Nyquist-Shannon interpolation formula
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
where is the sinc function.
Note that this form is a convolution sum of
and - .
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.
See also Aliasing, Anti-aliasing, Anti-aliasing filter Fourier transform Nyquist-Shannon sampling theorem Rectangular function Sampling (information theory) Signal (information theory) Sinc function, Sinc filter
|
|
