Adaptive Filters

Summary and Notes

The chapters in this part describe a notion of robustness and a procedure for designing robust filters by relying on indefinite-least-squares theory.

SUMMARY OF MAIN RESULTS

1. A recursive procedure similar to RLS is derived for time-updating the solution of an indefinite least-squares problem. Algorithm 44.1 does so for the general case in which the observation data and the regression data are allowed to be vectors and matrices, respectively. In contrast to the block RLS solution of Prob. VII.36, for example, we now require the successive P_i to be positive-definite in order for the successive weight vectors to correspond to global minima.
2. There are many notions of robustness. The one described in Chapter 45 requires that we bound the energy gain from disturbances to estimation errors at each iteration. This formulation reduces the design of robust filters to the solution of indefinite least-squares problems.
3. Two kinds of robust filters are designed: a posteriori filters and a priori filters. It turns out that ϵ-NLMS is a special case of the former while LMS is a special case of the latter. It is further argued in the text and in Prob. XI.5 that LMS and ϵ –NLMS are optimal robust filters.
4. Chapter 46 develops a framework for robustness analysis by using the same energy conser vation arguments that we employed earlier in Parts IV (Mean-Square Performance) and V (Transient Performance). This framework allows us to examine the robustness of a larger class of adaptive ...