Improving the tracking capability of adaptive filters via convex combination

As is well known, Hessian-based adaptive filters (such as the recursive-least squares algorithm (RLS) for supervised adaptive filtering, or the Shalvi-Weinstein algorithm (SWA) for blind equalization) converge much faster than gradient-based algorithms [such as the least-mean-squares algorithm (LMS) or the constant-modulus algorithm (CMA)]. However, when the problem is tracking a time-variant filter, the issue is not so clear-cut: there are environments for which each family presents better performance. Given this, we propose the use of a convex combination of algorithms of different families to obtain an algorithm with superior tracking capability. We show the potential of this combination and provide a unified theoretical model for the steady-state excess mean-square error for convex combinations of gradient- and Hessian-based algorithms, assuming a random-walk model for the parameter variations. The proposed model is valid for algorithms of the same or different families, and for supervised (LMS and RLS) or blind (CMA and SWA) algorithms.