https://en.lntwww.de/index.php?title=Channel_Coding/Soft-in_Soft-Out_Decoder&feed=atom&action=history
Channel Coding/Soft-in Soft-Out Decoder - Revision history
2024-03-28T21:27:28Z
Revision history for this page on the wiki
MediaWiki 1.34.1
https://en.lntwww.de/index.php?title=Channel_Coding/Soft-in_Soft-Out_Decoder&diff=51767&oldid=prev
Guenter at 16:21, 12 January 2023
2023-01-12T16:21:30Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 16:21, 12 January 2023</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT= </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT= </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>$\text{Definition:}$&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; <del class="diffchange diffchange-inline">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: </del>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:<del class="diffchange diffchange-inline">'''</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$\text{Definition:}$&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:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<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 ]}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<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 ]}</div></td></tr>
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Guenter
https://en.lntwww.de/index.php?title=Channel_Coding/Soft-in_Soft-Out_Decoder&diff=51457&oldid=prev
Hwang at 21:08, 1 January 2023
2023-01-01T21:08:54Z
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|Untermenü=Iterative Decoding Methods </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|Untermenü=Iterative Decoding Methods </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>|Vorherige Seite=Distance Characteristics and Error Probability <del class="diffchange diffchange-inline">Barriers</del></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>|Vorherige Seite=Distance Characteristics and Error Probability <ins class="diffchange diffchange-inline">Bounds</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|Nächste Seite=The Basics of Product Codes</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|Nächste Seite=The Basics of Product Codes</div></td></tr>
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Hwang
https://en.lntwww.de/index.php?title=Channel_Coding/Soft-in_Soft-Out_Decoder&diff=51456&oldid=prev
Hwang at 20:58, 1 January 2023
2023-01-01T20:58:26Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:58, 1 January 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l369" >Line 369:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\underline{L} = (L(1), \ \text{...}\hspace{0.05cm} , \ L(n)).$$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>:$$\underline{L} = (L(1), \ \text{...}\hspace{0.05cm} , \ L(n)).$$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:EN_KC_T_4_1_S6.png|right|frame|Parallel concatenated codes|class=fit]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:EN_KC_T_4_1_S6.png|right|frame|Parallel concatenated codes|class=fit]]</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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#<del class="diffchange diffchange-inline">Second_Requirement_f.C3.BCr_Turbocodes</del>:_Interleaving| $\rm interleaver$]],&nbsp; which is mandatory e.g. for turbo codes.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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#<ins class="diffchange diffchange-inline">Second_requirement_for_turbo_codes</ins>:_Interleaving| $\rm interleaver$]],&nbsp; which is mandatory e.g. for turbo codes.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># 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. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># 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. </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># 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></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div># 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></div></td></tr>
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Hwang
https://en.lntwww.de/index.php?title=Channel_Coding/Soft-in_Soft-Out_Decoder&diff=51455&oldid=prev
Hwang at 20:56, 1 January 2023
2023-01-01T20:56:42Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:56, 1 January 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l360" >Line 360:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*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 $0.421$.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*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 $0.421$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*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#<del class="diffchange diffchange-inline">Basic_structure_of_a_Turbo_Code</del>|$\text{turbo code}$]]&nbsp; $($by this one understands per se the concatenation of two convolutional encoders$)$.&nbsp; </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*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#<ins class="diffchange diffchange-inline">Basic_structure_of_a_turbo_code</ins>|$\text{turbo code}$]]&nbsp; $($by this one understands per se the concatenation of two convolutional encoders$)$.&nbsp; </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*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></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*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></div></td></tr>
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Hwang
https://en.lntwww.de/index.php?title=Channel_Coding/Soft-in_Soft-Out_Decoder&diff=51454&oldid=prev
Hwang at 20:55, 1 January 2023
2023-01-01T20:55:58Z
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To begin with,&nbsp; here are a few examples from mobile communications:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To begin with,&nbsp; here are a few examples from mobile communications:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*In&nbsp; [[Examples_of_Communication_Systems/General_Description_of_GSM#.<del class="diffchange diffchange-inline">23_.C3.9CBERBLICK_To_the_PRINCIPAL_CHAPTER_</del>.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 $0.421$.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*In&nbsp; [[Examples_of_Communication_Systems/General_Description_of_GSM#.<ins class="diffchange diffchange-inline">23_OVERVIEW_OF_THE_THIRD_MAIN_CHAPTER_</ins>.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 $0.421$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*In the 3G cellular system&nbsp; [[Examples_of_Communication_Systems/General_Description_of_UMTS#.<del class="diffchange diffchange-inline">23_.C3.9CBERBLICK_To_the_Fourth_Main_Chapter_</del>.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; [[<del class="diffchange diffchange-inline">Channel_Coding</del>/<del class="diffchange diffchange-inline">Basics_of_Convolutional_Coding</del>#.<del class="diffchange diffchange-inline">23_.C3.9CBERBLICK_To_the_PRINCIPAL_CHAPTER_</del>.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; </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*In the 3G cellular system&nbsp; [[Examples_of_Communication_Systems/General_Description_of_UMTS#.<ins class="diffchange diffchange-inline">23_OVERVIEW_OF_THE_FOURTH_MAIN_CHAPTER_</ins>.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; [[<ins class="diffchange diffchange-inline">Examples_of_Communication_Systems</ins>/<ins class="diffchange diffchange-inline">General_Description_of_GSM</ins>#.<ins class="diffchange diffchange-inline">23_OVERVIEW_OF_THE_THIRD_MAIN_CHAPTER_</ins>.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; </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*In 4G mobile communication systems&nbsp; [[<del class="diffchange diffchange-inline">Mobile_Communications</del>/<del class="diffchange diffchange-inline">General_Information_on_the_LTE_Mobile_Communications_Standard</del>#.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></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*In 4G mobile communication systems&nbsp; [[<ins class="diffchange diffchange-inline">Examples_of_Communication_Systems</ins>/<ins class="diffchange diffchange-inline">General_Description_of_UMTS</ins>#.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></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Channel_Coding/Soft-in_Soft-Out_Decoder&diff=51452&oldid=prev
Hwang at 20:45, 1 January 2023
2023-01-01T20:45:33Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:45, 1 January 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l132" >Line 132:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> L(y = -1\hspace{0.05cm}\vert\hspace{0.05cm}x) = -2.197\hspace{0.05cm}.</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> L(y = -1\hspace{0.05cm}\vert\hspace{0.05cm}x) = -2.197\hspace{0.05cm}.</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*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|<del class="diffchange diffchange-inline">$\text{</del>Exercise 4.1Z<del class="diffchange diffchange-inline">}$</del>]],&nbsp; the BEC&ndash;model is described in a similar way.}}<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*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|<ins class="diffchange diffchange-inline">"</ins>Exercise 4.1Z<ins class="diffchange diffchange-inline">"</ins>]],&nbsp; the BEC&ndash;model is described in a similar way.}}<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{GraueBox|TEXT= </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{GraueBox|TEXT= </div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l181" >Line 181:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:EN_KC_T_4_1_S4.png|right|frame|Model of bit-wise soft-in soft-out decoding|class=fit]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:EN_KC_T_4_1_S4.png|right|frame|Model of bit-wise soft-in soft-out decoding|class=fit]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*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|$\text{ <del class="diffchange diffchange-inline">"</del>block-wise MAP<del class="diffchange diffchange-inline">"</del>}$]]</b>&nbsp; &ndash; very elaborate.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*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.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*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; $\rm (APP)$.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*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; $\rm (APP)$.</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l290" >Line 290:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> \hspace{0.05cm}.</math></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> \hspace{0.05cm}.</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*As will be shown in the&nbsp; [[Aufgaben:Aufgabe_4.4:_Extrinsische_L–Werte_beim_SPC|<del class="diffchange diffchange-inline">$\text{</del>Exercise 4.4<del class="diffchange diffchange-inline">}$</del>]],&nbsp; it is also possible to write for this:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*As will be shown in the&nbsp; [[Aufgaben:Aufgabe_4.4:_Extrinsische_L–Werte_beim_SPC|<ins class="diffchange diffchange-inline">"</ins>Exercise 4.4<ins class="diffchange diffchange-inline">"</ins>]],&nbsp; it is also possible to write for this:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<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 ]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<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 ]</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l335" >Line 335:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*While Viterbi estimates the total sequence &nbsp; &#8658; &nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Definitions_of_the_different_optimal_receivers| <del class="diffchange diffchange-inline">"</del>block&ndash;wise maximum likelihood<del class="diffchange diffchange-inline">"</del>]],&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|<del class="diffchange diffchange-inline">"</del>bit&ndash;wise MAP<del class="diffchange diffchange-inline">"</del>]].<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*While Viterbi estimates the total sequence &nbsp; &#8658; &nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Definitions_of_the_different_optimal_receivers|<ins class="diffchange diffchange-inline">$\text{</ins>block&ndash;wise maximum likelihood<ins class="diffchange diffchange-inline">}$</ins>]],&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|<ins class="diffchange diffchange-inline">$\text{</ins>bit&ndash;wise MAP<ins class="diffchange diffchange-inline">}$</ins>]].<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*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></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*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></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l343" >Line 343:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:EN_KC_T_4_1_S5.png|right|frame|Comparison of Viterbi and BCJR algorithm|class=fit]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:EN_KC_T_4_1_S5.png|right|frame|Comparison of Viterbi and BCJR algorithm|class=fit]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*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 $. We refer to the sample solution to&nbsp; [[Aufgaben:Exercise_3.09Z:_Viterbi_Algorithm_again|<del class="diffchange diffchange-inline">$\text{</del>Exercise 3.9Z<del class="diffchange diffchange-inline">}$</del>]].<br></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*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|<ins class="diffchange diffchange-inline">"</ins>Exercise 3.9Z<ins class="diffchange diffchange-inline">"</ins>]].<br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*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:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*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:</div></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Channel_Coding/Soft-in_Soft-Out_Decoder&diff=51451&oldid=prev
Hwang at 20:42, 1 January 2023
2023-01-01T20:42:19Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:42, 1 January 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l64" >Line 64:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**$x_3 = -1$ &nbsp; &#8658; &nbsp; $y_1 = +0.1,$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**$x_3 = -1$ &nbsp; &#8658; &nbsp; $y_1 = +0.1,$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*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; </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*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|<ins class="diffchange diffchange-inline">$\text{</ins>&raquo;block-wise maximum-likelihood&laquo;<ins class="diffchange diffchange-inline">}$</ins>]]&nbsp; $\text{(ML)}$,&nbsp; distinguishing between&nbsp; </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**"hard decision" &nbsp; $\rm {(ML&ndash;HD)}$,&nbsp; and&nbsp; </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**"hard decision" &nbsp; $\rm {(ML&ndash;HD)}$,&nbsp; and&nbsp; </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**"soft decision" &nbsp; $\rm {(ML&ndash;SD)}$&nbsp;.<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**"soft decision" &nbsp; $\rm {(ML&ndash;SD)}$&nbsp;.<br></div></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Channel_Coding/Soft-in_Soft-Out_Decoder&diff=51450&oldid=prev
Hwang at 20:41, 1 January 2023
2023-01-01T20:41:55Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-marker" />
<col class="diff-content" />
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:41, 1 January 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l64" >Line 64:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**$x_3 = -1$ &nbsp; &#8658; &nbsp; $y_1 = +0.1,$</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**$x_3 = -1$ &nbsp; &#8658; &nbsp; $y_1 = +0.1,$</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*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 <del class="diffchange diffchange-inline"> </del>maximum-likelihood&laquo;]]&nbsp; $\text{(ML)}$,&nbsp; distinguishing between&nbsp; </div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*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; </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**"hard decision" &nbsp; $\rm {(ML&ndash;HD)}$,&nbsp; and&nbsp; </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**"hard decision" &nbsp; $\rm {(ML&ndash;HD)}$,&nbsp; and&nbsp; </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**"soft decision" &nbsp; $\rm {(ML&ndash;SD)}$&nbsp;.<br></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>**"soft decision" &nbsp; $\rm {(ML&ndash;SD)}$&nbsp;.<br></div></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Channel_Coding/Soft-in_Soft-Out_Decoder&diff=51449&oldid=prev
Hwang at 20:38, 1 January 2023
2023-01-01T20:38:48Z
<p></p>
<table class="diff diff-contentalign-left" data-mw="interface">
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<col class="diff-marker" />
<col class="diff-content" />
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:38, 1 January 2023</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{GraueBox|TEXT= </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{GraueBox|TEXT= </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>$\text{Example 6:}$&nbsp; We assume a&nbsp; "Single <del class="diffchange diffchange-inline">Parity</del>&ndash;check code"&nbsp; with&nbsp; $n = 3, \ k = 2$ &nbsp; &#8658; &nbsp; briefly&nbsp; ${\rm SPC} \ (3, \ 2, \ 2)$.</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$\text{Example 6:}$&nbsp; We assume a&nbsp; "Single <ins class="diffchange diffchange-inline">parity</ins>&ndash;check code"&nbsp; with&nbsp; $n = 3, \ k = 2$ &nbsp; &#8658; &nbsp; briefly&nbsp; ${\rm SPC} \ (3, \ 2, \ 2)$.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*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; $x_i &#8712; \{&plusmn;1\}$:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*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; $x_i &#8712; \{&plusmn;1\}$:</div></td></tr>
</table>
Hwang
https://en.lntwww.de/index.php?title=Channel_Coding/Soft-in_Soft-Out_Decoder&diff=51448&oldid=prev
Hwang at 20:37, 1 January 2023
2023-01-01T20:37:44Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:37, 1 January 2023</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT= </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{BlaueBox|TEXT= </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>$\text{Definition:}$&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:</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>$\text{Definition:}$&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: <ins class="diffchange diffchange-inline">'''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:'''</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<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 ]}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>::<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 ]}</div></td></tr>
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Hwang