Nyquist rate

In signal processing, the Nyquist rate is two times the bandwidthbut this concept has two rather different meanings: as a lower bound for the sample rate for alias-free signal sampling, and as an upper bound for the signaling rate across a bandwidth-limited channel such as a telegraph line.

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In signal processing, the Nyquist rate is two times the bandwidth—but this concept has two rather different meanings: as a lower bound for the sample rate for alias-free signal sampling, and as an upper bound for the signaling rate across a bandwidth-limited channel such as a telegraph line.

Nyquist rate sampling

The Nyquist rate is the minimum sampling rate required to avoid aliasing Aliasing

In statistics [i], signal processing [i], and related disciplines, aliasing is an effect that causes dif ...

when sampling a continuous signal. In other words, the Nyquist rate is the minimum sampling rate required to allow unambiguous reconstruction of a bandlimited continuous signal from its samples. If the input signal is real and bandlimited Bandlimited

A bandlimited signal is a deterministic [i] or stochastic [i] signal whose Fourier transform [i], or pow...

, the Nyquist rate is simply twice the highest frequency contained within the signal. In other words, the Nyquist rate is equal to the two-sided bandwidth Bandwidth

Bandwidth is a measure of frequency [i] range and is typically measured in hertz [i].
...

of the signal.

where is the highest frequency component contained in the signal.

To avoid aliasing, the sampling rate must exceed the Nyquist rate:

Signaling at the Nyquist rate

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 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, 2B was given by Nyquist as the maximum number of code elements per second that could be unambiguously resolved, assuming the peak interference is less half a quantum step. This rate is generally referred to as signaling at the Nyquist rate and 1/ has been termed a Nyquist interval."

According the the OED Oxford English Dictionary

The Oxford English Dictionary is a dictionary [i] published by the Oxford University Press [i] , an ...

, this is may be the origin of the term Nyquist rate.

Nyquist's famous 1928 paper was a study on how many pulses could be transmitted per second, and recovered, through a channel of limited bandwidth. 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 Nyquist–Shannon sampling theorem

The NyquistShannon sampling theorem is a fundamental result in the field of information theory [i], in p...

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 pointed out that, if the function is substantially limited to the time interval T, 2BT values are sufficient to specify the function, basing his conclusions on a Fourier series representation of the function over the time interval T."

Reference

Black, H. S., Modulation Theory, v. 65, 1953, cited in OED Oxford English Dictionary