Nonlinear adaptive algorithms for robust signal processing and communications in impulsive environments

Statistical signal processing and communication theory have traditionally been dominated by the Gaussian distribution model for the underlying random processes. However, a large number of physical processes are impulsive in nature, and are more accurately modeled by distributions with heavier-than-Gaussian tails in their probability density functions. Linear signal processing techniques, optimized under the Gaussian assumption, perform poorly in impulsive environments. There is, therefore, a strong motivation for the development of robust nonlinear signal processing techniques for impulsive non-Gaussian environments.Weighted Myriad Smoothers have been proposed recently as a class of robust, nonlinear filters based on the so-called alpha-stable distributions, which have heavier density tails than those of the Gaussian distribution, and include the Gaussian distribution as a special limiting case. However, weighted myriad smoothers are severely limited by their constraint of non-negative filter weights, which makes them unusable in bandpass and highpass filtering applications requiring negative weights. In this dissertation, we generalize the weighted myriad smoother into a class of Myriad Filters that admit real-valued weights. We also develop Fast Computation Algorithms that accurately generate the outputs of the different myriad filters. These filters constitute a robust generalization of linear filtering, with efficient performance in impulsive noise environments.The central theme of this dissertation is the development of efficient techniques for the optimization of the parameters (weights) of the different myriad filters. The filters are designed to optimally estimate a desired signal according to some statistical error criterion we have considered the mean square error (MSE) as well as the mean absolute error (MAE) criteria. We first derive necessary conditions to be satisfied by the optimal filter weights. We then develop nonlinear adaptive algorithms for the optimization of the filter weights. An issue of great concern in any adaptive filtering algorithm is the design of the adaptation step-size parameter that controls the convergence rate and steady-state filtering error of the algorithm. We develop a general class of nonlinear normalized adaptive filtering algorithms that are applicable to a wide variety of nonlinear filters. These algorithms have fast convergence properties, while eliminating the problem of step-size design inherent in any adaptive algorithm.