Sampling Theorem for Spline Signal Space of Arbitrary Degree

Mamoru IWAKI  Kazuo TORAICHI  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E77-A    No.5    pp.810-817
Publication Date: 1994/05/25
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Signal Processing and System Theory)
digital signal processing,  information theory,  coding theory,  spline functions,  

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In the band-limited signal space, the mutual relation between continuous time signal and discrete time signal is expressed by the sampling theorem of Whittaker-Someya-Shannon. This theorem consists of an orthonormal expansion formula using sinc functions. In that formula, the expansion coefficients are identical to the sample values of signals. In general, the bandlimited signal space is not always suited to model the signals in nature. The authors have proposed a new model for signal processing based on finite times continuously differentiable functions. This paper aims to complete the sampling theorem for the spline signal spaces, which corresponds to the sampling theorem of Whittaker-Someya-Shannon in the band-limited signal space. Since the obtained sampling theorem gives the simplest representation of signals, it is considered to be the most fundamental characterization of spline functions used for signal processing. The biorthonormal basis derived in this paper is considered to be a system of delta functions at sampling points with some continuous differentiability.