Recent Advances in Adaptive Filtering

Adaptive filters are inherently nonlinear and time-variant devices that adjust themselves to an ever-changing environment; the structure of an adaptive system changes in such a way that its performance improves through a continuing interaction with its surroundings.

The performance of an adaptive filter is generally measured in terms of its transient behavior and its steady-state behavior. The former provides information about the stability and the convergence rate of an adaptive filter, while the latter provides information about the mean-square-error of the filter once it reaches steady-state.

There have been numerous ingenious works in the literature on the performance analysis of adaptive filters. In general, these works tend to study different adaptive filters separately due to the variations that exist in their update equations. Such distinct treatments tend to obscure commonalities that exist among algorithms.

This talk provides an overview of an energy-based approach to the performance analysis of adaptive filters. The framework is based on studying the energy flow through each iteration of an adaptive filter and establishing a fundamental energy conservation relation. By specializing the relation to the cases of steady-state operation, tracking, and transient operation, and by formulating state-space models in terms of moment variables, steady-state results as well as stability and robustness results can be obtained uniformly across algorithms. This point of view not only allows a unified treatment, but it also allows to obtain performance results independent of input signal statistics and to study algorithms for which earlier analysis are not complete.