Cryptography Reference
In-Depth Information
When Gray mapping rules are used, experience shows that we can reduce the
complexity of the demapping by ignoring the
a priori
information coming from
the decoder in the equations above
10
, without really affecting the performance
of the device. On the other hand, this simplification no longer applies when
we consider other mapping rules, like the
Set Partitioning
rule used in coded
modulation schemes. This point has been particularly well highlighted in [11.14]
and [11.30].
This completes the description of the soft-input soft-output linear MMSE
equalizer. Finally, we can note that unlike the BCJR-MAP equalizer, the com-
plexity of the SISO mapping and demapping operations increases linearly (and
not exponentially) as a function of size
M
of the constellation and of the number
L
of taps in the impulse response of the discrete-time equivalent channel model.
Adaptive implementation of the equalizer
Historically, the first MMSE turbo equalizer was proposed in 1997, directly in
adaptive form [11.20, 11.32]. The closed-form expression (11.40) enabling the
computation of the equalizer coecients from the knowledge of the channel im-
pulse response was not known at that time. The chosen solution thus involved
determining the filter coecients in an adaptive manner, using stochastic gra-
dient descent algorithms aiming at minimizing the mean square error between
the transmitted data and the equalizer output. As we shall see in the following,
when evaluating performance, the adaptive MMSE turbo equalizer remains a
very interesting solution for time-invariant or slowly time-varying channels. The
purpose of this section is to show that, for such channels, the adaptive MMSE
equalizer and the MMSE equalizer proposed in (11.40) have similar performance
and characteristics.
The structure of the considered equalizer is shown in Figure 11.15 (imple-
mentation (2)). An adaptive procedure is used to obtain the filters' coecients.
This adaptive algorithm is composed of two distinct phases: the training phase
and the tracking phase. The training phase makes use of sequences known by the
receiver (training sequences) to initialize the equalizer coecients. Next, during
the tracking period, the coecients are continuously updated in a decision-
directed manner, based on the receiver estimate of the transmitted sequence.
Adaptive algorithms involve determining, for each symbol entering the equal-
izer, output
z
i
from the following relation:
z
i
=
f
i
y
i
−
g
i
x
i
(11.55)
where
y
i
=(
y
i
+
F
,...,y
i−F
)
T
is the vector of channel output samples and
x
i
=(
x
i
+
G
,...,x
i−
Δ+1
,
0
, x
i−
Δ
−
1
,...,x
i−G
)
T
is the vector of estimated sym-
bols, with respective lengths
2
F
+1
and
2
G
+1
. Note that the coordinate relative
10
This amounts to assuming the transmitted symbols to be equiprobable, i.e. to putting
P
a
(
X
l
)=1
/M
whatever the symbol and the iteration considered.
