Digital Signal Processing, 10th Edition

14Adaptive Filtering

Adaptive filtering is used when we need to realize, simulate, or model a system whose characteristics develop over time. It leads to the use of filters whose coefficients change with time. The variations in the coefficients are defined by an optimization criterion and are realized according to an adaptation algorithm, both of which are determined depending on the application. There are many different possible criteria and algorithms [1-4]. This chapter examines the simple but, in practice, most important case in which the criterion of mean square error minimization is associated with the gradient algorithm.

While fixed coefficient filtering is generally associated with specifications in the frequency domain, adaptive filtering corresponds to specifications in time. It is natural to introduce this subject by considering the calculation of filter coefficients in these conditions. We begin by examining FIR filters.

14.1 Principle of Adaptive Filtering

The principle of adaptive filtering is illustrated in Figure 14.1. It consists of processing the input signal x(n) to produce an output , whose difference with the reference y(n) is minimized. For every new set of data, reference, and input signal, the coefficients of the filter, assumed to be of FIR type and represented by the vector H(n), are updated.

At time n, assuming n data have been received, the cost ...