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The **Nyquist frequency**, named after electronic engineer Harry Nyquist, is half of the sampling rate of a discrete signal processing system.^{ [1] }^{ [2] } It is sometimes known as the folding frequency of a sampling system.^{ [3] } An example of folding is depicted in Figure 1, where f_{s} is the sampling rate and 0.5 f_{s} is the corresponding Nyquist frequency.^{ [note 1] } The black dot plotted at 0.6 f_{s} represents the amplitude and frequency of a sinusoidal function whose frequency is 60% of the sample-rate (f_{s}). 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 f_{s} is referred to as *folding*.

**Harry Nyquist** was a Swedish-born American electronic engineer who made important contributions to communication theory.

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 f_{s} is 1.2 f_{s}, which means that at the f_{s} 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.^{ [4] }^{ [5] }

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:

- as a lower bound for the sample rate for alias-free signal sampling and
- 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 (10^{3} Hz, kHz), megahertz (10^{6} Hz, MHz), gigahertz (10^{9} Hz, GHz), terahertz (10^{12} Hz, THz), petahertz (10^{15} Hz, PHz), and exahertz (10^{18} Hz, EHz).

Referring again to Figure 1, undersampling of the sinusoid at 0.6 f_{s} 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 f_{s} sinusoid would produce a 0.4 f_{s} sinusoid instead. If the true frequency was 0.4 f_{s}, 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.

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;^{ [6] }^{ [7] } this is a distinctly minority usage, and the frequency at twice the signal bandwidth is otherwise commonly referred to as the Nyquist rate.

- ↑ In this context, the factor of ½ has units of
*cycles per sample*, as explained at Aliasing#Sampling sinusoidal functions. - ↑ 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.

- ↑ Grenander, Ulf (1959).
*Probability and Statistics: The Harald Cramér Volume*. Wiley.The Nyquist frequency is that frequency whose period is two sampling intervals.

- ↑ Harry L. Stiltz (1961).
*Aerospace Telemetry*. Prentice-Hall.the existence of power in the continuous signal spectrum at frequencies higher than the Nyquist frequency is the cause of aliasing error

- ↑ Thomas Zawistowski; Paras Shah. "An Introduction to Sampling Theory" . Retrieved 17 April 2010.
Frequencies "fold" around half the sampling frequency - which is why [the Nyquist] frequency is often referred to as the folding frequency.

- ↑ James J. Condon & Scott M. Ransom (2016).
*Essential Radio Astronomy*. Princeton University Press. pp. 280–281. ISBN 9781400881161. - ↑ John W. Leis (2011).
*Digital Signal Processing Using MATLAB for Students and Researchers*. John Wiley & Sons. p. 82. ISBN 9781118033807.The

*Nyquist rate*is twice the bandwidth of the signal ... The*Nyquist frequency*or*folding frequency*is half the sampling rate and corresponds to the highest frequency which a sampled data system can reproduce without error. - ↑ Jonathan M. Blackledge (2003).
*Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications*. Horwood Publishing. ISBN 1-898563-48-9. - ↑ Paulo Sergio Ramirez Diniz, Eduardo A. B. Da Silva, Sergio L. Netto (2002).
*Digital Signal Processing: System Analysis and Design*. Cambridge University Press. ISBN 0-521-78175-2.CS1 maint: Multiple names: authors list (link)

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**Bandwidth** is the difference between the upper and lower frequencies in a continuous band of frequencies. It is typically measured in hertz, and depending on context, may specifically refer to *passband bandwidth* or *baseband bandwidth*. Passband bandwidth is the difference between the upper and lower cutoff frequencies of, for example, a band-pass filter, a communication channel, or a signal spectrum. Baseband bandwidth applies to a low-pass filter or baseband signal; the bandwidth is equal to its upper cutoff frequency.

In electronics, an **analog-to-digital converter** is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal. An ADC may also provide an isolated measurement such as an electronic device that converts an input analog voltage or current to a digital number representing the magnitude of the voltage or current. Typically the digital output is a two's complement binary number that is proportional to the input, but there are other possibilities.

**Pulse width modulation** (**PWM**), or **pulse-duration modulation** (**PDM**), is a method of reducing the average power delivered by an electrical signal, by effectively chopping it up into discrete parts. The average value of voltage fed to the load is controlled by turning the switch between supply and load on and off at a fast rate. The longer the switch is on compared to the off periods, the higher the total power supplied to the load. Along with MPPT maximum power point tracking, it is one of the primary methods of reducing the output of solar panels to that which can be utilized by a battery. PWM is particularly suited for running inertial loads such as motors, which are not as easily affected by this discrete switching. Because they have inertia they react slower. The PWM switching frequency has to be high enough not to affect the load, which is to say that the resultant waveform perceived by the load must be as smooth as possible.

The **Whittaker–Shannon interpolation formula** or **sinc interpolation** is a method to construct a continuous-time bandlimited function from a sequence of real numbers. The formula dates back to the works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M. Whittaker in 1935, and in the formulation of the Nyquist–Shannon sampling theorem by Claude Shannon in 1949. It is also commonly called **Shannon's interpolation formula** and **Whittaker's interpolation formula**. E. T. Whittaker, who published it in 1915, called it the **Cardinal series**.

In signal processing and related disciplines, **aliasing** is an effect that causes different signals to become indistinguishable when sampled. It also refers to the distortion or artifact that results when the signal reconstructed from samples is different from the original continuous signal.

Sound can be recorded and stored and played using either digital or analog techniques. Both techniques introduce errors and distortions in the sound, and these methods can be systematically compared. Musicians and listeners have argued over the superiority of digital versus analog sound recordings. Arguments for analog systems include the absence of fundamental error mechanisms which are present in digital audio systems, including aliasing and quantization noise. Advocates of digital point to the high levels of performance possible with digital audio, including excellent linearity in the audible band and low levels of noise and distortion.

**Filter design** is the process of designing a signal processing filter that satisfies a set of requirements, some of which are contradictory. The purpose is to find a realization of the filter that meets each of the requirements to a sufficient degree to make it useful.

In signal processing, **sampling** is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of samples.

**Frequency response** is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. In simplest terms, if a sine wave is injected into a system at a given frequency, a linear system will respond at that same frequency with a certain magnitude and a certain phase angle relative to the input. Also for a linear system, doubling the amplitude of the input will double the amplitude of the output. In addition, if the system is time-invariant, then the frequency response also will not vary with time. Thus for LTI systems, the frequency response can be seen as applying the system's transfer function to a purely imaginary number argument representing the frequency of the sinusoidal excitation.

In signal processing, **undersampling** or **bandpass sampling** is a technique where one samples a bandpass-filtered signal at a sample rate below its Nyquist rate, but is still able to reconstruct the signal.

In digital signal processing, **downsampling** and **decimation** are terms associated with the process of resampling in a multi-rate digital signal processing system. Both terms are used by various authors to describe the entire process, which includes lowpass filtering, or just the part of the process that does not include filtering.
When downsampling (decimation) is performed on a sequence of samples of a *signal* or other continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a lower rate. The *decimation factor* is usually an integer or a rational fraction greater than one. This factor multiplies the sampling interval or, equivalently, divides the sampling rate. For example, if compact disc audio at 44,100 samples/second is decimated by a factor of 5/4, the resulting sample rate is 35,280. A system component that performs decimation is called a *decimator*.

In digital signal processing, **upsampling**, **expansion**, and **interpolation** are terms associated with the process of resampling in a multi-rate digital signal processing system. *Upsampling* can be synonymous with *expansion*, or it can describe an entire process of *expansion* and filtering (*interpolation*). When upsampling is performed on a sequence of samples of a *signal* or other continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a higher rate. For example, if compact disc audio at 44,100 samples/second is upsampled by a factor of 5/4, the resulting sample-rate is 55,125.

In signal processing, **oversampling** is the process of sampling a signal at a sampling frequency significantly higher than the Nyquist rate. Theoretically, a bandwidth-limited signal can be perfectly reconstructed if sampled at the Nyquist rate or above it. The Nyquist rate is defined as twice the highest frequency component in the signal. Oversampling is capable of improving resolution, reducing noise and can be helpful in avoiding aliasing and phase distortion by relaxing anti-aliasing filter performance requirements.

In a mixed-signal system, a **reconstruction filter** is used to construct a smooth analog signal from a digital input, as in the case of a digital to analog converter (DAC) or other sampled data output device.

A **Bitcrusher** is a lo-fi digital audio effect, which produces a distortion by the reduction of the resolution or bandwidth of digital audio data. The resulting quantization noise may produce a “warmer” sound impression, or a harsh one, depending on the amount of reduction.

**Normalized frequency** is a unit of measurement of frequency equivalent to *cycles/sample*.
In digital signal processing (DSP), the continuous time variable, **t**, with units of *seconds*, is replaced by the discrete integer variable, **n**, with units of *samples*. More precisely, the time variable, in *seconds*, has been normalized (divided) by the sampling interval, **T** (*seconds/sample*), which causes time to have convenient integer values at the moments of sampling. This practice is analogous to the concept of natural units, meaning that the natural unit of time in a DSP system is *samples*.

**Frequency ambiguity resolution** is used to find true target velocity for medium pulse repetition frequency (PRF) radar systems. This is used with pulse-Doppler radar.