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Whittaker–Shannon interpolation formula

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The Whittaker–Shannon interpolation formula or sinc interpolation is a method to reconstruct a continuous-time bandlimited

Bandlimited

Bandlimiting is the limiting of a deterministic or stochastic signal's Fourier transform or power spectral density to zero above a certain finite frequency...

signal from a set of equally spaced samples.

Definition

The interpolation formula, as it is commonly called, dates back to the works of E. Borel in 1898, and E. T. Whittaker

E. T. Whittaker

Edmund Taylor Whittaker FRS FRSE was an English mathematician who contributed widely to applied mathematics, mathematical physics and the theory of special functions. He had a particular interest in numerical analysis, but also worked on celestial mechanics and the history of physics...

in 1915, and was cited from works of J. M. Whittaker in 1935, and in the formulation of the Nyquist–Shannon sampling theorem

Nyquist–Shannon sampling theorem

The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...

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/f_s is the sampling interval, f_s is the sampling rate
Sampling rate
The sampling rate, sample rate, or sampling frequency defines the number of samples per unit of time taken from a continuous signal to make a discrete signal. For time-domain signals, the unit for sampling rate is hertz , sometimes noted as Sa/s...

, 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

the function x(t) is bandlimited to bandwidth B; that is, it has a Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

for |f| > B; and
the sampling rate
Sampling rate
The sampling rate, sample rate, or sampling frequency defines the number of samples per unit of time taken from a continuous signal to make a discrete signal. For time-domain signals, the unit for sampling rate is hertz , sometimes noted as Sa/s...

, f_s, exceeds the Nyquist rate
Nyquist rate
In signal processing, the Nyquist rate, named after Harry Nyquist, is two times the bandwidth of a bandlimited signal or a bandlimited channel...

, twice the bandwidth: f_s > 2B. Equivalently:

then the interpolation formula will exactly reconstruct the original x(t) from its samples. Otherwise, aliasing may occur; that is, frequencies at or above f_s/2 may be erroneously reconstructed. See Aliasing
Aliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...

for further discussion on this point.

Interpolation as convolution sum

The interpolation formula is derived in the Nyquist–Shannon sampling theorem

Nyquist–Shannon sampling theorem

The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...

article, which points out that it can also be expressed as the convolution

Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

of an infinite impulse train

Dirac comb

In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions...

with a sinc function:

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

Low-pass filter

A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...

.

Convergence

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

By the Hölder inequality this is satisfied if the sequence

belongs to any of the

spaces

Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector 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 any stationary process

Stationary process

In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...

, in which case the sample sequence is not square summable, and is not in any

space.

Stationary random processes

If x[n] is an infinite sequence of samples of a sample function of a wide-sense stationary process

Stationary process

In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...

, then it is not a member of any

or L^p space

Lp space

In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...

, with probability 1; that is, the infinite sum of samples raised to a power p 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 an autocorrelation function and hence a spectral density

Spectral density

In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...

according to the Wiener–Khinchin theorem

Wiener–Khinchin theorem

The Wiener–Khinchin theorem states that the power spectral density of a wide–sense stationary random process is the Fourier transform of the corresponding autocorrelation function.-History:Norbert Wiener first published the result in...