Adaptive Filters

CHAPTER 22 Weighted Energy Conservation

As is evident by now, adaptive filters are time-variant and nonlinear stochastic systems with inherent learning and tracking abilities. The success of their learning mechanism can be measured in terms of how well they learn the underlying signal statistics given sufficient time (i.e., in terms of their steady-state performance) and in terms of how fast and how stably they adapt to changes in the signal statistics (i.e., in terms of their transient and convergence performance). For this reason, it is customary to study the performance of adaptive filters by examining their transient performance and their steady-state performance. The former is concerned with the stability and convergence rate of an adaptive scheme, while the latter is concerned with the mean-square error that remains in steady-state.

In Part IV (Mean-Square Performance) we focused on the steady-state performance of adaptive filters in both stationary and nonstationary environments, as well as on the performance degradation that occurs in finite-precision implementations. In this part we turn our attention to the transient performance of adaptive filters. We continue to rely on the energy conservation arguments of Chapter 15 and show how these same arguments can be used to perform transient analysis in a uniform manner across different algorithms. Compared with the derivations in Chapters l5, and for reasons explained below, it will turn out that transient analysis is more ...