Cryptography Reference
In-Depth Information
⎛
⎞
m
+
m
M
i
(
s
)=
⎝
M
i
+1
(
s
)
⎠
u
i,l
+2
L
i
(
d
(
s, s
)
min
s
=0
−
v
i,l
·
(7.45)
···
2
ν
−
1
l
=1
Applying the Max-Log-MAP logarithm in fact amounts to performing two
Viterbi decodings, in the forward and backward directions. That is the rea-
son why it is also called the
dual Viterbi
algorithm.
If the initial state of the encoder,
S
0
, is known, then
M
0
(
S
0
)=0
and
M
0
(
s
)=+
for any other state, otherwise all the
M
0
(
s
)
are initialized to the
same value. The same rule is applied for the final state. For circular codes, all
the metrics are initialized to the same value at the beginning of the prologue.
Finally, taking into account (7.38) and replacing
M
i
(
s
,s
)
by its expression
(7.43), we obtain:
∞
⎛
⎞
m
+
m
⎝
M
i
+1
(
s
)+
M
i
(
s
)
⎠
+2
L
i
(
j
)
(7.46)
A
i
(
j
)=
min
(
s
,s
)
/
d
i
(
s
,s
)
≡j
−
v
i,l
·
u
i,l
l
=1
The hard decision taken by the decoder is the value of
j
,
j
=0
···
2
m
1
,which
minimizes
A
i
(
j
)
. Let us denote this value
j
0
. According to (7.40),
L
i
(
j
)
can be
written:
−
L
i
(
j
)=
1
2
m
2
[
A
i
(
j
)
−
A
i
(
j
0
)]
pour
j
=0
···
−
1
(7.47)
We note that the presence of coecient
σ
2
in definition (7.32) of
L
i
(
j
)
allows us
to ignore the knowledge of this parameter for computing the metrics and hence
for all the decoding. This is an important advantage of the Max-Log-MAP
method over the MAP method.
In the context of iterative decoding, term
L
i
(
j
)
is modified in order to take
into account extrinsic information
L
i
(
j
)
coming from the other elementary de-
coder:
L
i
(
j
)=
L
i
(
j
)+
L
i
(
j
)
(7.48)
On the other hand, the extrinsic information produced by the decoder is obtained
by eliminating in
L
i
(
j
)
the terms containing the direct information about
d
i
,
that is, the intrinsic and
a priori
information:
M
i
+1
(
s
)+
M
i
(
s
)
m
+
m
L
i
(
j
)=
2
min
(
s
,s
)
/
d
i
(
s
,s
)
≡j
−
v
i,l
·
u
i,l
l
=
m
+1
M
i
+1
(
s
)+
M
i
(
s
)
(7.49)
m
+
m
−
min
(
s
,s
)
/
d
i
(
s
,s
)
≡j
0
−
v
i,l
·
u
i,l
l
=
m
+1
The expression of
L
i
(
j
)
can then be formulated as follows:
m
u
i,l
|
d
i
≡j
0
]+
L
i
(
j
)
L
i
(
j
0
)
(7.50)
L
i
(
j
)=
L
i
(
j
)+
1
v
i,l
·
[
u
i,l
|
d
i
≡j
−
−
2
l
=1
