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Categories: Digital signal processing | Signal processing

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 $s(t) \$ has a Fourier transform $\mathcal{F} \{s(t) \} = S(f) = 0 \$ for $|f| \ge W \$ , then $s(t) \$ can be recovered from its samples $s_n = s(n/(2 W)) \$ by the formula

$s(x) = \sum_{n=-\infty}^\infty s_n \frac {\sin \left(\pi (2 W t - n)\right)} {\pi (2 W t - n)} = \sum_{n=-\infty}^{\infty} s_n {\rm sinc}\left(\pi (2 W t - n)\right)$

where ${\rm sinc}(x) \$ is the sinc function. Note that this form is a convolution sum of

$\sum_{n=-\infty}^{\infty} s_n \delta \left( t - \frac{n}{2 W}\right)$

and

${\rm sinc}\left(\pi (2 W t )\right)$ .

It then follows that multiplication by the sinc function's Fourier transform with

$\sum_{k=-\infty}^\infty S(f - 2 k W) \$

has the same result. The Fourier transform of a sinc function is the rectangular function. If $\mathcal{F} \{s(t) \} = S(f) = 0 \$ for $|f| \ge W \$ , then this multiplication results in $S(f) \$ , removing all other shifted copies of $S(f) \$ .

This ideal interpolation filter is an ideal brick-wall low-pass filter. The Nyquist-Shannon interpolation will always recover the original signal, $s(t) \$ , as long as the sampling criterion, $\mathcal{F} \{s(t) \} = S(f) = 0 \$ for $|f| \ge W \$ , is held to. If not, aliasing will occur, where frequencies higher than $W \$ are folded back to aliased frequencies less than $W \$ . See Aliasing#Caveats for further discussion on this point.

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Nyquist-Shannon interpolation formula

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