Adaptive Filters

CHAPTER 45 Robust Adaptive Filters

We now formulate a robust design criterion and proceed to devise adaptive filters that meet the desired robustness performance. The derivations and arguments in this chapter are similar in nature to the arguments we employed in Chapters 29 and 30 while studying recursive least-squares problems. The main distinction will be in the use of quadratic cost functions with indefinite weighting matrices, as opposed to positive-definite weights. A key conclusion from the discussion here will be that some of the adaptive filters that we encountered before, e.g., LMS and –NLMS, and which were derived in Chapters 10 and 11 by appealing to stochastic-gradient approximations, will now be shown to satisfy the adopted robustness measure. Actually, the arguments in the current chapter will allow us to motivate and derive these algorithms as optimal, as opposed to approximate, recursive solutions to well-defined optimization problems, in much the same way as the RLS algorithm was derived in Chapter 30 as the optimal recursive solution to a regularized least-squares problem.

45.1 A POSTERIORI-BASED ROBUST FILTERS

Thus, consider measurements {d(i)} that are related to an unknown vector w° via

(45.1)

where v(i) denotes an unknown disturbance and u_i is a row vector. ...