Adaptive Filters

CHAPTER 46 Robustness Properties

This his chapter deals with robustness analysis, as opposed to robust filter design. While the presentation in Chapter 45 was concerned with a framework for designing robust filters and applying it to the study of LMS and ∈-NLMS, the results obtained therein are not immediately applicable to studying the robustness performance of other adaptive filters. To do so, in this chapter we resort to the same energy-conservation arguments that we employed in Parts IV {Mean-Square Performance) and V {Transient Performance) while studying the performance of adaptive filters. As a byproduct, we shall gain further insights into the ro bustness performance not only of LMS and ∈–NLMS, but of other adaptive filters as well. Besides providing a more intuitive route to the robustness results of Chapter 45, the energy arguments also lead to tighter robustness bounds. The discussion in this chapter is self-contained and it approaches the subject of robustness from first principles; the presentation does not rely on the indefinite least-squares theory developed in the previous two chapters.

46.1 ROBUSTNESS OF LMS

Let us start with the LMS algorithm in order to illustrate the main ideas. Consider again measurements {d(i)} that arise from a model of the form

(46.1)

for some unknown weight vector w° and unknown disturbance v(i). The LMS algorithm estimates w° recursively ...