Nyquist rate

Not to be confused withNyquist frequency.

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).

Insignal processing,theNyquist rate,named afterHarry Nyquist,is a value equal to twice the highest frequency (bandwidth) of a given function or signal. It has units ofsamplesper unit time, conventionally expressed as samples per second, orhertz(Hz).When the signal is sampled at a highersample rate(see§ Critical frequency), the resultingdiscrete-timesequence is said to be free of the distortion known asaliasing.Conversely, for a given sample rate the correspondingNyquist frequencyis one-half the sample rate. Note that theNyquist rateis a property of acontinuous-time signal,whereasNyquist frequencyis a property of a discrete-time system.

The termNyquist rateis 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 thesymbol rateacross a bandwidth-limitedbasebandchannel such as atelegraph lineorpassbandchannel such as a limited radio frequency band or afrequency division multiplexchannel.

Relative to sampling

Fig 2: Fourier transform of a bandlimited function (amplitude vs frequency)

When a continuous function,$x(t),$is sampled at a constant rate,$f_{s}$samples/second,there is always an unlimited number of other continuous functions that fit the same set of samples. But only one of them isbandlimitedto${\tfrac {1}{2}}f_{s}$cycles/second(hertz),which means that itsFourier transform,$X(f),$is$0$for all$|f|\geq {\tfrac {1}{2}}f_{s}.$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,$x(t),$is bandlimited to${\tfrac {1}{2}}f_{s},$which is called theNyquist criterion,then it is the one unique function the interpolation algorithms are approximating. In terms of a function's ownbandwidth$(B),$as depicted here, theNyquist criterionis often stated as$f_{s}>2B.$And$2B$is called theNyquist ratefor functions with bandwidth$B.$When the Nyquist criterion is not met$($say,$B>{\tfrac {1}{2}}f_{s}),$a condition calledaliasingoccurs, which results in some inevitable differences between$x(t)$and a reconstructed function that has less bandwidth. In most cases, the differences are viewed as distortion.

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 article:Undersampling

Figure 3 depicts a type of function calledbaseband or lowpass,because its positive-frequency range of significant energy is [0,B). When instead, the frequency range is (A,A+B), for someA>B,it is calledbandpass,and a common desire (for various reasons) is to convert it to baseband. One way to do that is frequency-mi xing (heterodyne) the bandpass function down to the frequency range (0,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 frequencyAthat meets the baseband Nyquist criterion: f_s> 2B.For a more general discussion, seebandpass sampling.

Relative to signaling

Long beforeHarry Nyquisthad his name associated with sampling, the termNyquist ratewas used differently, with a meaning closer to what Nyquist actually studied. QuotingHarold S. Black's1953 bookModulation Theory,in the sectionNyquist Intervalof the opening chapterHistorical Background:

"If the essential frequency range is limited toBcycles per second, 2Bwas 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 assignaling at the Nyquist rateand 1/(2B) has been termed aNyquist interval."(bold added for emphasis; italics from the original)

According to theOED,Black's statement regarding 2Bmay be the origin of the termNyquist rate.

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. Signaling at the Nyquist ratemeant putting as many code pulses through a telegraph channel as its bandwidth would allow. Shannon used Nyquist's approach when he proved thesampling theoremin 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 intervalT,2BTvalues are sufficient to specify the function, basing his conclusions on a Fourier series representation of the function over the time intervalT."

See also

• Nyquist frequency
• Nyquist ISI criterion
• Nyquist–Shannon sampling theorem
• Sampling (signal processing)