# Nyquist–Shannon sampling theorem

sampling theoremNyquist-Shannon sampling theoremNyquist theoremsampling theoryNyquistShannon sampling theoremNyquist sampling theoremNyquist's theoremsamplingcritical frequency
In the field of digital signal processing, the sampling theorem is a fundamental bridge between continuous-time signals and discrete-time signals.wikipedia
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### Compressed sensing

compressive sensingcompressed sensing techniquesCompressed-Sensing
(See below and compressed sensing.) In some cases (when the sample-rate criterion is not satisfied), utilizing additional constraints allows for approximate reconstructions.
This is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by the Nyquist–Shannon sampling theorem.

### Digital signal processing

DSPsignal processingdigital
In the field of digital signal processing, the sampling theorem is a fundamental bridge between continuous-time signals and discrete-time signals.
The Nyquist–Shannon sampling theorem states that a signal can be exactly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency component in the signal.

### Sampling (signal processing)

sampling ratesamplingsample rate
It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth.
The approximately double-rate requirement is a consequence of the Nyquist theorem.

### Bandwidth (signal processing)

bandwidthbandwidthssignal bandwidth
It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth.
In the context of, for example, the sampling theorem and Nyquist sampling rate, bandwidth typically refers to baseband bandwidth.

### Sinc function

sinccardinal sine functioncardinal sine
A mathematically ideal way to interpolate the sequence involves the use of sinc functions. Practical digital-to-analog converters produce neither scaled and delayed sinc functions, nor ideal Dirac pulses.
It is used in the concept of reconstructing a continuous bandlimited signal from uniformly spaced samples of that signal.

### Bandlimiting

bandlimitedband-limitedband limited
The sampling theorem introduces the concept of a sample rate that is sufficient for perfect fidelity for the class of functions that are bandlimited to a given bandwidth, such that no actual information is lost in the sampling process.
This result, usually attributed to Nyquist and Shannon, is known as the Nyquist–Shannon sampling theorem.

### Anti-aliasing filter

anti-aliasinganti-aliasing (AA) filteroptical low-pass filter
The type of filter required is a lowpass filter, and in this application it is called an anti-aliasing filter.
An anti-aliasing filter (AAF) is a filter used before a signal sampler to restrict the bandwidth of a signal to approximately or completely satisfy the Nyquist–Shannon sampling theorem over the band of interest.

### Discrete-time Fourier transform

convolution theoremDFTDTFT &sect; Properties
which is a periodic function and its equivalent representation as a Fourier series, whose coefficients are This function is also known as the discrete-time Fourier transform (DTFT) of the sample sequence.
Under certain theoretical conditions, described by the sampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples.

### Nyquist frequency

Nyquist limitNyquistN/2 different frequencies
The threshold f s /2 is called the Nyquist frequency and is an attribute of the sampling equipment.
The Nyquist frequency should not be confused with the Nyquist rate, the latter is the minimum sampling rate that satisfies the Nyquist sampling criterion for a given signal or family of signals.

### Whittaker–Shannon interpolation formula

interpolation/sampling theoryreconstructingsinc interpolation
The inverse transform of both sides produces the Whittaker–Shannon interpolation formula:
The formula dates back to the works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M. Whittaker in 1935, and in the formulation of the Nyquist–Shannon sampling theorem by Claude Shannon in 1949.

### Aliasing

aliasaliasedtemporal aliasing
When the bandlimit is too high (or there is no bandlimit), the reconstruction exhibits imperfections known as aliasing.
is met for the highest frequency component of the original signal, then it is met for all the frequency components, a condition called the Nyquist criterion.

### Zero-order hold

ZOH
Instead they produce a piecewise-constant sequence of scaled and delayed rectangular pulses (the zero-order hold), usually followed by a lowpass filter (called an "anti-imaging filter") to remove spurious high-frequency replicas (images) of the original baseband signal.
The filter can then be analyzed in the frequency domain, for comparison with other reconstruction methods such as the Whittaker–Shannon interpolation formula suggested by the Nyquist–Shannon sampling theorem, or such as the first-order hold or linear interpolation between sample values.

### Harry Nyquist

NyquistNyquist, Harry
The name Nyquist–Shannon sampling theorem honours Harry Nyquist and Claude Shannon albeit the fact that it had already been discovered in 1933 by Vladimir Kotelnikov.
This rule is essentially a dual of what is now known as the Nyquist–Shannon sampling theorem.

### Poisson summation formula

Poisson series
the Poisson summation formula indicates that the samples, x(nT), of x(t) are sufficient to create a periodic summation of X(f).
For band-limited functions, choosing the sampling rate 2f_o guarantees that no information is lost: since \hat f can be reconstructed from these sampled values, then, by Fourier inversion, so can f. This leads to the Nyquist–Shannon sampling theorem.

### Optical transfer function

modulation transfer functionMTFcontrast transfer function
Each of these components is characterized by a modulation transfer function (MTF), representing the precise resolution (spatial bandwidth) available in that component.
As explained by the Nyquist–Shannon sampling theorem, to match the optical resolution of the given example, the pixels of each color channel should be separated by 1 micrometer, half the period of 500 cycles per millimeter.

### Nonuniform sampling

The sampling theory of Shannon can be generalized for the case of nonuniform sampling, that is, samples not taken equally spaced in time.
Nonuniform sampling is a branch of sampling theory involving results related to the Nyquist–Shannon sampling theorem.

### Nyquist rate

Nyquist sampling rateNyquist limitNyquist
The threshold 2B is called the Nyquist rate and is an attribute of the continuous-time input x(t) to be sampled.
Shannon used Nyquist's approach when he proved the sampling theorem in 1948, but Nyquist did not work on sampling per se.

### E. T. Whittaker

Edmund Taylor WhittakerSir Edmund Taylor WhittakerEdmund Whittaker
The theorem was also discovered independently by E. T. Whittaker and by others.

### Digital-to-analog converter

DACDACsD/A
Practical digital-to-analog converters produce neither scaled and delayed sinc functions, nor ideal Dirac pulses.

### Spatial anti-aliasing

anti-aliasinganti-aliasedantialiasing
When the area of the sampling spot (the size of the pixel sensor) is not large enough to provide sufficient spatial anti-aliasing, a separate anti-aliasing filter (optical low-pass filter) may be included in a camera system to reduce the MTF of the optical image.
The goal of an anti-aliasing filter is to greatly reduce frequencies above a certain limit, known as the Nyquist frequency, so that the signal will be accurately represented by its samples, or nearly so, in accordance with the sampling theorem; there are many different choices of detailed algorithm, with different filter transfer functions.

### Sinc filter

brick-wallbrick-wall filterbrickwall filter
The corresponding interpolation function is the impulse response of an ideal brick-wall bandpass filter (as opposed to the ideal brick-wall 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:
Real-time filters can only approximate this ideal, since an ideal sinc filter (a.k.a. rectangular filter) is non-causal and has an infinite delay, but it is commonly found in conceptual demonstrations or proofs, such as the sampling theorem and the Whittaker–Shannon interpolation formula.

### Dirac comb

Sampling functionimpulse traininfinite impulse train
This leads to a natural formulation of the Nyquist–Shannon sampling theorem.

### Claude Shannon

Claude E. ShannonShannonClaude Elwood Shannon
The name Nyquist–Shannon sampling theorem honours Harry Nyquist and Claude Shannon albeit the fact that it had already been discovered in 1933 by Vladimir Kotelnikov.
He is also credited with the introduction of sampling theory, which is concerned with representing a continuous-time signal from a (uniform) discrete set of samples.