Channel Coding/The Basics of Product Codes - Revision history

Guenter at 16:26, 16 January 2023

2023-01-16T16:26:38Z

Hwang at 21:34, 1 January 2023

2023-01-01T21:34:05Z

Hwang at 21:33, 1 January 2023

2023-01-01T21:33:13Z

Hwang at 21:25, 1 January 2023

2023-01-01T21:25:51Z

Hwang at 21:12, 1 January 2023

2023-01-01T21:12:56Z

Guenter at 13:51, 20 December 2022

2022-12-20T13:51:26Z

Guenter at 13:49, 20 December 2022

2022-12-20T13:49:38Z

Guenter at 16:50, 19 December 2022

2022-12-19T16:50:51Z

Guenter at 16:32, 19 December 2022

2022-12-19T16:32:08Z

Guenter at 10:24, 19 December 2022

2022-12-19T10:24:37Z

@@ Line 162: / Line 162: @@
-[[File:EN_KC_T_4_2_S3_v1.png|right|frame|Bit and block error probability of a&nbsp; $(1024, 676)$&nbsp; product code at AWGN '''KORREKTUR: exakt'''|class=fit]]
+[[File:EN_KC_T_4_2_S3_v3.png|right|frame|Bit and block error probability of a&nbsp; $(1024, 676)$&nbsp; product code at AWGN |class=fit]]
 The graph shows for a&nbsp; $(1024, 676)$&nbsp; product code, based on the&nbsp; extended Hamming code&nbsp; ${\rm eHC} \ (32, 26)$&nbsp; as component codes,
 *on the left,&nbsp; the bit error probability as  function of the AWGN parameter&nbsp; $10 \cdot {\rm lg} \, (E_{\rm B}/N_0)$&nbsp; the number of iterations&nbsp; $(I)$,

@@ Line 191: / Line 191: @@
 == Exercises for the chapter==
 <br>
-[[Aufgaben:Exercise_4.6:_Product_Code_Generation|"Exercise 4.6: Product Code Generation"]]
+[[Aufgaben:Exercise_4.6:_Product_Code_Generation|Exercise 4.6: Product Code Generation]]
-[[Aufgaben:Exercise_4.6Z:_Basics_of_Product_Codes|"Exercise 4.6Z: Basics of Product Codes"]]
+[[Aufgaben:Exercise_4.6Z:_Basics_of_Product_Codes|Exercise 4.6Z: Basics of Product Codes]]
-[[Aufgaben:Exercise_4.7:_Product_Code_Decoding|"Exercise 4.7: Product Code Decoding"]]
+[[Aufgaben:Exercise_4.7:_Product_Code_Decoding|Exercise 4.7: Product Code Decoding]]
-[[Aufgaben:Exercise_4.7Z:_Principle_of_Syndrome_Decoding|"Exercise 4.7Z: Principle of Syndrome Decoding"]]
+[[Aufgaben:Exercise_4.7Z:_Principle_of_Syndrome_Decoding|Exercise 4.7Z: Principle of Syndrome Decoding]]
 ==References==

@@ Line 162: / Line 162: @@
-[[File:EN_KC_T_4_2_S3_v1.png|right|frame|Bit and block error probability of a&nbsp; $(1024, 676)$&nbsp; product code at AWGN|class=fit]]
+[[File:EN_KC_T_4_2_S3_v1.png|right|frame|Bit and block error probability of a&nbsp; $(1024, 676)$&nbsp; product code at AWGN '''KORREKTUR: exakt'''|class=fit]]
 The graph shows for a&nbsp; $(1024, 676)$&nbsp; product code, based on the&nbsp; extended Hamming code&nbsp; ${\rm eHC} \ (32, 26)$&nbsp; as component codes,
 *on the left,&nbsp; the bit error probability as  function of the AWGN parameter&nbsp; $10 \cdot {\rm lg} \, (E_{\rm B}/N_0)$&nbsp; the number of iterations&nbsp; $(I)$,

@@ Line 30: / Line 30: @@
 {{BlaueBox|TEXT=
-$\text{Conclusions:}$&nbsp;  '''All product codes'''&nbsp; according to the above graph have the following properties:
+$\text{Conclusions:}$&nbsp;  &raquo;'''All product codes'''&laquo;&nbsp; according to the above graph have the following properties:
 *For linear component codes&nbsp; $\mathcal{C}_1$&nbsp; and&nbsp; $\mathcal{C}_2$&nbsp; the product code&nbsp; $\mathcal{C} = \mathcal{C}_1 &times; \mathcal{C}_2$&nbsp; is also linear.<br>
@@ Line 83: / Line 83: @@
-&rArr; &nbsp; The&nbsp; <b>row-by-row syndrome decoding</b>&nbsp; is done via the syndrome&nbsp; $\underline{s} = \underline{y} \cdot \mathbf{H}_1^{\rm T}$&nbsp; with
+&rArr; &nbsp; The&nbsp; &raquo;<b>row-by-row syndrome decoding</b>&laquo;&nbsp; is done via the syndrome&nbsp; $\underline{s} = \underline{y} \cdot \mathbf{H}_1^{\rm T}$&nbsp; with
 :$$\boldsymbol{\rm H}_1^{\rm T} =
    \begin{pmatrix}
@@ Line 126: / Line 126: @@
 \hspace{0.05cm}.</math>
-&rArr; &nbsp; The&nbsp; <b>column-by-column syndrome decoding</b>&nbsp; removes all single errors in columns&nbsp; 1,&nbsp; 2,&nbsp; 3,&nbsp; 4&nbsp; and&nbsp; 7.
+&rArr; &nbsp; The&nbsp; &raquo;<b>column-by-column syndrome decoding</b>&laquo;&nbsp; removes all single errors in columns&nbsp; 1,&nbsp; 2,&nbsp; 3,&nbsp; 4&nbsp; and&nbsp; 7.
 *<b>Column 5</b> &nbsp;$($contains two errors$)$ &nbsp; &#8658; &nbsp; Error correction concerning bit&nbsp; '''4''':
@@ Line 144: / Line 144: @@
 \hspace{0.05cm}.</math>
-&rArr; &nbsp; The remaining three errors are corrected by decoding the&nbsp; <b>second row iteration loop</b>&nbsp; $($line-by-line$)$.<br>
+&rArr; &nbsp; The remaining three errors are corrected by decoding the&nbsp; &raquo;<b>second row iteration loop</b>&laquo;&nbsp; $($line-by-line$)$.<br>
 Whether all errors of a block are correctable depends on the error pattern.&nbsp; Here we refer to&nbsp; [[Aufgaben:Aufgabe_4.7:_Decodierung_von_Produktcodes|"Exercise 4.7"]].}}<br>

@@ Line 9: / Line 9: @@
 <br>
 The graphic shows the principle structure of&nbsp; &raquo;product codes&laquo;,&nbsp; which were already introduced in 1954 by&nbsp; [https://en.wikipedia.org/wiki/Peter_Elias $\text{Peter Elias}$].&nbsp;
-*The&nbsp; '''two-dimensional product code'''&nbsp; $\mathcal{C} = \mathcal{C}_1 &times; \mathcal{C}_2$&nbsp; shown here is based on the two linear and binary block codes with parameters&nbsp; $(n_1, \ k_1)$&nbsp; and&nbsp; $(n_2, \ k_2)$ respectively.
+*The&nbsp; &raquo;'''two-dimensional product code'''&laquo;&nbsp; $\mathcal{C} = \mathcal{C}_1 &times; \mathcal{C}_2$&nbsp; shown here is based on the two linear and binary block codes with parameters&nbsp; $(n_1, \ k_1)$&nbsp; and&nbsp; $(n_2, \ k_2)$ respectively.
 *The code word length is&nbsp; $n = n_1 \cdot n_2$.
@@ Line 23: / Line 23: @@
 *The upper right matrix&nbsp; $\mathbf{P}^{(1)}$&nbsp; with dimension&nbsp; $k_2 &times; m_1$&nbsp; contains the parity bits with respect to the code&nbsp; $\mathcal{C}_1$.
-*In each of the&nbsp; $k_2$&nbsp; rows,&nbsp; $m_1 = n_1 - k_1$&nbsp; check bits are added to the&nbsp; $k_1$&nbsp;  information bits as described in an earlier chapter using the example of&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes|"Hamming codes"]]&nbsp;.<br>
+*In each of the&nbsp; $k_2$&nbsp; rows,&nbsp; $m_1 = n_1 - k_1$&nbsp; check bits are added to the&nbsp; $k_1$&nbsp;  information bits as described in an earlier chapter using the example of&nbsp; [[Channel_Coding/Examples_of_Binary_Block_Codes#Hamming_Codes|$\text{Hamming codes}$]]&nbsp;.<br>
 *The lower left matrix&nbsp; $\mathbf{P}^{(2)}$&nbsp; of dimension&nbsp; $m_2 &times; k_1$&nbsp; contains the check bits for the second component code&nbsp; $\mathcal{C}_2$.&nbsp; Here the encoding&nbsp; $($and also the decoding$)$&nbsp; is done line by line: &nbsp; In each of the&nbsp; $k_1$&nbsp; columns,&nbsp;  the&nbsp; $k_2$&nbsp; information bits are still supplemented by&nbsp; $m_2 = n_2 -k_2$&nbsp; check bits.<br>
@@ Line 146: / Line 146: @@
 &rArr; &nbsp; The remaining three errors are corrected by decoding the&nbsp; <b>second row iteration loop</b>&nbsp; $($line-by-line$)$.<br>
-Whether all errors of a block are correctable depends on the error pattern.&nbsp; Here we refer to&nbsp; [[Aufgaben:Aufgabe_4.7:_Decodierung_von_Produktcodes|$\text{Exercise 4.7}$]].}}<br>
+Whether all errors of a block are correctable depends on the error pattern.&nbsp; Here we refer to&nbsp; [[Aufgaben:Aufgabe_4.7:_Decodierung_von_Produktcodes|"Exercise 4.7"]].}}<br>
 == Performance of product codes ==
@@ Line 157: / Line 157: @@
 Usually one uses very long product codes &nbsp; $($large&nbsp; $n = n_1 \cdot n_2)$ &nbsp; with the following consequence:
-*For effort reasons,&nbsp; the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB|"maximum likelihood decoding at block level"]]&nbsp; is not applicable for the component codes&nbsp; $\mathcal{C}_1$&nbsp; and&nbsp; $\mathcal{C}_2$&nbsp; nor the&nbsp; [[Channel_Coding/Decoding_of_Linear_Block_Codes#Principle_of_syndrome_decoding| "syndrome decoding"]],&nbsp; which is after all a realization form of maximum likelihood decoding.
+*For effort reasons,&nbsp; the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB|$\text{maximum likelihood decoding at block level}$]]&nbsp; is not applicable for the component codes&nbsp; $\mathcal{C}_1$&nbsp; and&nbsp; $\mathcal{C}_2$&nbsp; nor the&nbsp; [[Channel_Coding/Decoding_of_Linear_Block_Codes#Principle_of_syndrome_decoding|$\text{syndrome decoding}$]],&nbsp; which is after all a realization form of maximum likelihood decoding.
-*Applicable,&nbsp; on the other hand,&nbsp; even with large&nbsp; $n$&nbsp; is the&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Bit-wise_soft-in_soft-out_decoding|"iterative symbol-wise MAP decoding"]].&nbsp; The exchange of extrinsic and a-priori&ndash;information happens here between the two component codes.&nbsp; More details on this can be found in&nbsp; [Liv15]<ref name='Liv15'>Liva, G.:&nbsp; Channels Codes for Iterative Decoding.&nbsp; Lecture manuscript, Department of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2015.</ref>.<br>
+*Applicable,&nbsp; on the other hand,&nbsp; even with large&nbsp; $n$&nbsp; is the&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Bit-wise_soft-in_soft-out_decoding|$\text{iterative symbol-wise MAP decoding}$]].&nbsp; The exchange of extrinsic and a-priori&ndash;information happens here between the two component codes.&nbsp; More details on this can be found in&nbsp; [Liv15]<ref name='Liv15'>Liva, G.:&nbsp; Channels Codes for Iterative Decoding.&nbsp; Lecture manuscript, Department of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2015.</ref>.<br>

← Older revision		Revision as of 13:51, 20 December 2022
Line 80:		Line 80:
	*the code matrix  $\mathbf{X}$  was chosen to be a  $6 × 7$ zero matrix,		*the code matrix  $\mathbf{X}$  was chosen to be a  $6 × 7$ zero matrix,

−	*so ~~that~~ the eight  "ones"  in  $\mathbf{Y}$  represent transmission errors<br>	+	*so the eight  "ones"  in  $\mathbf{Y}$  represent transmission errors.<br>

← Older revision		Revision as of 13:49, 20 December 2022
Line 80:		Line 80:
	*the code matrix  $\mathbf{X}$  was chosen to be a  $6 × 7$ zero matrix,		*the code matrix  $\mathbf{X}$  was chosen to be a  $6 × 7$ zero matrix,

−	*so that the ~~nine~~  "ones"  in  $\mathbf{Y}$  represent transmission errors<br>	+	*so that the eight  "ones"  in  $\mathbf{Y}$  represent transmission errors<br>

@@ Line 72: / Line 72: @@
 [[File:EN_KC_T_4_2_S2a_v1.png|right|frame|Syndrome decoding of the&nbsp; $(42, 12)$&nbsp; product code|class=fit]]
-[[File:EN_KC_T_4_2_S2b_v1.png|right|frame|Syndrome table for code $\mathcal{C}_1$]]
+[[File:EN KC T 4 2 S2b v2 neu.png|right|frame|Syndrome table for code $\mathcal{C}_1$]]
 [[File:EN_KC_T_4_2_S2c_v2.png|right|frame|Syndrome table for the code $\mathcal{C}_2$]]