SEQUENCES WITH POSITIVE SEMIDEFINITE FOURIER-TRANSFORMS

A sequence is said to be positive if its Fourier transform exclusively takes on real nonnegative values as a function of frequency. Some fundamental positive sequence properties found in dispersed sources are first reviewed. Several new properties are then developed and used in turn 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.