TheInfoList

In
signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetis ...

, sampling is the reduction of a continuous-time signal to a
discrete-time signalIn mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time. Discrete time Discrete sampled signal Discrete time views values of variables as occurring at disti ...
. A common example is the conversion of a
sound wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

(a continuous signal) to a sequence of samples (a discrete-time signal). A sample is a value or set of values at a point in time and/or space. A sampler is a subsystem or operation that extracts samples from a
continuous signal In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which to model variables that evolve over time. Discrete time Discrete time views values of variables as occurring at distinct, separate "points ...
. A theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous signal at the desired points. The original signal is retrievable from a sequence of samples, up to the
Nyquist limit In signal processing, the Nyquist frequency (or folding frequency), named after Harry Nyquist, is a characteristic of a Sampling (signal processing), sampler, which converts a continuous function or signal into a discrete sequence. In units of ...
, by passing the sequence of samples through a type of
low pass filter A low-pass filter (LPF) is a filter that passes signals with a frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial freq ...

called a
reconstruction filter Reconstruction may refer to: Politics, history, and sociology * Reconstruction (law), the transfer of a company's (or several companies') business to a new company *''Perestroika ''Perestroika'' postage stamp, 1988 Perestroika (; russian: Đ ...
.

# Theory

Sampling can be done for functions varying in space, time, or any other dimension, and similar results are obtained in two or more dimensions. For functions that vary with time, let ''s''(''t'') be a continuous function (or "signal") to be sampled, and let sampling be performed by measuring the value of the continuous function every ''T'' seconds, which is called the sampling interval or the sampling period.  Then the sampled function is given by the sequence: :''s''(''nT''),   for integer values of ''n''. The sampling frequency or sampling rate, ''fs'', is the average number of samples obtained in one second, thus ''fs = 1/T''. Its units are samples per second or
hertz The hertz (symbol: Hz) is the unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action ...

e.g. 48 kHz is 48,000 samples per second. Reconstructing a continuous function from samples is done by interpolation algorithms. The
Whittakerâ€“Shannon interpolation formula The Whittakerâ€“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 i ...
is mathematically equivalent to an ideal
low-pass filter A low-pass filter is a filter Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ...

whose input is a sequence of
Dirac delta functions File:Dirac function approximation.gif, frame, The Dirac delta function as the limit as a \rightarrow 0 (in the sense of distribution (mathematics), distributions) of the sequence of zero-centered normal distributions \delta_a(x) = \frac \mathrm ...
that are modulated (multiplied) by the sample values. When the time interval between adjacent samples is a constant (''T''), the sequence of delta functions is called a
Dirac comb In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

. Mathematically, the modulated Dirac comb is equivalent to the product of the comb function with ''s''(''t''). That purely mathematical abstraction is sometimes referred to as ''impulse sampling''. Most sampled signals are not simply stored and reconstructed. But the fidelity of a theoretical reconstruction is a customary measure of the effectiveness of sampling. That fidelity is reduced when ''s''(''t'') contains frequency components whose periodicity is smaller than two samples; or equivalently the ratio of cycles to samples exceeds Â˝ (see
Aliasing In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Sig ...
). The quantity Â˝ ''cycles/sample'' Ă— fs ''samples/sec'' = fs/2 ''cycles/sec'' (
hertz The hertz (symbol: Hz) is the unit Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action ...

) is known as the
Nyquist frequency In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signa ...
of the sampler. Therefore, ''s''(''t'') is usually the output of a
low-pass filter A low-pass filter is a filter Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ...

, functionally known as an ''anti-aliasing filter''. Without an anti-aliasing filter, frequencies higher than the Nyquist frequency will influence the samples in a way that is misinterpreted by the interpolation process.

# Practical considerations

In practice, the continuous signal is sampled using an
analog-to-digital converter In electronics, an analog-to-digital converter (ADC, A/D, or A-to-D) 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 (signal processing), digit ...
(ADC), a device with various physical limitations. This results in deviations from the theoretically perfect reconstruction, collectively referred to as
distortion Distortion is the alteration of the original shape (or other characteristic) of something. In communications Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an apparent answer to the painful di ...

. Various types of distortion can occur, including: *
Aliasing In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Sig ...

. Some amount of aliasing is inevitable because only theoretical, infinitely long, functions can have no frequency content above the Nyquist frequency. Aliasing can be made
arbitrarily smallIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
by using a
sufficiently large In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Î•Î»Î»Î¬Î´Î±, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population i ...
order of the anti-aliasing filter. * Aperture error results from the fact that the sample is obtained as a time average within a sampling region, rather than just being equal to the signal value at the sampling instant. In a
capacitor A capacitor is a device that stores electric charge in an electric field. It is a passivity (engineering), passive electronic component with two terminal (electronics), terminals. The effect of a capacitor is known as capacitance. While som ...

-based
sample and hold In electronics Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter. It uses active devices to control electron flow by amplifier, amplif ...

circuit, aperture errors are introduced by multiple mechanisms. For example, the capacitor cannot instantly track the input signal and the capacitor can not instantly be isolated from the input signal. *
Jitter In electronics Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter. It uses active devices to control electron flow by amplifier, amplific ...

or deviation from the precise sample timing intervals. *
Noise Noise is unwanted sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and t ...
, including thermal sensor noise,
analog circuit Analogue electronics ( en-US, analog electronics) are electronics, electronic systems with a continuous function, continuously variable signal, in contrast to digital electronics where signals usually take binary code, only two levels. The term "an ...
noise, etc. *
Slew rate In electronics Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter. It uses active devices to control electron flow by amplifier, amplifica ...
limit error, caused by the inability of the ADC input value to change sufficiently rapidly. * Quantization as a consequence of the finite precision of words that represent the converted values. * Error due to other
non-linear In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
effects of the mapping of input voltage to converted output value (in addition to the effects of quantization). Although the use of
oversampling In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signal ...
can completely eliminate aperture error and aliasing by shifting them out of the passband, this technique cannot be practically used above a few GHz, and may be prohibitively expensive at much lower frequencies. Furthermore, while oversampling can reduce quantization error and non-linearity, it cannot eliminate these entirely. Consequently, practical ADCs at audio frequencies typically do not exhibit aliasing, aperture error, and are not limited by quantization error. Instead, analog noise dominates. At RF and microwave frequencies where oversampling is impractical and filters are expensive, aperture error, quantization error and aliasing can be significant limitations. Jitter, noise, and quantization are often analyzed by modeling them as random errors added to the sample values. Integration and zero-order hold effects can be analyzed as a form of
low-pass filter A low-pass filter is a filter Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass ...

ing. The non-linearities of either ADC or DAC are analyzed by replacing the ideal
linear function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

mapping with a proposed
nonlinear function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

.

# Applications

## Audio sampling

Digital audio Digital audio is a representation of sound recorded in, or converted into, Digital signal (signal processing), digital form. In digital audio, the sound wave of the audio signal is typically encoded as numerical Sampling (signal processing), s ...
uses
pulse-code modulation Pulse-code modulation (PCM) is a method used to Digital signal (signal processing), digitally represent sampled analog signals. It is the standard form of digital audio in computers, compact discs, digital telephony and other digital audio ap ...
(PCM) and digital signals for sound reproduction. This includes analog-to-digital conversion (ADC), digital-to-analog conversion (DAC), storage, and transmission. In effect, the system commonly referred to as digital is in fact a discrete-time, discrete-level analog of a previous electrical analog. While modern systems can be quite subtle in their methods, the primary usefulness of a digital system is the ability to store, retrieve and transmit signals without any loss of quality. When it is necessary to capture audio covering the entire 20â€“20,000 Hz range of human hearing, such as when recording music or many types of acoustic events, audio waveforms are typically sampled at 44.1 kHz (), 48 kHz, 88.2 kHz, or 96 kHz. The approximately double-rate requirement is a consequence of the Nyquist theorem. Sampling rates higher than about 50 kHz to 60 kHz cannot supply more usable information for human listeners. Early
professional audio Professional audio, abbreviated as pro audio, refers to both an activity and a category of high quality, studio-grade audio equipment. Typically it encompasses sound recording, sound reinforcement system setup and audio mixing, and music product ...
equipment manufacturers chose sampling rates in the region of 40 to 50 kHz for this reason. There has been an industry trend towards sampling rates well beyond the basic requirements: such as 96 kHz and even 192 kHz Even though frequencies are inaudible to humans, recording and mixing at higher sampling rates is effective in eliminating the distortion that can be caused by . Conversely, ultrasonic sounds may interact with and modulate the audible part of the frequency spectrum (
intermodulation distortion plot showing intermodulation between two injected signals at 270 and 275 MHz (the large spikes). Visible intermodulation products are seen as small spurs at 280 MHz and 265 MHz. Intermodulation (IM) or intermodulation distortion (IMD) is the a ...
), ''degrading'' the fidelity. One advantage of higher sampling rates is that they can relax the low-pass filter design requirements for ADCs and DACs, but with modern oversampling sigma-delta converters this advantage is less important. The
Audio Engineering Society The Audio Engineering Society (AES) is a professional body for engineers, scientists, other individuals with an interest or involvement in the professional audio Professional audio, abbreviated as pro audio, refers to both an activity and a ...
recommends 48 kHz sampling rate for most applications but gives recognition to 44.1 kHz for
Compact Disc The compact disc (CD) is a digital Digital usually refers to something using digits, particularly binary digits. Technology and computing Hardware *Digital electronics Digital electronics is a field of electronics Electronics compri ...

(CD) and other consumer uses, 32 kHz for transmission-related applications, and 96 kHz for higher bandwidth or relaxed
anti-aliasing filter An anti-aliasing filter (AAF) is a filter Filter, filtering or filters may refer to: Science and technology Device * Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the f ...
ing. Both Lavry Engineering and J. Robert Stuart state that the ideal sampling rate would be about 60 kHz, but since this is not a standard frequency, recommend 88.2 or 96 kHz for recording purposes. A more complete list of common audio sample rates is:

### Bit depth

Audio is typically recorded at 8-, 16-, and 24-bit depth, which yield a theoretical maximum signal-to-quantization-noise ratio (SQNR) for a pure
sine wave A sine wave or sinusoid is any of certain mathematical curves that describe a smooth periodic oscillation Oscillation is the repetitive variation, typically in time, of some measure about a central value (often a point of Mechanical equilib ...

of, approximately, 49.93 , 98.09 dB and 122.17 dB. CD quality audio uses 16-bit samples.
Thermal noise A thermal column (or thermal) is a column of rising air File:Atmosphere gas proportions.svg, Composition of Earth's atmosphere by volume, excluding water vapor. Lower pie represents trace gases that together compose about 0.043391% of the ...
limits the true number of bits that can be used in quantization. Few analog systems have signal to noise ratios (SNR) exceeding 120 dB. However,
digital signal processing Digital signal processing (DSP) is the use of digital processing Digital data, in information theory and information systems, is information represented as a string of discrete symbols each of which can take on one of only a finite number of ...
operations can have very high dynamic range, consequently it is common to perform mixing and mastering operations at 32-bit precision and then convert to 16- or 24-bit for distribution.

### Speech sampling

Speech signals, i.e., signals intended to carry only human
speech Speech is human vocal communication Communication (from Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''c ...
, can usually be sampled at a much lower rate. For most
phoneme In phonology and linguistics, a phoneme is a unit of sound that distinguishes one word from another in a particular language. For example, in most List of dialects of English, dialects of English, with the notable exception of the West Midlan ...
s, almost all of the energy is contained in the 100 Hzâ€“4 kHz range, allowing a sampling rate of 8 kHz. This is the
sampling rate In , sampling is the reduction of a to a . A common example is the conversion of a (a continuous signal) to a sequence of samples (a discrete-time signal). A sample is a value or set of values at a point in time and/or space. A sampler is a su ...
used by nearly all
telephony Telephony ( ) is the field of technology involving the development, application, and deployment of telecommunication services for the purpose of electronic transmission of voice, fax, or data, between distant parties. The history of telephony is i ...
systems, which use the
G.711 G.711 is a narrowband {{refimprove, date=March 2011 In radio communications, a narrowband channel is a channel in which the bandwidth of the message does not significantly exceed the channel's coherence bandwidth. In the study of wired chan ...
sampling and quantization specifications.

## Video sampling

Standard-definition television Standard-definition television (SDTV, SD, often shortened to standard definition) is a television system which uses a resolution that is not considered to be either high-definition television, high or enhanced-definition television, enhanced ...
(SDTV) uses either 720 by 480
pixels In digital imaging, a pixel, pel, or picture element is the smallest addressable element in a Raster graphics, raster image, or the smallest addressable element in an all points addressable display device; so it is the smallest controllable elem ...

(US
NTSC The National Television System Committee (NTSC)National Television System Committee (1951â€“1953), Report and Reports of Panel No. 11, 11-A, 12-19, with Some supplementary references cited in the Reports, and the Petition for adoption of transm ...
525-line) or 720 by 576
pixels In digital imaging, a pixel, pel, or picture element is the smallest addressable element in a Raster graphics, raster image, or the smallest addressable element in an all points addressable display device; so it is the smallest controllable elem ...

(UK
PAL PAL 300px, upright=1.2, Analog television encoding systems by nation; SECAM.html"_;"title="NTSC_(green),_SECAM">NTSC_(green),_SECAM_(orange),_and_PAL_(blue). Phase_Alternating_Line_(PAL)_is_a_colour_encoding_system_for_Analog_television.html" ...
625-line) for the visible picture area.
High-definition television High-definition television (HD or HDTV) describes a television system providing a substantially higher image resolution Image resolution is the detail an holds. The term applies to s, film images, and other types of images. Higher resolution m ...
(HDTV) uses
720p 720p (1280Ă—720 px; also called HD ready or standard HD) is a progressive HDTV High-definition television (HD or HDTV) describes a television system providing a substantially higher image resolution Image resolution is the detail an image hold ...
(progressive),
1080i 1080i (also known as Full HD or BT.709 Rec. 709, also known as Rec.709, BT.709, and ITU 709, is a standard developed by ITU-R The ITU Radiocommunication Sector (ITU-R) is one of the three sectors (divisions or units) of the International T ...
(interlaced), and
1080p 1080p (1920Ă—1080 Progressive_scan, progressively displayed pixels; also known as Full HD or FHD, and Rec. 709, BT.709) is a set of High-definition television, HDTV high-definition video modes characterized by 1,920 pixels displayed across the scr ...
(progressive, also known as Full-HD). In
digital video Digital video is an electronic representation of moving visual images (video) in the form of encoded digital data. This is in contrast to analog video, which represents moving visual images in the form of analog signals. Digital video comprises ...
, the temporal sampling rate is defined the
frame rate Frame rate (expressed in or FPS) is the frequency Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Time is the indefinite continued sequence, progress of existence an ...
or rather the
field rate The refresh rate (or "vertical refresh rate", "vertical scan rate", terminology originating with the cathode ray tube File:CRT monochrome.png, 250px, Cutaway rendering of a monochrome CRT: 1. Deflection coils2. Electron beam3.  ...
rather than the notional pixel clock. The image sampling frequency is the repetition rate of the sensor integration period. Since the integration period may be significantly shorter than the time between repetitions, the sampling frequency can be different from the inverse of the sample time: * 50 Hz â€“
PAL PAL 300px, upright=1.2, Analog television encoding systems by nation; SECAM.html"_;"title="NTSC_(green),_SECAM">NTSC_(green),_SECAM_(orange),_and_PAL_(blue). Phase_Alternating_Line_(PAL)_is_a_colour_encoding_system_for_Analog_television.html" ...
video * 60 / 1.001 Hz ~= 59.94 Hz â€“
NTSC The National Television System Committee (NTSC)National Television System Committee (1951â€“1953), Report and Reports of Panel No. 11, 11-A, 12-19, with Some supplementary references cited in the Reports, and the Petition for adoption of transm ...
video Video
digital-to-analog converter In , a digital-to-analog converter (DAC, D/A, D2A, or D-to-A) is a system that converts a into an . An (ADC) performs the reverse function. There are several DAC ; the suitability of a DAC for a particular application is determined by ...
s operate in the megahertz range (from ~3 MHz for low quality composite video scalers in early games consoles, to 250 MHz or more for the highest-resolution VGA output). When analog video is converted to
digital video Digital video is an electronic representation of moving visual images (video) in the form of encoded digital data. This is in contrast to analog video, which represents moving visual images in the form of analog signals. Digital video comprises ...
, a different sampling process occurs, this time at the pixel frequency, corresponding to a spatial sampling rate along
scan line A scan line (also scanline) is one line, or row, in a raster scan A raster scan, or raster scanning, is the rectangular pattern of image capture and reconstruction in television. By analogy, the term is used for raster graphics file:Rgb-raste ...
s. A common
pixel In digital imaging Digital imaging or digital image acquisition is the creation of a representation of the visual characteristics of an object, such as a physical scene or the interior structure of an object. The term is often assumed to imp ...

sampling rate is: * 13.5 MHz â€“
CCIR 601 ITU-R The ITU Radiocommunication Sector (ITU-R) is one of the three sectors (divisions or units) of the International Telecommunication Union 260px, ITU Monument, Bern The International Telecommunication Union is a specialized agency of ...
, D1 video Spatial sampling in the other direction is determined by the spacing of scan lines in the
raster Raster may refer to: * Raster graphics, graphical techniques using arrays of pixel values * Raster graphics editor, a computer program * Raster scan, the pattern of image readout, transmission, storage, and reconstruction in television and computer ...
. The sampling rates and resolutions in both spatial directions can be measured in units of lines per picture height. Spatial
aliasing In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Sig ...

of high-frequency luma or chroma video components shows up as a
moirĂ© pattern In mathematics, physics, and art, moirĂ© patterns ( , , ) or moirĂ© fringes are large-scale interference patterns that can be produced when an opaque ruled pattern with transparent gaps is overlaid on another similar pattern. For the moirĂ© ...

.

## 3D sampling

The process of
volume rendering In scientific visualization Surface rendering of '' confocal_microscope..html" ;"title="onfocal_microscopy.html" ;"title="pollen.html" ;"title="Arabidopsis thaliana'' Arabidopsis_thaliana''_pollen_grains_with_Confocal_microscopy">confocal ...
samples a 3D grid of
voxel In 3D computer graphics 3D computer graphics, sometimes called CGI, 3DCG or three-dimensional computer graphics (in contrast to 2D computer graphics 2D computer graphics is the Computer-generated imagery, computer-based generation of digita ...

s to produce 3D renderings of sliced (tomographic) data. The 3D grid is assumed to represent a continuous region of 3D space. Volume rendering is common in medical imaging,
X-ray computed tomography An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Moti ...
(CT/CAT),
magnetic resonance imaging Magnetic resonance imaging (MRI) is a medical imaging Medical imaging is the technique and process of imaging Imaging is the representation or reproduction of an object's form; especially a visual representation (i.e., the formation of a ...
(MRI),
positron emission tomography Positron emission tomography (PET) is a functional imaging technique that uses radioactive substances known as radiotracers to visualize and measure changes in Metabolism, metabolic processes, and in other physiological activities including blo ...
(PET) are some examples. It is also used for
seismic tomography Seismic tomography is a technique for imaging the subsurface of the Earth with seismic waves produced by earthquakes or explosions. P-, S-, and surface waves can be used for tomographic models of different resolutions based on seismic wavelength, ...
and other applications.

# Undersampling

When a
bandpass File:Bandpass_Filter.svg, 300px, A medium-complexity example of a band-pass filter. A band-pass filter or bandpass filter (BPF) is a device that passes frequency, frequencies within a certain range and rejects (attenuates) frequencies outside t ...

signal is sampled slower than its
Nyquist rate In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signa ...
, the samples are indistinguishable from samples of a low-frequency
alias An alias is an alternative name, such as a pseudonym. Alias or aliasing may also refer to: Arts and entertainment Comics * Alias (comics), ''Alias'' (comics), an American comic book * Alias Enterprises, a United States publishing company, includ ...

of the high-frequency signal. That is often done purposefully in such a way that the lowest-frequency alias satisfies the Nyquist criterion, because the bandpass signal is still uniquely represented and recoverable. Such
undersampling In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signal ...
is also known as ''bandpass sampling'', ''harmonic sampling'', ''IF sampling'', and ''direct IF to digital conversion.''

# Oversampling

Oversampling is used in most modern analog-to-digital converters to reduce the distortion introduced by practical
digital-to-analog converter In , a digital-to-analog converter (DAC, D/A, D2A, or D-to-A) is a system that converts a into an . An (ADC) performs the reverse function. There are several DAC ; the suitability of a DAC for a particular application is determined by ...
s, such as a
zero-order hold The zero-order hold (ZOH) is a mathematical model of the practical signal reconstruction done by a conventional digital-to-analog converter In electronics Electronics comprises the physics, engineering, technology and applications tha ...
Whittakerâ€“Shannon interpolation formula The Whittakerâ€“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 i ...
.

# Complex sampling

''Complex sampling'' (I/Q sampling) is the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
.  When one waveform$, \hat s\left(t\right),$  is the
Hilbert transform In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...

of the other waveform$, s\left(t\right),\,$  the complex-valued function,  $s_a\left(t\right) \triangleq s\left(t\right) + i\cdot \hat s\left(t\right),$  is called an
analytic signal Generally speaking, analytic (from el, áĽ€Î˝Î±Î»Ď…Ď„ÎąÎşĎŚĎ‚, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic can also have the following meanings: Natural sciences Chemistry ...
,  whose Fourier transform is zero for all negative values of frequency. In that case, the
Nyquist rate In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signa ...
for a waveform with no frequencies â‰Ą ''B'' can be reduced to just ''B'' (complex samples/sec), instead of 2''B'' (real samples/sec). More apparently, the equivalent baseband waveform,  $s_a\left(t\right)\cdot e^,$  also has a Nyquist rate of ''B'', because all of its non-zero frequency content is shifted into the interval [-B/2, B/2). Although complex-valued samples can be obtained as described above, they are also created by manipulating samples of a real-valued waveform. For instance, the equivalent baseband waveform can be created without explicitly computing $\hat s\left(t\right),$  by processing the product sequence$, \left \left[s\left(nT\right)\cdot e^\right \right],$  through a digital low-pass filter whose cutoff frequency is B/2. Computing only every other sample of the output sequence reduces the sample-rate commensurate with the reduced Nyquist rate. The result is half as many complex-valued samples as the original number of real samples. No information is lost, and the original s(t) waveform can be recovered, if necessary.

* Crystal oscillator frequencies *
DownsamplingIn digital signal processing, downsampling, compression, and decimation are terms associated with the process of sample rate conversion, ''resampling'' in a multi-rate digital signal processing system. Both ''downsampling'' and ''decimation'' can be ...
*
Upsampling 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 ...
* Multidimensional sampling *
Sample rate conversion Sample-rate conversion is the process of changing the sampling rate of a discrete signal to obtain a new discrete representation of the underlying continuous signal. Application areas include image scaling and audio/visual systems, where differe ...
*
Digitizing DigitizationDefinition of digitization
at WhatIs.com
is the process of converting ...
*
Sample and hold In electronics Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter. It uses active devices to control electron flow by amplifier, amplif ...

* Beta encoder * Kell factor * Bit rate

# References

* Matt Pharr, Wenzel Jakob and Greg Humphreys, ''Physically Based Rendering: From Theory to Implementation, 3rd ed.'', Morgan Kaufmann, November 2016. . The chapter on sampling
available online
is nicely written with diagrams, core theory and code sample.