20 Facts About Sampling theorem

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.

Strictly speaking, the Sampling theorem only applies to a class of mathematical functions having a Fourier transform that is zero outside of a finite region of frequencies.

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.

The Sampling theorem leads to a formula for perfectly reconstructing the original continuous-time function from the samples.

Sampling theorem is a process of converting a signal into a sequence of values .

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The Sampling theorem is applicable to functions of other domains, such as space, in the case of a digitized image.

Shannon's proof of the Sampling 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.

Sampling theorem is usually formulated for functions of a single variable.

Consequently, the Sampling theorem is directly applicable to time-dependent 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.

Sampling theorem applies to camera systems, where the scene and lens constitute an analog spatial signal source, and the image sensor is a spatial sampling device.

Sampling theorem applies to post-processing digital images, such as to up or down sampling.

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 it is generally safe to assume that if the sampling theorem is not satisfied then information will most likely be lost.

Sampling theorem proved that the average sampling rate must be twice the occupied bandwidth of the signal, assuming it is a priori known what portion of the spectrum was occupied.

Nyquist–Shannon sampling theorem provides a sufficient condition for the sampling and reconstruction of a band-limited signal.

Sampling theorem was implied by the work of Harry Nyquist in 1928, 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.

In Shannon 1948 the sampling theorem is formulated as “Theorem 13”: Let f contain no frequencies over W Then.

In later years it became known that the sampling theorem had been presented before Shannon to the Russian communication community by Kotel'nikov [173].

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.

The term Nyquist Sampling Theorem appeared as early as 1959 in a book from his former employer, Bell Labs, and appeared again in 1963, and not capitalized in 1965.