Affine combinations of adaptive filters

We extend the analysis presented in for the affine combination of two least mean-square (LMS) filters to allow for colored inputs and nonstationary environments. Our theoretical model deals, in a unified way, with any combinations based on the following algorithms: LMS, normalized LMS (NLMS), and recursive-least squares (RLS). Through the analysis, we observe that the affine combination of two algorithms of the same family with close adaptation parameters (step-sizes or forgetting factors) provides a 3 dB gain in relation to its best component filter. We study this behavior in stationary and nonstationary environments. Good agreement between analytical and simulation results is always observed. Furthermore, a simple geometrical interpretation of the affine combination is investigated. A model for the transient and steady-state behavior of two possible algorithms for estimation of the mixing parameter is proposed. The model explains situations in which adaptive combination algorithms may achieve good performance.