nyquist-shannon interpolation formula

The Nyquist-Shannon interpolation formula is used in conjuction with the Nyquist-Shannon sampling theorem that states that if a function

s(x)

has a Fourier transform

F s(x) = S(f) = 0

for

|f| \ge W

, then

s(x)

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

where sinc is the sinc function. Note that this form is a convolution sum of

s_x

and

{\rm sinc}\left(\pi 2 W x\right)

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

S(f)

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.

Nyquist-shannon Interpolation Formula

See also