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TheWhittaker–Shannon interpolation formula orsinc interpolation is a method to construct acontinuous-timebandlimited function from a sequence of real numbers. The formula dates back to the works ofE. Borel in 1898, andE. T. Whittaker in 1915, and was cited from works ofJ. M. Whittaker in 1935, and in the formulation of theNyquist–Shannon sampling theorem byClaude Shannon in 1949. It is also commonly calledShannon's interpolation formula andWhittaker's interpolation formula. E. T. Whittaker, who published it in 1915, called it theCardinal series.

Given a sequence of real numbers,${\displaystyle x[n]=x(nT)}$, representing a sequence of samples at time intervals of${\displaystyle T}$ seconds, consider the following continuous function:

${\displaystyle x(t)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}x[n]\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left(\frac{t-nT}{T}\right),}$

where${\displaystyle \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}(\mathrm{t})}$ denotes thenormalized sinc function${\displaystyle \frac{\sin (\pi t)}{\pi t}}$.

${\displaystyle x(t)}$ has aFourier transform${\displaystyle X(f)}$, whose non-zero values are confined to the region${\displaystyle |f|\le \frac{1}{2T}}$, and abandlimit of${\displaystyle 1/(2T)}$ cycles/sec (hertz). The quantity${\displaystyle f_s=1/T}$ is known as thesample rate, and${\displaystyle f_s/2}$ is the correspondingNyquist frequency. When the sampled function has a bandlimit less than the Nyquist frequency,${\displaystyle x(t)}$ is a perfect reconstruction of the original function (seesampling theorem). Otherwise, the frequency components above the Nyquist frequency will "fold" into the sub-Nyquist region of${\displaystyle X(f)}$, resulting in distortion.

The interpolation formula is derived in theNyquist–Shannon sampling theorem article, which points out that it can also be expressed as theconvolution of aninfinite impulse train with asinc function:

${\displaystyle x(t)=\left(\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}T\cdot \underset{x[n]}{\underbrace{x(nT)}}\cdot \delta \left(t-nT\right)\right)⊛\left(\frac{1}{T}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left(\frac{t}{T}\right)\right).}$

This is equivalent to filtering the impulse train with an ideal (brick-wall)low-pass filter with gain of 1 (or 0 dB) in the passband. If the sample rate is sufficiently high, this means that the baseband image (the original signal before sampling) is passed unchanged and the other images are removed by the brick-wall filter.

The interpolation formula always convergesabsolutely andlocally uniformly as long as

By theHölder inequality this is satisfied if the sequence${\displaystyle (x[n]{)}_{n\in \mathbb{Z}}}$ belongs to any of the${\displaystyle {\ell }^p(\mathbb{Z},\mathbb{C})}$spaces with 1 ≤ p < ∞, that is

This condition is sufficient, but not necessary. For example, the sum will generally converge if the sample sequence comes from sampling almost anystationary process, in which case the sample sequence is not square summable, and is not in any${\displaystyle {\ell }^p(\mathbb{Z},\mathbb{C})}$ space.

Ifx[n] is an infinite sequence of samples of a sample function of a wide-sensestationary process, then it is not a member of any${\displaystyle {\ell }^p}$ orL^p space, with probability 1; that is, the infinite sum of samples raised to a powerp does not have a finite expected value. Nevertheless, the interpolation formula converges with probability 1. Convergence can readily be shown by computing the variances of truncated terms of the summation, and showing that the variance can be made arbitrarily small by choosing a sufficient number of terms. If the process mean is nonzero, then pairs of terms need to be considered to also show that the expected value of the truncated terms converges to zero.

Since a random process does not have a Fourier transform, the condition under which the sum converges to the original function must also be different. A stationary random process does have anautocorrelation function and hence aspectral density according to theWiener–Khinchin theorem. A suitable condition for convergence to a sample function from the process is that the spectral density of the process be zero at all frequencies equal to and above half the sample rate.

• Digital signal processing
• Signal processing
• Fourier analysis
• E. T. Whittaker

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