A switched variable step size NLMS adaptive filter

Abstract

This paper studies the statistical behavior of a switched variable step size Normalized Least Mean Square adaptive filter. The purpose of the switching is to obtain a Normalized Least Mean Square adaptive filter combination with fast convergence and small steady-state mean-square weight deviation (MSD). Previous works studied variable step-size Least Mean Square (LMS) and Normalized Mean Square (NLMS) adaptive filters using a measured quantity (say instantaneous squared error) in a nonlinear recursion for varying the step size. Here, it is proposed to avoid the nonlinear recursion in a novel way that is based upon a simple physical premise: For several practical applications, what is required is a faster initial convergence rate phase, followed by a slower convergence rate period that allows for a desired steady-state performance. The fast NLMS transient behavior is obtained with a fixed step size. After convergence, the step size is reduced to a smaller value to satisfy a desired tradeoff between the second transient and the desired steady-state MSD. Thus, during the first transient, the Switched VSS-NLMS adaptive weights will converge faster or as fast as any VSS-NLMS algorithm. Of course, during the second phase, some variable step size algorithms may eventually outperform the Switched VSS-NLMS algorithm. However, in many cases, this can be a small price to pay in exchange for the theoretical and computational simplicity. No new theory is needed to predict the behavior of the nonlinear recursion for the step size. Furthermore, the analytical theory for the MSD of the new scheme is well-known.