In signal processing, the Nyquist rate is two times the bandwidth of a bandlimited signal or a bandlimited channel. This term is used to mean two different things under two different circumstances:
Nyquist rate relative to sampling
The Nyquist rate is the minimum sampling rate required to avoid aliasing, equal to twice the highest frequency contained within the signal.
where is the highest frequency at which the signal can have nonzero energy.
To avoid aliasing, the sampling rate must exceed the Nyquist rate:
A signal whose positive-frequency range of significant energy is (0,B), as depicted above, is called baseband or lowpass. But when the frequency range is (A,B), for some A > B-A, it is called bandpass. In that case, aliasing is not necessarily detrimental, and sampling below the Nyquist rate, called bandpass sampling, is sometimes done. With careful design, a rate as low as 2(B-A) may be achievable, and it is equivalent to mixing (heterodyne) the signal into the frequency range (0,B-A), whose Nyquist rate is 2(B-A). An even lower Nyquist rate can be achieved for a bandpass signal, such as amplitude modulation, whose energy distribution in (A,B) is symmetrical. In that case, the homodyne mixer translates the signal frequencies by (A+B)/2, with a synchronized phase, which moves the highest component to (B-A)/2 and the Nyquist rate to just (B-A).
Nyquist rate 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 S. Black's 1953 book Modulation Theory, in the section Nyquist Interval of the opening chapter Historical Background:
According to the OED, this may be the origin of the term Nyquist 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 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:
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