Sampling Theorem in the Signal Space Spanned by Spline Functions of Degree 2

Kazuo TORAICHI  Masaru KAMADA  Ryoichi MORI  

IEICE TRANSACTIONS (1976-1990)   Vol.E68   No.10   pp.660-666
Publication Date: 1985/10/25
Online ISSN: 
Print ISSN: 0000-0000
Type of Manuscript: PAPER
Category: Signal Analysis

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The present paper derives a sampling basis in the signal space spanned by spline functions of degree 2 with equidistantly spaced knots. It also analyzes properties of the sampling basis. The spline signal space is defined as the space spanned by B-spline basis composed of normalized B-splines. As the norm is , L norm is adopted. The existence of a sampling basis is examined by functional analytic approach. The discrete signal space K[b] corresponding to the B-spline basis and the discrete signal space K[s] corresponding to the sampling basis are defined. As the norm in K[b] and K[s], l norm is adopted. B-spline transform B is defined as the operator which transforms a discrete signal in K[s] into one in K[b]. It is shown that the sampling basis exists in if and only if B has a bounded unit-pulse response in the sense of l1 norm. The sampling basis in is derived using the unit-pulse response of B which is derived by z-transform. As properties of the sampling basis in the time domain, its symmetric property, its shift-invariant property and the exponential attenuation of its amplitude are shown. As properties in the frequency domain, the frequency response of the interpolation by spline functions of degree 2 is shown by the Fourier transform of the sampling basis. It is clarified that the frequency response is of low-pass type, and it has no phase delay in the pass band and the transition band.