Generalizations of the sampling theorem: Seven decades after Nyquist

@article{Vaidyanathan2001GeneralizationsOT,
  title={Generalizations of the sampling theorem: Seven decades after Nyquist},
  author={Palghat P. Vaidyanathan},
  journal={IEEE Transactions on Circuits and Systems I-regular Papers},
  year={2001},
  volume={48},
  pages={1094-1109}
}
  • P. Vaidyanathan
  • Published 1 September 2001
  • Computer Science
  • IEEE Transactions on Circuits and Systems I-regular Papers
The sampling theorem is one of the most basic and fascinating topics in engineering sciences. The most well-known form is Shannon's uniform-sampling theorem for bandlimited signals. Extensions of this to bandpass signals and multiband signals, and to nonuniform sampling are also well-known. The connection between such extensions and the theory of filter banks in DSP has been well established. This paper presents some of the less known aspects of sampling, with special emphasis on non… 
Beyond Bandlimited Sampling: Nonlinearities, Smoothness and Sparsity
TLDR
Several extensions of the Shannon theorem are presented, which treat a wide class of input signals as well as nonideal sampling and nonlinear distortions, and can be used to uniformly sample non-bandlimited signals, and to perfectly compensate for nonlinear effects.
Effective Approximation of Bandlimited Signals and Their Samples
  • H. Boche, Ullrich J. Mönich
  • Computer Science
    ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
  • 2020
TLDR
This paper provides a simple necessary and sufficient condition for the computability of the continuous-time signal, and a simple canonical algorithm that can be used for the computation.
The role of biorthogonal partners in sampling theory for non bandlimited signals: a review
  • B. Vrcelj, P. Vaidyanathan
  • Computer Science
    Conference Record of the Thirty-Sixth Asilomar Conference on Signals, Systems and Computers, 2002.
  • 2002
TLDR
This paper presents biorthogonal partners, which unify many aspects of wavelet theory into the same framework, and discusses spline interpolation, fractionally spaced equalization of noisy channels and more recently, rational FSEs.
SAMPLING THEOREMS FOR NON BANDLIMITED SIGNALS : THEORETICAL IMPACT AND PRACTICAL APPLICATIONS
TLDR
This lecture gives a brief overview of some of the ideas in digital signal processing applications such as image interpolation, equalization of communication channels, and in multiresolution computation extended to the case of non bandlimited signals.
Reconstruction and processing of bandlimited signals based on their discrete values
TLDR
This dissertation analyzes the interplay between the analog and the digital worlds, and the classical and distributional convergence behavior of different convolution-type system representations is analyzed.
Periodic Non Uniform Sampling of Non Bandlimited Signals
  • J. Tuqan
  • Computer Science, Mathematics
    2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings
  • 2006
TLDR
This work considers the periodic non uniform sampling of a class of continuous time non bandlimited signals and proposes a number of ways to retrieve the signal and derive necessary and sufficient conditions for signal reconstruction with FIR digital filters.
A Generalized Sampling Theorem for Frequency Localized Signals
TLDR
A generalized sampling theorem for frequency localized signals is presented and an approximate reconstruction from sample values rather than a perfect one is obtained (though the former might be "practically perfect" in many cases).
Sampling Theorems for Nonbandlimited Signals
TLDR
This chapter gives a brief overview of some of the ideas for bandlimited sampling extensions that have been found to be useful in digital signal processing applications such as image interpolation, equalization of communication channels, and multiresolution computation.
A Theory for Sampling Signals From a Union of Subspaces
  • Yue M. Lu, M. Do
  • Mathematics, Computer Science
    IEEE Transactions on Signal Processing
  • 2008
TLDR
A general sampling framework where sampled signals come from a known union of subspaces and the sampling operator is linear is studied, which can serve as a guideline for designing new algorithms for various applications in signal processing and inverse problems.
A new approach to spectral estimation from irregular sampling
This article addresses the problem of signal reconstruction, spectral estimation and linear filtering directly from irregularly-spaced samples of a continuous signal (or autocorrelation function in
...
...

References

SHOWING 1-10 OF 42 REFERENCES
Classical sampling theorems in the context of multirate and polyphase digital filter bank structures
TLDR
Novel theorems for the subsampling of sequences are derived by direct use of the digital-filter-bank framework and are related to the theory of perfect reconstruction in maximally decimated digital- filter-bank systems.
Sampling-50 years after Shannon
  • M. Unser
  • Computer Science
    Proceedings of the IEEE
  • 2000
TLDR
The standard sampling paradigm is extended for a presentation of functions in the more general class of "shift-in-variant" function spaces, including splines and wavelets, and variations of sampling that can be understood from the same unifying perspective are reviewed.
Filterbank reconstruction of bandlimited signals from nonuniform and generalized samples
TLDR
A filterbank interpretation of various sampling strategies, which leads to efficient interpolation and reconstruction methods and an identity is developed that leads to new sampling strategies including an extension of Papoulis' (1977) generalized sampling expansion.
Reconstruction of sequences from nonuniform samples
If a discrete time signal x(n) is obtained as the output of an interpolation filter F(z), it is natural to expect that it can be recovered from the decimated samples x(Mn), even though the signal is
Generalized sampling theorems in multiresolution subspaces
TLDR
This paper extends the existing sampling theory for wavelet subspaces in several directions, and extends the sampling theory to random processes, where it turns out that one cannot recover random processes themselves but only their power spectral density functions.
The Shannon sampling theorem—Its various extensions and applications: A tutorial review
It has been almost thirty years since Shannon introduced the sampling theorem to communications theory. In this review paper we will attempt to present the various contributions made for the sampling
Generalized sampling without bandlimiting constraints
  • M. Unser, J. Zerubia
  • Computer Science
    1997 IEEE International Conference on Acoustics, Speech, and Signal Processing
  • 1997
We investigate the problem of the reconstruction of a continuous-time function f(x)/spl isin//spl Hscr/ from the responses of m linear shift-invariant systems sampled at 1/m the reconstruction rate,
Sampling procedures in function spaces and asymptotic equivalence with shannon's sampling theory
We view Shannon's sampling procedure as a problem of approximation in the space S = {s: s (x) = (c * sinc)(x)c e l 2}. We show that under suitable conditions on a generating function λ e L 2, the
Discrete time signals which can be recovered from samples
TLDR
This work considers both uniform and nonuniform decimation of this kind and explores some applications, especially in noise shaping and in /spl Sigma/-/spl Delta/ modulator type of architectures.
A sampling theorem for wavelet subspaces
  • G. Walter
  • Mathematics
    IEEE Trans. Inf. Theory
  • 1992
TLDR
The classical Shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory, and is first shown to have a Riesz basis formed from the reproducing kernels.
...
...