Editing Nyquist–Shannon sampling theorem (section)

{{distinguish|Shannon–Hartley theorem}}

The '''Nyquist–Shannon sampling theorem''' is an essential principle for [[digital signal processing]] linking the [[frequency range]] of a signal and the [[sample rate]] required to avoid a type of [[distortion]] called [[aliasing]]. The theorem states that the sample rate must be at least twice the [[Bandwidth (signal processing)|bandwidth]] of the signal to avoid aliasing. In practice, it is used to select [[band-limiting]] filters to keep aliasing below an acceptable amount when an analog signal is sampled or when sample rates are changed within a digital signal processing function.

[[File:Bandlimited.svg|thumb|250px|Example of magnitude of the Fourier transform of a bandlimited function]]

The Nyquist–Shannon sampling theorem is a theorem in the field of [[signal processing]] which serves as a fundamental bridge between [[continuous-time signal]]s and [[discrete-time signal]]s. 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 (signal processing)|bandwidth]].

Strictly speaking, the theorem only applies to a class of [[mathematical function]]s having a [[continuous Fourier transform|Fourier transform]] that is zero outside of a finite region of frequencies. Intuitively we expect that when one reduces a continuous function to a discrete sequence and [[interpolates]] back to a continuous function, the fidelity of the result depends on the density (or [[Sampling (signal processing)|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 functions that are [[bandlimiting|band-limited]] to a given bandwidth, such that no actual information is lost in the sampling process. It expresses the sufficient sample rate in terms of the bandwidth for the class of functions. The theorem also leads to a formula for perfectly reconstructing the original continuous-time function from the samples.

Perfect reconstruction may still be possible when the sample-rate criterion is not satisfied, provided other constraints on the signal are known (see {{section link||Sampling of non-baseband signals}} below and [[compressed sensing]]). In some cases (when the sample-rate criterion is not satisfied), utilizing additional constraints allows for approximate reconstructions. The fidelity of these reconstructions can be verified and quantified utilizing [[Bochner's theorem]].<ref>{{cite arXiv |last1=Nemirovsky|first1=Jonathan|last2=Shimron|first2=Efrat|title=Utilizing Bochners Theorem for Constrained Evaluation of Missing Fourier Data|eprint=1506.03300 |class=physics.med-ph |date=2015 }}</ref>

The name ''Nyquist–Shannon sampling theorem'' honours [[Harry Nyquist]] and [[Claude Shannon]], but the theorem was also previously discovered by [[E. T. Whittaker]] (published in 1915), and Shannon cited Whittaker's paper in his work. The theorem is thus also known by the names ''Whittaker–Shannon sampling theorem'', ''Whittaker–Shannon'', and ''Whittaker–Nyquist–Shannon'', and may also be referred to as the ''cardinal theorem of interpolation''.