On Characterizing Discrete Signals in Additive Noise - A Unified Treatment

摘要

A characterization and classification result is established which applies to Binomial, Negative Binomial or Poisson signals in additive noise. The result unifies and generalizes three separate characterization results appearing in the recent literature. The distributions of discrete signals in additive noise have been characterized via systems of differential equations satisfied by their probability mass functions in a series of recent papers. These papers have dealt with signal distributions belonging to various discrete exponential families. The present result relies on a new and general parametrization of a discrete family of distributions which includes all discrete convolutions of Binomial, Negative Binomial (Pascal) and Poisson distributions as special cases. The proof of our characterization and classification theorem differs radically from the individual proofs of the characterization results in the papers cited. Moreover, the theorem requires somewhat weaker assumptions than cummulatively contained in previous results. In particular, no moment conditions are required in the present result.