Cryptography Reference
In-Depth Information
β
i
+1
(
s
)
α
i
(
s
)
g
i
(
s
,s
)
(
s
,s
)
/
d
i
(
s
,s
)
≡
j
Pr
ex
(
d
i
≡
j
|
v
)=
(7.30)
β
i
+1
(
s
)
α
i
(
s
)
g
i
(
s
,s
)
(
s
,s
)
The terms
g
i
(
s
,s
)
are non-zero if
s
→
s
corresponds to a transition of the
trellis and are then inferred from the expression of
p
(
v
i
|
u
i
)
by eliminating the
systematic part of the information. In the case of a transmission over a Gaussian
channel with binary inputs and starting from the simplified expression (7.26) of
p
(
v
i
|
u
i
)
,wehave:
⎛
⎝
⎞
⎠
m
+
m
v
i,l
u
i,l
l
=
m
+1
g
i
(
s
,s
)=exp
(7.31)
σ
2
The simplified Max-Log-MAP or SubMAP algorithm
Decoding following the MAP criterion requires a large number of operations,
including calculating exponentials and multiplications. Re-writing the decoding
algorithm in the logarithmic domain simplifies the processing. The weighted es-
timations provided by the decoder are then values proportional to the logarithms
of the APPs, called Log-APP logarithms, denoted
L
:
σ
2
2
2
m
L
i
(
j
)=
−
ln
Pr
(
d
i
≡
j
|
v
)
,
j
=0
···
−
1
(7.32)
We define
M
i
(
s
)
and
M
i
(
s
)
the forward and backward metrics relative to node
s
at instant
i
,and
M
i
(
s
,s
)
, the branch metric relative to the
s
→
s
transition
of the trellis at instant
i
by:
M
i
(
s
)=
σ
2
ln
α
i
(
s
)
−
M
i
(
s
)=
σ
2
ln
β
i
(
s
)
M
i
(
s
,s
)=
−
(7.33)
σ
2
ln
g
i
(
s
,s
)
−
Introduce values
A
i
(
j
)
and
B
i
calculated as:
⎡
⎤
σ
2
ln
⎣
β
i
+1
(
s
)
α
i
(
s
)
g
i
(
s
,s
)
⎦
A
i
(
j
)=
−
(7.34)
(
s
,s
)
/
d
i
(
s
,s
)
≡j
⎡
⎣
(
s
,s
)
⎤
σ
2
ln
β
i
+1
(
s
)
α
i
(
s
)
g
i
(
s
,s
)
⎦
B
i
=
−
(7.35)
