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In
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a
sound wave 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 their ''perception'' by the ...
to a sequence of "samples". A sample is a value of the
signal In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
at a point in time and/or space; this definition differs from the usage in statistics, which refers to a set of such values. 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 variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
. A theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous signal at the desired points. The original signal can be reconstructed from a sequence of samples, up to the Nyquist limit, by passing the sequence of samples through a type of
low-pass filter A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
called a
reconstruction filter In a mixed-signal system ( analog and digital), a reconstruction filter, sometimes called an anti-imaging 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 samp ...
.


Theory

Functions of space, time, or any other dimension can be sampled, and similarly 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 sampling period.  Then the sampled function is given by the sequence: :''S''(''nT''),   for integer values of ''n''. The sampling frequency or sampling rate, ''f''s, is the average number of samples obtained in one second, thus . Its unit is sample per second or
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
e.g. 48 kHz is 48,000 samples per second. Reconstructing a continuous function from samples is done by interpolation algorithms. The
Whittaker–Shannon interpolation formula 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 i ...
is mathematically equivalent to an ideal
low-pass filter A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
whose input is a sequence of
Dirac delta functions In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
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, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula \operatorname_(t) \ := \sum_^ \delta(t - k T) for some given period T. Here ''t'' is a real variable and th ...
. Mathematically, the modulated Dirac comb is equivalent to the product of the comb function with ''s''(''t''). That mathematical abstraction is sometimes referred to as ''impulse sampling''. Most sampled signals are not simply stored and reconstructed. The fidelity of a theoretical reconstruction is a common measure of the effectiveness of sampling. That fidelity is reduced when ''s''(''t'') contains frequency components whose period is less than double the sampling interval (see ''
Aliasing In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or ''aliases'' of one another) when sampled. It also often refers to the distortion or artifact that results when ...
''). The quantity  cycle/sample × ''f''s sample/sec = ''f''s/2 cycles/sec (
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
) is known as the
Nyquist frequency 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. In units of cycles per second ( Hz), it ...
of the sampler. Therefore, ''s''(''t'') is usually the output of a
low-pass filter A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
, 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. An ADC may also provide ...
(ADC), a device with various physical limitations. This results in deviations from the theoretically perfect reconstruction, collectively referred to as
distortion In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio signa ...
. Various types of distortion can occur, including: *
Aliasing In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or ''aliases'' of one another) when sampled. It also often refers to the distortion or artifact that results when ...
. 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 small by using a
sufficiently large In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will after some instances have pa ...
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 electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
-based
sample and hold In electronics, a sample and hold (also known as sample and follow) circuit is an analog device that samples (captures, takes) the voltage of a continuously varying analog signal and holds (locks, freezes) its value at a constant level for ...
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 or deviation from the precise sample timing intervals. *
Noise Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arise ...
, including thermal sensor noise,
analog circuit Analogue electronics ( en-US, analog electronics) are electronic systems with a continuously variable signal, in contrast to digital electronics where signals usually take only two levels. The term "analogue" describes the proportional relat ...
noise, etc. *
Slew rate In electronics, slew rate is defined as the change of voltage or current, or any other electrical quantity, per unit of time. Expressed in SI units, the unit of measurement is volts/second or amperes/second, but is usually expressed in terms of m ...
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 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, 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 o ...
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 that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
ing. The non-linearities of either ADC or DAC are analyzed by replacing the ideal
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...
mapping with a proposed
nonlinear function In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
.


Applications


Audio sampling

Digital audio Digital audio is a representation of sound recorded in, or converted into, digital form. In digital audio, the sound wave of the audio signal is typically encoded as numerical samples in a continuous sequence. For example, in CD audio, sa ...
uses
pulse-code modulation Pulse-code modulation (PCM) is a method used to digitally represent sampled analog signals. It is the standard form of digital audio in computers, compact discs, digital telephony and other digital audio applications. In a PCM stream, the ...
(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 20–20,000 Hz range of
human hearing Hearing, or auditory perception, is the ability to perceive sounds through an organ, such as an ear, by detecting vibrations as periodic changes in the pressure of a surrounding medium. The academic field concerned with hearing is auditory ...
, such as when recording music or many types of acoustic events, audio waveforms are typically sampled at 44.1 kHz ( CD), 48 kHz, 88.2 kHz, or 96 kHz. The approximately double-rate requirement is a consequence of the
Nyquist theorem Nyquist may refer to: * Nyquist (surname) *Nyquist (horse), winner of the 2016 Kentucky Derby * Nyquist (programming language), computer programming language for sound synthesis and music composition See also *Johnson–Nyquist noise, thermal nois ...
. 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 studio mus ...
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
ultrasonic Ultrasound is sound waves with frequencies higher than the upper audible limit of human hearing. Ultrasound is not different from "normal" (audible) sound in its physical properties, except that humans cannot hear it. This limit varies fr ...
frequencies are inaudible to humans, recording and mixing at higher sampling rates is effective in eliminating the distortion that can be caused by foldback aliasing. Conversely, ultrasonic sounds may interact with and modulate the audible part of the frequency spectrum (
intermodulation distortion Intermodulation (IM) or intermodulation distortion (IMD) is the amplitude modulation of Signal (electrical engineering), signals containing two or more different frequencies, caused by non-linear, nonlinearities or time variance in a system. ...
), ''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 converter Delta-sigma (ΔΣ; or sigma-delta, ΣΔ) modulation is a method for encoding analog signals into Digital signal (signal processing), digital signals as found in an analog-to-digital converter (ADC). It is also used to convert high bit-count, l ...
s 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 industry. The membership largely comprises engineers developing devices or products ...
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 media, digital optical disc data storage format that was co-developed by Philips and Sony to store and play digital audio recordings. In August 1982, the first compact disc was manufactured. It was then rele ...
(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 used before a signal sampler to restrict the bandwidth of a signal to satisfy the Nyquist–Shannon sampling theorem over the band of interest. Since the theorem states that unambiguous reconstruct ...
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 Signal-to-quantization-noise ratio (SQNR or SNqR) is widely used quality measure in analysing digitizing schemes such as pulse-code modulation (PCM). The SQNR reflects the relationship between the maximum nominal signal strength and the quantizati ...
(SQNR) for a pure
sine wave A sine wave, sinusoidal wave, or just sinusoid is a curve, mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph of a function, graph. It is a type of continuous wave and also a Smoothness, smooth p ...
of, approximately, 49.93  dB, 98.09 dB and 122.17 dB. CD quality audio uses 16-bit samples.
Thermal noise A thermal column (or thermal) is a rising mass of buoyant air, a convective current in the atmosphere, that transfers heat energy vertically. Thermals are created by the uneven heating of Earth's surface from solar radiation, and are an example ...
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, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...
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 a human vocal communication using language. Each language uses Phonetics, phonetic combinations of vowel and consonant sounds that form the sound of its words (that is, all English words sound different from all French words, even if ...
, can usually be sampled at a much lower rate. For most
phoneme In phonology and linguistics, a phoneme () is a unit of sound that can distinguish one word from another in a particular language. For example, in most dialects of English, with the notable exception of the West Midlands and the north-west o ...
s, almost all of the energy is contained in the 100 Hz – 4 kHz range, allowing a sampling rate of 8 kHz. This is the
sampling rate 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". A sample is a value of the signal at a point in time and/or spac ...
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 audio codec originally designed for use in telephony that provides toll-quality audio at 64 kbit/s. G.711 passes audio signals in the range of 300–3400 Hz and samples them at the rate of 8,000 samples per second ...
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 or enhanced definition. "Standard" refers to it being the prevailing sp ...
(SDTV) uses either 720 by 480
pixels In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device. In most digital display devices, pixels are the sm ...
(US
NTSC The first American standard for analog television broadcast was developed by National Television System Committee (NTSC)National Television System Committee (1951–1953), Report and Reports of Panel No. 11, 11-A, 12–19, with Some supplement ...
525-line) or 720 by 576
pixels In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device. In most digital display devices, pixels are the sm ...
(UK
PAL Phase Alternating Line (PAL) is a colour encoding system for analogue television. It was one of three major analogue colour television standards, the others being NTSC and SECAM. In most countries it was broadcast at 625 lines, 50 fields (25 ...
625-line) for the visible picture area.
High-definition television High-definition television (HD or HDTV) describes a television system which provides a substantially higher image resolution than the previous generation of technologies. The term has been used since 1936; in more recent times, it refers to the g ...
(HDTV) uses
720p 720p (1280×720 px; also called HD ready, standard HD or just HD) is a progressive HDTV signal format with 720 horizontal lines/1280 columns and an aspect ratio (AR) of 16:9, normally known as widescreen HDTV (1.78:1). All major HDTV broadcast ...
(progressive),
1080i 1080i (also known as Full HD or BT.709) is a combination of frame resolution and scan type. 1080i is used in high-definition television (HDTV) and high-definition video. The number "1080" refers to the number of horizontal lines on the screen. ...
(interlaced), and
1080p 1080p (1920×1080 progressively displayed pixels; also known as Full HD or FHD, and BT.709) is a set of HDTV high-definition video modes characterized by 1,920 pixels displayed across the screen horizontally and 1,080 pixels down the screen vert ...
(progressive, also known as Full-HD). In digital video, the temporal sampling rate is defined the frame rate or rather the
field rate The refresh rate (or "vertical refresh rate", "vertical scan rate", terminology originating with the cathode ray tubes) is the number of times per second that a Raster scan, raster-based display device displays a new image. This is independent fro ...
rather than the notional
pixel clock In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device. In most digital display devices, pixels are the sm ...
. 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 –
PAL Phase Alternating Line (PAL) is a colour encoding system for analogue television. It was one of three major analogue colour television standards, the others being NTSC and SECAM. In most countries it was broadcast at 625 lines, 50 fields (25 ...
video * 60 / 1.001 Hz ~= 59.94 Hz –
NTSC The first American standard for analog television broadcast was developed by National Television System Committee (NTSC)National Television System Committee (1951–1953), Report and Reports of Panel No. 11, 11-A, 12–19, with Some supplement ...
video Video
digital-to-analog converter In electronics, a digital-to-analog converter (DAC, D/A, D2A, or D-to-A) is a system that converts a digital signal into an analog signal. An analog-to-digital converter (ADC) performs the reverse function. There are several DAC archit ...
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, 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 scanning pattern, such as a line of video on a cathode ray tube (CRT) display of a television set or computer monitor. On CRT screens the horizontal scan lines are visually discernible ...
s. A common
pixel In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device. In most digital display devices, pixels are the ...
sampling rate is: * 13.5 MHz –
CCIR 601 ITU-R Recommendation BT.601, more commonly known by the abbreviations Rec. 601 or BT.601 (or its former name CCIR 601) is a standard originally issued in 1982 by the Comité consultatif international pour la radio, CCIR (an organization, ...
,
D1 video D-1 or 4:2:2 Component Digital is an SMPTE digital recording video standard, introduced in 1986 through efforts by SMPTE engineering committees. It started as a Sony and Bosch - BTS product and was the first major professional digital 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 compu ...
. The sampling rates and resolutions in both spatial directions can be measured in units of lines per picture height. Spatial
aliasing In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or ''aliases'' of one another) when sampled. It also often refers to the distortion or artifact that results when ...
of high-frequency
luma Luma or LUMA may refer to: Arts * La Trobe University Museum of Art, Melbourne, Australia * LUMA Projection Arts Festival, an annual event featuring building-scale projection mapping and light installations in Binghamton, NY * LUMA Foundation, ...
or chroma video components shows up as a
moiré pattern In mathematics, physics, and art, moiré patterns ( , , ) or moiré 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 moiré ...
.


3D sampling

The process of
volume rendering In scientific visualization and computer graphics, volume rendering is a set of techniques used to display a 2D projection of a 3D discretely sampled data set, typically a 3D scalar field. A typical 3D data set is a group of 2D slice imag ...
samples a 3D grid of
voxel In 3D computer graphics, a voxel represents a value on a regular grid in three-dimensional space. As with pixels in a 2D bitmap, voxels themselves do not typically have their position (i.e. coordinates) explicitly encoded with their values. I ...
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. Most X-rays have a wavelength ranging from 10 picometers to 10 nanometers, corresponding to frequencies in the range 30  ...
(CT/CAT),
magnetic resonance imaging Magnetic resonance imaging (MRI) is a medical imaging technique used in radiology to form pictures of the anatomy and the physiological processes of the body. MRI scanners use strong magnetic fields, magnetic field gradients, and radio wave ...
(MRI), positron emission tomography (PET) are some examples. It is also used for seismic tomography and other applications.


Undersampling

When a
bandpass A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range. Description In electronics and signal processing, a filter is usually a two-po ...
signal is sampled slower than its
Nyquist rate In signal processing, the Nyquist rate, named after Harry Nyquist, is a value (in units of samples per second or hertz, Hz) equal to twice the highest frequency ( bandwidth) of a given function or signal. When the function is digitized at a hi ...
, the samples are indistinguishable from samples of a low-frequency
alias Alias may refer to: * Pseudonym * Pen name * Nickname Arts and entertainment Film and television * ''Alias'' (2013 film), a 2013 Canadian documentary film * ''Alias'' (TV series), an American action thriller series 2001–2006 * ''Alias the ...
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, undersampling or bandpass sampling is a technique where one samples a bandpass-filtered signal at a sample rate below its Nyquist rate (twice the upper cutoff frequency), but is still able to reconstruct the signal. When ...
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 electronics, a digital-to-analog converter (DAC, D/A, D2A, or D-to-A) is a system that converts a digital signal into an analog signal. An analog-to-digital converter (ADC) performs the reverse function. There are several DAC archit ...
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 (DAC). That is, it describes the effect of converting a discrete-time signal to a continuous-time sign ...
instead of idealizations like the
Whittaker–Shannon interpolation formula 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 i ...
.


Complex sampling

Complex sampling (or 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, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
.  When one waveform, \hat s(t),  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(t),\,  the complex-valued function,  s_a(t) \triangleq s(t) + i\cdot \hat s(t),  is called an
analytic signal In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components.  The real and imaginary parts of an analytic signal are real-valued functions related to each other by the Hil ...
,  whose Fourier transform is zero for all negative values of frequency. In that case, the
Nyquist rate In signal processing, the Nyquist rate, named after Harry Nyquist, is a value (in units of samples per second or hertz, Hz) equal to twice the highest frequency ( bandwidth) of a given function or signal. When the function is digitized at a hi ...
for a waveform with no frequencies ≥ ''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(t)\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(t),  by processing the product sequence, \left [s(nT)\cdot e^\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.


See also

* Crystal oscillator frequencies *
Downsampling In digital signal processing, downsampling, compression, and decimation are terms associated with the process of ''resampling'' in a multi-rate digital signal processing system. Both ''downsampling'' and ''decimation'' can be synonymous with ''com ...
*
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 a ...
* Multidimensional sampling *
Sample rate conversion Sample-rate conversion, sampling-frequency conversion or resampling is the process of changing the sampling rate or sampling frequency of a discrete signal to obtain a new discrete representation of the underlying continuous signal. Application ...
*
Digitizing DigitizationTech Target. (2011, April). Definition: digitization. ''WhatIs.com''. Retrieved December 15, 2021, from https://whatis.techtarget.com/definition/digitization is the process of converting information into a digital (i.e. computer- ...
*
Sample and hold In electronics, a sample and hold (also known as sample and follow) circuit is an analog device that samples (captures, takes) the voltage of a continuously varying analog signal and holds (locks, freezes) its value at a constant level for ...
* Beta encoder *
Kell factor The Kell factor, named after RCA engineer Raymond D. Kell, is a parameter used to limit the bandwidth of a sampled image signal to avoid the appearance of beat frequency patterns when displaying the image in a discrete display device, usually t ...
* Bit rate * Normalized frequency


Notes


References


Further reading

* 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.


External links


Journal devoted to Sampling Theory

I/Q Data for Dummies
a page trying to answer the question ''Why I/Q Data?''
Sampling of analog signals
an interactive presentation in a web-demo at the Institute of Telecommunications, University of Stuttgart {{DEFAULTSORT:Sampling Rate Digital signal processing Signal processing