# Nyquist frequency

Last updated Fig 1. The black dots are aliases of each other. The solid red line is an example of adjusting amplitude vs frequency. The dashed red lines are the corresponding paths of the aliases.

The Nyquist frequency, named after electronic engineer Harry Nyquist, is half of the sampling rate of a discrete signal processing system.   It is sometimes known as the folding frequency of a sampling system.    An example of folding is depicted in Figure 1, where fs is the sampling rate and 0.5 fs is the corresponding Nyquist frequency. [note 1]   The black dot plotted at 0.6 fs represents the amplitude and frequency of a sinusoidal function whose frequency is 60% of the sample-rate (fs). 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. The symmetry about 0.5 fs is referred to as folding. Harry Nyquist was a Swedish-born American electronic engineer who made important contributions to communication theory.

## Contents

The Nyquist frequency should not be confused with the Nyquist rate , the latter is the minimum sampling rate that satisfies the Nyquist sampling criterion for a given signal or family of signals. The Nyquist rate is twice the maximum component frequency of the function being sampled. For example, the Nyquist rate for the sinusoid at 0.6 fs is 1.2 fs, which means that at the fs rate, it is being undersampled. Thus, Nyquist rate is a property of a continuous-time signal, whereas Nyquist frequency is a property of a discrete-time system. In signal processing, the Nyquist rate, named after Harry Nyquist, is twice the bandwidth of a bandlimited function or a bandlimited channel. This term means two different things under two different circumstances:

1. as a lower bound for the sample rate for alias-free signal sampling and
2. as an upper bound for the symbol rate across a bandwidth-limited baseband channel such as a telegraph line or passband channel such as a limited radio frequency band or a frequency division multiplex channel. In the field of digital signal processing, the sampling theorem is a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth.

When the function domain is time, sample rates are usually expressed in samples per second, and the unit of Nyquist frequency is cycles per second (hertz). When the function domain is distance, as in an image sampling system, the sample rate might be dots per inch and the corresponding Nyquist frequency would be in cycles/inch. The hertz (symbol: Hz) is the derived unit of frequency in the International System of Units (SI) and is defined as one cycle per second. It is named for Heinrich Rudolf Hertz, the first person to provide conclusive proof of the existence of electromagnetic waves. Hertz are commonly expressed in multiples: kilohertz (103 Hz, kHz), megahertz (106 Hz, MHz), gigahertz (109 Hz, GHz), terahertz (1012 Hz, THz), petahertz (1015 Hz, PHz), and exahertz (1018 Hz, EHz).

## Aliasing

Referring again to Figure 1, undersampling of the sinusoid at 0.6 fs is what allows there to be a lower-frequency alias, which is a different function that produces the same set of samples. That condition is usually described as aliasing. The mathematical algorithms that are typically used to recreate a continuous function from its samples will misinterpret the contributions of undersampled frequency components, which causes distortion. Samples of a pure 0.6 fs sinusoid would produce a 0.4 fs sinusoid instead. If the true frequency was 0.4 fs, there would still be aliases at 0.6, 1.4, 1.6, etc., [note 2] but the reconstructed frequency would be correct.

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.

An anti-aliasing filter (AAF) is a filter used before a signal sampler to restrict the bandwidth of a signal to approximately or completely satisfy the Nyquist–Shannon sampling theorem over the band of interest. Since the theorem states that unambiguous reconstruction of the signal from its samples is possible when the power of frequencies above the Nyquist frequency is zero, a real anti-aliasing filter trades off between bandwidth and aliasing. A realizable anti-aliasing filter will typically either permit some aliasing to occur or else attenuate some in-band frequencies close to the Nyquist limit. For this reason, many practical systems sample higher than would be theoretically required by a perfect AAF in order to ensure that all frequencies of interest can be reconstructed, a practice called oversampling.

In applications where the sample-rate is pre-determined, the filter is chosen based on the Nyquist frequency, rather than vice versa. For example, audio CDs have a sampling rate of 44100 samples/sec. The Nyquist frequency is therefore 22050 Hz. The anti-aliasing filter must adequately suppress any higher frequencies but negligibly affect the frequencies within the human hearing range. A filter that preserves 0–20 kHz is more than adequate for that. Compact disc (CD) is a digital optical disc data storage format that was co-developed by Philips and Sony and released in 1982. The format was originally developed to store and play only sound recordings (CD-DA) but was later adapted for storage of data (CD-ROM). Several other formats were further derived from these, including write-once audio and data storage (CD-R), rewritable media (CD-RW), Video Compact Disc (VCD), Super Video Compact Disc (SVCD), Photo CD, PictureCD, CD-i, and Enhanced Music CD. The first commercially available audio CD player, the Sony CDP-101, was released October 1982 in Japan.

Hearing range describes the range of frequencies that can be heard by humans or other animals, though it can also refer to the range of levels. The human range is commonly given as 20 to 20,000 Hz, although there is considerable variation between individuals, especially at high frequencies, and a gradual loss of sensitivity to higher frequencies with age is considered normal. Sensitivity also varies with frequency, as shown by equal-loudness contours. Routine investigation for hearing loss usually involves an audiogram which shows threshold levels relative to a normal.

## Other meanings

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;   this is a distinctly minority usage, and the frequency at twice the signal bandwidth is otherwise commonly referred to as the Nyquist rate.

1. In this context, the factor of ½ has units of cycles per sample, as explained at Aliasing#Sampling sinusoidal functions.
2. As previously mentioned, these are the frequencies of other sinusoids that would produce the same set of samples as the one that was actually sampled.