Sliding-Window RLS Low-Cost Implementation of Proportionate Affine Projection Algorithms

Abstract

This paper addresses adaptive filtering for sparse identification. Proportionate affine projection algorithms (PAPAs) are known to be efficient techniques for this purpose. We show that the PAPA performance may improve with an increase in the projection order M (for example, such as M = 512), which, however, also results in an increased complexity; the complexity is in general O(M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> N) or at least O(MN) operations per sample, where N is the filter length. We show that PAPAs are equivalent to specific sliding-window recursive least squares (SRLS) adaptive algorithms with time-varying and tap-varying diagonal loading (SRLS-VDLs). We then propose an approximation to the SRLS-VDLs based on dichotomous coordinate descent (DCD) iterations with a complexity of O(N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> N), which does not depend on M; it depends on the number of DCD iterations N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> , which as we show can be significantly smaller than M, thus allowing a low-complexity implementation of PAPA adaptive filters.