Hybrid LDPC codes and iterative decoding methods - i3s

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Hybrid LDPC codes and iterative decoding methods - i3s

3.2 Machine Learning Methods for Code Design 99

check node c to variable node v. The Logarithmic Likelihood Ratio (LLR) corresponding

to the observation of the n th bit is denoted by LLR(n). V(v) denotes the first level

neighborhood of variable node v, i.e., all the check nodes which are connected to v by a

single edge. The same holds for V(c) of check node c. The BP equations corresponding

to update these messages can then be expressed by

m (t)

vc = LLR(v) + ∑

tanh( p(t) cv

2 ) = ∏

u∈V(c)\v

d∈V(v)\c

p (t−1)

dv

(3.1)

tanh( m(t) uc

2 ) (3.2)

As seen in the previous section, many inputs correspond to each neuron, but only one

output does. Hence, the ANN, modeling the BP decoding, is made of summator and

polynomial neurons and corresponds to unfold the decoding iterations. However, one

pattern of the network, corresponding to one iteration, is not a copy of the factor graph.

This is illustrated on figure 3.4.

Factor−graph

00 11

11 00

11 00

000 111 000 111

000 111 000 111

000 111 000 111

Corresponding neural network

w (0) w (1) w (2N iter −1)

111 000

111 000

111 000

111 000

111 000

111 000

111 000

111 000

111 000

111 000

111 000

111 000

111 000

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

Input layer

111 000

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

111 000

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

000 111

Output layer

111 000

111 000

111 000

One pattern = One decoding iteration

As many patterns as iterations

Figure 3.4 : A factor graph and its corresponding neural network. Each neuron corresponds

to an edge of the factor graph, hence there are 2.N edge .N iter neurons in the network.

In this network, the number of layers is double the number of decoding iterations,

and the number of nodes on each layer is equal to the number of edges in the Tanner

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