Cryptography Reference
In-Depth Information
Decoding following the Maximum
A Posteriori
(MAP) criterion
At each instant
i
, the weighted (probabilistic) estimates provided by the
MAP decoder are the
2
m
a posteriori
probabilities (APP)
Pr(
d
i
≡
j
|
v
)
,
1
. The corresponding hard decision,
d
i
, is the binary repre-
sentation of value
j
that maximizes the APP.
Each APP can be expressed as a function of the joint likelihoods
p
(
d
i
≡
2
m
j
=0
···
−
j,
v
)
:
v
)=
p
(
d
i
≡
j,
v
)
p
(
v
)
p
(
d
i
≡
j,
v
)
Pr(
d
i
≡
j
|
=
(7.22)
2
m
l
=0
−
1
p
(
d
i
≡
l,
v
)
2
m
In practice, we calculate the joint likelihoods
p
(
d
i
≡
j,
v
)
for
j
=0
···
−
1
then each APP is obtained by normalization.
The trellis representative of a code with memory
ν
has
2
ν
states, taking
their scalar value
s
in (
0
,
2
ν
1
). The joint likelihoods are calculated from
the recurrent
forward
α
i
(
s
)
and
backward
probabilities
β
i
(
s
)
and the branch
likelihoods
g
i
(
s
,s
)
:
−
β
i
+1
(
s
)
α
i
(
s
)
g
i
(
s
,s
)
p
(
d
i
≡
j,
v
)=
(7.23)
(
s
,s
)
/
d
i
(
s
,s
)
≡j
where
(
s
,s
)
/
d
i
(
s
,s
)
j
denotes the set of transitions from state to state
s
→
s
associated with the
m
-binary
j
. This set is, of course, always the same in a trellis
that is invariant over time.
The value
g
i
(
s
,s
)
is expressed as:
≡
g
i
(
s
,s
)=
Pr
a
(
d
i
≡
j,
d
i
(
s
,s
)
≡
j
)
.p
(
v
i
|
u
i
)
(7.24)
where
u
i
is the set of systematic and redundant information symbols associated
with transition
s
→
s
of the trellis at instant
i
and Pr
a
(
d
i
≡
j,
d
i
(
s
,s
)
j
)
is the
a priori
probability of transmitting the
m
-tuple of information and that
this would correspond to transition
s
→
≡
s
at instant
i
. If transition
s
→
s
does
j
,thenPr
a
(
d
i
≡
j,
d
i
(
s
,s
)
not exist in the trellis for
d
i
≡
j
)=0
,otherwise
the transition is given by the source statistics (usually uniform, in practice).
In the case of a Gaussian channel with binary inputs, value
p
(
v
i
|
≡
u
i
)
can
be written:
σ
√
2
π
exp
m
+
m
u
i,l
)
2
2
σ
2
1
(
v
i,l
−
p
(
v
i
|
u
i
)=
−
(7.25)
l
=1
where
σ
2
is the variance of the additive white Gaussian noise. In practice, we
keep only the terms that are specific to the transition considered and that are
