Cryptography Reference
In-Depth Information
Figure 11.15 - Practical implementation of the equalizer using transverse filters.
decoder. Generally, we can always decompose the expression of
z
i
as the sum of
two quantities:
z
i
=
g
Δ
x
i−
Δ
+
ν
i
(11.46)
The term
g
Δ
x
i−
Δ
represents the useful signal up to a constant factor
g
Δ
.We
recall that this factor quite simply corresponds to the central coecient of the
cascading of the channel with the equalizer. The term
ν
i
accounts for both
residual interference and noise at the output of the equalizer. In order to perform
the demapping operation, we make the hypothesis
9
that interference term
ν
i
follows a complex Gaussian distribution, with zero mean and total variance
σ
ν
.
Parameters
g
Δ
and
σ
ν
are easy to deduce from the knowledge of the set of
equalizer coecients. We can thus show ([11.51, 11.33]) that we have:
g
Δ
=
f
T
h
Δ
and
σ
ν
=
E
2
=
σ
x
g
Δ
(1
|
z
i
−
g
Δ
x
i−
Δ
|
−
g
Δ
)
(11.47)
Starting from these hypotheses, the demapping module calculates the
aposte-
riori
LLR on the coded interleaved bits, denoted
L
(
x
i,j
)
and defined as follows:
L
(
x
i,j
)=ln
Pr(
x
i,j
=1
|
z
i
)
(11.48)
Pr(
x
i,j
=0
|
z
i
)
The values present in the numerator and denominator can be evaluated by sum-
ming the
a posteriori
probability
Pr(
x
i
=
X
l
|
z
i
)
of having transmitted a par-
ticular symbol
X
l
of the constellation on all the symbols for which the
j
-th bit
making up this symbol takes the value
0
or
1
respectively. Thus, we can write:
9
This hypothesis rigorously only holds on condition that the equalizer suppresses all the
ISI, which assumes perfect knowledge of the transmitted data. Nevertheless, it is a good
approximation in practice, particularly in a turbo equalization context where the reliability
of the decisions at the output of the decoder increases along the iterations which, in its turn,
improves the equalization operation.
