In the field of digital signal processing, the sampling theorem is a fundamental bridge between continuous signals (analog domain) and discrete signals (digital domain). Strictly speaking, it only applies to a class of mathematical functions whose Fourier transforms are zero outside of a finite region of frequencies (see Fig 1). The analytical extension to actual signals, which can only approximate that condition, is provided by the discrete-time Fourier transform, a version of the Poisson summation formula. Intuitively we expect that when one reduces a continuous function to a discrete sequence (called samples) 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 bandlimited functions. And it expresses the sample-rate in terms of the function's bandwidth. Thus no actual "information" is lost during the sampling process. The theorem also leads to a formula for the mathematically ideal interpolation algorithm.
The theorem does not preclude the possibility of perfect reconstruction under special circumstances that do not satisfy the sample-rate criterion. (See Sampling of non-baseband signals below, and Compressed sensing.)
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