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{{Use American English|date = March 2019}}▼
{{Short description|Signal (re-)construction algorithm}}
{{
The '''Whittaker–Shannon interpolation formula''' or '''sinc interpolation''' is a method to construct a [[continuous-time]] [[bandlimited]] function from a sequence of real numbers. The formula dates back to the works of [[E. Borel]] in 1898, and [[E. T. Whittaker]] in 1915, and was cited from works of [[J. M. Whittaker]] in 1935, and in the formulation of the [[Nyquist–Shannon sampling theorem]] by [[Claude Shannon]] in 1949. It is also commonly called '''Shannon's interpolation formula''' and '''Whittaker's interpolation formula'''. E. T. Whittaker, who published it in 1915, called it the '''Cardinal series'''.
==Definition==
[[File:Nyquist sampling.gif|500px|thumb|right|
Given a sequence of real numbers, ''x''[''n''], the continuous function
:<math>x(t) = \sum_{n=-\infty}^{\infty} x[n] \, {\rm sinc}\left(\frac{t - nT}{T}\right)\,</math>
(where "sinc" denotes the [[normalized sinc function]]) has a [[Fourier transform]], ''X''(''f''), whose non-zero values are confined to the region |''f''| ≤ 1/(2''T'').
==Equivalent formulation: convolution/lowpass filter==
The interpolation formula is derived in the [[Nyquist–Shannon sampling theorem]] article, which points out that it can also be expressed as the [[convolution]] of an [[Dirac comb|infinite impulse train]] with a [[sinc function]]:
:<math> x(t) = \left( \sum_{n=-\infty}^{\infty}
\left( \frac{1}{T}{\rm sinc}\left(\frac{t}{T}\right) \right). </math>
This is equivalent to filtering the impulse train with an ideal (''brick-wall'') [[low-pass filter]] with gain of 1 (or 0
==Convergence==
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:<math>\sum_{n\in\Z,\,n\ne 0}\left|\frac{x[n]}n\right|<\infty.</math>
By the [[Hölder inequality]] this is satisfied if the sequence <math>
:<math>\sum_{n\in\Z}\left|x[n]\right|^p<\infty.</math>
This condition is sufficient, but not necessary. For example, the sum will generally converge if the sample sequence comes from sampling almost any [[stationary process]], in which case the sample sequence is not square summable, and is not in any <math>
==Stationary random processes==
If ''x''[''n''] is an infinite sequence of samples of a sample function of a wide-sense [[stationary process]], then it is not a member of any <math>
Since a random process does not have a Fourier transform, the condition under which the sum converges to the original function must also be different. A stationary random process does have an [[autocorrelation function]] and hence a [[spectral density]] according to the [[Wiener–Khinchin theorem]]. A suitable condition for convergence to a sample function from the process is that the spectral density of the process be zero at all frequencies equal to and above half the sample rate.
==See also==
{{cols}}
* [[Aliasing]], [[Anti-aliasing filter]], [[Spatial anti-aliasing]]
* [[Rectangular function]]
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* [[Sinc function]], [[Sinc filter]]
* [[Lanczos resampling]]
{{colend}}
{{Use dmy dates|date=May 2014}}
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*http://www.stanford.edu/class/ee104/shannonpaper.pdf-->
{{DEFAULTSORT:Whittaker-Shannon
[[Category:Digital signal processing]]
[[Category:Signal processing]]
[[Category:Fourier analysis]]
[[Category:E. T. Whittaker]]
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