Filter Cost Function Filter

This report addresses the problem of computation of the output of the Weighted Myriad Filter. Weighted Myriad Filters form a large and important class of nonlinear lters that includes linear FIR lters, with several potential applications in robust signal processing and communications in impulsive non-Gaussian environments. Just as the weighted mean and the weighted median are optimized for the Gaussian and Laplacian distributions, respectively, the weighted myriad is derived based on the properties of-stable distributions, which have been shown to accurately model practically occurring impulsive processes. The weighted myriad is an M-estimator that is deened in an implicit manner; no closed-form expression exists for its computation, and its direct computation is a non-trivial and prohibitively expensive task. In this report, the weighted myriad is formulated as one of the xed points of a certain mapping. An iterative algorithm is proposed to compute these xed points, and its convergence is proved rigourously. Fast algorithms for the weighted myriad are then developed, incorporating these xed point iterations. Numerical simulations serve to demonstrate that these algorithms compute the weighted myriad with a very high degree of accuracy, at a relatively low computational cost.