Maximal gap of a sampling set for the exact iterative reconstruction algorithm in shift invariant spaces

A conventional A/D converter prefilters a signal by an ideal lowpass filter and performs sampling for bandlimited signals by the Nyquist sampling rate. Recent research reveals that A/D conversion in a shift invariant space provides more flexible choices for designing a practical A/D conversion system of high accuracy. This paper focuses on the maximal gap of a sampling set for the iterative algorithm in shift invariant spaces, which provides an explicit formula to calculate the maximal gap of a sampling set in terms of a generator of the undertaken shift invariant spaces.

[1]  J. Lei.  Lp-Approximation by Certain Projection Operators , 1994 .

[2]  Xiang-Gen Xia,et al.  On sampling theorem, wavelets, and wavelet transforms , 1993, IEEE Trans. Signal Process..

[3]  Wenchang Sun,et al.  Reconstruction of functions in spline subspaces from local averages , 2003 .

[4]  Michael Unser,et al.  A general sampling theory for nonideal acquisition devices , 1994, IEEE Trans. Signal Process..

[5]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[6]  Karlheinz Gröchenig,et al.  On Landau's Necessary Density Conditions for Sampling and Interpolation of Band-Limited Functions , 1996 .

[7]  Amara Lynn Graps,et al.  An introduction to wavelets , 1995 .

[8]  L. Squire,et al.  Amalgams of Lᴾ and ℓ^q , 1984 .

[9]  J. A. Parker,et al.  Comparison of Interpolating Methods for Image Resampling , 1983, IEEE Transactions on Medical Imaging.

[10]  R. Jia,et al.  Approximation by multiinteger translates of functions having global support , 1993 .

[11]  H. Landau.  Necessary density conditions for sampling and interpolation of certain entire functions , 1967 .

[12]  C.E. Shannon,et al.  Communication in the Presence of Noise , 1949, Proceedings of the IRE.

[13]  Bin Han,et al.  Estimate of aliasing error for non-smooth signals prefiltered by quasi-projections into shift-invariant spaces , 2005, IEEE Transactions on Signal Processing.

[14]  Richard H. Sherman,et al.  Chaotic communications in the presence of noise , 1993, Optics & Photonics.

[15]  Shuichi Itoh,et al.  A sampling theorem for shift-invariant subspace , 1998, IEEE Trans. Signal Process..

[16]  R. DeVore,et al.  The Structure of Finitely Generated Shift-Invariant Spaces in , 1992 .

[17]  Shuichi Itoh,et al.  Oversampling Theorem for Wavelet Subspace , 1998 .

[18]  H. Feichtinger,et al.  Iterative reconstruction of multivariate band-limited functions from irregular sampling values , 1992 .

[19]  M. Unser,et al.  Approximation Error for Quasi-Interpolators and (Multi-)Wavelet Expansions , 1999 .

[20]  Augustus J. E. M. Janssen,et al.  The Zak transform and sampling theorems for wavelet subspaces , 1993, IEEE Trans. Signal Process..

[21]  K. Gröchenig,et al.  Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces , 2000 .

[22]  Gilbert G. Walter,et al.  A sampling theorem for wavelet subspaces , 1992, IEEE Trans. Inf. Theory.

[23]  A. Aldroubi,et al.  Sampling procedures in function spaces and asymptotic equivalence with shannon's sampling theory , 1994 .

[24]  Jean-Baptiste Lully,et al.  The collected works , 1996 .

[25]  Shuichi Itoh,et al.  Irregular Sampling Theorems for Wavelet Subspaces , 1998, IEEE Trans. Inf. Theory.

[26]  David J. DeFatta,et al.  Digital Signal Processing: A System Design Approach , 1988 .

[27]  A. Aldroubi.  Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces , 2002 .

[28]  H. Feichtinger,et al.  Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: The $L^p$-theory , 1998 .

[29]  H. Feichtinger.  Generalized Amalgams, With Applications to Fourier Transform , 1990, Canadian Journal of Mathematics.

[30]  K. Gröchenig.  RECONSTRUCTION ALGORITHMS IN IRREGULAR SAMPLING , 1992 .

[31]  Bin Han,et al.  On simple oversampled A/D conversion in shift-invariant spaces , 2005, IEEE Transactions on Information Theory.

[32]  Youming Liu.  Irregular Sampling for Spline Wavelet Subspaces , 1996, IEEE Trans. Inf. Theory.

[33]  Cormac Herley,et al.  Minimum rate sampling and reconstruction of signals with arbitrary frequency support , 1999, IEEE Trans. Inf. Theory.

[34]  R. Jia.  Shift-invariant spaces and linear operator equations , 1998 .

[35]  E. Cheney,et al.  Approximation from shift-invariant spaces by integral operators , 1997 .

[36]  D. Slepian,et al.  On bandwidth , 1976, Proceedings of the IEEE.

[37]  J. Stewart,et al.  Amalgams of $L^p$ and $l^q$ , 1985 .

[38]  Akram Aldroubi,et al.  Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces , 2001, SIAM Rev..

[39]  Shuichi Itoh,et al.  On sampling in shift invariant spaces , 2002, IEEE Trans. Inf. Theory.