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49.3: Decoding 583
(b)
(a)
1
0.1
0.01
0.001
0.0001
1e-05
9999 3000
N=30000
total
undetected
1
N=204
1 2 3 4 5
This graph is a factor graph for the prior probability over codewords,
with the circles being binary variable nodes, and the squares representing
two types of factor nodes. As usual, each contributes a factor of
the form
[ x=0 mod 2]; each contributes a factor of the form [x1 =x2 =x3].
49.3 Decoding
The repeat–accumulate code is normally decoded using the sum–product algorithm
on the factor graph depicted in figure 49.1b. The top box represents the
trellis of the accumulator, including the channel likelihoods. In the first half
of each iteration, the top trellis receives likelihoods for every transition in the
trellis, and runs the forward–backward algorithm so as to produce likelihoods
for each variable node. In the second half of the iteration, these likelihoods
are multiplied together at the nodes to produce new likelihood messages to
send back to the trellis.
As with Gallager codes and turbo codes, the stop-when-it’s-done decoding
method can be applied, so it is possible to distinguish between undetected
errors (which are caused by low-weight codewords in the code) and detected
errors (where the decoder gets stuck and knows that it has failed to find a
valid answer).
Figure 49.2 shows the performance of six randomly-constructed repeat–
accumulate codes on the Gaussian channel. If one does not mind the error
floor which kicks in at about a block error probability of 10 −4 , the performance
is staggeringly good for such a simple code (cf. figure 47.17).
816
408
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
Figure 49.1. Factor graphs for a
repeat–accumulate code with rate
1/3. (a) Using elementary nodes.
Each white circle represents a
transmitted bit. Each
constraint forces the sum of the 3
bits to which it is connected to be
even. Each black circle represents
an intermediate binary variable.
Each constraint forces the three
variables to which it is connected
to be equal.
(b) Factor graph normally used
for decoding. The top rectangle
represents the trellis of the
accumulator, shown in the inset.
Figure 49.2. Performance of six
rate- 1/3 repeat–accumulate codes
on the Gaussian channel. The
blocklengths range from N = 204
to N = 30 000. Vertical axis:
block error probability; horizontal
axis: Eb/N0. The dotted lines
show the frequency of undetected
errors.