Hybrid LDPC codes and iterative decoding methods - i3s

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Hybrid LDPC codes and iterative decoding methods - i3s

88 Chapitre 2 : Hybrid LDPC Codes

We finally obtain

This completes the proof.

P(X = x|A) = P(Y ×B−1 = x)

= P(Y ×A−1 = x)

= P(X = x)


Lemma 11 The product of two LM-invariant random probability-vectors is LM-invariant.

Proof: Let U and V be two LM-invariant random LDR-vectors of size q 2 . Let A and

B be any two linear maps from G(q 1 ) to G(q 2 ). Since U is LM-invariant, U ×A−1 and

U ×B−1 are identically distributed, by definition of LM-invariance. The same holds for

V. U ×A−1 V ×A−1 and U ×B−1 V ×B−1 are therefore identically distributed. Moreover, it

is clear that U ×A−1 V ×A−1 = UV ×A−1 , for any A. Hence, UV ×A−1 and UV ×B−1 is

LM-invariant. This completes the proof.

2.7.4 Proof of Theorem 3

X (k) denotes a probability-vector random variable of size q k . The j th component of the

random truncation of X (k) is denoted by

rt . The j th component of the random extension

of X (k) is denoted by

re

X (k)

j

X (k)

j

random truncation of X (k) is denoted by rt+re

X (k)

j

We define the operator D a by:

. The j th component of the random extension followed by a

D a (X (l) ) =

.


q

1

l −1




E⎝

q l − 1

j=1

√ X(l) j

X (l)

0

The following equalities are hence deduced from the previous definitions:




re

X

E⎝√

(k)

j


re

= ∑ q k −1


1 ∑

Π(l|k) E⎝√ X(k) i ⎠

q

X (k)

l − 1

l

i=1 X (k)

0

0





rt

q E⎝√


1

l −1



= E



X(l) j ⎟


q l − 1


E⎝√

X (l)

i

rt

X (l)

0

rt+re

X (k)

i

rt+re

X (k)

0


j=1

= D a (X (l) )

⎠ = ∑ l

1

Π(l|k)

q l − 1

X (l)

0

q k −1


i=1






E⎝

√ X(k) i

X (k)

0



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