Nyquist–Shannon sampling theorem

Sampling converts a continuous-time signal into a sequence of measurements taken at discrete, usually uniform, time intervals. Suppose a baseband signal contains no frequency components at or above B hertz, and let its sampling frequency be f_s samples per second. The Nyquist–Shannon sampling theorem states that the original signal can be reconstructed exactly from its samples in the ideal model when f_s > 2B. Equivalently, the sampling interval must be shorter than 1/(2B). The result depends on the signal being band-limited and on uniform sampling; it is not a guarantee of exact recovery for arbitrary signals or noisy measurements.

In theory, an appropriately band-limited signal can be recovered by interpolation from its samples; ideal reconstruction can be expressed as a sum of shifted sinc functions weighted by the sample values. The theorem's frequency limit matters because sampling makes frequency components separated by whole multiples of the sampling frequency indistinguishable in the sampled sequence. If the rate is too low, distinct high-frequency signals can therefore produce the same samples, and high-frequency content can appear as lower-frequency content. This effect is called aliasing. Once different components have overlapped in the sampled data, the original signal generally cannot be uniquely recovered from those samples alone.

For a baseband signal limited to B hertz, 2B is commonly called its Nyquist rate: the theoretical lower sampling-rate bound associated with that bandwidth. For a system sampling at f_s, the Nyquist frequency is f_s/2, half the sampling frequency. These terms are related but are not interchangeable. Technical writing has sometimes used “Nyquist rate” inconsistently for the half-sampling-frequency limit, so stating the actual frequency or sampling rate avoids ambiguity. Because the ideal theorem assumes no signal energy at or above the limiting frequency, practical systems place a margin between the signal band edge and half the sampling rate rather than relying on a component exactly at that boundary.

Real signals and measuring equipment do not meet the theorem's ideal assumptions perfectly. An analog low-pass anti-aliasing filter is commonly placed before an analogue-to-digital converter to attenuate frequencies that would exceed the intended band. Since physical filters cannot change from passing to rejecting frequencies instantaneously, designers typically allow a transition band and choose a sampling rate with practical margin. Digital reconstruction and filtering also approximate ideal operations. These measures reduce aliasing; simply increasing a sampling rate after aliasing has occurred cannot restore information that the original sampling discarded.

The theorem underpins the conversion of sound, radio, sensor measurements, and other time-varying physical signals into digital data. Related sampling principles also apply to spatial signals, including images, where inadequate sampling can produce visible patterns or detail that was not present in the scene. It specifies when samples contain enough information for ideal reconstruction, not how accurate or useful a complete recording or measurement will be: noise, quantization, finite recording duration, imperfect filters, and other engineering constraints still matter.

The theorem's familiar name reflects work by more than one researcher and builds on earlier mathematical results. In 1928, Harold Nyquist published an analysis of the frequency range needed to transmit telegraph signals at a given signaling speed. Claude Shannon's later work helped bring the sampling theorem into modern communication theory. Historical accounts therefore treat Nyquist's paper as an important precursor rather than reducing the theorem's development to a single publication or person.