Multichannel L filters based on marginal data ordering

The extension of single-channel nonlinear filters whose output is a linear combination of the order statistics of the input samples to the multichannel case is presented in the paper. The subordering principle of marginal ordering (M-ordering) is used for multivariate data ordering. Assuming a multichannel signal corrupted by additive white multivariate noise whose components are generally correlated, the coefficients of the multichannel L filter based on marginal ordering are chosen to minimize the output mean-squared-error (MSE) either subject to the constraints of unbiased or location-invariant estimation or without imposing any constraint. Both the case of a constant multichannel signal corrupted by additive white multivariate noise as well as the case of a nonconstant signal is considered. In order to test the performance of the designed multichannel marginal L filters, long-tailed multivariate distributions are required. The derivation and design of such a distribution, namely, the Laplacian (biexponential) distribution that belongs to Morgenstern's family in the 2D case is discussed. It is shown by simulation that the proposed multichannel L filters perform better than other multichannel nonlinear filters such as the vector median, the marginal α-trimmed mean, the marginal-median, the multichannel modified trimmed mean, the multichannel double-window trimmed mean, and the multivariate ranked-order estimator R_E proposed elsewhere as well as their single-channel counterparts