# Nyquist–Shannon sampling theorem

In the field of digital signal processing, the sampling theorem is a fundamental bridge between continuous signals (analog domain) and discrete signals (digital domain). Strictly speaking, it only applies to a class of mathematical functions whose Fourier transforms are zero outside of a finite region of frequencies (see Fig 1). The analytical extension to actual signals, which can only approximate that condition, is provided by the discrete-time Fourier transform, a version of the Poisson summation formula.  Intuitively we expect that when one reduces a continuous function to a discrete sequence (called samples) and interpolates back to a continuous function, the fidelity of the result depends on the density (or sample-rate) of the original samples. The sampling theorem introduces the concept of a sample-rate that is sufficient for perfect fidelity for the class of bandlimited functions; no actual "information" is lost during the sampling process. It expresses the sample-rate in terms of the function's bandwidth. The theorem also leads to a formula for the mathematically ideal interpolation algorithm.

The theorem does not preclude the possibility of perfect reconstruction under special circumstances that do not satisfy the sample-rate criterion. (See Sampling of non-baseband signals below, and compressed sensing.)

The name Nyquist–Shannon sampling theorem honors Harry Nyquist and Claude Shannon. The theorem was also discovered independently by E. T. Whittaker, by Vladimir Kotelnikov, and by others. So it is also known by the names Nyquist–Shannon–Kotelnikov, Whittaker–Shannon–Kotelnikov, Whittaker–Nyquist–Kotelnikov–Shannon, and cardinal theorem of interpolation.

## Introduction

Sampling is the process of converting a signal (for example, a function of continuous time or space) into a numeric sequence (a function of discrete time or space). Shannon's version of the theorem states:

If a function x(t) contains no frequencies higher than B cps, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.

A sufficient sample-rate is therefore 2B samples/second, or anything larger. Conversely, for a given sample rate fs  the bandlimit for perfect reconstruction is B ≤ fs/2 . When the bandlimit is too high (or there is no bandlimit), the reconstruction exhibits imperfections known as aliasing. Modern statements of the theorem are sometimes careful to explicitly state that x(t) must contain no sinusoidal component at exactly frequency B, or that B must be strictly less than ½ the sample rate. The two thresholds, 2B and fs/2 are respectively called the Nyquist rate and Nyquist frequency. And respectively, they are attributes of x(t) and of the sampling equipment. The condition described by these inequalities is called the Nyquist criterion, or sometimes the Raabe condition. The theorem is also applicable to functions of other domains, such as space, in the case of a digitized image. The only change, in the case of other domains, is the units of measure applied to t, fs, and B. Fig. 2: The normalized sinc function: sin(πx) / (πx) ... showing the central peak at x= 0, and zero-crossings at the other integer values of x.

The symbol T = 1/fs is customarily used to represent the interval between samples and is called the sample period or sampling interval. And the samples of function x(t) are commonly denoted by x[n] = x(nT) (alternatively "xn" in older signal processing literature), for all integer values of n. The mathematically ideal way to interpolate the sequence involves the use of sinc functions, like those shown in Fig 2. Each sample in the sequence is replaced by a sinc function, centered on the time axis at the original location of the sample, nT, with the amplitude of the sinc function scaled to the sample value, x[n]. Subsequently, the sinc functions are summed into a continuous function. A mathematically equivalent method is to convolve one sinc function with a series of Dirac delta pulses, weighted by the sample values. Neither method is numerically practical. Instead, some type of approximation of the sinc functions, finite in length, is used. The imperfections attributable to the approximation are known as interpolation error.

Practical digital-to-analog converters produce neither scaled and delayed sinc functions, nor ideal Dirac pulses. Instead they produce a piecewise-constant sequence of scaled and delayed rectangular pulses (the zero-order hold), usually followed by an "anti-imaging filter" to clean up spurious high-frequency content.

## Aliasing

{{#invoke:main|main}} Fig. 3: The samples of several different sine waves can be identical, when at least one of them is at a frequency above half the sample rate.

Let X(f) be the Fourier transform of the bandlimited function x(t):

$X(f)\ {\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }x(t)\ e^{-i2\pi ft}\ {\rm {d}}t,$ and $X(f)=0\quad$ for all  $|f|>B.\,$ The Poisson summation formula shows that the samples, x(nT), of x(t) are sufficient to create a periodic summation of X(f). The result is:

Template:NumBlk which is a periodic function and its equivalent representation as a Fourier series, whose coefficients are Tx(nT). This function is also known as the discrete-time Fourier transform (DTFT) of the sequence Tx(nT), for integers n. As depicted in Figures 4 and 5, copies of X(f) are shifted by multiples of fs and combined by addition.

If the Nyquist criterion is not satisfied, adjacent copies overlap, and it is not possible in general to discern an unambiguous X(f). Any frequency component above fs/2 is indistinguishable from a lower-frequency component, called an alias, associated with one of the copies. In such cases, the customary interpolation techniques produce the alias, rather than the original component. When the sample-rate is pre-determined by other considerations (such as an industry standard), x(t) is usually filtered to reduce its high frequencies to acceptable levels before it is sampled. The type of filter required is a lowpass filter, and in this application it is called an anti-aliasing filter. Fig. 4: X(f) (top blue) and XA(f) (bottom blue) are continuous Fourier transforms of two different functions, x(t) and xA(t) (not shown). When the functions are sampled at rate fs, the images (green) are added to the original transforms (blue) when one examines the discrete-time Fourier transforms (DTFT) of the sequences. In this hypothetical example, the DTFTs are identical, which means the sampled sequences are identical, even though the original continuous pre-sampled functions are not. If these were audio signals, x(t) and xA(t) might not sound the same. But their samples (taken at rate fs) are identical and would lead to identical reproduced sounds; thus xA(t) is an alias of x(t) at this sample rate. In this example (of a bandlimited function), such aliasing can be prevented by increasing fs such that the green images in the top figure do not overlap the blue portion. Fig. 5: Spectrum, Xs(f), of a properly sampled bandlimited signal (blue) and the adjacent DTFT images (green) that do not overlap. A brick-wall low-pass filter, H(f), removes the images, leaves the original spectrum, X(f), and recovers the original signal from its samples.

## Derivation as a special case of Poisson summation

From Figure 5, it is apparent that when there is no overlap of the copies (aka "images") of X(f), the k = 0 term of Xs(f) can be recovered by the product:

$X(f)=H(f)\cdot X_{s}(f),\,$ where:
$H(f)\ {\stackrel {\mathrm {def} }{=}}\ {\begin{cases}1&|f|f_{s}-B.\end{cases}}$ At this point, the sampling theorem is proved, since X(f) uniquely determines x(t).

All that remains is to derive the formula for reconstruction. H(f) need not be precisely defined in the region [B, fsB] because Xs(f) is zero in that region. However, the worst case is when B = fs/2, the Nyquist frequency. A function that is sufficient for that and all less severe cases is:

$H(f)={\mathrm {rect} }\left({\frac {f}{f_{s}}}\right)={\begin{cases}1&|f|<{\frac {f_{s}}{2}}\\0&|f|>{\frac {f_{s}}{2}},\end{cases}}$ where rect(•) is the rectangular function.  Therefore:

$X(f)={\mathrm {rect} }\left({\frac {f}{f_{s}}}\right)\cdot X_{s}(f)\$ $={\mathrm {rect} }(Tf)\cdot \sum _{n=-\infty }^{\infty }T\cdot x(nT)\ e^{-i2\pi nTf}$ (from  Template:EquationNote, above).
$=\sum _{n=-\infty }^{\infty }x(nT)\cdot \underbrace {T\cdot {\mathrm {rect} }(Tf)\cdot e^{-i2\pi nTf}} _{{\mathcal {F}}\left\{{\mathrm {sinc} }\left({\frac {t-nT}{T}}\right)\right\}}.$ The inverse transform of both sides produces the Whittaker–Shannon interpolation formula:

$x(t)=\sum _{n=-\infty }^{\infty }x(nT)\cdot {\mathrm {sinc} }\left({\frac {t-nT}{T}}\right),$ which shows how the samples, x(nT), can be combined to reconstruct x(t).

## Shannon's original proof

Poisson shows that the Fourier series in Template:EquationNote produces the periodic summation of X(f), regardless of fs and B. Shannon, however, only derives the series coefficients for the case fs = 2B. Virtually quoting Shannon's original paper:

Let $X(\omega )$ be the spectrum of $x(t).$ Then
since $X(\omega )$ is assumed to be zero outside the band $|{\frac {\omega }{2\pi }}| . If we let
$t={n \over {2B}}\,$ where n is any positive or negative integer, we obtain
$x\left({\tfrac {n}{2B}}\right)={1 \over 2\pi }\int _{-2\pi B}^{2\pi B}X(\omega )e^{i\omega {n \over {2B}}}\;{\rm {d}}\omega .$ On the left are values of $x(t)$ at the sampling points. The integral on the right will be recognized as essentially[note 1] the nth coefficient in a Fourier-series expansion of the function $X(\omega ),$ taking the interval –B to B as a fundamental period. This means that the values of the samples $x(n/2B)$ determine the Fourier coefficients in the series expansion of $X(\omega ).$ Thus they determine $X(\omega ),$ since $X(\omega )$ is zero for frequencies greater than B, and for lower frequencies $X(\omega )$ is determined if its Fourier coefficients are determined. But $X(\omega )$ determines the original function $x(t)$ completely, since a function is determined if its spectrum is known. Therefore the original samples determine the function $x(t)$ completely.

Shannon's proof of the theorem is complete at that point, but he goes on to discuss reconstruction via sinc functions, what we now call the Whittaker–Shannon interpolation formula as discussed above. He does not derive or prove the properties of the sinc function, but these would have been familiar to engineers reading his works at the time, since the Fourier pair relationship between rect (the rectangular function) and sinc was well known.

Let $x_{n}$ be the nth sample. Then the function $x(t)$ is represented by:
$x(t)=\sum _{n=-\infty }^{\infty }x_{n}{\sin \pi (2Bt-n) \over \pi (2Bt-n)}.\,$ As in the other proof, the existence of the Fourier transform of the original signal is assumed, so the proof does not say whether the sampling theorem extends to bandlimited stationary random processes.