Finite-memory denoising in impulsive noise using Gaussian mixture models

We propose an efficiently structured nonlinear finite-memory filter for denoising (filtering) a Gaussian signal contaminated by additive impulsive colored noise. The noise is modeled as a zero-mean Gaussian mixture (ZMGM) process. We first derive the optimal estimator for the static case, in which a Gaussian random variable (RV) is contaminated by an impulsive ZMGM RV. We provide an analytical derivation of the resulting mean-squared error (MSE), and compare the performance to that of the optimal linear estimator, identifying cases of significant improvement. Building upon these results, we develop a suboptimal finite-memory filter for the dynamic case, which is nearly optimal in the minimum MSE sense. The resulting filter is a nonlinearly weighted combination of a fixed number of linear filters, for which a computationally efficient architecture is proposed. Substantial improvement in performance over the optimal linear filter is demonstrated using simulation results.