightarrow \underline{v}$,&nbsp; but an estimate of&nbsp; $\underline{v}$&nbsp; based on maximum match &nbsp; ("mimimum error probability").<br>
@@ Line 274: / Line 274: @@
 This code would be unsuitable for the purpose of error detection or error correction.&nbsp; But it is constructed in such a way that it clearly illustrates the calculation of important descriptive variables:
-*Here,&nbsp; each individual code word&nbsp; $\underline{u}$&nbsp; is described by five bits.&nbsp; Throughout the book,&nbsp; we express this fact by the&nbsp; '''code word length'''&nbsp;  $n = 5$.
+*Here,&nbsp; each individual code word&nbsp; $\underline{u}$&nbsp; is described by five bits.&nbsp; Throughout the book,&nbsp; we express this fact by the&nbsp; '''&raquo;code word length&laquo;'''&nbsp;  $n = 5$.
-*The above code contains four elements.&nbsp; Thus the&nbsp; '''code size'''&nbsp;  $|C| = 4$.&nbsp; Accordingly,&nbsp; there are also four unique mappings between&nbsp; $\underline{u}$&nbsp; and&nbsp; $\underline{x}$.
+*The above code contains four elements.&nbsp; Thus the&nbsp; '''&raquo;code size&laquo;'''&nbsp;  $|C| = 4$.&nbsp; Accordingly,&nbsp; there are also four unique mappings between&nbsp; $\underline{u}$&nbsp; and&nbsp; $\underline{x}$.
-*The length of an information block&nbsp; $\underline{u}$ &nbsp; &rArr; &nbsp; '''information block length'''&nbsp; is denoted by&nbsp; $k$.&nbsp; Since for all binary codes&nbsp; $|C| = 2^k$&nbsp; holds,&nbsp; it follows from&nbsp; $|C| = 4$&nbsp; that&nbsp; $k = 2$.&nbsp; The assignments between&nbsp; $\underline{u}$&nbsp; and&nbsp; $\underline{x}$&nbsp; in the above code&nbsp; $C$&nbsp; are:
+*The length of an information block&nbsp; $\underline{u}$ &nbsp; &rArr; &nbsp; '''&raquo;information block length&laquo;'''&nbsp; is denoted by&nbsp; $k$.&nbsp; Since for all binary codes&nbsp; $|C| = 2^k$&nbsp; holds,&nbsp; it follows from&nbsp; $|C| = 4$&nbsp; that&nbsp; $k = 2$.&nbsp; The assignments between&nbsp; $\underline{u}$&nbsp; and&nbsp; $\underline{x}$&nbsp; in the above code&nbsp; $C$&nbsp; are:
 ::<math>\underline{u_0} = (0, 0) \hspace{0.2cm}\leftrightarrow \hspace{0.2cm}(0, 0, 0, 0, 0) = \underline{x_0}\hspace{0.05cm}, \hspace{0.8cm}
@@ Line 286: / Line 286: @@
 \underline{u_3} = (1, 1) \hspace{0.2cm} \leftrightarrow \hspace{0.2cm}(1, 1, 1, 1, 1) = \underline{x_3}\hspace{0.05cm}.</math>
-*The code has the&nbsp; '''code rate'''&nbsp; $R = k/n = 2/5$.&nbsp; Accordingly,&nbsp; its redundancy is&nbsp; $1-R$,&nbsp; that is&nbsp; $60\%$.&nbsp; Without error protection&nbsp; $(n = k)$&nbsp; the code rate&nbsp; $R = 1$.<br>
+*The code has the&nbsp; '''&raquo;code rate&laquo;'''&nbsp; $R = k/n = 2/5$.&nbsp; Accordingly,&nbsp; its redundancy is&nbsp; $1-R$,&nbsp; that is&nbsp; $60\%$.&nbsp; Without error protection&nbsp; $(n = k)$&nbsp; the code rate&nbsp; $R = 1$.<br>
 *A small code rate indicates that of the&nbsp; $n$&nbsp; bits of a code word, very few actually carry information.&nbsp; A repetition code&nbsp; $(k = 1,\ n = 10)$&nbsp; has the code rate&nbsp; $R = 0.1$.<br>
-*The&nbsp; '''Hamming weight'''&nbsp; $w_{\rm H}(\underline{x})$&nbsp; of the code word&nbsp; $\underline{x}$&nbsp; indicates the number of code word elements&nbsp; $x_i \in  \{0, \, 1\}$.&nbsp; For a binary code &nbsp; &rArr; &nbsp; $w_{\rm H}(\underline{x})$&nbsp; is equal to the sum&nbsp; $x_1 + x_2 + \hspace{0.05cm}\text{...} \hspace{0.05cm}+ x_n$.&nbsp; In the example:
+*The&nbsp; '''&raquo;Hamming weight&laquo;'''&nbsp; $w_{\rm H}(\underline{x})$&nbsp; of the code word&nbsp; $\underline{x}$&nbsp; indicates the number of code word elements&nbsp; $x_i \in  \{0, \, 1\}$.&nbsp; For a binary code &nbsp; &rArr; &nbsp; $w_{\rm H}(\underline{x})$&nbsp; is equal to the sum&nbsp; $x_1 + x_2 + \hspace{0.05cm}\text{...} \hspace{0.05cm}+ x_n$.&nbsp; In the example:
 ::<math>w_{\rm H}(\underline{x}_0) = 0\hspace{0.05cm}, \hspace{0.4cm}w_{\rm H}(\underline{x}_1) = 2\hspace{0.05cm}, \hspace{0.4cm} w_{\rm H}(\underline{x}_2) = 3\hspace{0.05cm}, \hspace{0.4cm}w_{\rm H}(\underline{x}_3) = 5\hspace{0.05cm}.
 </math>
-*The&nbsp; '''Hamming distance'''&nbsp; $d_{\rm H}(\underline{x}, \ \underline{x}\hspace{0.03cm}')$&nbsp; between the code words&nbsp; $\underline{x}$&nbsp; and&nbsp; $\underline{x}\hspace{0.03cm}'$&nbsp; denotes the number of bit positions in which the two code words differ:
+*The&nbsp; '''&raquo;Hamming distance&laquo;'''&nbsp; $d_{\rm H}(\underline{x}, \ \underline{x}\hspace{0.03cm}')$&nbsp; between the code words&nbsp; $\underline{x}$&nbsp; and&nbsp; $\underline{x}\hspace{0.03cm}'$&nbsp; denotes the number of bit positions in which the two code words differ:
 ::<math>d_{\rm H}(\underline{x}_0, \hspace{0.05cm}\underline{x}_1) = 2\hspace{0.05cm}, \hspace{0.4cm}
@@ Line 304: / Line 304: @@
 d_{\rm H}(\underline{x}_2, \hspace{0.05cm}\underline{x}_3) = 2\hspace{0.05cm}.</math>
-*An important property of a code&nbsp; $C$&nbsp; that significantly affects its ability to be corrected is the&nbsp; '''minimum distance'''&nbsp; between any two code words:
+*An important property of a code&nbsp; $C$&nbsp; that significantly affects its ability to be corrected is the&nbsp; '''&raquo;minimum distance&laquo;'''&nbsp; between any two code words:
 ::<math>d_{\rm min}(\mathcal{C}) =

@@ Line 39: / Line 39: @@
 $\text{Definitions:}$
-#&nbsp;'''Error Detection''': &nbsp; The decoder checks the integrity of the received blocks and marks any errors found.&nbsp;  If necessary,&nbsp;  the receiver informs the transmitter about erroneous blocks via the return channel,&nbsp;  so that the transmitter sends the corresponding block again.<br><br>
+#&nbsp;&raquo;'''Error Detection&laquo;''': &nbsp; The decoder checks the integrity of the received blocks and marks any errors found.&nbsp;  If necessary,&nbsp;  the receiver informs the transmitter about erroneous blocks via the return channel,&nbsp;  so that the transmitter sends the corresponding block again.<br><br>
-#&nbsp;'''Error Correction''': &nbsp; The decoder detects one&nbsp;  (or more)&nbsp; bit errors and provides further information for them,&nbsp;  for example their positions in the transmitted block.&nbsp;  In this way,&nbsp;  it may be possible to completely correct the errors that have occurred.<br><br>
+#&nbsp;&raquo;'''Error Correction&laquo;''': &nbsp; The decoder detects one&nbsp;  (or more)&nbsp; bit errors and provides further information for them,&nbsp;  for example their positions in the transmitted block.&nbsp;  In this way,&nbsp;  it may be possible to completely correct the errors that have occurred.<br><br>
-#&nbsp;'''Channel Coding'''&nbsp; includes both,&nbsp; procedures for&nbsp; '''error detection'''&nbsp; as well as those for&nbsp; '''error correction'''.<br>}}
+#&nbsp;&raquo;'''Channel Coding&laquo;'''&nbsp; includes both,&nbsp; procedures for&nbsp; '''&raquo;error detection&laquo;'''&nbsp; as well as those for&nbsp; '''&raquo;error correction&laquo;'''.<br>}}
@@ Line 237: / Line 237: @@
 [[File:EN_KC_T_1_1_S3c_v3.png|left|frame|Image transmission with source coding,&nbsp; but without channel coding|class=fit]]
 <br><br>
-'''Source coding but no channel coding'''
+'''&raquo;Source coding but no channel coding&laquo;'''
 Now let's consider the case of directly transferring the compressed data (e.g. JPG) without error-proofing measures.&nbsp; Then:

@@ Line 111: / Line 111: @@
 $\text{Example 4: &nbsp; Two-dimensional bar codes for online tickets}$
-If you book a railway ticket online and print it out,&nbsp; you will find an example of a two-dimensional bar code,&nbsp; namely the&nbsp; [https://en.wikipedia.org/wiki/Aztec_Code Aztec code]&nbsp; developed in 1995 by Andy Longacre at the Welch Allyn company in the USA,&nbsp; with which amounts of data up to&nbsp; $3000$&nbsp; characters can be encoded.&nbsp; Due to the&nbsp; [[Channel_Coding/Definition_und_Eigenschaften_von_Reed–Solomon–Codes|$\text{Reed&ndash;Solomon error correction}$]],&nbsp; reconstruction of the data content is still possible even if up to&nbsp; $40\%$&nbsp; of the code has been destroyed,&nbsp; for example by bending the ticket or by coffee stains.<br>
+If you book a railway ticket online and print it out,&nbsp; you will find an example of a two-dimensional bar code,&nbsp; namely the&nbsp; [https://en.wikipedia.org/wiki/Aztec_Code $\text{Aztec code}$]&nbsp; developed in 1995 by Andy Longacre at the Welch Allyn company in the USA,&nbsp; with which amounts of data up to&nbsp; $3000$&nbsp; characters can be encoded.&nbsp; Due to the&nbsp; [[Channel_Coding/Definition_und_Eigenschaften_von_Reed–Solomon–Codes|$\text{Reed&ndash;Solomon error correction}$]],&nbsp; reconstruction of the data content is still possible even if up to&nbsp; $40\%$&nbsp; of the code has been destroyed,&nbsp; for example by bending the ticket or by coffee stains.<br>
 On the right you see a $\rm QR\ code$&nbsp; ("Quick Response")&nbsp; with associated content.
@@ Line 150: / Line 150: @@
 $\text{Example 7: &nbsp; Error protection of the compact disc}$
-For a compact disc&nbsp; $\rm (CD)$,&nbsp; one uses&nbsp; "cross-interleaved"&nbsp; [[Channel_Coding/Definition_und_Eigenschaften_von_Reed–Solomon–Codes|$\text{Reed&ndash;Solomon codes}$]]&nbsp; $\rm (RS)$&nbsp; and then a so-called&nbsp; [https://en.wikipedia.org/wiki/Eight-to-fourteen_modulation Eight&ndash;to&ndash;Fourteen modulation].&nbsp; Redundancy is used for error detection and correction.&nbsp; This coding scheme shows the following characteristics:
+For a compact disc&nbsp; $\rm (CD)$,&nbsp; one uses&nbsp; "cross-interleaved"&nbsp; [[Channel_Coding/Definition_und_Eigenschaften_von_Reed–Solomon–Codes|$\text{Reed&ndash;Solomon codes}$]]&nbsp; $\rm (RS)$&nbsp; and then a so-called&nbsp; [https://en.wikipedia.org/wiki/Eight-to-fourteen_modulation $\text{Eight&ndash;to&ndash;Fourteen modulation}$].&nbsp; Redundancy is used for error detection and correction.&nbsp; This coding scheme shows the following characteristics:
 *The common code rate of the two RS&ndash;component codes is&nbsp; $R_{\rm RS} = 24/28 &middot; 28/32 = 3/4$.&nbsp; Through the 8&ndash;to&ndash;14 modulation and some control bits,&nbsp; one arrives at the total code rate&nbsp; $R &asymp; 1/3$.<br>
@@ Line 165: / Line 165: @@
 == The "Slit CD" - a demonstration by the LNT of TUM ==
 <br>
-At the end of the 1990s,&nbsp; members of the&nbsp;  [https://www.ce.cit.tum.de/en/lnt/home/ "Institute for Communications Engineering"]&nbsp; of the  [https://www.tum.de/en/about-tum/ "Technical University of Munich"] &nbsp; led by Prof.&nbsp; [[Biographies_and_Bibliographies/Lehrstuhlinhaber_des_LNT#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|$\text{Joachim Hagenauer}$]]&nbsp; eliberately damaged a music&ndash;CD by cutting a total of three slits,&nbsp; each more than one millimetre wide.&nbsp; With each defect,&nbsp; almost&nbsp; $4000$&nbsp; consecutive bits of audio coding are missing.<br>
+At the end of the 1990s,&nbsp; members of the&nbsp;  [https://www.ce.cit.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]&nbsp; of the  [https://www.tum.de/en/about-tum/ $\text{Technical University of Munich}$] &nbsp; led by Prof.&nbsp; [[Biographies_and_Bibliographies/Lehrstuhlinhaber_des_LNT#Prof._Dr.-Ing._Dr.-Ing._E.h._Joachim_Hagenauer_.281993-2006.29|$\text{Joachim Hagenauer}$]]&nbsp; eliberately damaged a music&ndash;CD by cutting a total of three slits,&nbsp; each more than one millimetre wide.&nbsp; With each defect,&nbsp; almost&nbsp; $4000$&nbsp; consecutive bits of audio coding are missing.<br>
 [[File:P ID2333 KC T 1 1 S2b.png|right|frame|"Slit CD"&nbsp;  of the&nbsp; $\rm LNT/TUM$]]
@@ Line 256: / Line 256: @@
 *Each information block&nbsp; $\underline{u}$&nbsp; is represented by a&nbsp; '''code word'''&nbsp; (or&nbsp; "code block")&nbsp; $\underline{x} = (x_1, x_2, \text{. ..} \hspace{0.05cm}, x_n)$ &nbsp; with &nbsp; $n \ge k$,&nbsp; $x_i \in \{0, \, 1\}.$&nbsp; One then speaks of a binary&nbsp; $(n, k)$&nbsp; block code&nbsp; $C$.&nbsp; We denote the assignment by&nbsp; $\underline{x} = {\rm enc}(\underline{u})$,&nbsp; where&nbsp; "enc"&nbsp; stands for&nbsp; "encoder function".<br>
-*The&nbsp; '''received word'''&nbsp;  $\underline{y}$&nbsp; results from the code word&nbsp; $\underline{x}$&nbsp; by&nbsp;  [https://en.wikipedia.org/wiki/Modular_arithmetic as the  modulo&ndash;2 sum]&nbsp; with the likewise binary error vector&nbsp; $\underline{e} = (e_1, e_2, \text{. ..} \hspace{0.05cm}, e_n)$,&nbsp; where&nbsp; "$e= 1$"&nbsp; represents a transmission error and&nbsp; "$e= 0$"&nbsp; indicates that the&nbsp; $i$&ndash;th bit of the code word was transmitted correctly.&nbsp; The following therefore applies:
+*The&nbsp; '''received word'''&nbsp;  $\underline{y}$&nbsp; results from the code word&nbsp; $\underline{x}$&nbsp; by&nbsp;  [https://en.wikipedia.org/wiki/Modular_arithmetic $\text{as the  modulo&ndash;2 sum}$]&nbsp; with the likewise binary error vector&nbsp; $\underline{e} = (e_1, e_2, \text{. ..} \hspace{0.05cm}, e_n)$,&nbsp; where&nbsp; "$e= 1$"&nbsp; represents a transmission error and&nbsp; "$e= 0$"&nbsp; indicates that the&nbsp; $i$&ndash;th bit of the code word was transmitted correctly.&nbsp; The following therefore applies:
 ::<math>\underline{y} = \underline{x} \oplus \underline{e} \hspace{0.05cm}, \hspace{0.5cm} y_i  =   x_i \oplus e_i \hspace{0.05cm}, \hspace{0.2cm} i = 1, \text{...} \hspace{0.05cm}  , n\hspace{0.05cm}, </math>

@@ Line 324: / Line 324: @@
 {{GraueBox|TEXT=
 $\text{Example 10:}$&nbsp; &nbsp; &nbsp;$\text{(4, 2, 2) block code}$
-[[File:EN_KC_T_1_1_S5a_v2.png|right|frame|$\rm (4, 2, 2)$&nbsp; block code for error detection '''KORREKTUR: channel encoding, received words?'''|class=fit]]
+[[File:EN_KC_T_1_1_S5a_v3.png|right|frame|$\rm (4, 2, 2)$&nbsp; block code for error detection|class=fit]]
 In the graphic,&nbsp; the arrows

@@ Line 114: / Line 114: @@
 On the right you see a $\rm QR\ code$&nbsp; ("Quick Response")&nbsp; with associated content.
-[[File:KC_T_1_1_S2a.png|right|frame|Two-dimensional bar codes:&nbsp; Aztec and QR code '''KORREKTUR: Aztec Code, "", phone'''|class=fit]]
+[[File:EN_KC_T_1_1_S2a_v2.png|right|frame|Two-dimensional bar codes:&nbsp; Aztec and QR code|class=fit]]
 *The QR code was developed in 1994 for the automotive industry in Japan to mark components and also allows error correction.
@@ Line 211: / Line 211: @@
 {{GraueBox|TEXT=
 $\text{Example 8:}$&nbsp; For the graph,&nbsp; it was assumed,&nbsp; as an example and for the sake of simplicity,&nbsp; that
-[[File:EN_KC_T_1_1_S3a.png|right|frame|Image transmission with source and channel coding  '''KORREKTUR: the'''|class=fit]]
+[[File:EN_KC_T_1_1_S3a_v3.png|right|frame|Image transmission with source and channel coding|class=fit]]
 *the source symbol sequence has only the length&nbsp; $40$,&nbsp;
@@ Line 225: / Line 225: @@
 {{GraueBox|TEXT=
-[[File:EN_KC_T_1_1_S3b_v2.png|right|frame|Image transmission without source and channel coding '''KORREKTUR: the'''|class=fit]]
+[[File:EN_KC_T_1_1_S3b_v3.png|right|frame|Image transmission without source and channel coding|class=fit]]
 $\text{Example 9:}$&nbsp;
 <br><br><br><br>
@@ Line 235: / Line 235: @@
 i.e. transmit the BMP data directly without error protection,&nbsp; the result would be extremely poor despite&nbsp; $($by a factor&nbsp; $40/24)$&nbsp; greater bit rate.
 <br clear=all>
-[[File:EN_KC_T_1_1_S3c_v2.png|left|frame|Image transmission with source coding,&nbsp; but without channel coding '''KORREKTUR: the'''|class=fit]]
+[[File:EN_KC_T_1_1_S3c_v3.png|left|frame|Image transmission with source coding,&nbsp; but without channel coding|class=fit]]
 <br><br>
 '''Source coding but no channel coding'''

@@ Line 363: / Line 363: @@
 == On the nomenclature in this book ==
 <br>
-One of the objectives of our learning tutorial $\rm LNTwww$&nbsp; was to describe the entire field of Communications Engineering and the associated basic subjects with uniform nomenclature.&nbsp; In this most recently tackled book&nbsp; "Channel Coding"&nbsp; some changes have to be made with regard to the nomenclature after all. The reasons for this are:
+One of the objectives of our learning tutorial&nbsp; $\rm LNTwww$&nbsp; was to describe the entire field of Communications Engineering and the associated basic subjects with uniform nomenclature.&nbsp; In this most recently tackled book&nbsp; "Channel Coding"&nbsp; some changes have to be made with regard to the nomenclature after all. The reasons for this are:
 *Coding theory is a largely self-contained subject and few authors of relevant reference books on the subject attempt to relate it to other aspects of digital signal transmission.
@@ Line 369: / Line 369: @@
-Some nomenclature changes compared to the other&nbsp; $\rm LNTwww$ books&nbsp; shall be mentioned here:
+Some nomenclature changes compared to the other&nbsp; $\rm LNTwww$&nbsp; books&nbsp; shall be mentioned here:
 #All signals are represented by sequences of symbols in vector notation.&nbsp; For example,&nbsp; $\underline{u} = (u_1, u_2, \hspace{0.05cm}\text{...}\hspace{0.05cm}, u_k)$&nbsp; is the &nbsp; &nbsp; "source symbol sequence and&nbsp; $\underline{v} = (v_1, v_2, \hspace{0.05cm}\text{...}\hspace{0.05cm}, v_k)$&nbsp; the&nbsp; "sink symbol sequence".&nbsp; Previously,&nbsp; these symbol sequences were designated&nbsp; $\langle q_\nu \rangle$&nbsp; and&nbsp; $\langle v_\nu \rangle$,&nbsp; respectively.<br><br>
 #The vector $\underline{x} = (x_1, x_2, \hspace{0.05cm}\text{...} \hspace{0.05cm}, x_n)$ &nbsp; now denotes the discrete-time equivalent to the transmitted signal&nbsp; $s(t)$, while the received  signal&nbsp; $r(t)$ is described by the vector&nbsp; $\underline{y} = (y_1, y_2, \hspace{0.05cm}\text{...}\hspace{0.05cm}, y_n)$&nbsp;. The code rate is the quotient </i>&nbsp; $R=k/n$ &nbsp; with &nbsp; $0 \le R \le 1$ and the number of check bits is given by&nbsp; $m = n-k$.<br><br>
-#In the first main chapter,&nbsp; the elements&nbsp; $u_i$&nbsp; and&nbsp; $v_i$&nbsp; &nbsp;$($each with index&nbsp; $i = 1, \hspace{0.05cm}\text{...} \hspace{0.05cm}, k)$ &nbsp; of the vectors &nbsp; $\underline{u}$ &nbsp; and &nbsp; $\underline{v}$ &nbsp; are always binary&nbsp; $(0$&nbsp; or &nbsp;$1)$,&nbsp; as are the&nbsp; $n$&nbsp; elements&nbsp; $x_i$&nbsp; of the code word&nbsp; $\underline{x}$. &nbsp; For digital channel model&nbsp; ([[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Channel_.E2.80.93_BSC|BSC]],&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Erasure_Channel_.E2.80.93_BEC|BEC]],&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Error_.26_Erasure_Channel_.E2.80.93_BSEC|BSEC]])&nbsp; this also applies to the&nbsp; $n$&nbsp; received values&nbsp; $y_i \in \{0, 1\}$.<br><br>
+#In the first main chapter,&nbsp; the elements&nbsp; $u_i$&nbsp; and&nbsp; $v_i$&nbsp; &nbsp;$($each with index&nbsp; $i = 1, \hspace{0.05cm}\text{...} \hspace{0.05cm}, k)$ &nbsp; of the vectors &nbsp; $\underline{u}$ &nbsp; and &nbsp; $\underline{v}$ &nbsp; are always binary&nbsp; $(0$&nbsp; or &nbsp;$1)$,&nbsp; as are the&nbsp; $n$&nbsp; elements&nbsp; $x_i$&nbsp; of the code word&nbsp; $\underline{x}$. &nbsp; For digital channel model&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Channel_.E2.80.93_BSC|$\text{(BSC}$]],&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Erasure_Channel_.E2.80.93_BEC|$\text{BEC}$]],&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Error_.26_Erasure_Channel_.E2.80.93_BSEC|$\text{BSEC})$]]&nbsp; this also applies to the&nbsp; $n$&nbsp; received values&nbsp; $y_i \in \{0, 1\}$.<br><br>
 #The&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input|$\text{AWGN channel}$]]&nbsp; is characterised by real-valued output values&nbsp; $y_i$.&nbsp; The&nbsp; "code word estimator"&nbsp; in this case extracts from the vector&nbsp;$\underline{y} = (y_1, y_2, \hspace{0.05cm}\text{...}\hspace{0.05cm}, y_n)$ &nbsp; the binary vector&nbsp; $\underline{z} = (z_1, z_2, \hspace{0.05cm}\text{...} \hspace{0.05cm}, z_n)$&nbsp; to be compared with the code word&nbsp; $\underline{x}$.<br><br>
 #The transition from &nbsp; $\underline{y}$ &nbsp; to &nbsp; $\underline{z}$ &nbsp; is done by threshold decision &nbsp; &#8658; &nbsp; "Hard Decision"&nbsp; or according"&nbsp; to the MAP criterion &nbsp; &#8658; &nbsp; "Soft Decision".&nbsp; For equally likely input symbols,&nbsp; the&nbsp; "maximum Likelihood estimation also leads to the minimum error rate.<br><br>