Constrained adaptive LMS L-filters

Abstract

Two novel adaptive nonlinear filter structures are proposed which are based on linear combinations of order statistics. These adaptive schemes are modifications of the standard LMS algorithm and have the ability to incorporate constraints imposed on coefficients in order to permit location-invariant and unbiased estimation of a constant signal in the presence of additive white noise. The convergence in the mean and in the mean square of the proposed adaptive nonlinear filters is studied. The rate of convergence is also considered. It is verified by simulations that the independence theory provides useful bounds on the rate of convergence. The extreme eigenvalues of the matrix which controls the performance of the location-invariant adaptive LMS L-filter are related to the extreme eigenvalues of the correlation matrix of the ordered noise samples which controls the performance of other adaptive LMS L-filters proposed elsewhere. The proposed filters can adapt well to a variety of noise probability distributions ranging from the short-tailed ones (e.g. uniform distribution) to long-tailed ones (e.g. Laplacian distribution).