Zero-order statistics: a signal processing framework for very impulsive processes

Techniques based on conventional higher-order statistics fail when the underlying processes become impulsive. Although methods based on fractional lower-order statistics (FLOS) have proven successful in dealing with heavy-tailed processes, they fail in general when the noise distribution has very heavy algebraic tails, i.e., when the algebraic tail constant is close to zero. In this paper we introduce a signal processing framework that we call zero-order statistics (ZOS). ZOS are well defined for any process with algebraic or lighter tails, including the full class of α-stable distributions. We introduce zero-order scale and location statistics and study several of their properties. The intimate link between ZOS and FLOS is presented. We also show that ZOS are the optimal framework when the underlying processes are very impulsive. All figures, simulations and source code utilized in this paper are reproducible and freely accessible in the Internet at http://www.ee.udel./edu/^~gonzalez/PUBS/HOS97a