Cryptography Reference
In-Depth Information
at instant
i
, on all of the transitions between instants
i
1
and
i
for which the
j
-th bit making up the symbol associated with these transitions equals 0 or 1.
Thus,
−
⎛
⎞
Pr(
s
,s,
y
)
⎝
⎠
(
s
,s
)
/x
i,j
=1
L
(
x
i,j
)=ln
(11.7)
Pr(
s
,s,
y
)
(
s
,s
)
/x
i,j
=0
Adopting a similar approach now to the one presented in the original article
by Bahl
et al
. [11.3], we can show that the joint probability
Pr(
s
,s,
y
)
associated
with each transition considered can be decomposed into a product of 3 terms:
Pr(
s
,s,
y
)=
α
i−
1
(
s
)
γ
i−
1
(
s
,s
)
β
i
(
s
)
(11.8)
Figure 11.12 shows the conventions of notation used here.
Figure 11.12 - Conventions of notation used to describe the MAP equalizer.
Forward and backward state probabilities
α
i−
1
(
s
)
and
β
i
(
s
)
can be calcu-
lated recursively for each state and at each instant in the trellis, by applying the
following update equations:
α
i
(
s
)=
(
s
,s
)
α
i−
1
(
s
)
γ
i−
1
(
s
,s
)
(11.9)
β
i
(
s
)=
(
s
,s
)
γ
i
(
s
,s
)
β
i
+1
(
s
)
(11.10)
These two steps are called
forward recursion
and
backward recursion
, respec-
tively. Summations are performed over all the couples of states with indices (
s
,
s
) for which there is a valid transition between two consecutive instants in the
trellis. Forward recursion uses the following initial condition:
α
0
(0) = 1
,
α
0
(
s
)=1
for
s
=0
(11.11)
This condition translates the fact that the initial state in the trellis (with
index
0
, by convention) is perfectly known. Concerning the backward recursion,
