Nyquist-Shannon interpolation formula
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The Nyquist-Shannon interpolation formula or Cardinal series dates back to works of E. Borel in 1898 and was cited from works of J. M. Whittaker in 1935 in the formulation of the Nyquist-Shannon sampling theorem by C. E. Shannon in 1949. The latter states that if a function <math>s(t) \ <math> has a Fourier transform <math>\mathcal{F} \{s(t) \} = S(f) = 0 \ <math> for <math>|f| \ge W \ <math>, then <math>s(t) \ <math> can be recovered from its samples <math>s_n = s(n/(2 W)) \ <math> by the formula
- <math>s(t) = \sum_{n=-\infty}^\infty s_n \frac {\sin \left(\pi (2 W t - n)\right)} {\pi (2 W t - n)} = \sum_{n=-\infty}^{\infty} s_n {\rm sinc}\left(\pi (2 W t - n)\right)<math>
where <math>{\rm sinc}(x) \ <math> is the sinc function or Sinus cardinalis. It always converges as long as <math>\sum_{n\in\Z,\,n\ne 0}\left|\frac{s_n}n\right|<\infty<math>.
Note that this form is a convolution sum of
- <math>\sum_{n=-\infty}^{\infty} s_n \delta \left( t - \frac{n}{2 W}\right) <math>
and
- <math> {\rm sinc}\left(\pi (2 W t )\right)<math>.
It then follows by the Poisson summation formula that multiplication by the sinc function's Fourier transform with
- <math>\sum_{k=-\infty}^\infty S(f - 2 k W) \ <math>
has the same result. The Fourier transform of a sinc function is the rectangular function. If <math>\mathcal{F} \{s(t) \} = S(f) = 0 \ <math> for <math>|f| \ge W \ <math>, then this multiplication results in <math> S(f) \ <math>, removing all other shifted copies of <math> S(f) \ <math>.
This ideal interpolation filter is an ideal brick-wall low-pass filter. The Nyquist-Shannon interpolation will always recover the original signal, <math>s(t) \ <math>, as long as the sampling criterion, <math>\mathcal{F} \{s(t) \} = S(f) = 0 \ <math> for <math>|f| \ge W \ <math>, is held to. If not, aliasing will occur, where frequencies higher than <math> W \ <math> are folded back to aliased frequencies less than <math> W \ <math> . See Aliasing#Caveats for further discussion on this point.