Aufgaben:Exercise 4.12: Regular and Irregular Tanner Graph

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Tanner graph for code  $\rm A$

Shown is a Tanner graph of a code  $\rm A$  with

• the  variable nodes  $V_1, \hspace{0.05cm} \text{...} \hspace{0.05cm} , \ V_6$, . where  $V_i$  denotes the  $i$^th  code word bit  $($whether information bit or parity bit$)$  and corresponds to the  $i$^th  column of the parity-check matrix;

• the  check nodes  $C_1, \hspace{0.05cm} \text{...} \hspace{0.05cm} , \ C_3$,  which represent the rows of the  $\mathbf{H}_{\rm A}$  matrix and hence the parity-check equations.

An edge between  $V_i$  and  $C_j$  indicates that the  $i$^th  code word symbol is involved in the  $j$^th  parity-check equation.  In this case,  the element  $h_{j,\hspace{0.05cm}i}$  of the parity-check matrix is equal  $1$.

In this exercise, the relation between the above Tanner graph  $($valid for the code  $\rm A)$  and the matrix  $\mathbf{H}_{\rm A}$  shall be given. 

In addition,  the Tanner graph to a parity-check matrix  $\mathbf{H}_{\rm B}$  shall be set up,  resulting from  $\mathbf{H}_{\rm A}$  adding another row.  This is to be determined so that the associated code  $\rm B$  is regular.  This means:

• From all  variable nodes  $V_i$  $($with  $1 ≤ i ≤ n)$  go off an equal number of edges, 
  likewise from all  check nodes  $C_j$  $($with  $1 ≤ j ≤ m)$.

• The Hamming weights of all rows of  $\mathbf{H}_{\rm B}$  are each said to be equal  $(w_{\rm R})$,  as are the Hamming weights of all columns  $(w_{\rm C})$.

• For the rate of the regular code $\rm B$  to be constructed,  the following lower bound then applies:

$$R \ge 1 - \frac{w_{\rm C}}{w_{\rm R}}\hspace{0.05cm}.$$

Hints:

• This exercise belongs to the chapter  "Basics of Low–density Parity–check Codes".

• Reference is made in particular to the section  "Two-part LDPC graph representation – Tanner graph".

Questions

1 How many rows  $(m)$  and columns  $(n)$  does the parity-check matrix  $\mathbf{H}_{\rm A}$ have?

$m \hspace{0.18cm} = \ $

$n \hspace{0.3cm} = \ $

2 Which statements are true based on the Tanner graph?

	The row 1 of the  $\mathbf{H}_{\rm A}$  matrix is  "$1 \ 1 \ 0 \ 1 \ 0 \ 0$".
	The row 2 of the  $\mathbf{H}_{\rm A}$  matrix is  "$1 \ 0 \ 1 \ 0 \ 0 \ 1$".
	The row 3 of the  $\mathbf{H}_{\rm A}$  matrix is  "$0 \ 1 \ 1 \ 0 \ 0 \ 1$".

3 What are the properties of code  $\rm A$ ?

	The code is systematic.
	The code is regular.
	The code rate is  $R = 1/2$.
	The code rate is  $R = 1/3$.

4 The matrix  $\mathbf{H}_{\rm B}$  is obtained from  $\mathbf{H}_{\rm A}$  by adding one more row.  By which fourth row does a regular code  $\rm B$  result?

	By adding "$0 \ 0 \ 0 \ 1 \ 1 \ 1$".
	By adding "$1 \ 1 \ 1 \ 1 \ 1 \ 1$".
	By adding any other row.

5 What are the properties of code  $\rm B$?

	The code is systematic.
	The code is regular.
	The code rate is  $R = 1/2$.
	The code rate is  $R = 1/3$.

Solutiojn

Solution

(1)  The number of  $\mathbf{H}_{\rm A}$  rows is equal to the number of check nodes  $C_j$  in the Tanner graph   ⇒   $\underline{m = 3}$, 
and the number  $\underline{n = 6}$  of variable nodes  $V_i$  is equal to the column number.

Underlying parity-check equations

(2)  Correct are the  answers 1 and 3  in contrast to statement 2:

• The second  $\mathbf{H}_{\rm A}$  row is rather  "$1 \ 0 \ 1 \ 0 \ 1 \ 0$".

• Thus,  this exercise is based on the following parity-check equation:

$${ \boldsymbol{\rm H}}_{\rm A} =\begin{pmatrix}1 &1 &0 &1 &0 &0\\1 &0 &1 &0 &1 &0\\0 &1 &1 &0 &0 &1\end{pmatrix}\hspace{0.05cm}.$$

• In the diagram,  the parity-check equations are illustrated as red  (row 1),  green  (row 2)  resp. blue  (row 3)  groupings.

(3)  Correct are the  solutions 1 and 3:

• The  $\mathbf{H}$  matrix ends with a  $3 &times 3$  diagonal matrix   ⇒   systematic code.

• Thus,  the Hamming weights of the last three columns are  $w_{\rm C}(4) = w_{\rm C}(5) = w_{\rm C}(6) = 1$.

• For the first three columns,  $w_{\rm C}(1) = w_{\rm C}(2) = w_{\rm C}(3) = 2$   ⇒   irregular code.

• The three matrix rows are linearly independent. 

• Thus  $k = n - m = 6 - 3 = 3$  and  $R = k/n = 1/2$  holds.

Modified Tanner graph for code  $\rm B$

(4)  Correct is the  proposed solution 1:

• Looking at the previous Tanner graph,  one can see the correctness of the proposed solution 1.

• By adding the row  "$0 \ 0 \ 0 \ 1 \ 1 \ 1$"  to the  $\mathbf{H}_{\rm A}$  matrix,  one obtains:

$${ \boldsymbol{\rm H}}_{\rm B} =\begin{pmatrix}1 &1 &0 &1 &0 &0\\1 &0 &1 &0 &1 &0\\0 &1 &1 &0 &0 &1\\0 &0 &0 &1 &1 &1\end{pmatrix}\hspace{0.05cm}.$$

The modifications are marked in red in the adjacent graphic.

• Due to the newly added check node  $C_4$  and the connections with  $V_4, \ V_5$ and $V_6$,  there are now

• from all variable nodes  $V_i$  two lines,  and

• from all check nodes  $C_j$  uniformly four.

• This is the condition for the code  $\rm B$  to be regular.

(5)  Correct are the  solutions 2 and 3:

• The construction in subtask  (4)  yields a regular code.

• The Hamming weights of the rows and columns,  respectively,  are $w_{\rm R} = 3$  and  $w_{\rm C} = 2$.
• This gives as lower bound for the code rate:

$$R \ge 1 - \frac{w_{\rm C}}{w_{\rm R}}= 1 - {2}/{3} = 1/3\hspace{0.05cm}.$$

• The  $\mathbf{H}$  manipulation does not change the generator matrix  $\mathbf{G}$.

• The same code is still sent with code rate  $R = 1/2$.