Signal Processing In Stable Noise Environments: A Least lp Norm Approach

This dissertation is concerned with the development of new optimal techniques for the solution of signal processing problems involving impulsive data. Although the signal processing and communications field has been dominated by the Gaussian distribution, it has been common knowledge that atmospheric noise, underwater acoustic noise, electro-magnetic disturbances on telephone lines and nancial time series showed an impulsive character which cannot be described by a Gaussian distribution. Recently, there has been great interest in the alpha-stable distribution. This thesis, in agreement with some of the recent work defends the alpha-stable model for impulsive data. Justications for the alpha-stable model are given and various analytical properties of these distributions are discussed. This discussion leads us to the minimum dispersion criterion which is the analogue of the minimum mean squared error criterion for alpha-stable distributed data. Based on the minimum dispersion criterion, we suggest new least lp-norm estimation techniques as opposed to least squares estimation for the estimation of linear AR alpha-stable processes. These techniques are also extended to polynomial estimation techniques which we have employed in nonlinear AR alpha-stable process estimation and impulsive noise cancellation. The main drawback of alpha-stable distributions is the fact that the alpha-stable probability density function does not have a compact analytical form, which has prevented the use of maximum-likelihood and Bayesian estimation techniques for a long time. In this thesis, we introduce the first numerically stable analytical model for the alpha-stable probability density function. We give an extensive list of possible applications opened up by this model. In particular, based on the new analytical representation, we introduce a near-optimal receiver for the detection of signals in apha-stable distributed noise. Simulation results demonstrate the impressive success of the new techniques and show that very signicant gains are obtained via nonlinear estimation techniques when compared to linear ones. This indicates the need for further research in the properties of nonlinear regressions of alpha-stable random variables. The thesis concludes with an account of future research directions in signal processing with alpha-stable distributions.