Editing Nyquist frequency

{{Use American English|date = March 2019}}
{{Short description|Maximum frequency of non-aliased component upon sampling}}
{{distinguish|Nyquist rate}}
[[File:Nyquist frequency & rate.svg|thumb|Typical example of Nyquist frequency and rate. To avoid aliasing, the sampling rate must be no less than the Nyquist rate of the signal; that is, the Nyquist rate of the signal must be under double the Nyquist frequency of the sampling.]]
In [[signal processing]], the '''Nyquist frequency''' (or '''folding frequency'''), named after [[Harry Nyquist]], is a characteristic of a [[Sampling (signal processing)|sampler]], which converts a continuous function or signal into a discrete sequence. For a given [[Sampling (signal processing)|sampling rate]] (''samples per second''), the Nyquist frequency ''([[cycles per second]]'') is the frequency whose cycle-length (or period) is twice the interval between samples, thus ''0.5 cycle/sample''.  For example, audio [[compact disc|CDs]] have a sampling rate of 44100 ''samples/second''.  At ''0.5 cycle/sample'', the corresponding Nyquist frequency is 22050 ''cycles/second'' ([[hertz|Hz]]).  Conversely, the Nyquist rate for sampling a 22050 Hz signal is 44100 ''samples/second''.<ref name="Grenlander" /><ref name="Leis" />{{efn-ua
|When the function domain is distance, as in an image sampling system, the sample rate might be dots per inch and the corresponding Nyquist frequency would be in cycles per inch.
}} 

When the highest frequency ([[Bandwidth (signal processing)|bandwidth]]) of a signal is less than the Nyquist frequency of the sampler, the resulting [[discrete-time]] sequence is said to be free of the distortion known as [[aliasing]], and the corresponding sample rate is said to be above the [[Nyquist rate]] for that particular signal.<ref name="Condon" /><ref name="Stiltz" />

In a typical application of sampling, one first chooses the highest frequency to be preserved and recreated, based on the expected content (voice, music, etc.) and desired fidelity. Then one inserts an [[anti-aliasing filter]] ahead of the sampler. Its job is to attenuate the frequencies above that limit. Finally, based on the characteristics of the filter, one chooses a sample rate (and corresponding Nyquist frequency) that will provide an acceptably small amount of [[aliasing]].  In applications where the sample rate is predetermined (such as the CD rate), the filter is chosen based on the Nyquist frequency, rather than vice versa.

== Folding frequency ==
{{main|Aliasing}}
[[File:Aliasing-folding-2.svg|thumb|400px|The black dots are aliases of each other. The solid red line is an <u>example</u> of amplitude varying with frequency. The dashed red lines are the corresponding paths of the aliases.]]

In this example, {{math|''f''{{sub|s}}}} is the sampling rate, and {{math|0.5 ''cycle/sample'' × ''f''{{sub|s}}}} is the corresponding Nyquist frequency. The black dot plotted at {{math|0.6 ''f''{{sub|s}}}} represents the amplitude and frequency of a sinusoidal function whose frequency is 60% of the sample rate. The other three dots indicate the frequencies and amplitudes of three other sinusoids that would produce the same set of samples as the actual sinusoid that was sampled.  Undersampling of the sinusoid at {{math|0.6 ''f''{{sub|s}}}} is what allows there to be a lower-frequency [[Aliasing#Sampling sinusoidal functions|alias]].  If the true frequency were {{math|0.4 ''f''{{sub|s}}}}, there would still be aliases at 0.6, 1.4, 1.6, etc.

The red lines depict the paths ([[wikt:loci|loci]]) of the 4 dots if we were to adjust the frequency and amplitude of the sinusoid along the solid red segment (between &nbsp;{{math|''f''{{sub|s}}/2}}&nbsp; and &nbsp;{{math|''f''{{sub|s}}}}).&nbsp; No matter what function we choose to change the amplitude vs frequency, the graph will exhibit symmetry between 0 and &nbsp;{{math|''f''{{sub|s}}.}}&nbsp;  This symmetry is commonly referred to as '''folding''', and another name for &nbsp;{{math|''f''{{sub|s}}/2}}&nbsp; (the Nyquist frequency) is '''folding frequency'''.<ref name=Zawistowski/>

==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 frequency;<ref name=Blackledge/><ref name=Diniz/> this is a distinctly minority usage, and the frequency at twice the signal bandwidth is otherwise commonly referred to as the [[Nyquist rate]].

==Notes==
{{notelist-ua}}

==References==
{{reflist|1|refs=
<ref name=Grenlander>
{{cite book|last=Grenander|first=Ulf|url=https://books.google.com/books?id=UPc0AAAAMAAJ&q=%22nyquist+frequency%22+date:0-1965|title=Probability and Statistics: The Harald Cramér Volume|publisher=Wiley|year=1959|quote=The Nyquist frequency is that frequency whose period is two sampling intervals.|author-link=Ulf Grenander
}}</ref>

<ref name=Stiltz>
{{cite book|author=Harry L. Stiltz|url=https://books.google.com/books?id=cro8AAAAIAAJ&q=%22nyquist+frequency%22+date:0-1965|title=Aerospace Telemetry|publisher=Prentice-Hall|year=1961| isbn=9780130182838 |quote=the existence of power in the continuous signal spectrum at frequencies higher than the Nyquist frequency is the cause of aliasing error
}}</ref>

<ref name=Condon>
{{cite book|author1=James J. Condon|url=https://books.google.com/books?id=Jg6hCwAAQBAJ&pg=PA281|title=Essential Radio Astronomy|author2=Scott M. Ransom|date=2016|publisher=Princeton University Press|isbn=9781400881161|pages=280–281|name-list-style=amp
}}</ref>

<ref name=Leis>
{{cite book|author=John W. Leis|url=https://books.google.com/books?id=Qtd-e1NtZVkC&pg=PA82|title=Digital Signal Processing Using MATLAB for Students and Researchers|date=2011|publisher=John Wiley & Sons|isbn=9781118033807|page=82|quote=The ''Nyquist rate'' is twice the bandwidth of the signal ... The ''Nyquist frequency'' or ''folding frequency'' is half the sampling rate and corresponds to the highest frequency which a sampled data system can reproduce without error.
}}</ref>

<ref name=Zawistowski>
{{Cite web
  |author1=Thomas Zawistowski |author2=Paras Shah | title = An Introduction to Sampling Theory
  | url = http://www2.egr.uh.edu/~glover/applets/Sampling/Sampling.html
  | access-date = 17 April 2010
  | quote = Frequencies "fold" around half the sampling frequency - which is why the [Nyquist] frequency is often referred to as the folding frequency.
}}</ref>

<ref name=Blackledge>
{{cite book | title = Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications | author = Jonathan M. Blackledge | publisher = Horwood Publishing | year = 2003 | isbn = 1-898563-48-9 | url = https://books.google.com/books?id=G_2Zh7ldQIIC&dq=intitle:digital+intitle:signal+intitle:processing+nyquist-frequency&pg=PA93
}}</ref>

<ref name=Diniz>
{{cite book | title = Digital Signal Processing: System Analysis and Design | author = Paulo Sergio Ramirez Diniz, Eduardo A. B. Da Silva, Sergio L. Netto | publisher = Cambridge University Press | year = 2002 | isbn = 0-521-78175-2 | url = https://books.google.com/books?id=L9ENNEPbZ8IC&dq=intitle:digital+intitle:signal+intitle:processing+bandwidth+nyquist-frequency&pg=PA24
}}</ref>

}}

== See also ==

* [[Nyquist–Shannon sampling theorem]]
{{DSP}}

[[Category:Digital signal processing]]