Aufgaben:Exercise 4.13: Decoding LDPC Codes

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Given LDPC parity-check matrix

The exercise deals with  "Iterative decoding of LDPC–codes"  according to the  Message passing algorithm.

The starting point is the presented  $9 &times 12$  parity-check matrix  $\mathbf{H}$,  which is to be represented as Tanner graph at the beginning of the exercise.  It should be noted:

1. The  "variable nodes"  $V_i$  denote the  $n$  bits of the code word.
2. The  "check nodes"  $C_j$  represent the  $m$  parity-check equations.
3. A connection between  $V_i$  and  $C_j$  indicates that the element of matrix  $\mathbf{H}$  $($in row  $j$, column  $i)$  is   $h_{j,\hspace{0.05cm} i} =1$.
4. For  $h_{j,\hspace{0.05cm}i} = 0$  there is no connection between  $V_i$  and  $C_j$.
5. The  "neighbors  $N(V_i)$  of  $V_i$"  is called the set of all  check nodes  $C_j$ connected to  $V_i$  in the Tanner graph.
6. Correspondingly,  to  $N(C_j)$  belong all variable nodes  $V_i$  with a connection to  $C_j$.

The decoding is performed alternately with respect to

• the  variable nodes   ⇒   "variable nodes decoder"  $\rm (VND)$,  and

• the  check nodes   ⇒   "check nodes decoder"  $\rm (CND)$.

This is referred to in subtasks  (5)  and  (6).

Hints:

• The exercise belongs to the chapter  "Basic information about Low–density Parity–check Codes".

• Reference is made in particular to the section  "Iterative decoding of LDPC codes".

Questions

1 How many  variable nodes  $(I_{\rm VN})$  and  check nodes  $(I_{\rm CN})$  are to be considered?

$I_{\rm VN} \ = \ $

$I_{\rm CN} \ = \ $

2 Which of the following  check nodes  and  variable nodes  are connected?

	$C_4$  and  $V_6$.
	$C_5$  and  $V_5$.
	$C_6$  and  $V_4$.
	$C_6$  and  $V_i$  for  $i > 9$.
	$C_j$  and  $V_{j-1}$  for  $j > 6$.

3 Which statements are true regarding neighbors   $N(V_i)$   and   $N(C_j)$ ?

	$N(V_1) = \{C_1, \ C_2, \ C_3, \ C_4\}$,
	$N(C_1) = \{V_1, \ V_2, \ V_3, \ V_4\}$,
	$N(V_9) = \{C_3, \ C_5, \, C_7\}$,
	$N(C_9) = \{V_3, \ V_5, \ V_7\}$.

4 Which statements are true for the  variable node decoder  $\rm (VND)$?

	At the beginning  $($iteration 0$)$  the  $L$–values of the nodes  $V_1, \hspace{0.05cm} \text{...} \hspace{0.05cm}, \ V_n$  are preassigned corresponding to the channel input values  $y_i$.
	For the VND represents  $L(C_j → V_i)$  a-priori information.
	There are analogies between the  "variable node decoder"  and the decoding of a single parity–check code.

5 Which statements are true for the  check node decoder  $\rm (CND)$?

	The CND returns at the end the desired a-posteriori  $L$–values.
	For the CND represents  $L(C_j → V_i)$  a-priori information.
	There are analogies between the  "check node decoder"  and the decoding of a single parity–check code.

Solution

Solution

(1)  The variable node  $V_i$  stands for the  $i$^th  code word bit,  so that  $I_{\rm VN}$  is  equal to the code word length  $n$.

• From the column number of the  $\mathbf{H}$  matrix,  we can see  $I_{\rm VN} = n \ \underline{= 12}$.

• For the set of all variable nodes,  one can thus write in general:  ${\rm VN} = \{V_1, \hspace{0.05cm} \text{...} \hspace{0.05cm} , V_i, \hspace{0.05cm} \text{...} \hspace{0.05cm} , \ V_n\}$.

• The check node  $ C_j$  represents the  $j$^th  parity-check equation,  and for the set of all check nodes:

$${\rm CN} = \{C_1, \hspace{0.05cm} \text{...} \hspace{0.05cm} , \ C_j, \hspace{0.05cm} \text{...} \hspace{0.05cm} , \ C_m\}.$$

• From the number of rows of the  $\mathbf{H}$  matrix we get  $I_{\rm CN} \ \underline {= m = 9}$.

Tanner graph for the present example

(2)  The results can be read from the Tanner graph sketched on the right.

Correct are the proposed solutions 1, 2 and 5:

• The element  $h_{5,\hspace{0.05cm}5}=1$   $($column 5, row 5$)$   ⇒   red edge.

• The element  $h_{4,\hspace{0.05cm} 6}=1$  $($column 4, row 6$)$   ⇒   blue edge.

• The element  $h_{6, \hspace{0.05cm}4}=0$  $($column 6, row 4$)$   ⇒   no edge.

• $h_{6,\hspace{0.05cm} 10} = h_{6,\hspace{0.05cm} 11} = 1$,  $h_{6,\hspace{0.05cm}12} = 0$   ⇒   not all three edges exist.

• It holds  $h_{7,\hspace{0.05cm}6} = h_{8,\hspace{0.05cm}7} = h_{9,\hspace{0.05cm}8} = 1$   ⇒   green edges.

(3)  It is a regular LDPC code with

• $w_{\rm R}(j) = 4 = w_{\rm R}$ for $1 ≤ j ≤ 9$,

• $w_{\rm C}(i) = 3 = w_{\rm C}$ for $1 ≤ i ≤ 12$.

The  answers 2 and 3  are correct,  as can be seen from the first row and ninth column,  respectively, of the parity-check matrix  $\mathbf{H}$.

The Tanner graph confirms these results:

• From  $C_1$  there are edges to  $V_1, \ V_2, \ V_3$, and $V_4$.

• From  $V_9$  there are edges to $C_3, \ C_5$, and $C_7$.

The answers 1 and 4 cannot be correct already because

• the neighborhood  $N(V_i)$  of each variable node  $V_i$  contains exactly  $w_{\rm C} = 3$  elements,  and

• the neighborhood  $N(C_j)$  of each check node  $C_j$  contains exactly  $w_{\rm R} = 4$  elements.

(4)  Correct are the  proposed solutions 1 and 2,  as can be seen from the  "corresponding theory page":

• At the start of decoding  $($so to speak at iteration  $I=0)$  the  $L$–values of the variable nodes   ⇒   $L(V_i)$ are preallocated with the channel input values.

• Later  $($from iteration $I = 1)$  the log likelihood ratio  $L(C_j → V_i)$  transmitted by the CND is considered in the VND as a-priori information.

• Answer 3 is wrong.  Rather,  the correct answer would be:  There are analogies between the VND algorithm and the decoding of a  "repetition code".

(5)  Correct is  only proposed solution 3  because

• the final a-posteriori  $L$–values are derived from the VND,  not from the CND;

• the  $L$–value  $L(C_j → V_i)$  represents extrinsic information for the CND;  and

• there are indeed analogies between the CND algorithm and SPC decoding.