Nyquist–Shannon sampling theorem: Difference between revisions

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The sampling theorem, essentially a [[duality (mathematics)|dual]] of Nyquist's result, was proved by [[Claude E. Shannon]].<ref name="Shannon49"/> The mathematician [[E. T. Whittaker]] published similar results in 1915,<ref>{{cite journal |last=Whittaker |first=E. T. |author-link=E. T. Whittaker |title=On the Functions Which are Represented by the Expansions of the Interpolation Theory |journal=Proceedings of the Royal Society of Edinburgh |volume=35 |pages=181–194 |date=1915 |doi=10.1017/s0370164600017806|url=https://zenodo.org/record/1428702 }} ({{lang|de|"Theorie der Kardinalfunktionen"}}).</ref> J. M. Whittaker in 1935,<ref>{{cite book | last=Whittaker |first=J. M. | author-link =J. M. Whittaker | title =Interpolatory Function Theory | url=https://archive.org/details/in.ernet.dli.2015.219870 | publisher =Cambridge University Press | date =1935 | ___location =Cambridge, England}}</ref> and [[Dennis Gabor|Gabor]] in 1946 ("Theory of communication").

In 1948 and 1949, Claude E. Shannon published the two revolutionary articles in which he founded the [[information theory]].<ref>{{cite journal |ref=refShannon48 |last=Shannon |first=Claude E. |author-link=Claude Shannon |title=A Mathematical Theory of Communication |journal=Bell System Technical Journal |volume=27 |issue=3 |pages=379–423 |date=July 1948 |doi=10.1002/j.1538-7305.1948.tb01338.x|hdl=11858/00-001M-0000-002C-4317-B |hdl-access=free }}</ref><ref>{{cite journal |ref=refShannon48oct |last=Shannon |first=Claude E. |author-link=Claude Shannon |title=A Mathematical Theory of Communication |journal=Bell System Technical Journal |volume=27 |issue=4 |pages=623–666 |date=October 1948 |doi=10.1002/j.1538-7305.1948.tb00917.x|hdl=11858/00-001M-0000-002C-4314-2 |hdl-access=free }}</ref><ref name="Shannon49"/> In Shannon's "[[#refShannon48oct|ShannonA 1948Mathematical Theory of Communication]]", the sampling theorem is formulated as "Theorem 13": Let <math>f(t)</math> contain no frequencies over W. Then

<math display="block">f(t) = \sum_{n=-\infty}^\infty X_n \frac{\sin \pi(2Wt - n)}{\pi(2Wt - n)},</math>

where <math>X_n = f\left(\frac n {2W} \right).</math>

It was not until these articles were published that the theorem known as "Shannon's sampling theorem" became common property among communication engineers, although Shannon himself writes that this is a fact which is common knowledge in the communication art.{{efn-ua|group=bottom|[[#refShannon49|Shannon 1949]], p. 448.}} A few lines further on, however, he adds: "but in spite of its evident importance, [it] seems not to have appeared explicitly in the literature of [[communication theory]]". Despite his sampling theorem being published at the end of the 1940s, Shannon had derived his sampling theorem as early as 1940.<ref>{{Cite conference |last1=Stanković |first1=Raromir S. |last2=Astola |first2=Jaakko T. |last3=Karpovsky |first3=Mark G. |date=September 2006 |title=Some Historic Remarks On Sampling Theorem |url=https://sites.bu.edu/mark/files/2018/02/196.pdf |conference=Proceedings of the 2006 International TICSP Workshop on Spectral Methods and Multirate Signal Processing}}</ref>

===Other discoverers===