Nyquist frequency
Encyclopedia
The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist
or the Nyquist–Shannon sampling theorem
, is half the sampling frequency of a discrete signal
processing system. It is sometimes known as the folding frequency of a sampling system.
The sampling theorem shows that aliasing
can be avoided if the Nyquist frequency is greater than the bandwidth, or maximum component frequency, of the signal being sampled.
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.
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 filter
s: 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 no higher than 20050 Hz to yield a signal with negligible power at frequencies of 22050 Hz and greater.
) as Nyquist frequency; this is a distinctly minority usage.
Harry Nyquist
Harry Nyquist was an important contributor to information theory.-Personal life:...
or the Nyquist–Shannon sampling theorem
Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...
, is half the sampling frequency of a discrete signal
Discrete signal
A discrete signal or discrete-time signal is a time series consisting of a sequence of qualities...
processing system. It is sometimes known as the folding frequency of a sampling system.
The sampling theorem shows that aliasing
Aliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...
can be avoided if the Nyquist frequency is greater than the bandwidth, or maximum component frequency, of the signal being sampled.
The Nyquist frequency should not be confused with the Nyquist rate
Nyquist rate
In signal processing, the Nyquist rate, named after Harry Nyquist, is two times the bandwidth of a bandlimited signal or a bandlimited channel...
, 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
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....
, this does not have to be the case; for example, an image sampling system has a Nyquist frequency expressed in units of reciprocal length
Reciprocal length
Reciprocal length or inverse length is a measurement used in several branches of science and mathematics. As the reciprocal of length, common units used for this measurement include the reciprocal metre or inverse metre , the reciprocal centimetre or inverse centimetre , and, in optics, the...
, 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
Oversampling
In signal processing, oversampling is the process of sampling a signal with a sampling frequency significantly higher than twice the bandwidth or highest frequency of the signal being sampled...
is necessary to accommodate the practical constraints on anti-aliasing filter
Anti-aliasing filter
An anti-aliasing filter is a filter used before a signal sampler, to restrict the bandwidth of a signal to approximately satisfy the sampling theorem....
s: instead of a brickwall, one has flat response in the passband
Passband
A passband is the range of frequencies or wavelengths that can pass through a filter without being attenuated.A bandpass filtered signal , is known as a bandpass signal, as opposed to a baseband signal....
up to a point called the cutoff frequency
Cutoff frequency
In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced rather than passing through.Typically in electronic systems such as filters and...
or corner frequency, (pass all frequencies below there unchanged), then gradual rolloff in a transition band
Transition band
The transition band is a range of frequencies, that allows a transition between a passband and a stopband of a signal processing filter. The transition band is defined by a passband and a stopband cutoff frequency or corner frequency....
, finally suppressing signals above a certain point completely or almost completely in the stopband
Stopband
A stopband is a band of frequencies, between specified limits, through which a circuit, such as a filter or telephone circuit, does not allow signals to pass, or the attenuation is above the required stopband attenuation level...
. 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
Compact Disc
The Compact Disc is an optical disc used to store digital data. It was originally developed to store and playback sound recordings exclusively, but later expanded to encompass data storage , write-once audio and data storage , rewritable media , Video Compact Discs , Super Video Compact Discs ,...
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
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
in this case) has a transition band
Transition band
The transition band is a range of frequencies, that allows a transition between a passband and a stopband of a signal processing filter. The transition band is defined by a passband and a stopband cutoff frequency or corner frequency....
of 2000 Hz, then the cut-off frequency should be no higher than 20050 Hz to yield a signal with negligible power at frequencies of 22050 Hz and greater.
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 rateNyquist rate
In signal processing, the Nyquist rate, named after Harry Nyquist, is two times the bandwidth of a bandlimited signal or a bandlimited channel...
) as Nyquist frequency; this is a distinctly minority usage.
See also
- Nyquist rateNyquist rateIn signal processing, the Nyquist rate, named after Harry Nyquist, is two times the bandwidth of a bandlimited signal or a bandlimited channel...
- Kell factorKell factorThe Kell factor, named after RCA engineer Raymond D. Kell, is a parameter used to limit the bandwidth of a sampled image signal to avoid the appearance of beat frequency patterns when displaying the image in a discrete display devices, usually taken to be 0.7. The number was first measured in 1934...
- Sampling frequency
- SuperoscillationSuperoscillationSuperoscillation is a phenomenon in which a signal which is globally band-limited can contain local segments that oscillate faster than its fastest Fourier components...
- Signal