Cryptography Reference
In-Depth Information
not eliminated by division in the expression (7.22):
⎛
⎞
m
+
m
l
=1
v
i,l
·
u
i,l
⎝
⎠
p
(
v
i
|
u
i
)=exp
(7.26)
σ
2
The forward and backward recurrent probabilities are calculated as follows:
2
ν
−
1
α
i−
1
(
s
)
g
i−
1
(
s
,s
)
α
i
(
s
)=
for
i
=1
···
k
(7.27)
s
=0
and:
2
ν
−
1
β
i
+1
(
s
)
g
i
(
s, s
)
β
i
(
s
)=
for
i
=
k
−
1
···
0
(7.28)
s
=0
To avoid problems of precision or of overflow in the representation of these
values, they have to be normalized regularly. The initialization of the recursions
depends on the knowledge or not of the state of the encoder at the beginning
and at the end of encoding. If the initial state
S
0
of the encoder is known,
then
α
0
(
S
0
)=1
and
α
0
(
s
)=0
foranyotherstate,otherwiseallthe
α
0
(
s
)
are
initialized to the same value. The same rule is applied for the final state
S
k
.For
circular codes, initialization is performed automatically after the prologue step,
which starts from identical values for all the states of the trellis.
In the context of iterative decoding, the composite decoder uses two ele-
mentary decoders exchanging
extrinsic probabilities
. Consequently, the basic
decoding brick described above must be reconsidered in order to:
j
1. take into account an extrinsic probability, Pr
ex
(
d
i
≡
v
)
, in expres-
sion (7.24), calculated by the other elementary decoder of the composite
decoder, from its own input sequence
v
,
j
2. produce its own extrinsic probability Pr
ex
(
d
i
≡
v
)
that will be used
by the other elementary decoder.
2
m
In practice, for each value of
j
,
j
=0
···
−
1
:
1. in expression (7.24), the
a priori
probability Pr
a
(
d
i
≡
j,
d
i
(
s
,s
)
≡
j
)
is
replaced by the modified
a
priori
probability Pr
@
(
d
i
≡
j,
d
i
(
s
,s
)
≡
j
)
,
having for its expression, to within one normalization factor:
Pr
@
(
d
i
≡
j
)=
Pr
a
(
d
i
≡
j
)
.
Pr
ex
(
d
i
≡
j,
d
i
(
s
,s
)
j,
d
i
(
s
,s
)
v
)
(7.29)
≡
≡
j
|
1. Pr
ex
(
d
i
≡
j
|
v
)
is given by:
