Decoding of LDPC Codes

Low-density Parity-check Codes

2.3 Decoding of LDPC Codes

sequence of vertices starting and ending at the same vertex with no repetitions of vertices and edges allowed. The presence of short cycles in the Tanner graph significantly degrades the performance of iterative decoders specifically for short-length LDPC codes [34, 51, 70]. The length of the shortest cycle/cycles is known as thegirth which partially reflects the severity of the performance degradation.

In Section 2.5, we shall review some well-established strategies to construct a Tanner graph by avoiding harmful short cycles. As an example of cycle, consider the Tanner graph shown in Figure 2.1. A cycle of length 6 is shown in thick blue lines. The blue-colored 1s of H in (2.1) correspond to the edges present in the cycle. There is no cycle of length 4 in the Tanner graph. Thus the girth is 6.

2.3 Decoding of LDPC Codes

The MAP decoding is performed block-wise. Although optimal, the MAP decoding is not feasible for the LDPC codes of practical lengths because of the enormous number of codewords involved. In the SPA, the decoding is carried out in a bitwise manner as shown below. For the bitxj, the decoder output is given by

ˆ

xj = arg max

xj

Pr (xj|y, all checks involvingxj are satisfied). (2.5)

As xj ∈ {0,1}, the decision aboutxj can be made from the a posteriori LLRLj defined by

Lj ,logPr (x_j = 0|y,all checks involvingx_j are satisfied)

Pr (x_j = 1|y,all checks involvingx_j are satisfied). (2.6) In the above expression, Pr (xj =u|y,all checks involvingxj are satisfied) is the conditional proba- bility that xj = u, u ∈ {0,1} given that (i) all the CNs ci, i ∈ N(j) are satisfied (i.e all such i^th components of the syndrome vector are zero) and (ii) the received sequence is y.

Now, (2.5) can be alternatively expressed as

ˆ

xj = 1−sgn (Lj)

2 . (2.7)

In the SPA,L_js are updated iteratively by exchanging messages between the VNsv_js and the CNs c_is. The extrinsic messages from v_j are initialized with the channel LLR (intrinsic message) L_chj for xj given by

L_chj = logPr (xj = 0|yj)

Pr (x_j = 1|y_j). (2.8)

We consider the transmission of the codewords over an AWGN channel with mean zero and variance σ_n². The codeword sequence x = [x₁. . . x_N]^T is BPSK modulated to tx = [t_x1. . . t_xN]^T according to t_xj = 1−2x_j. After the transmission of tx over an AWGN channel, the received sequence is given by y=tx+n, where n = [n₁. . . n_N]^T and n_js are independent and identically distributed Gaussian random variables with mean zero and varianceσ²_n. Assumingxjs to be equally likely to take 0 and 1,

TH -1443_08610205

L_chj in (2.8) can now be expressed as

L_chj = log exp

−^(yj_2σ⁻¹⁾²_n ²

exp

−^(yj_2σ⁺¹⁾2 ² n

= 2yj

σ_n² . (2.9)

b b b b

| {z } {^{vj′:j′∈M(i)\{j}}} v_j^′

ci

Vj^′i

vj

Uij

(a)CN processing

b b b b

| {z }

{c_i′:i^′∈N(j)\{i}}

L_chj

Ui^′j

ci^′

vj

ci

Vji

(b)VN processing

Figure 2.3: Diagrammatic representation of the message passing in the SPA

The SPA in the LLR domain is described in Algorithm 2.1. At the beginning, the VN→CN messages are initialized with the channel LLR. The rest of the SPA can be presented in the following steps:

(i) CN processing: The messageUij sent from a CNci towards its neighboring VNvj is defined as

Uij ,logPr (checkci is satisfied|xj = 0,y) Pr (checkci is satisfied|xj = 1,y).

In the above expression, Pr (check ci is satisfied|xj =u,y) is the conditional probability that the i^th CN is satisfied (i.e. the i^th element of the syndrome vector is zero) given that (i) the value of xj isu and (ii) the received sequence is y. With the assumption that all the VNs connected

TH -1443_08610205

2.3 Decoding of LDPC Codes

Algorithm 2.1:Sum-Product Algorithm

input : A received sequencey= [y₁. . . yN]^T, the parity-check matrixH and the maximum number of iterations Im

output: Estimated codeword ˆx= [ˆx₁. . .xˆ_N]^T

Find the set of neighbors for each VN j ∈ {1, . . . , N},N(j) ={i:hij = 1}; Find the set of neighbors for each CN i∈ {1, . . . , M},M(i) ={j:hij = 1};

for j←1 to N do // initiliazation of the VN→CN messages L_chj = ^2y_σ2^j

n ;

Set V_ji =L_chj,∀i∈N(j);

end

for I ←1 toIm do // Loop for iterations

fori←1 toM do // CN processing

ComputeU_ij ∀j∈M(i):

U_ij = log

1+ Q

j′∈M(i)\{j}

tanh

_V

j′i 2

1− Q

j′∈M(i)\{j}

tanh

_V

j′i 2

end

for j←1 to N do // VN processing

ComputeVji ∀i∈N(j):

V_ji= ^P

i^′∈N(j)\{i}

U_i^′_j +L_chj end

forj←1 to N do // Computation of a posteriori LLR Lj = ^P

i^′∈N(j)

Ui^′j+Lchj

end

for j←1 to N do // Hard decision

ˆ

xj = ^(1−sgn(L₂ ^j)) end

if Hˆx=0 then // Stopping criterion

Stop;

end end

to c_i send independent messages towardsc_i,U_ij can be computed as

Uij = log

1 + ^Q

j^′∈M(i)\{j}

tanh^Vj₂^′i

1− ^Q

j^′∈M(i)\{j}

tanh^Vj₂^′i, (2.10)

whereM(i) ={j:hij = 1}is the set of the indices of the neighboring VNs connected toci and V_j^′_i is the message sent fromv_j^′ toci as described in the next item. The graphical representation of the CN processing atci is shown in Figure 2.3(a).

TH -1443_08610205

(ii) VN processing: The message Vji sent from a VNvj towards its neighboring CN ci is defined as

Vji,logPr (xj = 0|y,all checks involving xj except ci are satisfied) Pr (xj = 1|y,all checks involving xj except ci are satisfied).

With the assumption that all the checks connected tov_j are conditionally independent given the value of x_j,V_ji can be computed as

Vji = ^X

i^′∈N(j)\{i}

Ui^′j +L_chj, (2.11)

where N(j) is the set of the indices of the neighboring CNs connected to vj, i.e., N(j) = {i:hij = 1}. The graphical representation of the VN processing atvj is shown in Figure 2.3(b).

(iii) Computation of a posteriori LLR: Thea posteriori LLRL_j for the VNv_j as formulated in (2.6) can be computed as

Lj = ^X

i^′∈N(j)

U_i^′_j +L_chj. (2.12)

(iv) Hard decision and stopping criterion: Based on the sign of the a posteriori LLR, the hard decision is made according to (2.7). If a valid codeword is obtained, i.e., Hˆx=0, then the decoding process is stopped, otherwise it is continued till the maximum number of iterations are executed.