Nyquist rate

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

When a continuous function, ${\displaystyle x(t),}$ is sampled at a constant rate, ${\displaystyle f_{s}}$ samples/second, there is always an unlimited number of other continuous functions that fit the same set of samples. But only one of them is bandlimited to ${\displaystyle {\tfrac {1}{2}}f_{s}}$ cycles/second (hertz),[upper-alpha 1] which means that its Fourier transform, ${\displaystyle X(f),}$ is ${\displaystyle 0}$ for all ${\displaystyle |f|\geq {\tfrac {1}{2}}f_{s}.}$  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, ${\displaystyle x(t),}$ is bandlimited to ${\displaystyle {\tfrac {1}{2}}f_{s},}$ 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 ${\displaystyle (B),}$ as depicted here, the Nyquist criterion is often stated as ${\displaystyle f_{s}>2B.}$  And ${\displaystyle 2B}$ is called the Nyquist rate for functions with bandwidth ${\displaystyle B.}$ When the Nyquist criterion is not met ${\displaystyle (}$say, ${\displaystyle B>{\tfrac {1}{2}}f_{s}),}$ a condition called aliasing occurs, which results in some inevitable differences between ${\displaystyle x(t)}$ 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.

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 than half a quantum step. This rate is generally referred to as signaling at the Nyquist rate and 1/(2B) has been termed a Nyquist interval." (bold added for emphasis; italics from the original)

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. 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, 2BT values are sufficient to specify the function, basing his conclusions on a Fourier series representation of the function over the time interval T."

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