Real Domain Adaptive WOS Filtering using Neural Network Approximations - Nonlinear Digital Signal Processing, 1993. IEEE Winter Workshop on

The problem of optimal Weighted Order Statistics (WOS) filters design is first related to the optimal design in a larger class, called Variable Rank Order Statistics (VROS) filters, the results obtained p r e viding guidelines for the approach to be used in WOS filtering. Since adaptive methods applied directly to WOS filter models have to cope with a very ill conditioned problem, the adaptation will act on a model which belongs to Neural Networks (NN) class. This particular neural mcdel can be trained using an algorithm very similar to the classical Backpropagation algorithm. In the final stage of training, the neural model can be made arbitrarily close to a WOS filter. 1. WOS Filters and Optimal Design Problem 1.1 WOS and VROS filters The archetype of WOS filter is tlie order statistics (OS) filter, which processcs a t any time t the input values inside a window X ( t ) of length N = N1 + Nz + 1, including tlre c u i r c ~ t input z(6), X ( t ) = [z( t N , ) . . . z(6) . . . z(t + N2)] = [ X , s 2 . . . X N ] ( 1 ) (1) First, the values i n tlie window are ordered decreasingly, resulting the ordered vector 8x’(t) = [ X ( l ) . .x(N)](t) = [xi, . . . s , , ] ( f ) (2) where X ( k ) denotes tlie k’tli value in tlie ordered set. This operation can be thought of as being irnplemented by the ordering operator 0, using the permutation