Page 363 - Satellite Communications, Fourth Edition

Error Control Coding 343

Parity nodes
(Row no.)
0 1 2 3 4 5 6

C 0 C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 10 C 11 C 12 C 13 C 14 C 15
Figure 11.13 Illustrating a Tanner graph.

lines (technically referred to as edges) join the bit nodes to their respec-
tive parity check nodes. Edges occur wherever a 1 appears in the H
matrix. Thus for row 0, 1s appear in the c , c , c and c positions [(and
2
1
9
0
as shown by the first parity line in Eq. (11.26)]. In Fig. 11.13, only the
parity equations for rows 0, 1, and 3 are shown for clarity, but the com-
plete Tanner graph would show all the edges. Messages pass along the
edges. Initially, the output from the channel demodulator provides the
1
first “soft” estimate of a bit. If p is the probability that, the bit is a 1,
1
the probability that it is a zero is 1 p . These estimates are sent to their
respective parity check equations where the equation is used to derive
probability estimates for a bit. Considering the first equation for exam-
ple, an estimate for the probability that c is a 1 can be obtained from
0
the probabilities for c , c and c being 1. For the group c c c the com-
1 2 9
2,
1
9
binations that would result in c being 1 are 100, 010, 100, and 111. The
0
sum of the corresponding probabilities gives an estimate, p′ for the
0
probability of c being 1:
0
1
1
1
1
1
1
pr p Q1 p RQ1 p R p Q1 p RQ1 p R
9
1
9
1
2
2
0
(11.27)
1 1 1 1 1 1
2
1
p 9 Q1 p RQ1 p R p p p 9
2
1
The estimates from the parity nodes are returned to the respective bit
nodes. The bit node now has estimates from the parity check nodes and
from the channel, which enables a new estimate for probability to be cal-
culated. For example if bit node 0 receives estimates from parity check
1
1
1
1
nodes A, B, and C, denoted by p A , p B , p C respectively, and p CH from the
channel, the new estimates sent to these parity check nodes are the
1
1
1
1
1
1
products K(p CH p B p C ) to parity node A, K(p CH p A p C ) to parity node B, and
1
K(p CH p p ) to parity node C, where K is a normalizing constant. It will
B
A