Cryptography Reference
In-Depth Information
11.1.5 MAP turbo equalization
MAP turbo equalization corresponds to the turbo equalization scheme originally
introduced by Douillard
et al.
[11.12]. In this section, we first present the
equations for implementing the SISO equalizer. The good performance of the
MAP turbo equalizer is illustrated by simulation. We also introduce solutions
of less complexity derived from the MAP criterion. Finally, we examine the
problems encountered during a circuit implementation of the turbo equalizer, as
well as potential applications of this reception technique.
Implementation of the BCJR-MAP equalizer
The MAP equalizer shown in Figure 11.11 takes at its input vector
y
of the
discrete symbols observed at the output of the channel, as well as
a priori
information denoted
L
a
(
x
)
on the coded interleaved bits. This information
comes from the channel decoder and is produced at the previous iteration. In
the particular case of the first iteration, we do not generally have any
a priori
information other than the hypothesis of equiprobability on the bits transmitted,
which leads us to put
L
a
(
x
i,j
)=0
.
Figure 11.11 - Block diagram of the MAP equalizer.
The purpose of the MAP equalizer is to evaluate the
a posteriori
LLR
L
(
x
i,j
)
on each coded interleaved bit
x
i,j
, defined as follows:
L
(
x
i,j
)=ln
Pr(
x
i,j
=1
|
y
)
(11.5)
Pr(
x
i,j
=0
|
y
)
Using conventional results in detection theory, we can show that this equal-
izer is optimal in the sense of the minimization of the symbol error probability.
To calculate the
a posteriori
LLR
L
(
x
i,j
)
, we will use the trellis representa-
tion associated with transmission on the frequency selective channel. Applying
Bayes' relation, the previous relation can also be written:
L
(
x
i,j
)=ln
Pr(
x
i,j
=1
,
y
)
Pr(
x
i,j
=0
,
y
)
(11.6)
Among the set of possible sequences transmitted, each candidate sequence
traces a single path in the trellis. The joint probability
Pr(
x
i,j
=0
or
1
,
y
)
can
then be evaluated by summing the probability
Pr(
s
,s,
y
)
of passing through a
particular transition in the trellis linking a state
s
at instant
i
−
1
to a state
s
