Adaptive Filters

CHAPTER 20 Nonstationary Environments

A fundamental feature of adaptive filters is their ability to track variations in the underlying signal statistics. This is because by relying on instantaneous data, the statistical properties of the weight vector and error signals are able to react to changes in the input signal properties. The purpose of this chapter, and the next one, is to characterize the tracking ability of adaptive filters for nonstationary environments.

We shall continue to rely on the energy-conservation framework of Chapter 15 and use it to derive expressions for the excess mean-square error of an adaptive filter when the input signal properties vary with time. The presentation will reveal that there are actually minor differences between mean-square analysis and tracking analysis and that, in particular, tracking results can be obtained almost by inspection from the mean-square results of the prior chapters.

20.1 MOTIVATION

In order to motivate our setup for tracking analysis, we start by reviewing the basic linear least-mean-squares estimation problem of Sec. 15.1. Thus, let d and u be zero-mean random variables with second-order moments

The coefficient vector w° that estimates d from u optimally in the linear least-mean-squares sense, i.e., the vector that solves

(20.1)

is given by The corresponding minimum cost is

(20.2)

In Chapter 10 we developed stochastic-gradient ...