Nyquist rate

In signal processing, the Nyquist rate, named after Harry Nyquist, is a value equal to twice the highest frequency (bandwidth) of a given function or signal.

[1] When the signal is sampled at a higher sample rate (see § Critical frequency), the resulting discrete-time sequence is said to be free of the distortion known as aliasing.

Note that the Nyquist rate is a property of a continuous-time signal, whereas Nyquist frequency is a property of a discrete-time system.

In that context it is an upper bound for the symbol rate across a bandwidth-limited baseband channel such as a telegraph line[2] or passband channel such as a limited radio frequency band or a frequency division multiplex channel.

samples/second, there is always an unlimited number of other continuous functions that fit the same set of samples.

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.

which is called the Nyquist criterion, then it is the one unique function the interpolation algorithms are approximating.

is called the Nyquist rate for functions with bandwidth

a condition called aliasing occurs, which results in some inevitable differences between

Figure 3 depicts a type of function called baseband or lowpass, because its positive-frequency range of significant energy is [0, B).

When instead, the frequency range is (A, A+B), for some A > 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, 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:  fs > 2B.

Quoting Harold S. Black's 1953 book Modulation Theory, in the section Nyquist Interval of the opening chapter Historical Background: According to the OED, Black's statement regarding 2B may be the origin of the term Nyquist rate.

[3] 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.

[4] Signaling at the Nyquist rate meant putting as many code pulses through a telegraph channel as its bandwidth would allow.

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

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