Alpha-stable impulsive interference: Canonical statistical models and design and analysis of maximum likelihood and moment-based signal detection algorithms

Abstract

Symmetric, alpha-stable distributions and random processes have, recently, been receiving increasing attention from the signal processing and communication communities as statistical models for signals and noises that contain impulsive components. This Chapter is intended as a comprehensive review of the fundamental concepts and results of signal processing with alpha-stable processes with emphasis placed on acquainting its readers with this emerging discipline and revealing potential applications. More specifically, we start with summarizing the key definitions and properties of symmetric, alpha-stable distributions and the corresponding random processes and proceed to derive alpha-stable models for impulsive noise. The derivation serves as an illustration of the Generalized Central Limit Theorem, which states that the first-order distributions in all observed time series follow, to a higher or lesser degree, a stable law. We proceed to address two detection problems as examples of the methodologies required for designing robust processing algorithms for non-Gaussian, alpha-stable-distributed signals. These problems also build intuition on the differences between Gaussian and non-Gaussian, alpha-stable signal processing, as well as indicate the performance gains that are to be expected if the signal processing algorithms are designed on the basis of a non-Gaussian, alpha-stable assumption rather than on a Gaussianity assumption. A large number of references to the literature are included for the interested reader to study further.