Recent results in applications and processing of α-stable-distributed time series

Symmetric, α-stable random variables and 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 paper is intended as a comprehensive review of the fundamental concepts and results of signal processing with α-stable processes with emphasis placed on acquainting engineers with this emerging discipline in signal processing and revealing its potential applications. In particular, we start with deriving α-stable models for impulsive noise. This 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 present new, fast algorithms for estimation of the parameters of α-stable interference and address two signal detection problems. These problems also build intuition on the differences between Gaussian and non-Gaussian, α-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, α-stable assumption rather than on a Gaussianity assumption. In the paper, we follow a presentation style that emphasizes only the highlights of the field and omit the fine mathematical details. However, we have included a large number of references to the literature for the interested reader to study further.