PERMUTATION FILTERS - A CLASS OF NONLINEAR FILTERS BASED ON SET PERMUTATIONS

In this paper, we introduce and analyze a new class of nonlinear filters that have their roots in permutation theory. We show that a large body of nonlinear filters proposed to date constitute a proper subset of permutation filters (P filters). In particular, rank-order filters, weighted rank-order filters, and stack filters embody limited permutation transformations of a set. Indeed, by using the full potential of a permutation group transformation, we can design very efficient estimation algorithms. Permutation groups inherently utilize both rank-order and temporal-order information; thus, the estimation of nonstationary processes in Gaussian/nonGaussian environments with frequency selection can be effectively addressed. An adaptive design algorithm that minimizes the mean absolute error criterion is described as well as a more flexible adaptive algorithm that attains the optimal permutation filter under a deterministic least normed error criterion. Simulation results are presented to illustrate the performance of permutation filters in comparison with other widely used filters.