Reconstruction of bounded signals from pseudo-periodic, irregularly spaced samples

Abstract A sampling theorem for entire functions of exponential type, bounded on the real line, is established. The sampling points have to fulfill certain periodicity conditions, but they may be nonuniform and some of them may even lie arbitrarily close together. It is shown that one can select a subsequence of the partial sums of the sampling series which reconstructs the signal from its values at such sampling points. The proof uses contour integration and a combinatorial argument. The sampling series can be regarded as a generalization of the Lagrange interpolation formula. Several new concrete applications are given.

[1]  F. Stenger.  Numerical Methods Based on Whittaker Cardinal, or Sinc Functions , 1981 .

[2]  Lauwerens Kuipers,et al.  Uniform distribution of sequences , 1974 .

[3]  Paul L. Butzer,et al.  On Lagrange interpolation and Kramer-type sampling theorems associated with Sturm-Liouville problems , 1990 .

[4]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[5]  A. J. Jerri.  The Shannon sampling theorem—Its various extensions and applications: A tutorial review , 1977, Proceedings of the IEEE.

[6]  L. L. Campbell.  A Comparison of the Sampling Theorems of Kramer and Whittaker , 1964 .

[7]  H. Kramer,et al.  A Generalized Sampling Theorem , 1959 .

[8]  Eliahu Ibrahim Jury,et al.  Sampled-data control systems , 1977 .

[9]  J. R. Higgins.  A sampling theorem for irregularly spaced sample points (Corresp.) , 1976, IEEE Trans. Inf. Theory.

[10]  F. Beutler.  Error-Free Recovery of Signals from Irregularly Spaced Samples , 1966 .

[11]  A. Kohlenberg.  Exact Interpolation of Band‐Limited Functions , 1953 .

[12]  M. Nikolskii,et al.  Approximation of Functions of Several Variables and Embedding Theorems , 1971 .

[13]  J. M. Whittaker.  Interpolatory function theory , 1935 .

[14]  Farrokh Marvasti,et al.  A Unified Approach to Zero-Crossings and Nonuniform Sampling of Single and Multidimensional Signals and Systems , 1987 .

[15]  J. Yen.  On Nonuniform Sampling of Bandwidth-Limited Signals , 1956 .

[16]  Edmund Taylor Whittaker.  XVIII.—On the Functions which are represented by the Expansions of the Interpolation-Theory , 1915 .

[17]  K. Seip.  An irregular sampling theorem for functions bandlimited in a generalized sense , 1987 .

[18]  G. A. Watson.  A treatise on the theory of Bessel functions , 1944 .

[19]  J. Thomas,et al.  On Some Stability and Interpolatory Properties of Nonuniform Sampling Expansions , 1967, IEEE Transactions on Circuit Theory.

[20]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

[21]  A. Papoulis.  Signal Analysis , 1977 .

[22]  J. R. Higgins,et al.  Five short stories about the cardinal series , 1985 .

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

[24]  Paul L. Butzer,et al.  Shannon’s Sampling Theorem Cauchy’s Integral Formula, and Related Results , 1984 .