Digital Signal Processing Reference
In-Depth Information
2
2.5
c
5
c
1
SNR = 8 dB
2
d
=1
1.5
c
5
c
1
1.5
SNR = 3 dB
1
d
=2
c
7
1
c
3
c
7
c
3
0.5
0.5
0
0
c
6
c
2
c
2
c
6
−0.5
−0.5
−1
c
8
c
4
−1
c
8
c
4
SNR = 5 dB
−1.5
−1.5
−2
d
=0
SNR = 25 dB
−2
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−2
−1
0
1
2
(a)
x
(
n
)
(b)
x
(
n
)
FIGURE 7.5
Decision boundaries for different (a) SNRs and (b) equalization delays. The (
+
) sign indicates a
center associated with the transmission of a
−
1 symbol.
In a situation of this sort, the problem of building the optimal equalizer
can be directly understood in terms of finding the separating surface that
leads to the smallest possible SER. In fact, the nonlinear mapping previously
provided in Figure 7.4b corresponds to the Bayesian equalizer that recovers
the transmitted signal without delay and for SNR
15 dB.
In Figure 7.5a, we present the centers shown in Table 7.1, numbered and
labeled according to the choice
d
=
0, and the decision boundaries between
the two classes generated by the Bayesian equalizer. The boundary reveals
that for the adopted delay, an efficient equalizer must have a strongly non-
linear character. It is also interesting to notice that under the conditions in
question, the boundary, in some regions, lies approximately in the middle of
a pair of centers that are the nearest neighbors to each other. Another inter-
esting point is that the decision boundary does not seem to be very sensitive
to noise if SNR
=
8dB. There is a slight difference between the boundaries
corresponding to SNR
>
=
8dBandSNR
=
25dB, and they asymptotically tend
to a composition of line segments.
The decision boundaries for different equalization delays are shown in
Figure 7.5b. In this case, the decision boundaries are very different from each
other. For
d
0, the channel states can be correctly classified only if the deci-
sion boundary is nonlinear, while for
d
=
2, it should be possible
to classify the channel states by means of a linear boundary. This example
illustrates that even for a linear channel, equalization may be a nonlinear
classification problem.
The nonlinear nature of the Bayesian equalizer, even for linear channels,
highlights the fact that nonlinear filtering structures could be considered to
approximate the mapping provided by the optimal equalizer. One possibility
of this sort is to employ artificial neural networks, to be discussed in the
following.
=
1and
d
=
