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in the context of which it might be desirable to employ techniques with a
significant global search potential, e.g., genetic algorithms, instead of those
based on the derivatives of the MSE surface.
This more complicate scenario is, to a certain extent, avoided whenever
one chooses a nonlinear structure that is linear with respect to its free param-
eters. In such case, it is possible to employ the classical supervised tools and
results discussed in this chapter in a quite direct manner. On the other hand,
to build a device that is both linear with respect to the parameters and effi-
cient may require some structural choices that are not always simple to deal
with. A suitable model to be used in this context is, for instance, a Volterra
filter [201].
The problem of nonlinear filtering is discussed in more detail in
Chapter 7, in which the above mentioned topics will be revisited. For now,
such brief comments on the theme are useful to illustrate how the straight-
forward idea of MSE optimization becomes intricate with the introduction of
nonlinear devices.
3.7 Linear Filtering without a Reference Signal
Throughout this chapter, the linear filtering problem has been established in
a supervised context, which is characterized by the presence of a desired or
reference signal
d
that guides the process of optimization and/or adap-
tation of the system at hand. The existence of a reference signal, together
with an appropriate criterion, leads to a linear solution and to a convex
cost function.
However, the explicit use of reference signal is not necessarily easy, or
even feasible in some practical problems. Such problems can be separated in
two very distinct classes:
(
n
)
•
The reference signal is indeed desired but unavailable, so that we
must obtain a certain amount of a priori knowledge about its nature
as well as its statistic properties. This is the essence of unsupervised
or blind processing, which normally leads to nonlinear optimization
problems and multimodal cost functions. This topic deals with this
scenario from the next chapter on.
•
The reference signal is not desired per se or even unnecessary for a
given task. So, it can be purposely replaced by a set of constraints
on the filter coefficients that make possible the optimization and/or
adaptation process. Such procedure can be understood as a kind of
“missing link” between the “separated worlds” of supervised and
unsupervised filtering.
