Adaptive weighted myriad filter algorithms for robust signal processing in α-stable noise environments

Stochastic gradient-based adaptive algorithms are developed for the optimization of weighted myriad filters (WMyFs). WMyFs form a class of nonlinear filters, motivated by the properties of α-stable distributions, that have been proposed for robust non-Gaussian signal processing in impulsive noise environments. The weighted myriad for an N-long data window is described by a set of nonnegative weights {w_i }_i=l^N and the so-called linearity parameter K>0. In the limit, as K→∞, the filter reduces to the familiar weighted mean filter (which is a constrained linear FIR filter). Necessary conditions are obtained for optimality of the filter weights under the mean absolute error criterion. An implicit formulation of the filter output is used to find an expression for the gradient of the cost function. Using instantaneous gradient estimates, an adaptive steepest-descent algorithm is then derived to optimize the weights. This algorithm involves a very simple update term that is computationally comparable to the update in the classical LMS algorithm. The robust performance of this adaptive algorithm is demonstrated through a computer simulation example involving lowpass filtering of a one-dimensional chirp-type signal in impulsive noise