The basic bridge between continuous signal (usually called "analog signal") and discrete signal (usually called "digital signal"). It determines the upper limit of signal bandwidth, or the lower limit of sampling frequency allowed by discrete sampling signals that can capture all the information of continuous signals.

Strictly speaking, this theorem is only applicable to a kind of mathematical function with Fourier transform, that is, the frequency is zero outside the finite region. Discrete-time Fourier transform (a form of Poisson summation formula) provides an analytical continuation of the actual signal, but it can only approximate this situation.

Intuitively, we hope that when a continuous function is transformed into a discrete sequence of sampling values (called "samples") and interpolated into the continuous function, the fidelity of the result depends on the density (or sampling rate) of the original sample.

The sampling theorem introduces the concept of sampling rate, and the fidelity of sampling rate is complete enough for the function type with limited bandwidth. In the process of sampling, "information" is not lost. This theorem uses the bandwidth of a function to represent the sampling rate. The theorem also derives a mathematical ideal formula for reconstructing the original continuous signal.

This theorem does not rule out the possibility of complete reconstruction in some special cases that do not meet the sampling rate criterion. (See sampling and compressive sensing of non-baseband signals below. )

Non-uniform sampling

Shannon sampling theorem can be extended to non-uniform sampling, that is, the sampling interval is not constant. The sampling theorem of non-uniform sampling points out that the original signal can be completely reconstructed from the sampled signal as long as the average sampling frequency meets the Nyquist condition. Therefore, although uniform sampling is relatively simple in signal reconstruction algorithm, it is not a necessary condition for complete reconstruction.

The universal theory of non-baseband non-uniform sampling was put forward by Henry Landau in 1967. Simply put, the blue channel proves that the average sampling rate needs to be at least twice the bandwidth occupied by the signal, but only if the spectrum and bandwidth of the signal are known. ?

At the end of 1990, this research has been extended to the case that the bandwidth occupied by the signal is known, but the actual position in the spectrum is unknown. In 2000, a complete theory has been developed by using compressive sensing. This theory was written in the language of signal processing and published in a paper in 2009.

This paper proves that the sampling rate should be at least twice that of Nyquist criterion when the frequency position is unknown. In other words, because we don't know the position of the spectrum, we need to multiply the sampling rate by 2. Note that this minimum sampling rate requirement does not necessarily guarantee its numerical stability.