Sampling theorems using nonuniform sample points and interpolation functions with band‐limited continuous spectra

Abstract

AbstractThis paper presents a unified theory for the nonuniform sampling theorem for the low‐pass band‐limited waveform. In most practical cases, the mean interval of the sampling points is in the so‐called oversampling state. One can also assume that there exists a certain period in the sampling point sequence. Those properties of the sampling point sequence are assumed in this paper. Yen has derived a sampling theorem for the case where the sampling‐point sequence has a period. In his theorem, how ever, the interpolation function has a spectrum which is a stepwise discontinuous function, not suited to the realization of the interpolation function by a digital or other similar filters. One of the greatest features of the sampling theorem proposed in this paper is that the interpolation function has a continuous spectrum. the problem has been discussed by one of the authors for the case where the sampling frequency and the band‐end frequency of the waveform are in an integer ratio.In this paper, the sampling theorem is derived in a unified and integrated form without imposing such a condition. the variance of the approximation error is also derived for the case where a statistically independent additive error is superposed on the sampled value, and it is shown that the proposed sampling theorem is better in that respect. the discussion is extended to the multidimensional band‐limited waveform, and a sampling theorem is shown which has an interpolation function with a continuous spectrum.