Channel Coding/Soft-in Soft-Out Decoder - Revision history

Guenter at 16:21, 12 January 2023

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Hwang at 21:08, 1 January 2023

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Hwang at 20:58, 1 January 2023

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Hwang at 20:45, 1 January 2023

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Hwang at 20:38, 1 January 2023

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Hwang at 20:37, 1 January 2023

2023-01-01T20:37:44Z

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 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; Let the code word&nbsp; $\underline{x}^{(&ndash;i)}$&nbsp; contain all symbols except&nbsp; $x_i$ &nbsp; &#8658; &nbsp; vector of length&nbsp; $n&ndash;1$.&nbsp;  Thus,&nbsp;  the &nbsp; "extrinsic log likelihood ratio"&nbsp; with respect to the&nbsp; $i$&ndash;th symbol is,log likelihood ratiowhen&nbsp; $\underline{x}$&nbsp; was received: '''KORREKTUR: Thus,&nbsp;  the &nbsp; "extrinsic log likelihood ratio"&nbsp; will be with respect to the&nbsp; $i$&ndash;th symbol, when&nbsp; $\underline{x}$&nbsp; has been received:'''
+$\text{Definition:}$&nbsp; Let the code word&nbsp; $\underline{x}^{(&ndash;i)}$&nbsp; contain all symbols except&nbsp; $x_i$ &nbsp; &#8658; &nbsp; vector of length&nbsp; $n&ndash;1$.&nbsp;  Thus,&nbsp;  the &nbsp; "extrinsic log likelihood ratio"&nbsp; will be with respect to the&nbsp; $i$&ndash;th symbol, when&nbsp; $\underline{x}$&nbsp; has been received:
 ::<math>L_{\rm E}(i) = {\rm ln} \hspace{0.15cm}\frac{ {\rm Pr} \left [w_{\rm H}(\underline{x}^{(-i)})\hspace{0.15cm}{\rm is \hspace{0.15cm} even} \hspace{0.05cm} \vert\hspace{0.05cm}\underline{y} \hspace{0.05cm}\right ]}{ {\rm Pr} \left [w_{\rm H}(\underline{x}^{(-i)})\hspace{0.15cm}{\rm is \hspace{0.15cm} odd} \hspace{0.05cm} \vert \hspace{0.05cm}\underline{y} \hspace{0.05cm}\right ]}

@@ Line 2: / Line 2: @@
 {{Header
 |Untermenü=Iterative Decoding Methods
-|Vorherige Seite=Distance Characteristics and Error Probability Barriers
+|Vorherige Seite=Distance Characteristics and Error Probability Bounds
 |Nächste Seite=The Basics of Product Codes
 }}

@@ Line 369: / Line 369: @@
 :$$\underline{L} = (L(1), \ \text{...}\hspace{0.05cm} , \ L(n)).$$
 [[File:EN_KC_T_4_1_S6.png|right|frame|Parallel concatenated codes|class=fit]]
-So the calculation of all&nbsp; L&ndash;values is done on symbol level.&nbsp; Not shown here is the&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Second_Requirement_f.C3.BCr_Turbocodes:_Interleaving| $\rm interleaver$]],&nbsp; which is mandatory e.g. for turbo codes.
+So the calculation of all&nbsp; L&ndash;values is done on symbol level.&nbsp; Not shown here is the&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Second_requirement_for_turbo_codes:_Interleaving| $\rm interleaver$]],&nbsp; which is mandatory e.g. for turbo codes.
 # The sequences&nbsp; $\underline{x}_1$&nbsp; and&nbsp; $\underline{x}_2$&nbsp; are combined to form the vector&nbsp; $\underline{x}$&nbsp; for joint transmission over the channel by a multiplexer.
 # At the receiver, the sequence&nbsp; $\underline{y}$&nbsp; is again decomposed into the parts&nbsp; $\underline{y}_1$&nbsp; and&nbsp; $\underline{y}_2$.&nbsp;  From this the channel L&ndash;values&nbsp; $\underline{L}_{\rm K,\hspace{0.05cm}1}&nbsp;$ and&nbsp; $\underline{L}_{\rm K,\hspace{0.05cm}2}$&nbsp; are formed.<br>

@@ Line 360: / Line 360: @@
 *In&nbsp; [[Examples_of_Communication_Systems/General_Description_of_GSM#.23_OVERVIEW_OF_THE_THIRD_MAIN_CHAPTER_.23|$\rm GSM$]]&nbsp; $($"Global System for Mobile Communications",&nbsp; the second generation of mobile communication systems$)$,&nbsp; the data bit rate is first increased from&nbsp; $9. 6 \ \rm kbit/s$&nbsp; to&nbsp; $12 \ \rm kbit/s$&nbsp; in order to enable error detection in circuit-switched networks as well.&nbsp; This is followed by a punctured convolutional code with the output bit rate&nbsp; $22.8 \ \rm kbit/s$.&nbsp; The total code rate is thus about $0.421$.
-*In the 3G cellular system&nbsp; [[Examples_of_Communication_Systems/General_Description_of_UMTS#.23_OVERVIEW_OF_THE_FOURTH_MAIN_CHAPTER_.23|$\rm UMTS$]]&nbsp; $($"Universal Mobile Telecommunications System"),&nbsp; depending on the boundary conditions&nbsp; $($good/bad channel,&nbsp; few/many subscribers in the cell$)$&nbsp; one uses a&nbsp; [[Examples_of_Communication_Systems/General_Description_of_GSM#.23_OVERVIEW_OF_THE_THIRD_MAIN_CHAPTER_.23|$\text{convolutional code}$]]&nbsp; or a&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Basic_structure_of_a_Turbo_Code|$\text{turbo code}$]]&nbsp; $($by this one understands per se the concatenation of two convolutional encoders$)$.&nbsp;
+*In the 3G cellular system&nbsp; [[Examples_of_Communication_Systems/General_Description_of_UMTS#.23_OVERVIEW_OF_THE_FOURTH_MAIN_CHAPTER_.23|$\rm UMTS$]]&nbsp; $($"Universal Mobile Telecommunications System"),&nbsp; depending on the boundary conditions&nbsp; $($good/bad channel,&nbsp; few/many subscribers in the cell$)$&nbsp; one uses a&nbsp; [[Examples_of_Communication_Systems/General_Description_of_GSM#.23_OVERVIEW_OF_THE_THIRD_MAIN_CHAPTER_.23|$\text{convolutional code}$]]&nbsp; or a&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Basic_structure_of_a_turbo_code|$\text{turbo code}$]]&nbsp; $($by this one understands per se the concatenation of two convolutional encoders$)$.&nbsp;
 *In 4G mobile communication systems&nbsp; [[Examples_of_Communication_Systems/General_Description_of_UMTS#.23_OVERVIEW_OF_THE_FOURTH_MAIN_CHAPTER_.23|$\rm LTE$]]&nbsp; $($"Long Term Evolution"$)$,&nbsp; a convolutional code is used for short control signals,&nbsp; turbo code for the longer payload data.<br>

@@ Line 358: / Line 358: @@
 To begin with,&nbsp; here are a few examples from mobile communications:
-*In&nbsp; [[Examples_of_Communication_Systems/General_Description_of_GSM#.23_.C3.9CBERBLICK_To_the_PRINCIPAL_CHAPTER_.23|$\rm GSM$]]&nbsp; $($"Global System for Mobile Communications",&nbsp; the second generation of mobile communication systems$)$,&nbsp; the data bit rate is first increased from&nbsp; $9. 6 \ \rm kbit/s$&nbsp; to&nbsp; $12 \ \rm kbit/s$&nbsp; in order to enable error detection in circuit-switched networks as well.&nbsp; This is followed by a punctured convolutional code with the output bit rate&nbsp; $22.8 \ \rm kbit/s$.&nbsp; The total code rate is thus about $0.421$.
+*In&nbsp; [[Examples_of_Communication_Systems/General_Description_of_GSM#.23_OVERVIEW_OF_THE_THIRD_MAIN_CHAPTER_.23|$\rm GSM$]]&nbsp; $($"Global System for Mobile Communications",&nbsp; the second generation of mobile communication systems$)$,&nbsp; the data bit rate is first increased from&nbsp; $9. 6 \ \rm kbit/s$&nbsp; to&nbsp; $12 \ \rm kbit/s$&nbsp; in order to enable error detection in circuit-switched networks as well.&nbsp; This is followed by a punctured convolutional code with the output bit rate&nbsp; $22.8 \ \rm kbit/s$.&nbsp; The total code rate is thus about $0.421$.
-*In the 3G cellular system&nbsp; [[Examples_of_Communication_Systems/General_Description_of_UMTS#.23_.C3.9CBERBLICK_To_the_Fourth_Main_Chapter_.23|$\rm UMTS$]]&nbsp; $($"Universal Mobile Telecommunications System"),&nbsp; depending on the boundary conditions&nbsp; $($good/bad channel,&nbsp; few/many subscribers in the cell$)$&nbsp; one uses a&nbsp; [[Channel_Coding/Basics_of_Convolutional_Coding#.23_.C3.9CBERBLICK_To_the_PRINCIPAL_CHAPTER_.23|$\text{convolutional code}$]]&nbsp; or a&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Basic_structure_of_a_Turbo_Code|$\text{turbo code}$]]&nbsp; $($by this one understands per se the concatenation of two convolutional encoders$)$.&nbsp;
+*In the 3G cellular system&nbsp; [[Examples_of_Communication_Systems/General_Description_of_UMTS#.23_OVERVIEW_OF_THE_FOURTH_MAIN_CHAPTER_.23|$\rm UMTS$]]&nbsp; $($"Universal Mobile Telecommunications System"),&nbsp; depending on the boundary conditions&nbsp; $($good/bad channel,&nbsp; few/many subscribers in the cell$)$&nbsp; one uses a&nbsp; [[Examples_of_Communication_Systems/General_Description_of_GSM#.23_OVERVIEW_OF_THE_THIRD_MAIN_CHAPTER_.23|$\text{convolutional code}$]]&nbsp; or a&nbsp; [[Channel_Coding/The_Basics_of_Turbo_Codes#Basic_structure_of_a_Turbo_Code|$\text{turbo code}$]]&nbsp; $($by this one understands per se the concatenation of two convolutional encoders$)$.&nbsp;
-*In 4G mobile communication systems&nbsp; [[Mobile_Communications/General_Information_on_the_LTE_Mobile_Communications_Standard#.23_OVERVIEW_OF_THE_FOURTH_MAIN_CHAPTER_.23|$\rm LTE$]]&nbsp; $($"Long Term Evolution"$)$,&nbsp; a convolutional code is used for short control signals,&nbsp; turbo code for the longer payload data.<br>
+*In 4G mobile communication systems&nbsp; [[Examples_of_Communication_Systems/General_Description_of_UMTS#.23_OVERVIEW_OF_THE_FOURTH_MAIN_CHAPTER_.23|$\rm LTE$]]&nbsp; $($"Long Term Evolution"$)$,&nbsp; a convolutional code is used for short control signals,&nbsp; turbo code for the longer payload data.<br>

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   L(y = -1\hspace{0.05cm}\vert\hspace{0.05cm}x) = -2.197\hspace{0.05cm}.</math>
-*This example shows that the so-called&nbsp; &raquo;$\text{LLR algebra}$&laquo;  can also be applied to conditional probabilities. In &nbsp;[[Aufgaben:Exercise_4.1Z:_Log_Likelihood_Ratio_at_the_BEC_Model|$\text{Exercise 4.1Z}$]],&nbsp; the BEC&ndash;model is described in a similar way.}}<br>
+*This example shows that the so-called&nbsp; &raquo;$\text{LLR algebra}$&laquo;  can also be applied to conditional probabilities. In &nbsp;[[Aufgaben:Exercise_4.1Z:_Log_Likelihood_Ratio_at_the_BEC_Model|"Exercise 4.1Z"]],&nbsp; the BEC&ndash;model is described in a similar way.}}<br>
 {{GraueBox|TEXT=
@@ Line 181: / Line 181: @@
 [[File:EN_KC_T_4_1_S4.png|right|frame|Model of bit-wise soft-in soft-out decoding|class=fit]]
-*For long codes,&nbsp; a&nbsp; "maximum-a-posteriori block-level decision"&nbsp; $($for short: &nbsp;<b>[[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB|$\text{ "block-wise MAP"}$]]</b>&nbsp; &ndash; very elaborate.
+*For long codes,&nbsp; a&nbsp; "maximum-a-posteriori block-level decision"&nbsp; $($for short: &nbsp;<b>[[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB|$\text{ block-wise MAP}$]]</b>&nbsp; &ndash; very elaborate.
 *One would have to find among the&nbsp; $2^k&nbsp;$ allowable code words&nbsp; $\underline{x}_j &#8712; \mathcal{C}$&nbsp; the one with the greatest a-posteriori probability&nbsp; $\rm (APP)$.
@@ Line 290: / Line 290: @@
   \hspace{0.05cm}.</math>
-*As will be shown in the&nbsp; [[Aufgaben:Aufgabe_4.4:_Extrinsische_L–Werte_beim_SPC|$\text{Exercise 4.4}$]],&nbsp; it is also possible to write for this:
+*As will be shown in the&nbsp; [[Aufgaben:Aufgabe_4.4:_Extrinsische_L–Werte_beim_SPC|"Exercise 4.4"]],&nbsp; it is also possible to write for this:
 ::<math>L_{\rm E}(i) = 2 \cdot {\rm tanh}^{-1} \hspace{0.1cm} \left [  \prod\limits_{j \ne i}^{n} \hspace{0.15cm}{\rm tanh}(L_j/2) \right ]
@@ Line 335: / Line 335: @@
 <br>
 An example of iterative decoding of convolutional codes is the&nbsp; &raquo;<b>BCJR algorithm</b>&laquo;,&nbsp; named after its inventors L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv &nbsp; &#8658; &nbsp; [BCJR74]<ref name='BCJR74'>Bahl, L.R.; Cocke, J.; Jelinek, F.; Raviv, J.:&nbsp; Optimal Decoding of Linear Codes for Minimizing Symbol Error Rate.&nbsp; In: IEEE Transactions on Information Theory, Vol. IT-20, pp. 284-287, 1974.</ref>.&nbsp; The algorithm has many parallels to the seven-year older Viterbi decoding,&nbsp; but also some significant differences:
-*While Viterbi estimates the total sequence &nbsp; &#8658; &nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Definitions_of_the_different_optimal_receivers| "block&ndash;wise maximum likelihood"]],&nbsp; the BCJR&ndash; Algorithm minimizes the bit error probability &nbsp; &#8658; &nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Definitions_of_the_different_optimal_receivers|"bit&ndash;wise MAP"]].<br>
+*While Viterbi estimates the total sequence &nbsp; &#8658; &nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Definitions_of_the_different_optimal_receivers|$\text{block&ndash;wise maximum likelihood}$]],&nbsp; the BCJR&ndash; Algorithm minimizes the bit error probability &nbsp; &#8658; &nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Definitions_of_the_different_optimal_receivers|$\text{bit&ndash;wise MAP}$]].<br>
 *The Viterbi algorithm&nbsp; $($in its original form$)$&nbsp; cannot process soft information.&nbsp; In contrast,&nbsp; the BCJR algorithm specifies a reliability value for each individual bit at each iteration,&nbsp; which is taken into account in subsequent iterations.<br>
@@ Line 343: / Line 343: @@
 [[File:EN_KC_T_4_1_S5.png|right|frame|Comparison of Viterbi and BCJR algorithm|class=fit]]
-*Viterbi searches and finds the most likely path from&nbsp; ${\it \Gamma}_0(S_0)$&nbsp; to&nbsp; ${\it \Gamma}_5(S_0)$, namely&nbsp; $S_0 &#8594; S_1 &#8594; S_0 &#8594; S_0 &#8594; S_1&#8594; S_0 $. We refer to the sample solution to&nbsp; [[Aufgaben:Exercise_3.09Z:_Viterbi_Algorithm_again|$\text{Exercise 3.9Z}$]].<br>
+*Viterbi searches and finds the most likely path from&nbsp; ${\it \Gamma}_0(S_0)$&nbsp; to&nbsp; ${\it \Gamma}_5(S_0)$, namely&nbsp; $S_0 &#8594; S_1 &#8594; S_0 &#8594; S_0 &#8594; S_1&#8594; S_0 $. We refer to the sample solution to&nbsp; [[Aufgaben:Exercise_3.09Z:_Viterbi_Algorithm_again|"Exercise 3.9Z"]].<br>
 *The sketch for the BCJR algorithm illustrates the extraction of the extrinsic&nbsp; L&ndash;value for the third symbol &nbsp; &#8658; &nbsp; $L_{\rm E}(3)$.&nbsp; The relevant area in the trellis is shaded:

@@ Line 64: / Line 64: @@
 **$x_3 = -1$ &nbsp; &#8658; &nbsp; $y_1 = +0.1,$
-*Decoding is done according to the criterion&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB|&raquo;block-wise maximum-likelihood&laquo;]]&nbsp; $\text{(ML)}$,&nbsp; distinguishing between&nbsp;
+*Decoding is done according to the criterion&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB|$\text{&raquo;block-wise maximum-likelihood&laquo;}$]]&nbsp; $\text{(ML)}$,&nbsp; distinguishing between&nbsp;
 **"hard decision" &nbsp; $\rm {(ML&ndash;HD)}$,&nbsp; and&nbsp;
 **"soft decision" &nbsp; $\rm {(ML&ndash;SD)}$&nbsp;.<br>

@@ Line 298: / Line 298: @@
 {{GraueBox|TEXT=
-$\text{Example 6:}$&nbsp; We assume a&nbsp; "Single Parity&ndash;check code"&nbsp; with&nbsp; $n = 3, \ k = 2$ &nbsp; &#8658; &nbsp; briefly&nbsp; ${\rm SPC} \ (3, \ 2, \ 2)$.
+$\text{Example 6:}$&nbsp; We assume a&nbsp; "Single parity&ndash;check code"&nbsp; with&nbsp; $n = 3, \ k = 2$ &nbsp; &#8658; &nbsp; briefly&nbsp; ${\rm SPC} \ (3, \ 2, \ 2)$.
 *The &nbsp; $2^k = 4$ &nbsp; valid code words&nbsp; $\underline{x} = \{x_1,\ x_2,\ x_3\}$&nbsp; are for bipolar description &nbsp; &#8658; &nbsp; $x_i &#8712; \{&plusmn;1\}$: