# The Shannon sampling theorem—Its various extensions and applications: A tutorial review

@article{Jerri1977TheSS, title={The Shannon sampling theorem\&\#8212;Its various extensions and applications: A tutorial review}, author={Abdul J. Jerri}, journal={Proceedings of the IEEE}, year={1977}, volume={65}, pages={1565-1596} }

It has been almost thirty years since Shannon introduced the sampling theorem to communications theory. [... ] Key Method The extensions will include sampling for functions of more than one variable, random processes, nonuniform sampling, nonband-limited functions, implicit sampling, generalized functions (distributions), sampling with the function and its derivatives as suggested by Shannon in his original paper, and sampling for general integral transforms. Expand

## 1,337 Citations

Proofs of the Nyquist-Shannon Sampling Theorem

- Mathematics
- 2013

The sampling theorem states that a band limited function can be fully reconstructed by its discrete samples if they are close enough. Therefore, it is the basis for digitalization of continuous…

A sampling theorem for periodic functions with no minus frequency component and its application

- Mathematics2013 19th Asia-Pacific Conference on Communications (APCC)
- 2013

This paper introduces a sampling theorem for periodic functions, and gives its physical meaning; this theorem is effective for functions that can be expressed as a finite Fourier series in the same way as WOKSS's theorem deals with a band-limited signal.

Summation of certain series using the Shannon sampling theorem

- Mathematics
- 1990

In communication theory texts, it is usually observed that if the sampling theorem is uncritically applied to a pure sinusoidal signal sin 2 pi Wt using the Nyquist sampling rate of 2W samples/s,…

On quantization, truncation and jitter errors in the sampling theorem and its generalizations

- Computer Science, Mathematics
- 1980

Sampling-50 years after Shannon

- Computer ScienceProceedings of the IEEE
- 2000

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.

A New Interpretation of the Sampling Theorem and Its Extensions

- Mathematics
- 1997

We start with the classical sampling theorem for bandlimited signals and comment on its various extensions. In Section 2 we discuss the problem of sampling harmonic functions. In the subsequent…

A generalized sampling theorem

- Mathematics, Computer Science
- 1989

Using the theory of pseudo-biorthogonal base and the notion of reproducing kernel, a very broad generalized sampling theorem with real pulse is derived that is effective also for so-called undersampling whereby there are too few sampling points compared with the dimension of signal space, and for oversampling due to too many sampling points.

On a Non-Fourier Generalization of Shannon Sampling Theory

- Computer Science2007 10th Canadian Workshop on Information Theory (CWIT)
- 2007

New results are presented on an approach that makes use of powerful functional analytic methods to generalize Shannon sampling to allow varying Nyquist rates.

Generalizations of the sampling theorem: Seven decades after Nyquist

- Computer Science
- 2001

Some of the less known aspects of sampling are presented, with special emphasis on non bandlimited signals, pointwise stability of reconstruction, and reconstruction from nonuniform samples.

On sampling theorem, wavelets, and wavelet transforms

- MathematicsIEEE Trans. Signal Process.
- 1993

The authors study the properties of cardinal orthogonal scaling functions (COSF), which provide the standard sampling theorem in multiresolution spaces with scaling functions as interpolants, and present a family of COSF with exponential decay, which are generalizations of the Haar function.

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