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Applications in Adaptive Filtering 595
identification [21]; (3) adaptive noise canceling techniques, in which an
adaptive filter is used to estimate and eliminate a noise component in
some desired signal [27, 9, 15]; and (4) system modeling, in which an
adaptive filter is used as a model to estimate the characteristics of an unknown
system. These are just a few of the best known examples on the use
of adaptive filters.
Although both IIR and FIR filters have been considered for adaptive
filtering, the FIR filter is by far the most practical and widely used.
The reason for this preference is quite simple. The FIR filter has only
adjustable zeros, and hence it is free of stability problems associated with
adaptive IIR filters that have adjustable poles as well as zeros. We should
not conclude, however, that adaptive FIR filters are always stable. On the
contrary, the stability of the filter depends critically on the algorithm for
adjusting its coefficients.
Of the various FIR filter structures that we may use, the direct form
and the lattice form are the ones often used in adaptive filtering applications.
The direct form FIR filter structure with adjustable coefficients
h(0),h(1),...,h(N − 1) is illustrated in Figure 11.1. On the other hand,
the adjustable parameters in an FIR lattice structure are the reflection
coefficients K n shown in Figure 6.18.
An important consideration in the use of an adaptive filter is the
criterion for optimizing the adjustable filter parameters. The criterion
must not only provide a meaningful measure of filter performance, but it
must also result in a practically realizable algorithm.
One criterion that provides a good measure of performance in adaptive
filtering applications is the least-squares criterion, and its counterpart
in a statistical formulation of the problem, namely, the mean-square-error
(MSE) criterion. The least squares (and MSE) criterion results in a quadratic
performance index as a function of the filter coefficients, and hence
it possesses a single minimum. The resulting algorithms for adjusting the
coefficients of the filter are relatively easy to implement.
FIGURE 11.1
Direct-form adaptive FIR filter
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