In
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, sampling is the reduction of a
continuous-time 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 ...
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 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. 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 called a
reconstruction filter.
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 o ...
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 is mathematically equivalent to an ideal
low-pass filter 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. 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''). 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 o ...
) 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, 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 (ADC), a device with various physical limitations. This results in deviations from the theoretically perfect reconstruction, collectively referred to as
distortion.
Various types of distortion can occur, including:
*
Aliasing. 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 In mathematics, the phrases arbitrarily large, arbitrarily small and arbitrarily long are used in statements to make clear of the fact that an object is large, small and long with little limitation or restraint, respectively. The use of "arbitraril ...
by using a
sufficiently large 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 a ...
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 aris ...
, including thermal sensor noise,
analog circuit noise, etc.
*
Slew rate 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 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 filtering. The non-linearities of either ADC or DAC are analyzed by replacing the ideal
linear function mapping with a proposed
nonlinear function.
Applications
Audio sampling
Digital audio 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, 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. Sampling rates higher than about 50 kHz to 60 kHz cannot supply more usable information for human listeners. Early
professional audio 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 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), ''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 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 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 released in O ...
(CD) and other consumer uses, 32 kHz for transmission-related applications, and 96 kHz for higher bandwidth or relaxed
anti-aliasing filtering.
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 of, approximately, 49.93
dB, 98.09 dB and 122.17 dB. CD quality audio uses 16-bit samples.
Thermal noise 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, 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-wes ...
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
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
...
used by nearly all
telephony systems, which use the
G.711 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 (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 (UK
PAL 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 broadcas ...
(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 scree ...
(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 ve ...
(progressive, also known as Full-HD).
In
digital video, the temporal sampling rate is defined the
frame rate or rather the
field rate 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 –
PAL 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 converters 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 lines. 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,
D1 video
Spatial sampling in the other direction is determined by the spacing of scan lines in the
raster. The sampling rates and resolutions in both spatial directions can be measured in units of lines per picture height.
Spatial
aliasing of high-frequency
luma or
chroma video components shows up as a
moiré pattern.
3D sampling
The process of
volume rendering samples a 3D grid of
voxels 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 (CT/CAT),
magnetic resonance imaging (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-por ...
signal is sampled slower than its
Nyquist rate, 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 J ...
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 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 converters, 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.
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. When one waveform
is the
Hilbert transform of the other waveform
the complex-valued function,
is called an
analytic signal, whose Fourier transform is zero for all negative values of frequency. In that case, the
Nyquist rate 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,
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
by processing the product sequence
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
Crystal oscillators can be manufactured for oscillation over a wide range of frequencies, from a few kilohertz up to several hundred megahertz. Many applications call for a crystal oscillator frequency conveniently related to some other desired fr ...
*
Downsampling
*
Upsampling
*
Multidimensional sampling In digital signal processing, multidimensional sampling is the process of converting a function of a multidimensional variable into a discrete collection of values of the function measured on a discrete set of points. This article presents the basi ...
*
Sample rate conversion
*
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 a ...
*
Beta encoder A beta encoder is an analog-to-digital conversion (A/D) system in which a real number in the unit interval is represented by a finite representation of a sequence in ''base beta'', with beta being a real number between 1 and 2. Beta encoders are a ...
*
Kell factor
*
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 TheoryI/Q Data for Dummiesa page trying to answer the question ''Why I/Q Data?''
Sampling of analog signalsan interactive presentation in a web-demo at the Institute of Telecommunications, University of Stuttgart
{{DEFAULTSORT:Sampling Rate
Digital signal processing
Signal processing