Maximum energy window with constrained spectral values

Abstract

The optimum energy criterion continuous and discrete prolate spheroidal wave function windows had a profound effect on the basic understanding and practical optimization of many problems in communication theory, signal processing, and antenna theory. However, in many situations, there is interest in not only packing maximum window energy in some frequency interval, but also interest in imposing precise spectral window values and spectral nulls over some specified frequencies. We have formulated and solved such an optimization problem in an analytical sense as well as in an efficient computational sense. The problem is first expressed as a constrained maximization of a normalized quadratic form w′ Aw/ w′ w, where A is a positive-definite matrix specified by the window energy concentration interval, w is the window weighting vector, and the constraining subspace is specified by the spectral window values and locations. This problem is then transformed to a nonconstrained maximization of a normalized quadratic form w′ PAPw/ w′ w, where P is a projection operator onto the orthogonal complement of the original constraining subspace. Persymmetric properties of A and PAP are used to reduce the computational complexity of the solution of the optimum window. Numerical maximization can be readily performed using the iterative power method. Specific examples are presented.