Sequences with positive semidefinite Fourier transforms

Abstract

A sequence is said to be positive if its Fourier transform exclusively takes on real nonnegative values as a function of frequency. Positive sequences play a prominent role in contemporary signal processing and system theory. To illustrate this point, it is well known that the factorization theorem is extensively used in studies related to wide-sense stationary random signals and linear systems. The ability to appropriately factorize a Fourier transform is contingent on that transform being positive semidefinite. This paper is partially tutorial in that some fundamental positive sequence properties found in dispersed sources are first reviewed. This is followed by the development of several new properties. These properties are in turn used to develop an efficient algorithm for finding that positive sequence which lies closest to a given nonpositive sequence in the least-squares error sense. Interest in this approximation problem arises from the fact that although a given sequence may be theoretically positive, practical considerations often result in its realization being nonpositive. For instance, unbiased autocorrelation lag estimates can lead to nonpositive spectral density function estimates.