Nyquist–Shannon sampling theorem

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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 of the signal to avoid aliasing distortion. In practice, it is used to select band-limiting filters to keep aliasing distortion below an acceptable amount when an analog signal is sampled or when sample rates are changed within a digital signal processing function. The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. 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. Strictly speaking, the theorem only applies to a class of mathematical functions having a 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 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 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 below and compressed sensing). In some cases (when the sample-rate criterion is not satisfied), utilizing additional constraints allows for approximate reconstructions.

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Bandlimiting refers to a process which reduces the energy of a signal to an acceptably low level outside of a desired frequency range. Bandlimiting is an essential part of many applications in signal processing and communications. Examples include controlling interference between radio frequency communications signals, and managing aliasing distortion associated with sampling for digital signal processing. A bandlimited signal is, strictly speaking, a signal with zero energy outside of a defined frequency range.

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