Nonlinear filters with symmetry constraints

Nonrecursive digital filters are popular because they are inherently stable, unlike their recursive counterparts, and have been shown to be effective in a wide variety of situations. For example, Astola and Kuosmanen describe 20 different classes of nonrecursive filters [2], many of which are included in the general class of nonlinear smoothing filters considered by Mallows [11]. This paper defines three classes of nonrecursive filters based on symmetry restrictions and explores some of the consequences of these restrictions. Specific examples of each class are presented, general procedures are given for constructing new filters in each class from known examples, and the influence of these symmetries on root sequences is examined. 1. PROBLEM FORMULATION Consider the moving average filter of width 2K+1 centered at k defined by yk = (xk K ; : : : ; xk; : : : ; xk+K) (wk) (1) and define J as the (2K+1) (2K+1) permutation matrix with 1’s on the cross-diagonal (lower left to upper right). This paper considers the following three filter classes: PK : (Pwk) = (wk) for all permutationsP RK : (Jwk) = (wk) CK : (Jwk) = (wk). Note that Jwk represents a time-reversal of the data in wk, motivating the notation RK , and matrices A satisfying the condition JAJ = A are called centroskew, motivating the notation CK . Since filters in class PK are invariant under all permutations of the data window wk and filters in class RK are invariant under the specific permutation J, it follows immediately that PK RK . Conversely, note that if 2 RK \ CK , then (wk) = (wk) = 0 for all wk. Overall, the classes PK , RK and CK include many popular filter classes, either entirely or in part. 2. THE FILTER CLASS PK The class PK consists of the FIR filters of the form (1) that are symmetric with respect to all of the arguments xk j in the data window wk. Note that this class includes all Lfilters with constant weights fwig: