On the Multivariate Conditional Probability Density of a Vector Perturbed by Gaussian Noise

This correspondence examines the joint conditional probability density function (PDF) of the main variables (envelope, phase, and their eta-order time derivatives) of a time-varying random signal in the presence of additive Gaussian noise. The main variables are conditioned with respect to the given variables, which are the amplitude, phase, and their derivatives of the signal alone. We prove a theorem stating that some of the conditional PDFs of the main variables do not depend on some of the given variables. This theorem, together with Bayes's theorem, can substantially simplify the derivations of conditional PDFs and give alternative forms of them. Both theorems can also help in finding reasonable approximations, as we demonstrate for the phase and first time derivative of the envelope.

I. Introduction

Statistical properties of an information bearing narrowband signal perturbed by noise have been studied for decades beginning with the early works of Rice. In spite of this, as will become clear in the sequel, one of the important problems still remains unsolved.