Accurate approximation of a 0th order discrete prolate spheroidal sequence for filtering and data tapering

Abstract

The index-limited 0th order discrete prolate spheroidal sequence (DPSS) is very useful as both a data taper for spectral analysis and as a FIR filter since its frequency response has very low sidelobes. However its calculation is not straightforward. Kaiser produced a Bessel approximation to the continuous prolate spheroidal wave function. This note discusses how to sample the continuous Bessel expression in order to approximate the 0th order DPSS. The obvious approximation, with end point which involve the modified Bessel function evaluated at zero, is not the best. A much better result is obtained by slightly altering the sampling positions. A range of values for sample size and bandwidth are used to compare the recommended approximation with the actual 0th order DPSS. Differences are expressed in terms of sum of squared errors, by crossplotting corresponding sequence values, and by comparing magnitude squared transfer functions. In all cases the recommended approximation performs well.