Digital filtering and prolate functions

Abstract

A class of trigonometric polynomialsp(x) = \sum_{n=-N}^{N} a_{n} e^{j n \pi x}of unit energy is introduced such that their energy concentration\alpha = \int_{-e}^{e} p^{2}(x) dxin a specified interval(- \epsilon, \epsilon)is maximum. It is shown that the coefficientsa_{n}must be the eigenvectors of the system\sum_{m=-N}^{N} \frac{\sin (n - m)\pi \epsilon}{(n - m)\epsilon} a_{m} = \lambda a_{n}. corresponding to the maximum eigenvalue X. These polynomials are determined forN = 1, \cdots , 10and\epsilon = 0.025, \cdots , 0.5. The resulting family of periodic functions forms the discrete version of the familiar prolate spheroidal wave functions.