Whittaker–Shannon interpolation formula

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The Whittaker–Shannon interpolation formula is a method to reconstruct a continuous-time bandlimited signal from a set of equally spaced samples.

Contents

Definition

The interpolation formula, as it is commonly called, dates back to works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M. Whittaker in 1935 in the formulation of the Nyquist–Shannon sampling theorem by Claude Shannon in 1949. It is also commonly called Shannon's interpolation formula and Whittaker's interpolation formula. E. T. Whittaker, who published it in 1915, called it the Cardinal series.

The sampling theorem states that, under certain limiting conditions, a function x(t) can be recovered exactly from its samples,   x[n] = x(nT), by the Whittaker–Shannon interpolation formula:

where T = 1/fs is the sampling interval, fs is the sampling rate, and sinc(x) is the normalized sinc function.

Validity condition

If the function x(t) is bandlimited, and sampled at a high enough rate, the interpolation formula is guaranteed to reconstruct it exactly. Formally, if there exists some B ≥ 0 such that

then the interpolation formula will exactly reconstruct the original x(t) from its samples. Otherwise, aliasing may occur; that is, frequencies at or above fs/2 may be erroneously reconstructed. See Aliasing for further discussion on this point.

Interpolation as convolution sum

The interpolation formula is derived in the Nyquist–Shannon sampling theorem article, which points out that it can also be expressed as the convolution of an infinite impulse train with a sinc function:

This is equivalent to filtering the impulse train with an ideal (brick-wall) low-pass filter.

Convergence

The interpolation formula always converges absolutely and locally uniform as long as

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

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