Nyquist frequency

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Not to be confused with Nyquist rate.

The Nyquist frequency, named after Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system. It is sometimes called the critical frequency. The sampling theorem tells us that aliasing can be avoided if the Nyquist frequency is greater than the bandwidth, or maximum frequency, of the signal being sampled.

In principle, a Nyquist frequency just larger than the signal bandwidth is sufficient to allow perfect reconstruction of the signal from the samples. However, this reconstruction requires an unrealizable filter that passes some frequencies unchanged while suppressing all others completely (commonly called a brickwall filter). When realizable filters are used, some degree of oversampling is necessary to accommodate the practical constraints on anti-aliasing filters. That is, frequencies close to the Nyquist frequency may be distorted in the sampling and reconstruction process, so the bandwidth should be kept below the Nyquist frequency by some margin that depends on the actual filters used.

For example, audio CDs have a sampling frequency of 44,100 Hz. The Nyquist frequency is therefore 22,050 Hz, which is an upper bound on the highest frequency the data can unambiguously represent. If the chosen anti-aliasing filter (a low-pass filter in this case) has a transition band of 2,000 Hz, then the cut-off frequency should be no higher than 20,050 Hz to yield a signal with negligible power at frequencies of 22,050 Hz and greater.

It should be noted that the Nyquist frequency must be strictly greater than the maximum frequency component within the signal. If the signal contains a frequency component at precisely the Nyquist frequency then the corresponding component of the sample values can not have sufficient information to reconstruct the Nyquist component in the continuous-time signal because of phase ambiguity. In such a case, there would be an infinite number of possible and different sinusoids (of varying amplitude and phase) of the Nyquist frequency component that are represented by the discrete samples.

The Nyquist frequency is defined differently from the Nyquist rate, which is the minimum sampling frequency that meets the Nyquist sampling criterion for a given signal or family of signals. Nyquist rate, as commonly used with respect to sampling, is a property of a signal, not of a system.

Reference

Philosophical Transactions, Royal Society 1963, A. 255 512 (cited in OED)

Nyquist frequency

Reference

See also