The Statistical Properties of Filtered Signals

Abstract

In this paper, the statistical properties of filtered or convolved signals are considered by deriving the resulting density functions as well as the exact mean and variance expressions given a prior knowledge about the statistics of the individual signals in the filtering or convolution process. It is shown that the density function after linear convolution is a mixture density, where the number of density components is equal to the number of observations of the shortest signal. For circular convolution, the observed samples are characterized by a single density function, which is a sum of products. Keywords—Circular Convolution, linear Convolution, mixture density function. NOTATION A signal is a group of observations, and these are represented in vector form. For example, xi(n) = [xi(1), xi(2), · · · , xi(Ki)] is a vector for the i signal, for i [1, N ], whose observations are xi(n), for n [1,Ki], where Ki is the length of the i signal. Given a vector xi(n), the variable Xi (capital letter with a corresponding subscript) is defined where its possible values are the vector elements or observations, and for Xi we define the density function p(xi) summarizing the statistical properties of its observations.