Cryptography Reference
In-Depth Information
method for quantifying the sequence received,
r
, determines the choice of de-
coding algorithm.
Hard input algorithm
Quantization on one bit involves processing only the sign of the samples received.
Hard input decoding algorithms are based on the one proposed by Gallager under
the name of algorithm A [9.21]. These decoders of course offer lower performance
than those of soft input decoders. They are only implemented for very particular
applications like optical communications, for example [9.17]. These algorithms
will not be considered in the remainder of this chapter.
Belief propagation algorithm
When quantization is done on more than one bit, the decoding may use soft
inputs: the
a priori
probability of the received symbols. In the case of binary
codes and in the logarithm domain, we use the
a priori
log likelihood ratio
(LLR) of samples
r
j
:
c
j
)=ln
p
(
r
j
|
c
j
=0)
p
(
r
j
|
L
(
r
j
|
(9.19)
c
j
=1)
where
c
j
is the
j
-th bit of the codeword and
r
j
=2
c
j−
1
+
b
j
.Inthecaseof
the additive white Gaussian noise channel, the noise samples
b
j
follow a centred
Gaussian law with variance
σ
2
,thatis:
√
2
πσ
2
exp
(
r
j
−
1))
2
1
(2
c
j
−
p
(
r
j
|
c
j
)=
(9.20)
2
σ
2
Combining (9.19) and (9.20), intrinsic information
I
j
can be defined :
2
r
j
σ
2
I
j
=
L
(
r
j
|
c
j
)=
−
(9.21)
Each iteration of the BP algorithm is decomposed into two steps:
1. Processing the parities:
Z
j,p
=2tanh
−
1
⎡
⎤
tanh
L
j
,p
2
⎣
⎦
(9.22)
j
∈
J
(
p
)
/j
2. Processing the variables:
L
j,p
=
I
j
+
Z
j
,p
(9.23)
p
∈P
(
j
)
/p
