B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

858 ERROR CORRECTING CODES
Sum-Product Algorithm for LDPC Decoding
The sum-product algorithm (SPA) is the most commonly used LDPC decoding method. It is
an efficient soft-input, soft-output decoding algorithm based on iterative belief propagation.
SPA can be better interpreted via the Tanner graph. SPA is similar to a see-saw game. In one
step, every variable node passes information via its edges to its connected check nodes in
the top-down pass-flow. In the next step, every check node passes back information to all the
variable nodes it is connected to in a bottom-up pass-flow.
To understand SPA, let the parity matrix be H of size J x n where J = n - k for an (n, k)
LDPC block code. Let the codeword be represented by variable node bits {vj, j = 1, ... , n).
For the }th variable node V j , let
JLJ = {i : h u
= L l S i S J}
denote the set of variable nodes connected to V j , For the ith check node z;, let
u; = U : h u
= 1, 1 S j S n}
denote the set of variable nodes connected to z;.
First, define the probability of satisfying check node z; = 0 when Vj = u as
u = 0, 1 (14.72)
Let us denote the vector of variable bits as v. We can use the Bayes theorem on conditional
probability (Sec. 8.1) to show that
R;j(u) = L P[z; = 0lv] - P[v lvj = u]
V: Vj=U
L p [z; = 0lvj = u, {ve: e E <T; , e #J}] . p [{vt : e E <T;, e # J}lvj = u]
(14.73)
This is message passing in the bottom-up direction.
For the check node z; to estimate the probability P [ { ve : e E u;, e # j} I V j = u], the check
node must collect information from the variable node set u ; . Define the probability of v1 = x
obtained from its check nodes except for the ith one as
Q;,c (x) = P [ve = xi {zm = 0 : m E /Le , m # il] X = 0, 1 (14.74)
Furthermore, assume that the variable node probabilities are approximately independent. We
can estimate
p [{ve : e E <T ;, e # J}lvj = u] = n Q;,e (vc)
fE<T ;, ff.j
This means that the check nodes can update the message through
L p [z; = ojvj = u, {ve : f E <T;, e # J}] . n Q;,c (vc )
v,:: £E<T;,ff.j
(14.75)
(14.76)