The Nyquist–Shannon sampling theorem is a fundamental result in the field of information theory, in particular telecommunications and signal processing. 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:^{[1]}
If a function x(t) contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart.
The theorem is commonly called the Shannon sampling theorem, and is also known as Nyquist–Shannon–Kotelnikov, Whittaker–Shannon–Kotelnikov, Whittaker–Nyquist–Kotelnikov–Shannon, WKS, etc., sampling theorem, as well as the Cardinal Theorem of Interpolation Theory. It is often referred to as simply the sampling theorem.
In essence, the theorem shows that a bandlimited analog signal that has been sampled can be perfectly reconstructed from an infinite sequence of samples if the sampling rate exceeds 2B samples per second, where B is the highest frequency in the original signal. If a signal contains a component at exactly B hertz, then samples spaced at exactly 1/(2B) seconds do not completely determine the signal, Shannon's statement notwithstanding. This sufficient condition can be weakened, as discussed at Sampling of nonbaseband signals below.
More recent statements of the theorem are sometimes careful to exclude the equality condition; that is, the condition is if x(t) contains no frequencies higher than or equal to B; this condition is equivalent to Shannon's except when the function includes a steady sinusoidal component at exactly frequency B.
The theorem assumes an idealization of any realworld situation, as it only applies to signals that are sampled for infinite time; any timelimited x(t) cannot be perfectly bandlimited. Perfect reconstruction is mathematically possible for the idealized model but only an approximation for realworld signals and sampling techniques, albeit in practice often a very good one.
The theorem also leads to a formula for reconstruction of the original signal. The constructive proof of the theorem leads to an understanding of the aliasing that can occur when a sampling system does not satisfy the conditions of the theorem.
The sampling theorem provides a sufficient condition, but not a necessary one, for perfect reconstruction. The field of compressed sensing provides a stricter sampling condition when the underlying signal is known to be sparse. Compressed sensing specifically yields a subNyquist sampling criterion.
A signal or function is bandlimited if it contains no energy at frequencies higher than some bandlimit or bandwidth B. A signal that is bandlimited is constrained in how rapidly it changes in time, and therefore how much detail it can convey in an interval of time. The sampling theorem asserts that the uniformly spaced discrete samples are a complete representation of the signal if this bandwidth is less than half the sampling rate. To formalize these concepts, let represent a continuoustime signal and be the Fourier transform of that signal:
The signal is bandlimited to a onesided baseband bandwidth, B, if:
or, equivalently, supp(X)^{[2]} [B,B]. Then the sufficient condition for exact reconstructability from samples at a uniform sampling rate (in samples per unit time) is:
or equivalently:
is called the Nyquist rate and is a property of the bandlimited signal, while is called the Nyquist frequency and is a property of this sampling system.
The time interval between successive samples is referred to as the sampling interval:
and the samples of are denoted by:
The sampling theorem leads to a procedure for reconstructing the original from the samples and states sufficient conditions for such a reconstruction to be exact.
The theorem describes two processes in signal processing: a sampling process, in which a continuous time signal is converted to a discrete time signal, and a reconstruction process, in which the original continuous signal is recovered from the discrete time signal.
The continuous signal varies over time (or space in a digitized image, or another independent variable in some other application) and the sampling process is performed by measuring the continuous signal's value every T units of time (or space), which is called the sampling interval. In practice, for signals that are a function of time, the sampling interval is typically quite small, on the order of milliseconds, microseconds, or less. This results in a sequence of numbers, called samples, to represent the original signal. Each sample value is associated with the instant in time when it was measured. The reciprocal of the sampling interval (1/T) is the sampling frequency denoted f_{s}, which is measured in samples per unit of time. If T is expressed in seconds, then f_{s} is expressed in Hz.
Reconstruction of the original signal is an interpolation process that mathematically defines a continuoustime signal x(t) from the discrete samples x[n] and at times in between the sample instants nT.
If the original signal contains a frequency component equal to onehalf the sampling rate, the condition is not satisfied. The resulting reconstructed signal may have a component at that frequency, but the amplitude and phase of that component generally will not match the original component.
This reconstruction or interpolation using sinc functions is not the only interpolation scheme. Indeed, it is impossible in practice because it requires summing an infinite number of terms. However, it is the interpolation method that in theory exactly reconstructs any given bandlimited x(t) with any bandlimit B < 1/2T); any other method that does so is formally equivalent to it.
A few consequences can be drawn from the theorem:
The Poisson summation formula indicates that the samples of function x(t) are sufficient to create a periodic extension of function X(f). The result is:

( 
As depicted in Figures 3, 4, and 8, copies of X(f) are shifted by multiples of and combined by addition.
If the sampling condition is not satisfied, adjacent copies overlap, and it is not possible in general to discern an unambiguous X(f). Any frequency component above is indistinguishable from a lowerfrequency component, called an alias, associated with one of the copies. The reconstruction technique described below produces the alias, rather than the original component, in such cases.
For a sinusoidal component of exactly half the sampling frequency, the component will in general alias to another sinusoid of the same frequency, but with a different phase and amplitude.
To prevent or reduce aliasing, two things can be done:
The antialiasing filter is to restrict the bandwidth of the signal to satisfy the condition for proper sampling. Such a restriction works in theory, but is not precisely satisfiable in reality, because realizable filters will always allow some leakage of high frequencies. However, the leakage energy can be made small enough so that the aliasing effects are negligible.
The sampling theorem is usually formulated for functions of a single variable. Consequently, the theorem is directly applicable to timedependent signals and is normally formulated in that context. However, the sampling theorem can be extended in a straightforward way to functions of arbitrarily many variables. Grayscale images, for example, are often represented as twodimensional arrays (or matrices) of real numbers representing the relative intensities of pixels (picture elements) located at the intersections of row and column sample locations. As a result, images require two independent variables, or indices, to specify each pixel uniquely — one for the row, and one for the column.
Color images typically consist of a composite of three separate grayscale images, one to represent each of the three primary colors — red, green, and blue, or RGB for short. Other colorspaces using 3vectors for colors include HSV, LAB, XYZ, etc. Some colorspaces such as cyan, magenta, yellow, and black (CMYK) may represent color by four dimensions. All of these are treated as vectorvalued functions over a twodimensional sampled domain.
Similar to onedimensional discretetime signals, images can also suffer from aliasing if the sampling resolution, or pixel density, is inadequate. For example, a digital photograph of a striped shirt with high frequencies (in other words, the distance between the stripes is small), can cause aliasing of the shirt when it is sampled by the camera's image sensor. The aliasing appears as a moiré pattern. The "solution" to higher sampling in the spatial domain for this case would be to move closer to the shirt, use a higher resolution sensor, or to optically blur the image before acquiring it with the sensor.
Another example is shown to the left in the brick patterns. The top image shows the effects when the sampling theorem's condition is not satisfied. When software rescales an image (the same process that creates the thumbnail shown in the lower image) it, in effect, runs the image through a lowpass filter first and then downsamples the image to result in a smaller image that does not exhibit the moiré pattern. The top image is what happens when the image is downsampled without lowpass filtering: aliasing results.
The application of the sampling theorem to images should be made with care. For example, the sampling process in any standard image sensor (CCD or CMOS camera) is relatively far from the ideal sampling which would measure the image intensity at a single point. Instead these devices have a relatively large sensor area at each sample point in order to obtain sufficient amount of light. In other words, any detector has a finitewidth point spread function. The analog optical image intensity function which is sampled by the sensor device is not in general bandlimited, and the nonideal sampling is itself a useful type of lowpass filter, though not always sufficient to remove enough high frequencies to sufficiently reduce aliasing. When the area of the sampling spot (the size of the pixel sensor) is not large enough to provide sufficient antialiasing, a separate antialiasing filter (optical lowpass filter) is typically included in a camera system to further blur the optical image. Despite images having these problems in relation to the sampling theorem, the theorem can be used to describe the basics of down and up sampling of images.
When a signal is downsampled, the sampling theorem can be invoked via the artifice of resampling a hypothetical continuoustime reconstruction. The Nyquist criterion must still be satisfied with respect to the new lower sampling frequency in order to avoid aliasing. To meet the requirements of the theorem, the signal must usually pass through a lowpass filter of appropriate cutoff frequency as part of the downsampling operation. This lowpass filter, which prevents aliasing, is called an antialiasing filter.
To illustrate the necessity of f_{s} > 2B, consider the sinusoid:
With f_{s} = 2B or equivalently T = 1/(2B), the samples are given by:
Those samples cannot be distinguished from the samples of:
But for any θ such that sin(θ) ≠ 0, x(t) and y(t) have different amplitudes and different phase. This and other ambiguities are the reason for the strict inequality of the sampling theorem's condition.
From Figures 3 and 8, it is apparent that when there is no overlap of the copies (aka "images") of X(f), the k=0 term of X_{s}(f) can be recovered by the product:
H(f) need not be precisely defined in the region [B, f_{s}B], because X_{s}(f) is zero in that region. However, the worst case is when B = f_{s}/2, the Nyquist frequency. A function that is sufficient for that and all less severe cases is:
where is the rectangular function.
Therefore:
The original function that was sampled can be recovered by an inverse Fourier transform:
which is the Whittaker–Shannon interpolation formula. It shows explicitly how the samples, x(nT), can be combined to reconstruct x(t).
The original proof presented by Shannon is elegant and quite brief, but it offers less intuitive insight into the subtleties of aliasing, both unintentional and intentional. Quoting Shannon's original paper, which uses f for the function, F for the spectrum, and W for the bandwidth limit:
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. Quoting Shannon:
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.
As discussed by Shannon:^{[1]}
A similar result is true if the band does not start at zero frequency but at some higher value, and can be proved by a linear translation (corresponding physically to singlesideband modulation) of the zerofrequency case. In this case the elementary pulse is obtained from sinx / x by singlesideband modulation.
That is, a sufficient noloss condition for sampling signals that do not have baseband components exists that involves the width of the nonzero frequency interval as opposed to its highest frequency component. See Sampling (signal processing) for more details and examples.
A bandpass condition is that for all nonnegative outside the open band of frequencies:
for some nonnegative integer . This formulation includes the normal baseband condition as the case N=0.
The corresponding interpolation function is the impulse response of an ideal brickwall bandpass filter (as opposed to the ideal brickwall lowpass filter used above) with cutoffs at the upper and lower edges of the specified band, which is the difference between a pair of lowpass impulse responses:
Other generalizations, for example to signals occupying multiple noncontiguous bands, are possible as well. Even the most generalized form of the sampling theorem does not have a provably true converse. That is, one cannot conclude that information is necessarily lost just because the conditions of the sampling theorem are not satisfied; from an engineering perspective, however, it is generally safe to assume that if the sampling theorem is not satisfied then information will most likely be lost.
The sampling theory of Shannon can be generalized for the case of nonuniform samples, that is, samples not taken equally spaced in time. Shannon sampling theory for nonuniform sampling states that a bandlimited signal can be perfectly reconstructed from its samples if the average sampling rate satisfies the Nyquist condition^{[4]}. Therefore, although uniformly spaced samples may result in easier reconstruction algorithms, it is not a necessary condition for perfect reconstruction.
The Nyquist–Shannon sampling theorem is also known to be a sufficient condition (i.e., it is not necessary). The field of compressed sensing provides a stricter sampling condition when the underlying signal is known to be sparse. Compressed sensing specifically yields a subNyquist sampling criterion. The knowledge of the sparsity or compressibility of the signal provides some bounds on the number of samples needed to allow for reconstruction of the signal. In compressed sensing, the reconstruction of the original signal from the samples involves nonlinear solvers.
The sampling theorem was implied by the work of Harry Nyquist in 1928 ("Certain topics in telegraph transmission theory"), in which he showed that up to 2B independent pulse samples could be sent through a system of bandwidth B; but he did not explicitly consider the problem of sampling and reconstruction of continuous signals. About the same time, Karl Küpfmüller showed a similar result^{[5]}, and discussed the sincfunction impulse response of a bandlimiting filter, via its integral, the step response Integralsinus; this bandlimiting and reconstruction filter that is so central to the sampling theorem is sometimes referred to as a Küpfmüller filter (but seldom so in English).
The sampling theorem, essentially a dual of Nyquist's result, was proved by Claude E. Shannon in 1949 ("Communication in the presence of noise"). V. A. Kotelnikov published similar results in 1933 ("On the transmission capacity of the 'ether' and of cables in electrical communications", translation from the Russian), as did the mathematician E. T. Whittaker in 1915 ("Expansions of the InterpolationTheory", "Theorie der Kardinalfunktionen"), J. M. Whittaker in 1935 ("Interpolatory function theory"), and Gabor in 1946 ("Theory of communication").
Others who have independently discovered or played roles in the development of the sampling theorem have been discussed in several historical articles, for example by Jerri^{[6]} and by Lüke.^{[7]} For example, Lüke points out that H. Raabe, an assistant to Küpfmüller, proved the theorem in his 1939 Ph.D. dissertation; the term Raabe condition came to be associated with the criterion for unambiguous representation (sampling rate greater than twice the bandwidth).
Meijering^{[8]} mentions several other discoverers and names in a paragraph and pair of footnotes:
As pointed out by Higgins [135], the sampling theorem should really be considered in two parts, as done above: the first stating the fact that a bandlimited function is completely determined by its samples, the second describing how to reconstruct the function using its samples. Both parts of the sampling theorem were given in a somewhat different form by J. M. Whittaker [350, 351, 353] and before him also by Ogura [241, 242]. They were probably not aware of the fact that the first part of the theorem had been stated as early as 1897 by Borel [25].^{27} As we have seen, Borel also used around that time what became known as the cardinal series. However, he appears not to have made the link [135]. In later years it became known that the sampling theorem had been presented before Shannon to the Russian communication community by Kotel'nikov [173]. In more implicit, verbal form, it had also been described in the German literature by Raabe [257]. Several authors [33, 205] have mentioned that Someya [296] introduced the theorem in the Japanese literature parallel to Shannon. In the English literature, Weston [347] introduced it independently of Shannon around the same time.^{28}
^{27} Several authors, following Black [16], have claimed that this first part of the sampling theorem was stated even earlier by Cauchy, in a paper [41] published in 1841. However, the paper of Cauchy does not contain such a statement, as has been pointed out by Higgins [135].
^{28} As a consequence of the discovery of the several independent introductions of the sampling theorem, people started to refer to the theorem by including the names of the aforementioned authors, resulting in such catchphrases as “the WhittakerKotel’nikovShannon (WKS) sampling theorem" [155] or even "the WhittakerKotel'nikovRaabeShannonSomeya sampling theorem" [33]. To avoid confusion, perhaps the best thing to do is to refer to it as the sampling theorem, "rather than trying to find a title that does justice to all claimants" [136].
Exactly how, when, or why Harry Nyquist had his name attached to the sampling theorem remains obscure. The term Nyquist Sampling Theorem (capitalized thus) appeared as early as 1959 in a book from his former employer, Bell Labs,^{[9]} and appeared again in 1963,^{[10]} and not capitalized in 1965.^{[11]} It had been called the Shannon Sampling Theorem as early as 1954,^{[12]} but also just the sampling theorem by several other books in the early 1950s.
In 1958, Blackman and Tukey^{[13]} cited Nyquist's 1928 paper as a reference for the sampling theorem of information theory, even though that paper does not treat sampling and reconstruction of continuous signals as others did. Their glossary of terms includes these entries:
Exactly what "Nyquist's result" they are referring to remains mysterious.
When Shannon stated and proved the sampling theorem in his 1949 paper, according to Meijering^{[8]} "he referred to the critical sampling interval T = 1/2W as the Nyquist interval corresponding to the band W, in recognition of Nyquist’s discovery of the fundamental importance of this interval in connection with telegraphy." This explains Nyquist's name on the critical interval, but not on the theorem.
Similarly, Nyquist's name was attached to Nyquist rate in 1953 by Harold S. Black:^{[14]}
According to the OED, this may be the origin of the term Nyquist rate. In Black's usage, it is not a sampling rate, but a signaling rate.


