Nyquist frequency

The black dots are aliases of each other. The solid red line is an example of adjusting amplitude vs frequency. The dashed red lines are the corresponding paths of the aliases.

The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist, is half the sampling frequency of a discrete signal processing system.^[1][2] It is sometimes known as the folding frequency of a sampling system.^[3]

When the continuous function being sampled contains no frequencies equal or higher than the Nyquist frequency, all the aliases caused by sampling occur above the Nyquist frequency. The term aliasing usually refers to the case where some original frequency components have aliases below Nyquist. That often causes distortion when a continuous function is subsequently reconstructed from samples.

The Nyquist frequency should not be confused with the Nyquist rate, which is the lower bound of the sampling frequency that satisfies the Nyquist sampling criterion for a given signal or family of signals. This lower bound is twice the bandwidth or maximum component frequency of the signal. Nyquist rate, as commonly used with respect to sampling, is a property of a continuous-time signal, not of a system, whereas Nyquist frequency is a property of a discrete-time system, not of a signal. While the domain of the signals is commonly time, leading to a Nyquist frequency in Hertz, this does not have to be the case; for example, an image sampling system has a Nyquist frequency expressed in units of reciprocal length, such as cycles per meter.

The aliasing problem

In theory, a Nyquist frequency just larger than the signal bandwidth is sufficient to allow perfect reconstruction of the signal from the samples: see Sampling theorem: Critical frequency. However, this reconstruction requires an ideal filter that passes some frequencies unchanged while suppressing all others completely (commonly called a brick-wall filter). In practice, perfect reconstruction is unattainable. Some amount of aliasing is unavoidable.

Signal frequencies higher than the Nyquist frequency will encounter a "folding" about the Nyquist frequency, back into lower frequencies. For example, if the sample rate is 20 kHz, the Nyquist frequency is 10 kHz, and an 11 kHz signal will fold, or alias, to 9 kHz: . However, a 9 kHz signal can also fold up to 11 kHz in that case if the reconstruction filter is not adequate. Both types of aliasing can be important.

When attainable filters are used, some degree of oversampling is necessary to accommodate the practical constraints on anti-aliasing filters: instead of a brickwall, one has flat response in the passband up to a point called the cutoff frequency or corner frequency, (pass all frequencies below there unchanged), then gradual rolloff in a transition band, finally suppressing signals above a certain point completely or almost completely in the stopband. Thus, 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 (frequency headroom) that depends on the actual filters used.

For example, audio CDs have a sampling frequency of 44100 Hz. The Nyquist frequency is therefore 22050 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 2000 Hz, then the cut-off frequency should be ≤ 20050 Hz to yield a signal with negligible power at frequencies ≥ 22050 Hz and complete pass of frequencies ≤ 20 kHz (within the human hearing range).

Other meanings

Early uses of the term Nyquist frequency, such as those cited above, are all consistent with the definition presented in this article. Some later publications, including some respectable textbooks, call twice the signal bandwidth (the Nyquist rate) as Nyquist frequency;^[4][5] this is a distinctly minority usage.

See also

• Kell factor
• Sampling frequency
• Superoscillation
• Signal

References