A General Class of Nonlinear NormalizedAdaptive Filtering Algorithms 1

The Normalized Least Mean Square (NLMS) algorithm is an important variant of the classical LMS algorithm for adaptive linear ltering. It possesses many advantages over the LMS algorithm, including having a faster convergence and providing for an automatic time-varying choice of the LMS step-size parameter which aaects the stability , steady-state mean square error (MSE) and convergence speed of the algorithm. An auxiliary xed step-size that is often introduced in the NLMS algorithm has the advantage that its stability region (step-size range for algorithm stability) is independent of the signal statistics. In this paper, we generalize the NLMS algorithm by deriving a class of Nonlinear Normalized LMS-type (NLMS-type) Algorithms that are applicable to a wide variety of nonlinear lter structures. We obtain a general nonlinear NLMS-type algorithm by choosing an optimal time-varying step-size which minimizes the next-step MSE at each iteration of the general nonlinear LMS-type algorithm. As in the linear case, we introduce a dimensionless auxiliary step-size whose stability range is independent of the signal statistics. The stability region could therefore be determined empirically for any given nonlinear lter type. We present computer simulations of these algorithms for two speciic nonlinear lter structures: Volterra Filters, and the recently proposed class of Myriad Filters. These simulations indicate that the NLMS-type algorithms, in general, converge faster than their LMS-type counterparts.