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Kazuo TORAICHI, Masaru KAMADA, Ryoichi MORI, "Sampling Theorem in the Signal Space Spanned by Spline Functions of Degree 2" in IEICE TRANSACTIONS on transactions,
vol. E68-E, no. 10, pp. 660-666, October 1985, doi: .
Abstract: 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.
URL: https://global.ieice.org/en_transactions/transactions/10.1587/e68-e_10_660/_p
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@ARTICLE{e68-e_10_660,
author={Kazuo TORAICHI, Masaru KAMADA, Ryoichi MORI, },
journal={IEICE TRANSACTIONS on transactions},
title={Sampling Theorem in the Signal Space Spanned by Spline Functions of Degree 2},
year={1985},
volume={E68-E},
number={10},
pages={660-666},
abstract={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.},
keywords={},
doi={},
ISSN={},
month={October},}
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TY - JOUR
TI - Sampling Theorem in the Signal Space Spanned by Spline Functions of Degree 2
T2 - IEICE TRANSACTIONS on transactions
SP - 660
EP - 666
AU - Kazuo TORAICHI
AU - Masaru KAMADA
AU - Ryoichi MORI
PY - 1985
DO -
JO - IEICE TRANSACTIONS on transactions
SN -
VL - E68-E
IS - 10
JA - IEICE TRANSACTIONS on transactions
Y1 - October 1985
AB - 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.
ER -
English
