Channel Coding/Channel Models and Decision Structures - Revision history

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Hwang at 10:56, 17 November 2022

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@@ Line 62: / Line 62: @@
 The AWGN channel model is not a digital channel model as we have presupposed in the paragraph&nbsp;  [[Channel_Coding/Objective_of_Channel_Coding#Block_diagram_and_requirements|"Block diagram and prerequisities"]]&nbsp; for the introductory description of channel coding methods.&nbsp; However,&nbsp; if we take into account a hard decision,&nbsp; we arrive at the digital model&nbsp; "Binary Symmetric Channel"&nbsp; $\rm (BSC)$:<br>
-[[File:EN_KC_T_1_2_S2.png|right|frame|BSC model and relation with the AWGN model.&nbsp;<u>Note:</u>&nbsp; In the AWGN model,&nbsp; we have denoted the binary output as&nbsp; $z \in \{0, \hspace{0.05cm}1\}$.&nbsp; For the digital channel models&nbsp; (BSC, BEC, BSEC),&nbsp; we now denote the discrete value output again by&nbsp; $y$.&nbsp; To avoid confusion, we now call the output signal of the AWGN model&nbsp; $y_{\rm A}$&nbsp; and for the analog received signal then&nbsp; $y_{\rm A} = \tilde{x} +n$. '''KORREKTUR: ""'''|class=fit]]
+[[File:EN_KC_T_1_2_S2_neu.png|right|frame|BSC model and relation with the AWGN model.&nbsp;<u>Note:</u>&nbsp; In the AWGN model,&nbsp; we have denoted the binary output as&nbsp; $z \in \{0, \hspace{0.05cm}1\}$.&nbsp; For the digital channel models&nbsp; (BSC, BEC, BSEC),&nbsp; we now denote the discrete value output again by&nbsp; $y$.&nbsp; To avoid confusion, we now call the output signal of the AWGN model&nbsp; $y_{\rm A}$&nbsp; and for the analog received signal then&nbsp; $y_{\rm A} = \tilde{x} +n$|class=fit]]
 *We choose the falsification probabilities&nbsp; ${\rm Pr}(y = 1\hspace{0.05cm}|\hspace{0.05cm} x=0)$&nbsp; resp.&nbsp; ${\rm Pr}(y = 0\hspace{0.05cm}|\hspace{0.05cm} x=1)$&nbsp; to be
@@ Line 98: / Line 98: @@
 The&nbsp; "Binary Erasure Channel"&nbsp; $\rm (BEC)$&nbsp; provides such information.&nbsp; On the basis of the graph you can see:
-[[File:EN_KC_T_1_2_S3.png|right|frame|BEC and connection with the AWGN model '''KORREKTUR: "" '''|class=fit]]
+[[File:EN_KC_T_1_2_S3_neu.png|right|frame|BEC and connection with the AWGN model|class=fit]]
 *The input alphabet of the BEC model is binary &nbsp; &#8658; &nbsp; $x &#8712; \{0, \hspace{0.05cm}1\}$ and the output alphabet is ternary &nbsp; &#8658; &nbsp;  $y &#8712; \{0, \hspace{0.05cm}1, \hspace{0.05cm}\rm E\}$.
@@ Line 132: / Line 132: @@
 {{GraueBox|TEXT=
 $\text{Example 2:}$&nbsp;  We consider the BSEC model with the two decision lines&nbsp;
-[[File:EN_KC_T_1_2_S4.png|right|frame|BSEC and connection with the AWGN model '''KORREKTUR: "" '''|class=fit]]
+[[File:EN_KC_T_1_2_S4_neu.png|right|frame|BSEC and connection with the AWGN model|class=fit]]
 *$G_0 = G = 0.5$,
 *$G_1 = -G = -0.5$.

@@ Line 57: / Line 57: @@
 *$N_0$&nbsp; denotes the constant&nbsp; (one-sided)&nbsp; noise power density of the AWGN channel.<br><br>
-:Notes:&nbsp;The presented facts are clarified with the (German language) SWF applet&nbsp; [[Applets:Fehlerwahrscheinlichkeit|"Symbolfehlerwahrscheinlichkeit von Digitalsystemen"]] &nbsp; <br>&rArr; &nbsp; "Symbol error probability of digital systems"
+:<u>Notes:</u> &nbsp; The presented facts are clarified with the&nbsp; $($German language$)$&nbsp; SWF applet&nbsp; [[Applets:Fehlerwahrscheinlichkeit|"Symbolfehlerwahrscheinlichkeit von Digitalsystemen"]] &nbsp; <br>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &rArr;  "Symbol error probability of digital systems"
 == Binary Symmetric Channel – BSC ==
 <br>
-The AWGN channel model is not a digital channel model as we have presupposed in the paragraph&nbsp;  [[Channel_Coding/Objective_of_Channel_Coding#Block_diagram_and_requirements|$\text{Block diagram and prerequisities}$]]&nbsp; for the introductory description of channel coding methods.&nbsp; However,&nbsp; if we take into account a hard decision,&nbsp; we arrive at the digital model&nbsp; "Binary Symmetric Channel"&nbsp; $\rm (BSC)$:<br>
+The AWGN channel model is not a digital channel model as we have presupposed in the paragraph&nbsp;  [[Channel_Coding/Objective_of_Channel_Coding#Block_diagram_and_requirements|"Block diagram and prerequisities"]]&nbsp; for the introductory description of channel coding methods.&nbsp; However,&nbsp; if we take into account a hard decision,&nbsp; we arrive at the digital model&nbsp; "Binary Symmetric Channel"&nbsp; $\rm (BSC)$:<br>
 [[File:EN_KC_T_1_2_S2.png|right|frame|BSC model and relation with the AWGN model.&nbsp;<u>Note:</u>&nbsp; In the AWGN model,&nbsp; we have denoted the binary output as&nbsp; $z \in \{0, \hspace{0.05cm}1\}$.&nbsp; For the digital channel models&nbsp; (BSC, BEC, BSEC),&nbsp; we now denote the discrete value output again by&nbsp; $y$.&nbsp; To avoid confusion, we now call the output signal of the AWGN model&nbsp; $y_{\rm A}$&nbsp; and for the analog received signal then&nbsp; $y_{\rm A} = \tilde{x} +n$. '''KORREKTUR: ""'''|class=fit]]
@@ Line 67: / Line 67: @@
 ::<math>\varepsilon =  {\rm Q}(\sqrt{\rho})\hspace{0.05cm}.</math>
-*Then the connection to the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input|$\text{AWGN channel model}$]]&nbsp; is established.&nbsp;
+*Then the connection to the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input|$\text{AWGN}$]]&nbsp;  channel model&nbsp; is established.&nbsp;
 *The decision line is at&nbsp; $G = 0$,&nbsp; which also gives rise to the property&nbsp; "symmetrical".<br>
@@ Line 113: / Line 113: @@
 We refer here again to the following applets:
-*[[Applets:Fehlerwahrscheinlichkeit|$\text{Symbol error probability of digital systems}$]],
+*[[Applets:Fehlerwahrscheinlichkeit|"Symbol error probability of digital systems"]],
-*[[Applets:Complementary_Gaussian_Error_Functions|$\text{Complementary Gaussian Error Function}$]].
@@ Line 121: / Line 122: @@
 The BEC model is rather unrealistic &nbsp;$($error probability:&nbsp; $0)$&nbsp; and only an approximation for an extremely large signal&ndash;to&ndash;noise&ndash;power ratio&nbsp;  $\rho$.
-More severe disturbances '''KORREKTUR: noises'''&nbsp; $($a smaller&nbsp; $\rho)$&nbsp; would be better served by the&nbsp; "Binary Symmetric Error & Erasure Channel"&nbsp; $\rm (BSEC)$&nbsp; with the two parameters
+Stronger noise&nbsp; $($a smaller&nbsp; $\rho)$&nbsp; would be better served by the&nbsp; "Binary Symmetric Error & Erasure Channel"&nbsp; $\rm (BSEC)$&nbsp; with the two parameters
 *Falsification probability &nbsp; $\varepsilon = {\rm Pr}(y = 1\hspace{0.05cm}|\hspace{0.05cm} x=0)= {\rm Pr}(y = 0\hspace{0.05cm}|\hspace{0.05cm} x=1)$,<br>
@@ Line 152: / Line 153: @@
 == Criteria&nbsp; &raquo;Maximum-a-posteriori&laquo;&nbsp; and&nbsp; &raquo;Maximum-Likelihood&laquo;==
 <br>
-We now start from the model sketched below and apply the methods already described in chapter&nbsp; [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver|$\text{Structure of the optimal receiver}$]]&nbsp; of the book&nbsp; "Digital Signal Transmission"&nbsp; to the decoding process.<br>
+We now start from the model sketched below and apply the methods already described in chapter&nbsp; [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver|"Structure of the optimal receiver"]]&nbsp; of the book&nbsp; "Digital Signal Transmission"&nbsp; to the decoding process.<br>
 {{BlaueBox|TEXT=
@@ Line 197: / Line 198: @@
 ::<math>\underline{z} = {\rm arg} \max_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.03cm} \in \hspace{0.05cm} \mathcal{C} } \hspace{0.1cm} {\rm Pr}( \underline{x}_{\hspace{0.03cm}i} \vert\hspace{0.05cm} \underline{y} ) \hspace{0.05cm}.</math>
-*${\rm Pr}( \underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm}\vert \hspace{0.05cm} \underline{y} )$&nbsp; is the&nbsp; [[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Conditional_Probability|$\text{conditional probability}$]]&nbsp; that&nbsp; $\underline{x}_i$&nbsp; was sent,&ndash; when&nbsp; $\underline{y}$&nbsp; is received.}}<br>
+*${\rm Pr}( \underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm}\vert \hspace{0.05cm} \underline{y} )$&nbsp; is the&nbsp; [[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Conditional_Probability|"conditional probability"]]&nbsp; that&nbsp; $\underline{x}_i$&nbsp; was sent,&nbsp; when&nbsp; $\underline{y}$&nbsp; is received.}}<br>
 We now try to simplify this decision rule step by step.&nbsp; The inference probability can be transformed according to&nbsp; "Bayes rule"&nbsp; as follows:
@@ Line 288: / Line 289: @@
 Applications of ML/BSC decision can be found in the following sections:
-*[[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes|$\text{Single Parity&ndash;check Code}$]]&nbsp; $\rm (SPC)$.<br>
+*[[Channel_Coding/Examples_of_Binary_Block_Codes#Single_Parity-check_Codes|$\text{SPC}$]]&nbsp; $($"single parity&ndash;check code"$)$.<br>
-*[[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes|$\text{Repetition code}$]]&nbsp; $\rm (RC)$.<br>
+*[[Channel_Coding/Examples_of_Binary_Block_Codes#Repetition_Codes|$\text{RC}$]]&nbsp; $($"repetition code"$)$.<br>
 == Maximum-likelihood decision at the AWGN channel ==
@@ Line 346: / Line 347: @@
 ==Exercises for the chapter==
 <br>
-[[Aufgaben:Exercise_1.3:_Channel_Models_BSC_-_BEC_-_BSEC_-_AWGN|Exercise 1.3: Channel Models BSC - BEC - BSEC - AWGN]]
+[[Aufgaben:Exercise_1.3:_Channel_Models_BSC_-_BEC_-_BSEC_-_AWGN|Exercise 1.3: Channel Models&nbsp; "BSC - BEC - BSEC - AWGN"]]
 [[Aufgaben:Exercise_1.4:_Maximum_Likelihood_Decision|Exercise 1.4: Maximum Likelihood Decision]]
 {{Display}}

@@ Line 9: / Line 9: @@
 == AWGN channel at binary input ==
 <br>
-We consider the well-known discrete-time&nbsp; [[Modulation_Methods/Quality_Criteria#Some_remarks_on_the_AWGN_channel_model|$\text{AWGN channel model}$]]&nbsp; according to the left graph:
+We consider the well-known discrete-time&nbsp; [[Modulation_Methods/Quality_Criteria#Some_remarks_on_the_AWGN_channel_model|$\text{AWGN}$]]&nbsp;  channel model according to the left graph:
 *The binary and discrete-time message signal&nbsp; $x$&nbsp; takes the values&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; with equal probability:
 :$${\rm Pr}(x = 0) ={\rm Pr}(x = 1) = 1/2.$$
 *We now transform the unipolar variable&nbsp; $x \in \{0,\ 1 \}$&nbsp; into the bipolar variable&nbsp; $\tilde{x} \in \{+1, -1 \}$&nbsp; more suitable for signal transmission.&nbsp; Then holds:
-[[File:EN_KC_T_1_2_S1.png|right|frame|Probability density function&nbsp; (PDF)
+[[File:EN_KC_T_1_2_S1.png|right|frame|Probability density function&nbsp; $\rm (PDF)$&nbsp;
 of the AWGN channel|class=fit]]
 :$${\rm Pr}(\tilde{x} =+1) ={\rm Pr}(\tilde{x} =-1) = 1/2.$$
-*The transmission is affected by&nbsp; [[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Transmission_channel_and_interference| $\text{additive white Gaussian noise}$]]&nbsp; $\rm (AWGN)$&nbsp; $n$&nbsp; with&nbsp;  (normalized)&nbsp; noise power&nbsp; $\sigma^2 = N_0/E_{\rm B}$.&nbsp; The root mean square&nbsp; (rms value)&nbsp; of the Gaussian PDF is&nbsp; $\sigma$.<br>
+*The transmission is affected by&nbsp; [[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Transmission_channel_and_interference| "additive white Gaussian noise"]]&nbsp; $\rm (AWGN)$&nbsp; $n$&nbsp; with&nbsp;  (normalized)&nbsp; noise power&nbsp; $\sigma^2 = N_0/E_{\rm B}$.&nbsp; The root mean square&nbsp; (rms value)&nbsp; of the Gaussian PDF is&nbsp; $\sigma$.<br>
 *Because of the Gaussian PDF,&nbsp; the output signal&nbsp; $y = \tilde{x} +n$&nbsp; can take on any real value in the range&nbsp; $-\infty$&nbsp; to&nbsp; $+\infty$.&nbsp; That means:
@@ Line 22: / Line 22: @@
 *The AWGN output  value&nbsp; $y$&nbsp; is therefore discrete in time like&nbsp; $x$&nbsp; &nbsp;$($resp. $\tilde{x})$&nbsp; but in contrast to the latter it is continuous in value.<br>
 <br clear=all>
-The graph on the right shows&nbsp; (in blue color resp. red color)&nbsp; the conditional probability density functions:
+The graph on the right shows&nbsp; $($in blue resp. red color$)$&nbsp; the conditional probability density functions:
 ::<math>f_{y \hspace{0.03cm}| \hspace{0.03cm}x=0 } \hspace{0.05cm} (y \hspace{0.05cm}| \hspace{0.05cm}x=0 )\hspace{-0.1cm} =  \hspace{-0.1cm}
@@ Line 29: / Line 29: @@
 \frac {1}{\sqrt{2\pi} \cdot \sigma } \cdot {\rm e}^{ -  (y+1)^2/(2\sigma^2) }\hspace{0.05cm}.</math>
-Not shown is the total&nbsp; (unconditional)&nbsp; PDF,&nbsp; for which applies in the case of equally probable symbols:
+Not shown is the total&nbsp; $($unconditional$)$&nbsp; PDF,&nbsp; for which applies in the case of equally probable symbols:
 ::<math>f_y(y) = {1}/{2} \cdot  \big [ f_{y \hspace{0.03cm}| \hspace{0.03cm}x=0 } \hspace{0.05cm} (y \hspace{0.05cm}| \hspace{0.05cm}x=0 ) +
@@ Line 35: / Line 35: @@
 The two shaded areas &nbsp;$($each &nbsp;$\varepsilon)$&nbsp; mark decision errors under the condition&nbsp;
-*$x=0$ &nbsp; &#8658; &nbsp; $\tilde{x} = +1$&nbsp; (blue) &nbsp; respectively
+*$x=0$ &nbsp; &#8658; &nbsp; $\tilde{x} = +1$&nbsp; (blue),
 *$x=1$ &nbsp; &#8658; &nbsp; $\tilde{x} = -1$&nbsp; (red),
@@ Line 45: / Line 46: @@
 {\rm if} \hspace{0.15cm}y < 0\hspace{0.05cm}.\\ \end{array}</math>
-For equally probable input symbols, the bit error probability&nbsp; ${\rm Pr}(z \ne x)$&nbsp; is then also&nbsp; $\varepsilon$. With the&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability|$\text{complementary Gaussian error function}$]]&nbsp; ${\rm Q}(x)$&nbsp; the following holds:
+For equally probable input symbols, the bit error probability&nbsp; ${\rm Pr}(z \ne x)$&nbsp; is then also&nbsp; $\varepsilon$. With the&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability|"complementary Gaussian error function"]]&nbsp; ${\rm Q}(x)$&nbsp; the following holds:
 ::<math>\varepsilon = {\rm Q}(1/\sigma) = {\rm Q}(\sqrt{\rho}) =
@@ Line 51: / Line 52: @@
 \hspace{0.05cm}.</math>
-where&nbsp; $\rho = 1/\sigma^2 = 2 \cdot E_{\rm S}/N_0$&nbsp; denotes the signal&ndash;to&ndash;noise ratio&nbsp; $\rm (SNR)$&nbsp; before the decision device '''KORREKTUR: decision''', using the following system quantities:
+where&nbsp; $\rho = 1/\sigma^2 = 2 \cdot E_{\rm S}/N_0$&nbsp; denotes the signal&ndash;to&ndash;noise ratio&nbsp; $\rm (SNR)$&nbsp; before the decision,&nbsp; using the following system quantities:
 *$E_{\rm S}$&nbsp; is the signal energy per symbol&nbsp; (without coding: &nbsp;$E_{\rm S}=E_{\rm B}$,&nbsp; thus equal to the signal energy per bit),<br>
 *$N_0$&nbsp; denotes the constant&nbsp; (one-sided)&nbsp; noise power density of the AWGN channel.<br><br>
-Notes:&nbsp;The presented facts are clarified with the (German language) SWF applet<br> &nbsp; &nbsp; &nbsp; &nbsp; [[Applets:Fehlerwahrscheinlichkeit|$\text{Symbolfehlerwahrscheinlichkeit von Digitalsystemen}$]] &nbsp; &rArr; &nbsp; "Symbol error probability of digital systems"
+:Notes:&nbsp;The presented facts are clarified with the (German language) SWF applet&nbsp; [[Applets:Fehlerwahrscheinlichkeit|"Symbolfehlerwahrscheinlichkeit von Digitalsystemen"]] &nbsp; <br>&rArr; &nbsp; "Symbol error probability of digital systems"
 == Binary Symmetric Channel – BSC ==
 <br>

@@ Line 161: / Line 161: @@
 Formulated a little more precisely:
-*Minimizing the&nbsp; '''&raquo;block error probability&laquo;'''&nbsp;
+*Minimizing the&nbsp; &raquo;'''block error probability'''&laquo;&nbsp;
 :$${\rm Pr(block \:error)} = {\rm Pr}(\underline{v} \ne \underline{u}) $$
 :related to the vectors&nbsp; $\underline{u}$&nbsp; and&nbsp; $\underline{v}$&nbsp; of length&nbsp; $k$&nbsp;.
@@ Line 192: / Line 192: @@
 <br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The&nbsp; "block&ndash;wise maximum&ndash;a&ndash;posteriori receiver"&nbsp; &ndash; for short: &nbsp;<b>&raquo;block&ndash;wise MAP&laquo;</b> &ndash;  decides among the&nbsp; $2^k$&nbsp; code words&nbsp; $\underline{x}_i \in \mathcal{C}$&nbsp; for the code word with the highest a&ndash;posteriori probability:
+$\text{Definition:}$&nbsp; The&nbsp; "block&ndash;wise maximum&ndash;a&ndash;posteriori receiver"&nbsp; &ndash; for short: &nbsp;&raquo;<b>block&ndash;wise MAP</b>&laquo; &ndash;  decides among the&nbsp; $2^k$&nbsp; code words&nbsp; $\underline{x}_i \in \mathcal{C}$&nbsp; for the code word with the highest a&ndash;posteriori probability:
 ::<math>\underline{z} = {\rm arg} \max_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.03cm} \in \hspace{0.05cm} \mathcal{C} } \hspace{0.1cm} {\rm Pr}( \underline{x}_{\hspace{0.03cm}i} \vert\hspace{0.05cm} \underline{y} ) \hspace{0.05cm}.</math>
@@ Line 208: / Line 208: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp;  The&nbsp; "block&ndash;wise maximum&ndash;likelihood receiver"&nbsp; &ndash; for short: &nbsp;'''&raquo;block&ndash;wise ML&laquo;'''&nbsp; &ndash; decides among the&nbsp; $2^k$&nbsp; allowed code words&nbsp; $\underline{x}_i \in \mathcal{C}$&nbsp; for the code word with the highest transition probability:
+$\text{Definition:}$&nbsp;  The&nbsp; "block&ndash;wise maximum&ndash;likelihood receiver"&nbsp; &ndash; for short: &nbsp;&raquo;'''block&ndash;wise ML'''&laquo;&nbsp; &ndash; decides among the&nbsp; $2^k$&nbsp; allowed code words&nbsp; $\underline{x}_i \in \mathcal{C}$&nbsp; for the code word with the highest transition probability:
 ::<math>\underline{z} = {\rm arg} \max_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C} } \hspace{0.1cm} {\rm Pr}( \underline{y}  \hspace{0.05cm}\vert\hspace{0.05cm} \underline{x}_{\hspace{0.03cm}i} ) \hspace{0.05cm}.</math>
@@ Line 219: / Line 219: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp;  The&nbsp; "bit&ndash;wise maximum&ndash;a&ndash;posteriori receiver" &ndash; for short:&nbsp; '''&raquo;bit&ndash;wise MAP&laquo;'''&nbsp; &ndash; selects for each individual code bit&nbsp; $x_i$&nbsp; the value &nbsp;$(0$ or $1)$&nbsp; with the highest inference probability&nbsp; ${\rm Pr}( {x}_{\hspace{0.03cm}i}\vert \hspace{0.05cm} \underline{y} )$:
+$\text{Definition:}$&nbsp;  The&nbsp; "bit&ndash;wise maximum&ndash;a&ndash;posteriori receiver" &ndash; for short:&nbsp; &raquo;'''bit&ndash;wise MAP'''&laquo;&nbsp; &ndash; selects for each individual code bit&nbsp; $x_i$&nbsp; the value &nbsp;$(0$ or $1)$&nbsp; with the highest inference probability&nbsp; ${\rm Pr}( {x}_{\hspace{0.03cm}i}\vert \hspace{0.05cm} \underline{y} )$:
 ::<math>\underline{z} = {\rm arg}\hspace{-0.1cm}{ \max_{ {x}_{\hspace{0.03cm}i} \hspace{0.03cm} \in \hspace{0.05cm} \{0, 1\} } \hspace{0.03cm} {\rm Pr}( {x}_{\hspace{0.03cm}i}\vert \hspace{0.05cm} \underline{y} ) \hspace{0.05cm} }.</math>}}<br>
@@ Line 279: / Line 279: @@
 $\text{Maximum&ndash;likelihood decision at the BSC channel:}$&nbsp;
-Choose from the&nbsp; $2^k$&nbsp; allowed code words&nbsp; $\underline{x}_{\hspace{0.03cm}i}$&nbsp; the one with the&nbsp; '''&raquo;least Hamming distance&laquo;'''&nbsp; $d_{\rm H}(\underline{y}  \hspace{0.05cm}, \hspace{0.1cm}\underline{x}_{\hspace{0.03cm}i})$&nbsp; to the received vector&nbsp; $\underline{y}$:
+Choose from the&nbsp; $2^k$&nbsp; allowed code words&nbsp; $\underline{x}_{\hspace{0.03cm}i}$&nbsp; the one with the&nbsp; &raquo;'''least Hamming distance'''&laquo;&nbsp; $d_{\rm H}(\underline{y}  \hspace{0.05cm}, \hspace{0.1cm}\underline{x}_{\hspace{0.03cm}i})$&nbsp; to the received vector&nbsp; $\underline{y}$:
 ::<math>\underline{z} = {\rm arg} \min_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C} } \hspace{0.1cm}
@@ Line 337: / Line 337: @@
 $\text{Maximum&ndash;likelihood decision at the AWGN channel:}$&nbsp;
-Choose from the&nbsp; $2^k$&nbsp; allowed code words&nbsp; $\underline{x}_{\hspace{0.03cm}i}$&nbsp; the one with the&nbsp; '''&raquo;smallest Euclidean distance&laquo;'''&nbsp; $d_{\rm E}(\underline{y}  \hspace{0.05cm}, \hspace{0.1cm}\underline{x}_{\hspace{0.03cm}i})$&nbsp; to the received vector &nbsp;$\underline{y}$:
+Choose from the&nbsp; $2^k$&nbsp; allowed code words&nbsp; $\underline{x}_{\hspace{0.03cm}i}$&nbsp; the one with the&nbsp; &raquo;'''smallest Euclidean distance'''&laquo;&nbsp; $d_{\rm E}(\underline{y}  \hspace{0.05cm}, \hspace{0.1cm}\underline{x}_{\hspace{0.03cm}i})$&nbsp; to the received vector &nbsp;$\underline{y}$:
 ::<math>\underline{z} = {\rm arg} \min_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C} } \hspace{0.1cm}

@@ Line 311: / Line 311: @@
   \right ] \hspace{0.05cm}.</math>
-*Since&nbsp; $\underline{y}$&nbsp; is now value-continuous rather than value-discrete as in the BSCmodel,&nbsp; probability densities&nbsp; must now be examined according to the maximum-likelihood decision rule&nbsp; rather than probabilities. The optimal result is:
+*Since&nbsp; $\underline{y}$&nbsp; is now value-continuous rather than value-discrete as in the BSC model,&nbsp; probability densities&nbsp; must now be examined according to the maximum-likelihood decision rule&nbsp; rather than probabilities. The optimal result is:
 ::<math>\underline{z} = {\rm arg} \max_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm}

@@ Line 293: / Line 293: @@
 == Maximum-likelihood decision at the AWGN channel ==
 <br>
-The AWGN model for a&nbsp; $(n, k)$&nbsp; block code differs from the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input| $\text{model}$]]&nbsp; in the first chapter section in that for&nbsp; $x$, &nbsp;$\tilde{x}$&nbsp; and &nbsp;$y$&nbsp; now the corresponding vectors&nbsp; $\underline{x}$, &nbsp;$\underline{\tilde{x}}$&nbsp; and &nbsp;$\underline{y}$&nbsp; must be used,&nbsp; each consisting of&nbsp; $n$&nbsp; elements.
+The AWGN model for a&nbsp; $(n, k)$&nbsp; block code differs from the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input| $\text{model}$]]&nbsp; in the first chapter section in that for&nbsp; $x$, &nbsp;$\tilde{x}$&nbsp; and &nbsp;$y$&nbsp; now the corresponding vectors&nbsp; $\underline{x}$, &nbsp;$\underline{\tilde{x}}$&nbsp; and &nbsp;$\underline{y}$&nbsp; must be used,&nbsp; each consisting of&nbsp; $n$&nbsp; elements.
 The steps to derive the maximum&ndash;likelihood decision for AWGN are given below only in bullet form:

@@ Line 214: / Line 214: @@
 *The conditional probability&nbsp; ${\rm Pr}( \underline{y}  \hspace{0.05cm}\vert\hspace{0.05cm} \underline{x}_{\hspace{0.03cm}i} )$&nbsp; is now to be understood in the forward direction,&nbsp; namely as the probability that the vector&nbsp; $\underline{y}$&nbsp; is received when the code word&nbsp; $\underline{x}_i$&nbsp; has been sent.<br>
-*In the following, we always use the maximum&ndash;likelihood receiver at the block level.&nbsp; Due to the assumed equally likely information words,&nbsp; this also always provides the best possible decision}}.<br>
+*In the following, we always use the maximum&ndash;likelihood receiver at the block level.&nbsp; Due to the assumed equally likely information words,&nbsp; this also always provides the best possible decision.}}<br>
 However,&nbsp; the situation is different on the bit level.&nbsp; The goal of an iterative decoding is just to estimate probabilities for all code bits&nbsp; $x_i \in \{0, 1\}$&nbsp; and to pass these on to the next stage.&nbsp; For this one needs a MAP receiver.<br>

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 *$E_{\rm S}$&nbsp; is the signal energy per symbol&nbsp; (without coding: &nbsp;$E_{\rm S}=E_{\rm B}$,&nbsp; thus equal to the signal energy per bit),<br>
-*$N_0$&nbsp; denotes the constant&nbsp; (one-sided)&nbsp; noise power density of the AWGN&ndash;channel.<br><br>
+*$N_0$&nbsp; denotes the constant&nbsp; (one-sided)&nbsp; noise power density of the AWGN channel.<br><br>
 Notes:&nbsp;The presented facts are clarified with the (German language) SWF applet<br> &nbsp; &nbsp; &nbsp; &nbsp; [[Applets:Fehlerwahrscheinlichkeit|$\text{Symbolfehlerwahrscheinlichkeit von Digitalsystemen}$]] &nbsp; &rArr; &nbsp; "Symbol error probability of digital systems"
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-The parameters&nbsp; $\varepsilon$&nbsp; and&nbsp; $\lambda$&nbsp; are fixed by the&nbsp; $\rm SNR$&nbsp; $\rho=1/\sigma^2$&nbsp; of the comparable AWGN&ndash;channel.
+The parameters&nbsp; $\varepsilon$&nbsp; and&nbsp; $\lambda$&nbsp; are fixed by the&nbsp; $\rm SNR$&nbsp; $\rho=1/\sigma^2$&nbsp; of the comparable AWGN channel.
 *For&nbsp; $\sigma = 0.5$  &nbsp; &#8658; &nbsp; $\rho = 4$&nbsp; (assumed for the sketched PDF):