Aufgaben:Exercise 4.11: Analysis of Parity-check Matrices - Revision history

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	[[Category:Channel Coding: Exercises\|^4.4 Low–density Parity–check Codes^]]		[[Category:Channel Coding: Exercises\|^4.4 Low–density Parity–check Codes^]]
	[[de:Aufgaben:4.~~11_Analyse_von_Prüfmatrizen~~]]		[[de:Aufgaben:Aufgabe 4.11: Analyse von Prüfmatrizen]]

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	* From the parity-check matrix  $\mathbf{H}$  a lower bound for the code rate  $R$  can be given:		* From the parity-check matrix  $\mathbf{H}$  a lower bound for the code rate  $R$  can be given:
	:$$R \ge 1 - \frac{{\rm E}[w_{\rm C}]}{{\rm E}[w_{\rm R}]}		:$$R \ge 1 - \frac{{\rm E}[w_{\rm C}]}{{\rm E}[w_{\rm R}]}\hspace{0.5cm}{\rm with}\hspace{0.5cm}{\rm E}[w_{\rm C}] =\frac{1}{n} \cdot \sum_{i = 1}^{n}w_{\rm C}(i)\hspace{0.5cm}{\rm and}\hspace{0.5cm}{\rm E}[w_{\rm R}] =\frac{1}{m} \cdot \sum_{j = 1}^{ m}w_{\rm R}(j)\hspace{0.05cm}.$$
	\hspace{0.5cm}{\rm with}\hspace{0.5cm}
	{\rm E}[w_{\rm C}] =\frac{1}{n} \cdot \sum_{i = 1}^{n}w_{\rm C}(i)
	\hspace{0.5cm}{\rm and}\hspace{0.5cm}
	{\rm E}[w_{\rm R}] =\frac{1}{m} \cdot \sum_{j = 1}^{ m}w_{\rm R}(j)
	\hspace{0.05cm}.$$

	*This equation applies equally to regular and irregular LDPC codes,  where for regular codes holds  ${\rm E}[w_{\rm C}] = w_{\rm C}$  and  ${\rm E}[w_{\rm R}] = w_{\rm R}$.		*This equation applies equally to regular and irregular LDPC codes,  where for regular codes holds  ${\rm E}[w_{\rm C}] = w_{\rm C}$  and  ${\rm E}[w_{\rm R}] = w_{\rm R}$.
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	'''(1)'''  Correct are the  <u>solutions 1 and 3</u>:		'''(1)'''  Correct are the  <u>solutions 1 and 3</u>:
	*With the code word definition   $\underline{x} = (u_1, \, u_2, \, u_3, \, u_4, \, p_1, \, p_2, \, p_3, \, p_4)$,  the parity-check matrix  $\mathbf{H}_1$  denotes the following check equations:		*With the code word definition   $\underline{x} = (u_1, \, u_2, \, u_3, \, u_4, \, p_1, \, p_2, \, p_3, \, p_4)$,  the parity-check matrix  $\mathbf{H}_1$  denotes the following check equations:
	:$$u_1 \oplus u_2 \oplus p_1 = 0\hspace{0.05cm},\hspace{0.3cm}		:$$u_1 \oplus u_2 \oplus p_1 = 0\hspace{0.05cm},\hspace{0.3cm}u_3 \oplus u_4 \oplus p_2 = 0\hspace{0.05cm},\hspace{0.3cm}u_1 \oplus u_3 \oplus p_3 = 0\hspace{0.05cm},\hspace{0.3cm}u_2 \oplus u_4 \oplus p_4 = 0\hspace{0.05cm}.$$
	u_3 \oplus u_4 \oplus p_2 = 0\hspace{0.05cm},\hspace{0.3cm}
	u_1 \oplus u_3 \oplus p_3 = 0\hspace{0.05cm},\hspace{0.3cm}
	u_2 \oplus u_4 \oplus p_4 = 0\hspace{0.05cm}.$$
	*This corresponds exactly to the assumptions made above.		*This corresponds exactly to the assumptions made above.

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	With the same code word definition $\underline{x} = (u_1, \, p_1, \, u_2, \, p_2, \, u_3, \, p_3, \, u_4, \, p_4)$ the other parity-check matrices do not yield a meaningful set of equations:		With the same code word definition $\underline{x} = (u_1, \, p_1, \, u_2, \, p_2, \, u_3, \, p_3, \, u_4, \, p_4)$ the other parity-check matrices do not yield a meaningful set of equations:
	* According to parity-check matrix $\mathbf{H}_1$:		* According to parity-check matrix $\mathbf{H}_1$:
	:$$u_1 \oplus p_1 \oplus u_3 = 0\hspace{0.05cm},\hspace{0.3cm}		:$$u_1 \oplus p_1 \oplus u_3 = 0\hspace{0.05cm},\hspace{0.3cm}u_2 \oplus p_2 \oplus p_3 = 0\hspace{0.05cm},\hspace{0.3cm}u_1 \oplus u_2 \oplus u_4 = 0\hspace{0.05cm},\hspace{0.3cm}p_1 \oplus p_2 \oplus p_4 = 0\hspace{0.05cm};$$
	u_2 \oplus p_2 \oplus p_3 = 0\hspace{0.05cm},\hspace{0.3cm}
	u_1 \oplus u_2 \oplus u_4 = 0\hspace{0.05cm},\hspace{0.3cm}
	p_1 \oplus p_2 \oplus p_4 = 0\hspace{0.05cm};$$
	* corresponding to parity-check matrix $\mathbf{H}_3$:		* corresponding to parity-check matrix $\mathbf{H}_3$:
	:$$u_1 \hspace{-0.12cm}\oplus\hspace{-0.06cm} p_2 \hspace{-0.12cm}\oplus\hspace{-0.06cm} u_3 \hspace{-0.12cm}\oplus\hspace{-0.06cm} p_4 \hspace{-0.05cm} = \hspace{-0.05cm} 0\hspace{0.05cm},\hspace{0.15cm}		:$$u_1 \hspace{-0.12cm}\oplus\hspace{-0.06cm} p_2 \hspace{-0.12cm}\oplus\hspace{-0.06cm} u_3 \hspace{-0.12cm}\oplus\hspace{-0.06cm} p_4 \hspace{-0.05cm} = \hspace{-0.05cm} 0\hspace{0.05cm},\hspace{0.15cm}u_1 \hspace{-0.12cm}\oplus\hspace{-0.06cm} p_2 \hspace{-0.12cm}\oplus\hspace{-0.06cm} p_3 \hspace{-0.12cm}\oplus\hspace{-0.06cm}u_4 \hspace{-0.05cm} = \hspace{-0.05cm} 0\hspace{0.05cm},\hspace{0.15cm}p_1 \hspace{-0.12cm}\oplus\hspace{-0.06cm} u_2 \hspace{-0.12cm}\oplus\hspace{-0.06cm} u_3 \hspace{-0.12cm}\oplus\hspace{-0.06cm}u_4 \hspace{-0.05cm} = \hspace{-0.05cm} 0\hspace{0.05cm},\hspace{0.15cm}p_1 \hspace{-0.12cm}\oplus\hspace{-0.06cm} u_2 \hspace{-0.12cm}\oplus\hspace{-0.06cm} p_3 \hspace{-0.12cm}\oplus\hspace{-0.06cm} p_4 \hspace{-0.05cm} = \hspace{-0.05cm} 0\hspace{0.05cm}.$$
	u_1 \hspace{-0.12cm}\oplus\hspace{-0.06cm} p_2 \hspace{-0.12cm}\oplus\hspace{-0.06cm} p_3 \hspace{-0.12cm}\oplus\hspace{-0.06cm}u_4 \hspace{-0.05cm} = \hspace{-0.05cm} 0\hspace{0.05cm},\hspace{0.15cm}
	p_1 \hspace{-0.12cm}\oplus\hspace{-0.06cm} u_2 \hspace{-0.12cm}\oplus\hspace{-0.06cm} u_3 \hspace{-0.12cm}\oplus\hspace{-0.06cm}u_4 \hspace{-0.05cm} = \hspace{-0.05cm} 0\hspace{0.05cm},\hspace{0.15cm}
	p_1 \hspace{-0.12cm}\oplus\hspace{-0.06cm} u_2 \hspace{-0.12cm}\oplus\hspace{-0.06cm} p_3 \hspace{-0.12cm}\oplus\hspace{-0.06cm} p_4 \hspace{-0.05cm} = \hspace{-0.05cm} 0\hspace{0.05cm}.$$


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	* For an irregular code,  the lower bound for the code rate is:		* For an irregular code,  the lower bound for the code rate is:
	:$$R \ge 1 - \frac{{\rm E}[w_{\rm C}]}{{\rm E}[w_{\rm R}]}		:$$R \ge 1 - \frac{{\rm E}[w_{\rm C}]}{{\rm E}[w_{\rm R}]}= 1 - \frac{1.5}{3} = 1/2\hspace{0.05cm}.$$
	= 1 - \frac{1.5}{3} = 1/2
	\hspace{0.05cm}.$$

	* Because of the given code structure  $(k = 4$  information bits,  $m = 4$  check bits   ⇒   $n = 8$  encoded bits$)$  the code rate can also be given in the conventional form:  $R = k/n$   ⇒   Correct is solution 4 in contrast to answer 3.		* Because of the given code structure  $(k = 4$  information bits,  $m = 4$  check bits   ⇒   $n = 8$  encoded bits$)$  the code rate can also be given in the conventional form:  $R = k/n$   ⇒   Correct is solution 4 in contrast to answer 3.

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	[[Category:Channel Coding: Exercises\|^4.4 Low–density Parity–check Codes^]]		[[Category:Channel Coding: Exercises\|^4.4 Low–density Parity–check Codes^]]
			[[de:Aufgaben:4.11_Analyse_von_Prüfmatrizen]]

Guenter at 16:20, 19 December 2022

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	{{quiz-Header\|Buchseite=Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes}}		{{quiz-Header\|Buchseite=Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes}}

	[[File:~~P_ID3067__KC_A_4_11_v2~~.png\|right\|frame\|Product code described by the parity-check matrix  $\mathbf{H}$]]		[[File:EN_KC_A_4_11_v2.png\|right\|frame\|Product code described by the parity-check matrix  $\mathbf{H}$]]
	In the adjacent graphic,  a product code is indicated at the top,  which is characterized by the following parity-check equations:		In the adjacent graphic,  a product code is indicated at the top,  which is characterized by the following parity-check equations:
	:$$p_1 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} u_1 \oplus u_2\hspace{0.05cm},$$		:$$p_1 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} u_1 \oplus u_2\hspace{0.05cm},$$

Guenter at 17:03, 16 December 2022

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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  Correct are the <u>solutions 1 and 3</u>:		'''(1)'''  Correct are the  <u>solutions 1 and 3</u>:
	*With the code word definition $\underline{x} = (u_1, \, u_2, \, u_3, \, u_4, \, p_1, \, p_2, \, p_3, \, p_4)$, the parity-check matrix $\mathbf{H}_1$ denotes the following check equations:		*With the code word definition   $\underline{x} = (u_1, \, u_2, \, u_3, \, u_4, \, p_1, \, p_2, \, p_3, \, p_4)$,  the parity-check matrix  $\mathbf{H}_1$  denotes the following check equations:
	:$$u_1 \oplus u_2 \oplus p_1 = 0\hspace{0.05cm},\hspace{0.3cm}		:$$u_1 \oplus u_2 \oplus p_1 = 0\hspace{0.05cm},\hspace{0.3cm}
	u_3 \oplus u_4 \oplus p_2 = 0\hspace{0.05cm},\hspace{0.3cm}		u_3 \oplus u_4 \oplus p_2 = 0\hspace{0.05cm},\hspace{0.3cm}
	u_1 \oplus u_3 \oplus p_3 = 0\hspace{0.05cm},\hspace{0.3cm}		u_1 \oplus u_3 \oplus p_3 = 0\hspace{0.05cm},\hspace{0.3cm}
	u_2 \oplus u_4 \oplus p_4 = 0\hspace{0.05cm}.$$		u_2 \oplus u_4 \oplus p_4 = 0\hspace{0.05cm}.$$
	*This corresponds exactly to the assumptions made above. The same result is obtained for $\mathbf{H}_2$ and the code word definition $\underline{x} = (u_1, \, p_1, \, u_2, \, p_2, \, u_3, \, p_3, \, u_4, \, p_4)$.		*This corresponds exactly to the assumptions made above.

			*The same result is obtained for  $\mathbf{H}_2$  and the code word definition   $\underline{x} = (u_1, \, p_1, \, u_2, \, p_2, \, u_3, \, p_3, \, u_4, \, p_4)$.


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	'''(2)'''  Correct are the <u>proposed solutions 1 and 4</u>:		'''(2)'''  Correct are the  <u>proposed solutions 1 and 4</u>:
	* The code is systematic because $\mathbf{H}_1$ ends with a $4 &times 4$ diagonal matrix.		* The code is systematic because  $\mathbf{H}_1$  ends with a  $4 &times 4$  diagonal matrix.
	*For a regular (LDPC) code, there should be an equal number of ones in each row and in each column.
	*The first condition is satisfied $(w_{\rm Z} = 3)$, but not the second. Rather, there is (equally often) one one or two ones per column  ⇒  ${\rm E}[w_{\rm S}] = 1.5$.		*For a regular (LDPC) code,  there should be an equal number of  "ones"  in each row and in each column.
	* For an irregular code, the lower bound for the code rate is:
	:$$R \ge 1 - \frac{{\rm E}[w_{\rm S}]}{{\rm E}[w_{\rm Z}]}		*The first condition is satisfied  $(w_{\rm R} = 3)$,  but not the second.  Rather,  there is  (equally often)  one  "one"  or two  "ones"  per column   ⇒   ${\rm E}[w_{\rm C}] = 1.5$.

			* For an irregular code,  the lower bound for the code rate is:
			:$$R \ge 1 - \frac{{\rm E}[w_{\rm C}]}{{\rm E}[w_{\rm R}]}
	= 1 - \frac{1.5}{3} = 1/2		= 1 - \frac{1.5}{3} = 1/2
	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	* Because of the given code structure ($k = 4$ information bits, $m = 4$ check bits  ⇒  $n = 8$ ~~code~~ bits) the code rate can also be given in the conventional form: $R = k/n$   ⇒   Correct is solution 4 in contrast to answer 3.		* Because of the given code structure  $(k = 4$  information bits,  $m = 4$  check bits   ⇒   $n = 8$  encoded bits$)$  the code rate can also be given in the conventional form:  $R = k/n$   ⇒   Correct is solution 4 in contrast to answer 3.



			'''(3)'''  The  $\mathbf{H}_3$  rows result from linear combinations of  $\mathbf{H}_1$  rows:
			* The first  $\mathbf{H}_3$  row is the sum of row 1 and row 4.

			* The second  $\mathbf{H}_3$  row is the sum of row 2 and row 3.

			* The third  $\mathbf{H}_3$  row is the sum of row 1 and row 3.

			* The fourth  $\mathbf{H}_3$  row is the sum of row 2 and row 4.


			By linear combinations,  the four linearly independent equations with respect  to $\mathbf{H}_1$  now become four linearly independent equations with respect to  $\mathbf{H}_3$   <br>⇒   Therefore,  the  <u>proposed solutions 2 and 3</u>  are correct.

	~~'''(3)'''  The $\mathbf{H}_3$–rows result from linear combinations of $\mathbf{H}_1$–rows:~~
	* The first $\mathbf{H}_3$–row is the sum of row 1 and row 4.
	* The second $\mathbf{H}_3$–row is the sum of row 2 and row 3.
	* The third $\mathbf{H}_3$–row is the sum of row 1 and row 3.
	* The fourth $\mathbf{H}_3$–row is the sum of row 2 and row 4.

			'''(4)'''  Here,  <u>solutions 2, 3, and 4</u>  are correct:
			* If the code described by  $\mathbf{H}_3$  were systematic,  $\mathbf{H}_3$  should end with a  $4 &times 4$  diagonal matrix.  This is not the case here.

	~~By linear combinations, the four linearly independent equations with respect to~~ $\~~mathbf{H~~}_1$ ~~now become four linearly independent equations with respect to~~ $\~~mathbf{H~~}_3$   ~~<br>~~⇒   ~~Therefore,~~ the ~~<u>proposed solutions 2 and 3</u> are correct~~.		* The Hamming weights of all rows are equal  $(w_{\rm R} = 4)$  and also all columns each have the same Hamming weight  $(w_{\rm C} = 2)$   ⇒   the code is regular.

			* This gives  $R ≥ 1 - 2/4 = 1/2$  for the code rate.

	~~'''(4)'''  Here, <u>solutions 2, 3, and 4</u> are correct:~~		*But since the four rows of  $\mathbf{H}_3$  also describe four independent equations,  the equal sign   ⇒   $R = 1/2$  also holds.
	* ~~If the code described by $\mathbf{H}_3$ were systematic, $\mathbf{H}_3$ should end with a $4 &times 4$–diagonal matrix. This is not the case here.~~
	* The Hamming weights of all rows are equal $(w_{\rm Z} = 4)$ and also all columns each have the same Hamming weight $(w_{\rm S} = 2)$   ⇒   the code is regular.
	* This gives $R ≥ 1 - 2/4 = 1/2$ for the code rate. But since the four rows of $\mathbf{H}_3$ also describe four independent equations, the equal sign   ⇒   $R = 1/2$ also holds.
	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 16:41, 16 December 2022

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	<u>Hints:</u>		<u>Hints:</u>
	* This exercise belongs to the chapter  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|"Basics of the Low–density Parity–check"]].		* This exercise belongs to the chapter  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|"Basics of the Low–density Parity–check codes"]].

	* Reference is made in particular to the page  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes#Some_characteristics_of_LDPC_codes\|"Some characteristics of LDPC codes"]].		* Reference is made in particular to the page  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes#Some_characteristics_of_LDPC_codes\|"Some characteristics of LDPC codes"]].

Guenter at 16:41, 16 December 2022

2022-12-16T16:41:03Z

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	[[File:P_ID3067__KC_A_4_11_v2.png\|right\|frame\|Product code described by the parity-check matrix  $\mathbf{H}$]]		[[File:P_ID3067__KC_A_4_11_v2.png\|right\|frame\|Product code described by the parity-check matrix  $\mathbf{H}$]]
	In the adjacent graphic, a product code is indicated at the top, which is characterized by the following parity-check equations:		In the adjacent graphic,  a product code is indicated at the top,  which is characterized by the following parity-check equations:
	:$$p_1 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} u_1 \oplus u_2\hspace{0.05cm},~~\hspace{0.3cm}~~		:$$p_1 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} u_1 \oplus u_2\hspace{0.05cm},$$
	p_2 = u_3 \oplus u_4\hspace{0.05cm},~~\hspace{0.3cm}~~		:$$p_2 = u_3 \oplus u_4\hspace{0.05cm},$$
	p_3 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} u_1 \oplus u_3\hspace{0.05cm},~~\hspace{0.3cm}~~		:$$p_3 \hspace{-0.15cm} \ = \ \hspace{-0.15cm} u_1 \oplus u_3\hspace{0.05cm},$$
	p_4 = u_2 \oplus u_4\hspace{0.05cm}.$$		:$$p_4 = u_2 \oplus u_4\hspace{0.05cm}.$$

	Below are given the check matrices  $\mathbf{H}_1, \ \mathbf{H}_2$ and $\mathbf{H}_3$ . To be checked ~~is which~~ of the matrices ~~has~~ the given product code according to the equation  $\underline{x} = \underline{u} \cdot \mathbf{H}^{\rm T}$  ~~correctly describing the given product code,~~ assuming the following definitions:		Below are given the check matrices  $\mathbf{H}_1, \ \mathbf{H}_2$  and  $\mathbf{H}_3$.  To be checked:  Which of the matrices correctly describe the given product code according to the equation  $\underline{x} = \underline{u} \cdot \mathbf{H}^{\rm T}$  assuming the following definitions:
	* the code word  $\underline{x} = (u_1, \, u_2, \, u_3, \, u_4, \, p_1, \, p_2, \, p_3, \, p_4)$,		* the code word  $\underline{x} = (u_1, \, u_2, \, u_3, \, u_4, \, p_1, \, p_2, \, p_3, \, p_4)$,

	* the code word  $\underline{x} = (u_1, \, p_1, \, u_2, \, p_2, \, u_3, \, p_3, \, u_4, \, p_4)$.		* the code word  $\underline{x} = (u_1, \, p_1, \, u_2, \, p_2, \, u_3, \, p_3, \, u_4, \, p_4)$.


	All $\mathbf{H}$&~~ndash~~;matrices contain fewer ones than zeros. This is a characteristic of the so-called low–density parity–check codes  (short:  $\rm LDPC$ codes). In the case of LDPC codes relevant to practice, however, the share of ones is still significantly lower than in these examples.		All  $\mathbf{H}$  matrices contain fewer  "ones"  than  "zeros".  This is a characteristic of the so-called  "low–density parity–check codes"  $($short:  $\rm LDPC$ codes$)$.  In the case of LDPC codes relevant to practice,  however,  the share of  "ones"  is still significantly lower than in these examples.

	Furthermore, note for the exercise:		Furthermore, note for the exercise:
	* A  $(n, \ k)$&~~ndash~~;block code is systematic if the first  $k$  bits of the code word  $\underline{x}$  contains the information word  $\underline{u}$ . With the code word ~~definition~~  $\underline{x} = (u_1, \, u_2, \, u_3, \, u_4, \, p_1, \, p_2, \, p_3, \, p_4)$  the parity-check matrix  $\mathbf{H}$  must then end with a  $k &times k$ diagonal matrix.		* An  $(n, \ k)$  $\underline{x} = \underline{u} \cdot \mathbf{H}^{\rm T}$  block code is systematic if the first  $k$  bits of the code word  $\underline{x}$  contains the information word  $\underline{u}$.  With the code word  $\underline{x} = (u_1, \, u_2, \, u_3, \, u_4, \, p_1, \, p_2, \, p_3, \, p_4)$  the parity-check matrix  $\mathbf{H}$  must then end with a  $k &times k$ diagonal matrix.
	* A  <i>regular code</i>  (with respect to LDPC application) exists, if the Hamming–weight of all rows   ⇒   $w_{\rm Z}$ and the Hamming weight of all columns   ⇒   $w_{\rm S}$ are equal in each case. Otherwise, one speaks of an  <i>irregular LDPC code</i>.
	* The parity-check matrix  $\mathbf{H}$  of a conventional linear  $(n, \ k)$ block code consists of exactly  $m = n - k$  rows and  $n$  columns. For LDPC codes, on the other hand, the requirement is   $m ≥ n - k$. The equal sign is true if the  $m$  parity-check equations are statistically independent.
	* From the parity-check matrix  $\mathbf{H}$  a lower bound for the code rate  $R$  can be given:
	~~:$$R \ge 1 - \frac{{\rm E}[w_{\rm S}]}{{\rm E}[w_{\rm Z}]}~~
	~~\hspace{0.5cm}{\rm mit}\hspace{0.5cm}~~
	~~{\rm E}[w_{\rm S}] =\frac{1}{n} \cdot \sum_{i = 1}^{n}w_{\rm S}(i)~~
	~~\hspace{0.5cm}{\rm und}\hspace{0.5cm}~~
	~~{\rm E}[w_{\rm Z}] =\frac{1}{m} \cdot \sum_{j = 1}^{ m}w_{\rm Z}(j)~~
	~~\hspace{0.05cm}.$$~~

	*~~This equation applies equally~~ to ~~regular and irregular~~ LDPC&~~ndash~~;~~codes~~, ~~where~~ the ~~regular codes~~  $~~{\rm E}[~~w_{\rm ~~S}] = w_{\rm S~~}$  ~~and~~  $~~{\rm E}[~~w_{\rm ~~Z}] = w_{\rm Z~~}$  ~~can be considered~~.		* A  "regular code"  $($with respect to the LDPC application$)$  exists,  if the Hamming weight of all rows   ⇒   $w_{\rm R}$ and the Hamming weight of all columns   ⇒   $w_{\rm C}$ are equal in each case.  Otherwise,  one speaks of an  "irregular LDPC code".

			* The parity-check matrix  $\mathbf{H}$  of a conventional linear  $(n, \ k)$  block code consists of exactly  $m = n - k$  rows and  $n$  columns.  For LDPC codes,  the requirement is   $m ≥ n - k$.  The equal sign is true if the  $m$  parity-check equations are statistically independent.

			* From the parity-check matrix  $\mathbf{H}$  a lower bound for the code rate  $R$  can be given:
			:$$R \ge 1 - \frac{{\rm E}[w_{\rm C}]}{{\rm E}[w_{\rm R}]}
			\hspace{0.5cm}{\rm with}\hspace{0.5cm}
			{\rm E}[w_{\rm C}] =\frac{1}{n} \cdot \sum_{i = 1}^{n}w_{\rm C}(i)
			\hspace{0.5cm}{\rm and}\hspace{0.5cm}
			{\rm E}[w_{\rm R}] =\frac{1}{m} \cdot \sum_{j = 1}^{ m}w_{\rm R}(j)
			\hspace{0.05cm}.$$

			*This equation applies equally to regular and irregular LDPC codes,  where for regular codes holds  ${\rm E}[w_{\rm C}] = w_{\rm C}$  and  ${\rm E}[w_{\rm R}] = w_{\rm R}$.


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			<u>Hints:</u>
			* This exercise belongs to the chapter  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|"Basics of the Low–density Parity–check"]].

	~~Hints:~~
	* This exercise belongs to the chapter  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|"Basics of the Low–density Parity–check"]].
	* Reference is made in particular to the page  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes#Some_characteristics_of_LDPC_codes\|"Some characteristics of LDPC codes"]].		* Reference is made in particular to the page  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes#Some_characteristics_of_LDPC_codes\|"Some characteristics of LDPC codes"]].

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	\|type="[]"}		\|type="[]"}
	- No relation between  $\mathbf{H}_1$  and  $\mathbf{H}_3$  is discernible.		- No relation between  $\mathbf{H}_1$  and  $\mathbf{H}_3$  is discernible.
	+ The  $\mathbf{H}_3$&~~ndash~~;rows are linear combinations of two  $\mathbf{H}_1$ ;rows each.		+ The  $\mathbf{H}_3$  rows are linear combinations of two  $\mathbf{H}_1$  rows each.
	+ The four parity-check equations according to  $\mathbf{H}_3$  are linearly independent.		+ The four parity-check equations according to  $\mathbf{H}_3$  are linearly independent.

Noah at 19:09, 10 December 2022

2022-12-10T19:09:37Z

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Guenter: Guenter moved page Aufgaben:Exercise 4.11: Analysis of Parity-Check Matrices to Aufgaben:Exercise 4.11: Analysis of Parity-check Matrices

2022-11-22T13:02:32Z

Guenter moved page Aufgaben:Exercise 4.11: Analysis of Parity-Check Matrices to Aufgaben:Exercise 4.11: Analysis of Parity-check Matrices

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Guenter: Guenter moved page Aufgaben:Aufgabe 4.11: Analyse von Prüfmatrizen to Aufgaben:Exercise 4.11: Analysis of Parity-Check Matrices

2022-11-22T11:42:10Z

Guenter moved page Aufgaben:Aufgabe 4.11: Analyse von Prüfmatrizen to Aufgaben:Exercise 4.11: Analysis of Parity-Check Matrices

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