Hybrid LDPC codes and iterative decoding methods - i3s

Hybrid LDPC codes and iterative decoding methods - i3s

2.2 Asymptotic analysis of hybrid LDPC code ensembles 57

2.2.4 The Stability condition for hybrid LDPC Codes

The stability condition, introduced in [10], is a necessary and sufficient condition for the

error probability to converge to zero, provided it has already dropped below some value.

This condition must be satisfied by the SNR corresponding to the threshold of the code

ensemble. Therefore, ensuring this condition, when implementing an approximation of

the exact density evolution, helps to have a more accurate approximation of the exact

threshold.

In this paragraph, we generalize the stability condition to hybrid LDPC codes. Let

p(y|x) be the transition probabilities of the memoryless output symmetric channel and

c (k) be defined by

c (k) =

1

q k − 1

q k −1

i=1

∫ √p(y|i)p(y|0)dy

Let x be a positive real-valued vector of size the number of different group orders. Let us

define the g function by:

g(k, c (k) , Π,x) = c (k) Π(i = 2|k) ∑ j,l

Π(j, l|i, k)(j − 1) ∑ k ′ Π(k ′ |j, l) q k ′ − 1

q l − 1 x k ′

For more readable notations, we also define the vector output function G(x) by:

G(x) = {g(k, c (k) , Π,x)} k

which means that the p th component of G(x) is G p (x) = g(p, c (p) , Π,x). Let P (k)t

P e (R (k)

t ) be the error probability when deciding the value of a symbol in G(q k ) at iteration

t. The global error probability of decision is Pe t = ∑ Π(k)P e

(k)t . Let us denote the

k

convolution by ⊗. Then x ⊗n corresponds to the convolution of vector x by itself n times.

e =

Theorem 3 Consider a given hybrid LDPC code ensemble parametrized by Π(i, j, k, l).

If there exists a vector x with all positive components, such that, for all k,

lim g(k,

n→∞ c(k) , Π,G ⊗n (x)) = 0, then there exist t 0 and ǫ such that, if P t 0

e < ǫ, then Pe

t

converges to zero as t tends to infinity.

Proof of theorem 3 is given in section 2.7.4.

This theorem only gives a sufficient condition for stability of the code ensemble.

However, it may be possible to prove that this condition is also necessary by considering

the actual transmission channel as a degraded version of an erasurized channel, as

done in [48]. Indeed, all the necessary conditions to have such a proof, like, e.g., the

cyclic-symmetry of a symmetric channel, the binary symmetry of LM-invariant symmetric

messages or the equality between the random extended-truncated sum of messages

and the sum of extended-truncated messages can be easily shown. To do such a proof,

one must be careful to the fact that a node observes identically distributed messages, but

different kinds of nodes do not observe identically distributed messages. By lake of time,

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