IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS { II : ANALOG AND DIGITAL SIGNAL PROCESSING 1 Design of Permutation Order Statistic

| In this paper we develop a group theoretic method for reducing the class cardinality of Permutation (P) lters. P lters are a recently introduced class of non-linear lters that have their roots in permutation theory and are based on the mapping x 7 ! x r , where x and x r are temporal{ordered and rank{ordered observation vectors, respectively. While P lters contain as a proper subset a large body of non{linear lters proposed to date, and have shown to be eeective at estimating non{stationary processes in non{Gaussian environments, the large number of parameters required to deene a P lter limits the lter's practical window size. Also, the information required to formulate an acceptable estimate is often only a subset of that contained in the permutation mapping x 7 ! x r. To reduce in a exible and systematic way the information contained in the permutation mapping, the concept of coloring is introduced. A coloring is an equivalence relation applied to rank{ ordering, temporal{ordering, or both temporal{ and rank{ ordering. These equivalences lead to the deenitions of Rank colored Permutation (RP), Temporal colored Permutation (T P), and Temporal{Rank colored Permutation (T RP) l-ters, respectively. The rank{ and temporal{order equivalences are shown to induce equivalences on the group of permutations. The induced permutation equivalences form a quantization of the permutation group. A distance bound on this quantization is derived as well as the cardinalities of RP; T P, and T RP lter classes. Additionally, symmetry constraints are formally developed and the number of lters in an arbitrarily constrained class counted. Several simulations are provided showing the eeect of coloring and demonstrating the performance of colored P lters.