A common framework for transient analysis of adaptive filters

Employing a recently introduced framework in which a large number of adaptive filter algorithms can be viewed as special cases, we present a common framework for the analysis of their transient behavior. An important implication of this is that while the theoretical analysis can be performed for a generic filter coefficient update equation, the results are directly applicable to a large range of adaptive filter algorithms simply by specifying some parameters of the generic filter coefficient update equation. In particular we show that theoretical learning curves for the least mean square (LMS), normalized least mean square (NLMS), the affine projection algorithm (APA) and its relatives, as well as the recursive least squares (RLS) algorithm are obtained as special cases of a general result. The recently introduced fast Euclidean direction search (FEDS) algorithm is used as an example showing the usefulness and versatility of the proposed approach.