Cryptography Reference
In-Depth Information
we usually assign the same weight to each state at the end of the trellis since
the arrival state is generally not known
a priori
:
1
M
L−
1
β
N
(
s
)=
∀
s
(11.12)
In practice, we see that the dynamic of values
α
i−
1
(
s
)
and
β
i
(
s
)
increases
during the progression in the trellis. Consequently, these values must be normal-
ized at regular intervals in order to avoid overflow problems in the computations.
One natural solution involves dividing these metrics at each instant by constants
K
α
and
K
β
chosen in such a way as to satisfy the following normalization con-
dition:
K
α
s
K
β
s
1
1
α
i
(
s
)=1
and
β
i
(
s
)=1
(11.13)
the sums here concerning all possible states
s
of the trellis at instant
i
.
To complete the description of the algorithm, it remains for us to develop
the expression of the term
γ
i−
1
(
s
,s
)
. This term can be assimilated to a branch
metric. We can show that it is decomposed into a product with two terms:
γ
i−
1
(
s
,s
)=Pr(
s
s
)
P
(
y
i
|
s
,s
)
|
(11.14)
s
)
represents the
a priori
probability of going through the transi-
tion between state
s
and state
s
, that is to say, the
a priori
probability
P
a
(
X
l
)=Pr(
x
i
=
X
l
)
of having transmitted at time instant
i
the constel-
lation symbol
X
l
labeling the branch considered in the trellis. Owing to the
presence of the interleaver at transmission, bits
x
i,j
composing symbol
x
i
can
be assumed statistically independent. Consequently, probability
P
a
(
X
l
)
has the
following decomposition:
Pr(
s
|
m
P
a
(
X
l
)=Pr(
x
i
=
X
l
)=
P
a
(
X
l,j
)
(11.15)
j
=1
wherewehavewritten
P
a
(
X
l,j
)=Pr(
x
i,j
=
X
l,j
)
, binary element
X
l,j
taking the
value 0 or 1 according to the symbol
X
l
considered and the mapping rule. Within
the turbo equalization iterative process, the
a priori
probabilities
P
a
(
X
l,j
)
are
deduced from the
a priori
information available at the input of the equalizer.
From the initial definition (11.4) of the LLR, we can in particular show that
probability
P
a
(
X
l,j
)
and corresponding
a priori
LLR
L
a
(
x
i,j
)
are linked by the
following expression:
P
a
(
X
l,j
)=
K
exp (
X
l,j
L
a
(
x
i,j
))
with
X
l,j
∈{
0
,
1
}
(11.16)
The term
K
is a normalization constant that we can omit in the following
computations without compromising the final result in any way.
