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

14. 12 Low-Density Parity Check (LDPC) Codes 859
Note that the probability of P [zi = O !v1 = u, {ve : .C E <r ;, C i= j}] is either O or 1; that is, the
check node z; = 0 either succeeds or fails. The relationship Eq. (14.76) allows R;j(u) to be
updated when the ith check node receives Q;.e (ve).
Once R;j(u) have been updated, they can be passed to the variable nodes in the bottom-up
direction to update Qi,e(x). Again using Bayes' theorem (Sec. 8.1), we have
Q;,e(x) =
P [ve = x] P [ {Z m = 0 : m E /Le , m i= i}l ve = x]
P[{z m
= O : mEJLe ,mi=z}J
(14.77)
Once again assuming that each parity check is independent, we then write
P [ {Z m = 0 : m E /Le , mi= n l ve = x] =
TT R m,e(x)
mE/1, t
, m-/i
(14.78)
Now define the prior variable bit probability aspe (x) = P(ve = x). Let au be the normalization
factor such that Q;,e(l) + Qi.£ (0) = 1. We can update Qi,e(x) at the variable nodes based on
Eq. (14.76):
Q;,e(x) = au · Pe(x) TT R m,e(x) (14.79)
mEJL e
, m-li
This message will then be passed back in the top-down direction to the check nodes.
Figure 14.24 illustrates the basic operation of message passing in the bottom-up and the
top-down directions in the sum-product algorithm. The SPA can be summarized as follows.
Initialization: Let m = 0 and let m max be the maximum number of iterations. For every
h;,e = 1 in H, use prior probabilities to set
Step 1: Let the check node i update its information
Rjjl (l) = I: P [z; = O lv i
= 1, {ve }] . TT Qf7l (ve). (14.80a)
ve :fE<T;,£-1}
fE<T ;, f#}
R ( l) (O) =
ve :tE<T;,£-1}
lJ I: P [z; = O !v1 = 0, {ve }] . TT Q}7l (ve) (14.80b)
CE<r ;, E-/}
Fi g ure 14.24
Message passing
in the
sum-product
algorithm.
Variable nodes
Check nodes