Digital Signal Processing Reference
In-Depth Information
to illustrate the application of AG in an example related to the scope of
this topic. In this example, we employ a GA to perform the search for the
equalizer that minimizes the CM cost function, discussed in Chapter 4.
Example 8.1 Blind Equalization Using Genetic Algorithms
Let us consider two nonminimum phase channels
0.4
z
−1
0.9
z
−2
1.4
z
−3
h
1
(
z
)
=
1
+
+
+
(8.3)
and
1.2
z
−1
0.3
z
−2
0.8
z
−3
h
2
(
z
)
=
1
+
−
+
(8.4)
Each individual represents a possible equalizer, and real coding is employed.
The simulations were run using the following set of parameters:
•
Population size: 30 individuals
•
Number of crossovers per generation: 10
•
Probability of occurence of mutation: 0.1
•
Stopping criterion: 2000 generations
. Thus,
each individual in the population corresponds to a possible filter, and is repre-
sented by a vector with five elements. The solutions obtained in this scenario were
used to initialize a DD algorithm, which ideally converges to the closest Wiener
solution.
Notice that this test is carried out in a relatively small search space, and the
filter order cannot considerably reduce the MSE. Table 8.1 has been built from
the outcomes of 50 simulations. The frequency of each solution is also indicated,
i.e., the number of trials in which the algorithm converged to a given solution.
The results reveal that the method has always provided a solution rather close to
the global Wiener minimum, which clearly indicates a very good performance.
Let us now turn our attention to a larger and more complex search space with
an 8-tap filter, for which
Table 8.2
brings the corresponding results. We notice
that convergence to good minima was predominant, with an expressive global
convergence rate.
The last scenario is formed by
h
2
(
In a first test, a 5-tap filter is employed to the equalizer channel
h
1
(
z
)
and a 7-tap equalizer, an order high
enough to provide a condition close to the ZF one.
Table 8.3
shows the corre-
sponding results. Again, we have a good global convergence rate, thus revealing
once more the method efficiency.
z
)
GAs are probably the most widespread evolutionary approaches, but
there are many different paradigms of this class that can be efficiently used.
TABLE 8.1
Solution for a 5-Tap Equalizer for Channel
h
1
(
z
)
Solution
Residual MSE
Frequency
−
−
−
[0.2183,
0.1873,
0.0596,
0.2804,0.5892]
0.1751
100
