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The Nyquist-Shannon interpolation formula is used in conjuction with the Nyquist-Shannon sampling theorem that states that if a function <math>s(x)<math> has a Fourier transform <math>F[s(x)] = S(f) = 0<math> for <math>|f| \ge W<math>, then <math>s(x)<math> can be recovered from its samples <math>s_n = s(n/(2 W))<math> by the formula
- <math>s(x) = \sum_{n=-\infty}^\infty s_n \frac {\sin \left(\pi (2 W x - n)\right)} {\pi (2 W x - n)} = \sum_{n=-\infty}^{\infty} s_n sinc\left(\pi (2 W x - n)\right)<math>
where sinc is the sinc function.
Note that this form is a convolution sum of <math>s_x<math> and <math>sinc\left(\pi 2 W x\right)<math>.
It then follows that multiplication by the sinc function's fourier transform with <math>S(f)<math> has the same result.
The fourier transform of a sinc function is the rectangular function.
This interpolation filter can also be considered a perfect low-pass filter.
As such, the Nyquist-Shannon interpolator is not always satisfactory for reconstructing a signal.
Particularly in cases when the original signal is not low-frequencied like the frequency domain of the sinc function.
See Aliasing#Caveats for further discussion on this point.
See also
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