Some information theoretic and signal detection problems in the content of nearly Gaissian densities

Abstract

It is well known that the use of the Gaussian assoimption when the governing prohahility density function is only -nearly Gaussian causes serious errors in estimation and detection theory. Defining a nearly Gaussian pdf f(x) to "be of the foim f(x) = o£l.g(x) +ah(x) where 0<a<1, a + a = 1, g(x) is a pure Gaussian p.d.f. and h(x) some unknown p.d.f, v/e attempt to investigate the Gaussian assumption in Information Theory. The entropy of a random variable with such .continuous density f(x) is maximissed over sCLl densities h(x) which have specified variance. This result is used in determining the "behaviour of the capacity of an additive Gaussian channel whose input is from a nearly Gaussian source, and the Rate Distortion function of such a source. It is shown numerically that hoth of these show a significant decrease from the pvire Gaussian case. A rotust encoding scheme for discrete sources with in accurately known prohahilities of the formapj^ +a i=1 »...» N m where p^ and are probahilities, is presented and investigations of some other questions in Information Theory are made. The Fisher Information of a continuous nearly Gaussian density is minimized over all pdfs h(x) which have a specified variance. This result is used in the problem of detecting a cons-tant signal in nearly Gaussian noise. A non-linearity is obi^ajJied which is shown to provide an improvement of a known result in robist detection.