Editing Nyquist rate

{{Short description|Minimum sampling rate to avoid aliasing}}
{{distinguish|Nyquist frequency}}
{{Use American English|date = March 2019}}
[[File:Nyquist frequency & rate.svg|thumb|Fig 1: Typical example of Nyquist frequency and rate. They are rarely equal, because that would require over-sampling by a factor of 2 (i.e. 4 times the bandwidth).]] 

In [[signal processing]], the '''Nyquist rate''', named after [[Harry Nyquist]], is a value  equal to twice the highest frequency ([[Bandwidth (signal processing)|bandwidth]]) of a given function or signal. It has units of [[Sampling (signal processing)|samples]] per unit time, conventionally expressed as samples per second, or [[hertz]] (Hz).<ref name=Oppenheim/> When the signal is sampled at a higher [[Sampling (signal processing)|sample rate]] {{xref|(see {{Slink|Nyquist–Shannon_sampling_theorem|Critical_frequency|nopage=y}})}}, the resulting [[discrete-time]] sequence is said to be free of the distortion known as [[aliasing]].  Conversely, for a given sample rate the corresponding [[Nyquist frequency]] is one-half the sample rate.  Note that the ''Nyquist rate'' is a property of a [[continuous-time signal]], whereas ''Nyquist frequency'' is a property of a discrete-time system.

The term ''Nyquist rate'' is also used in a different context with units of symbols per second, which is actually the field in which Harry Nyquist was working.  In that context it is an upper bound for the [[symbol rate]] across a bandwidth-limited [[baseband]] channel such as a [[electrical telegraph|telegraph line]]<ref name=Freeman/> or [[passband]] channel such as a limited radio frequency band or a [[frequency division multiplex]] channel.

==Relative to sampling==
[[Image:Bandlimited.svg|thumb|Fig 2: Fourier transform of a bandlimited function (amplitude vs frequency)|right|240px]]
When a continuous function, <math>x(t),</math> is sampled at a constant rate, <math>f_s</math> ''samples/second'', there is always an unlimited number of other continuous functions that fit the same set of samples.  But only one of them is [[Bandlimiting|bandlimited]] to <math>\tfrac{1}{2}f_s</math> ''cycles/second'' ([[hertz]]),{{efn-ua
|The factor of <math>\tfrac{1}{2}</math> has the units ''cycles/sample'' (see [[Sampling (signal processing)#Theory|Sampling]] and [[Sampling theorem]]).
}} which means that its [[Fourier transform]], <math>X(f),</math> is <math>0</math> for all <math>|f| \ge \tfrac{1}{2}f_s.</math>&nbsp; The mathematical algorithms that are typically used to recreate a continuous function from samples create arbitrarily good approximations to this theoretical, but infinitely long, function.  It follows that if the original function, <math>x(t),</math> is bandlimited to <math>\tfrac{1}{2}f_s,</math> which is called the ''Nyquist criterion'', then it is the one unique function the interpolation algorithms are approximating.  In terms of a function's own [[bandwidth (signal processing)|bandwidth]] <math>(B), </math> as depicted here, the '''Nyquist criterion''' is often stated as <math>f_s > 2B.</math>&nbsp; And <math>2B</math> is called the '''Nyquist rate''' for functions with bandwidth <math>B.</math>  When the Nyquist criterion is not met {{nowrap|<math>(</math>say, <math>B > \tfrac{1}{2}f_s),</math>}} a condition called [[aliasing]] occurs, which results in some inevitable differences between <math>x(t)</math> and a reconstructed function that has less bandwidth.  In most cases, the differences are viewed as distortion.

[[File:Bandpass sampling depiction.svg|thumb|right|300px|Fig 3: The top 2 graphs depict Fourier transforms of 2 different functions that produce the same results when sampled at a particular rate.  The baseband function is sampled faster than its Nyquist rate, and the bandpass function is undersampled, effectively converting it to baseband.  The lower graphs indicate how identical spectral results are created by the aliases of the sampling process.]]

===Intentional aliasing===
{{main|Undersampling}}
Figure 3 depicts a type of function called [[Baseband|baseband or lowpass]], because its positive-frequency range of significant energy is [0,&nbsp;''B'').  When instead, the frequency range is (''A'',&nbsp;''A''+''B''), for some ''A''&nbsp;>&nbsp;''B'', it is called [[bandpass]], and a common desire (for various reasons) is to convert it to baseband.  One way to do that is frequency-mixing ([[heterodyne]]) the bandpass function down to the frequency range (0,&nbsp;''B'').  One of the possible reasons is to reduce the Nyquist rate for more efficient storage.  And it turns out that one can directly achieve the same result by sampling the bandpass function at a sub-Nyquist sample-rate that is the smallest integer-sub-multiple of frequency ''A'' that meets the baseband Nyquist criterion:&nbsp; f<sub>s</sub>&nbsp;>&nbsp;2''B''.  For a more general discussion, see [[Undersampling|bandpass sampling]].

==Relative to signaling==

Long before [[Harry Nyquist]] had his name associated with sampling, the term ''Nyquist rate'' was used differently, with a meaning closer to what Nyquist actually studied.  Quoting [[Harold Stephen Black|Harold S. Black's]] 1953 book ''Modulation Theory,'' in the section '''''Nyquist Interval''''' of the opening chapter ''Historical Background:''

:"If the essential frequency range is limited to ''B'' cycles per second, 2''B'' was given by Nyquist as the maximum number of code elements per second that could be unambiguously resolved, assuming the peak interference is less than half a quantum step.  This rate is generally referred to as '''signaling at the Nyquist rate''' and 1/(2''B'') has been termed a ''Nyquist interval''." (bold added for emphasis; italics from the original)

''B'' in this context, related to the [[Nyquist ISI criterion]], referring to the one-sided bandwidth rather than the total as considered in later usage.

According to the [[Oxford English Dictionary|OED]], Black's statement regarding 2''B'' may be the origin of the term ''Nyquist rate''.<ref name=Black/>

Nyquist's famous 1928 paper was a study on how many pulses (code elements) could be transmitted per second, and recovered, through a channel of limited bandwidth.<ref name=Nyquist/>
''Signaling at the Nyquist rate'' meant putting as many code pulses through a telegraph channel as its bandwidth would allow.  Shannon used Nyquist's approach when he proved the [[sampling theorem]] in 1948, but Nyquist did not work on sampling per se.

Black's later chapter on "The Sampling Principle" does give Nyquist some of the credit for some relevant math:

:"Nyquist (1928) pointed out that, if the function is substantially limited to the time interval ''T'', 2''BT'' values are sufficient to specify the function, basing his conclusions on a Fourier series representation of the function over the time interval ''T''."

==See also==
*[[Nyquist frequency]]
*[[Nyquist ISI criterion]]
*[[Nyquist–Shannon sampling theorem]]
*[[Sampling (signal processing)]]

== Notes ==
{{notelist-ua|1}}

== References ==
{{reflist|1|refs=
<ref name=Oppenheim>
{{Cite book |ref=Oppenheim |last1=Oppenheim |first1=Alan V. |authorlink=Alan V. Oppenheim |last2=Schafer |first2=Ronald W. |author2-link=Ronald W. Schafer |last3=Buck |first3=John R. |title=Discrete-time signal processing |year=1999 |publisher=Prentice Hall |location=Upper Saddle River, N.J. |isbn=0-13-754920-2 |edition=2nd |page=140 |url-access=registration |url=https://archive.org/details/discretetimesign00alan |quote=T is the sampling period, and its reciprocal, f<sub>s</sub>=1/T, is the sampling frequency, in samples per second.
}}
</ref>

<ref name=Freeman>
{{cite book | title = Telecommunication System Engineering | author = Roger L. Freeman | publisher = John Wiley & Sons | year = 2004 | isbn = 0-471-45133-9 | url = https://books.google.com/books?id=Ga7PYE7E8kQC&dq=nyquist-rate+define+bandwidth+symbols&pg=PA399| pages = 399
}}</ref>

<ref name=Black>
[[Harold Stephen Black|Black, H. S.]], ''Modulation Theory'', v. 65, 1953, cited in [[OED]]
</ref>

<ref name=Nyquist>
Nyquist, Harry. "Certain topics in telegraph transmission theory", Trans. AIEE, vol. 47, pp.&nbsp;617–644, Apr. 1928 [https://web.archive.org/web/20060706192816/http://www.loe.ee.upatras.gr/Comes/Notes/Nyquist.pdf Reprint as classic paper in: ''Proc. IEEE, Vol. 90, No. 2, Feb 2002''].
</ref>

}}

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[[Category:Digital signal processing]]
[[Category:Telecommunication theory]]
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