Unified approach to adaptive filters and their performance

A streamlined theory is presented for adaptive filters within which the major adaptive filter algorithms can be seen as special cases. The algorithm development part of the theory involves three ingredients: a preconditioned Wiener Hopf equation, its simplest possible iterative solution through the Richardson iteration, and an estimation strategy for the autocorrelation matrix, the cross-correlation vector and a preconditioning matrix. This results in a generalised adaptive filter in which intuitively plausible parameter selections give the major adaptive filter algorithms as special cases. This provides a setting where the similarities and differences between the many different adaptive filter algorithms are clearly and explicitly exposed. Based on the authors´ generalised adaptive filter, expressions for the learning curve, the excess mean square error and the mean square coefficient deviation are developed. These are general performance results that are directly applicable to the major families of adaptive filter algorithms through the selection of a few parameters. Finally, the authors demonstrate through simulations that these results are useful in predicting adaptive filter performance.