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This 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, however, 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. [ABSTRACT FROM AUTHOR]

An approximation method is proposed for a class of band-limited waveforms whose Fourier spectra F(ω) are confined in the interval -ω1 ≤ ω ≤ ω1. The side lobes of such waveforms are also small in the range of 0 < ω0 < ¦ω¦ < ω1. First, the approximation waveform is obtained from summation of the product of an interpolating function and the sampling value f(tk) at the sampling point tk (k = 0 ∼ K). One objective of the proposed method is to find an interpolating function that will minimize the envelope of a family of approximation waveforms, which is used as a measure of error associated with the approximation. The optimum interpolating function also satisfies the condition of discrete orthogonality. The proposed approximation technique also provides convergence to the actual waveform with respect to a set of sampling points determined by the conventional sampling theorem. The approximation error is evaluated and weighted in the frequency domain using Fourier series expansion. The lower and upper bounds of approximation error in the frequency domain are derived and their relationships to the original waveform are established. [ABSTRACT FROM AUTHOR]