Nyquist frequency
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In signal processing, the Nyquist frequency (or folding frequency), named after Harry Nyquist, is a characteristic of a sampler, which converts a continuous function or signal into a discrete sequence. For a given sampling rate (samples per second), the Nyquist frequency (cycles per second) is the frequency whose cycle-length (or period) is twice the interval between samples, thus 0.5 cycle/sample. For example, audio CDs have a sampling rate of 44100 samples/second. At 0.5 cycle/sample, the corresponding Nyquist frequency is 22050 cycles/second (Hz). Conversely, the Nyquist rate for sampling a 22050 Hz signal is 44100 samples/second.<ref name="Grenlander" /><ref name="Leis" />Template:Efn-ua
When the highest frequency (bandwidth) of a signal is less than the Nyquist frequency of the sampler, the resulting discrete-time sequence is said to be free of the distortion known as aliasing, and the corresponding sample rate is said to be above the Nyquist rate for that particular signal.<ref name="Condon" /><ref name="Stiltz" />
In a typical application of sampling, one first chooses the highest frequency to be preserved and recreated, based on the expected content (voice, music, etc.) and desired fidelity. Then one inserts an anti-aliasing filter ahead of the sampler. Its job is to attenuate the frequencies above that limit. Finally, based on the characteristics of the filter, one chooses a sample rate (and corresponding Nyquist frequency) that will provide an acceptably small amount of aliasing. In applications where the sample rate is predetermined (such as the CD rate), the filter is chosen based on the Nyquist frequency, rather than vice versa.
Folding frequencyEdit
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In this example, Template:Math is the sampling rate, and Template:Math is the corresponding Nyquist frequency. The black dot plotted at Template:Math represents the amplitude and frequency of a sinusoidal function whose frequency is 60% of the sample rate. The other three dots indicate the frequencies and amplitudes of three other sinusoids that would produce the same set of samples as the actual sinusoid that was sampled. Undersampling of the sinusoid at Template:Math is what allows there to be a lower-frequency alias. If the true frequency were Template:Math, there would still be aliases at 0.6, 1.4, 1.6, etc.
The red lines depict the paths (loci) of the 4 dots if we were to adjust the frequency and amplitude of the sinusoid along the solid red segment (between Template:Math and Template:Math). No matter what function we choose to change the amplitude vs frequency, the graph will exhibit symmetry between 0 and Template:Math This symmetry is commonly referred to as folding, and another name for Template:Math (the Nyquist frequency) is folding frequency.<ref name=Zawistowski/>
Other meaningsEdit
Early uses of the term Nyquist frequency, such as those cited above, are all consistent with the definition presented in this article. Some later publications, including some respectable textbooks, call twice the signal bandwidth the Nyquist frequency;<ref name=Blackledge/><ref name=Diniz/> this is a distinctly minority usage, and the frequency at twice the signal bandwidth is otherwise commonly referred to as the Nyquist rate.