The Nyquist-Shannon interpolation formula is used in conjunction with the Nyquist-Shannon sampling theorem that states that if a function has a Fourier transform for , then can be recovered from its samples by the formula
It then follows that multiplication by the sinc function's Fourier transform with
has the same result. The Fourier transform of a sinc function is the rectangular function. If for , then this multiplication results in , removing all other shifted copies of .
This ideal interpolation filter is an ideal brick-wall low-pass filter. The Nyquist-Shannon interpolation will always recover the original signal, , as long as the sampling criterion, for , is held to. If not, aliasing will occur, where frequencies higher than are folded back to aliased frequencies less than . See Aliasing#Caveats for further discussion on this point.