# Hybrid LDPC codes and iterative decoding methods - i3s Hybrid LDPC codes and iterative decoding methods - i3s

2.7 Proofs of theorems in Chapter 2 91

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 us denote

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

times. With these notations, we can write, for all n > 0:

D a (R (k)

t+n) ≤ g(k, c (k) , Π,G ⊗(n−1) ({D a (R (k′ )

t )} k ′)) + O(D a (R t ) 2 )

Let P e

(k)t = P e (R (k)

t

G(q k ) at iteration t. The global error probability of decision is Pe t = ∑ k

us recall lemma (34) in :

) be the error probability when deciding the value of a symbol in

. Let

Π(k)P

(k)t

e

1

D a (X (k) ) 2 ≤ P e (X (k) ) ≤ (q k − 1)D a (X (k) ) (2.40)

q 2 k

Let us consider a given k. If there exists a vector x such that lim

n→∞

g(k, c (k) , Π,G ⊗(n−1) (x)) =

0, then there exist α and n > 0 such that if ∀k, D a (R (k)

t 0

) < α, then

D a (R (k)

t 0 +n) < K k ′D a (R (k′ )

t 0

), ∀k ′ (2.41)

where, for all k ′ , K k ′ is a positive constant smaller than 1. If we consider P t 0

e < ξ such

that ∀k, P (k)t 0

√P

e < (q k α) 2 , then equation (2.40) ensures that ∀k, D a (R (k) e

t 0

) ≤

(k)t

q k

< α.

As previously explained, in this case, there exits n > 0 such that inequation (2.41) is

fulfilled. By induction, for all t > t 0 , there exists n > 0 such that

We have ∀(k, t), D a (R (k)

t

D a (R (k)

t+n) < K k ′D a (R (k′ )

t ), ∀k ′

) ≥ 0, therefore the sequence {D a (R (k) )} ∞ t=t 0

converges to zero

for all k. Finally, equation (2.40) ensures that, for all k, P e

(k)t converges to zero as t tends

to infinity. Thus, Pe, t the global error probability, averaged on all symbol sizes, converges

to zero as t tends to infinity.

This proves the sufficiency of the stability condition.

t

2.7.5 Information content Through Linear Maps

Lemma 12 Let x in denote the mutual information of a LDR-message v going out of a

G(q 1 ) variable node, and x out the mutual information of a LDR-message w going into

a G(q 2 ) check node. x in and x out are the input and output of the extension. They are

connected through the following expression, which is independent of the linear extension:

(1 − x in ) log 2 (q 1 ) = (1 − x out ) log 2 (q 2 ) (2.42)

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