Sampling (signal processing)

# Sampling (signal processing)

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In signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

, sampling is the reduction of a continuous signal
Continuous signal
A continuous signal or a continuous-time signal is a varying quantity whose domain, which is often time, is a continuum . That is, the function's domain is an uncountable set. The function itself need not be continuous...

to a discrete signal
Discrete signal
A discrete signal or discrete-time signal is a time series consisting of a sequence of qualities...

. A common example is the conversion of a sound wave (a continuous signal) to a sequence of samples (a discrete-time signal).

A sample refers to 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
Continuous signal
A continuous signal or a continuous-time signal is a varying quantity whose domain, which is often time, is a continuum . That is, the function's domain is an uncountable set. The function itself need not be continuous...

.
A theoretical ideal sampler
Ideal sampler
In signal processing, an ideal sampler is a theoretical operation whose input is a continuous signal and whose output is a sequence of instantaneous values of the signal at discrete moments of time, which is called a discrete signal....

produces samples equivalent to the instantaneous value of the continuous signal at the desired points.

## Theory

Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...

Sampling can be done for signals varying in space, time, or any other dimension, and similar results are obtained in two or more dimensions.

For signals that vary with time, let x(t) be a continuous signal to be sampled, and let sampling be performed by measuring the value of the continuous signal every T seconds, which is called the sampling interval. Thus, the sampled signal x[n] given by:
x[n] = x(nT), with n = 0, 1, 2, 3, ...

The sampling frequency or sampling rate fs is defined as the number of samples obtained in one second, or fs = 1/T.  The sampling rate is measured in hertz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

or in samples per second.

It is possible under some circumstances to reconstruct the original signal completely and exactly (perfect reconstruction).

The Nyquist–Shannon sampling theorem
Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...

provides a sufficient (but not always necessary) condition under which perfect reconstruction is possible. The sampling theorem guarantees that bandlimited
Bandlimited
Bandlimiting is the limiting of a deterministic or stochastic signal's Fourier transform or power spectral density to zero above a certain finite frequency...

signals (i.e., signals which have a maximum frequency) can be reconstructed perfectly from their sampled version, if the sampling rate is more than twice the maximum frequency. Reconstruction in this case can be achieved using the Whittaker–Shannon interpolation formula
Whittaker–Shannon interpolation formula
The Whittaker–Shannon interpolation formula or sinc interpolation is a method to reconstruct a continuous-time bandlimited signal from a set of equally spaced samples.-Definition:...

.

The frequency equal to one-half of the sampling rate is therefore a bound on the highest frequency that can be unambiguously represented by the sampled signal. This frequency (half the sampling rate) is called the Nyquist frequency
Nyquist frequency
The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system...

of the sampling system. Frequencies above the Nyquist frequency fN can be observed in the sampled signal, but their frequency is ambiguous. That is, a frequency component with frequency f cannot be distinguished from other components with frequencies NfN + f and NfNf for nonzero integers N. This ambiguity is called aliasing
Aliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...

. To handle this problem as gracefully as possible, most analog signals are filtered with an anti-aliasing
Anti-aliasing
In digital signal processing, spatial anti-aliasing is the technique of minimizing the distortion artifacts known as aliasing when representing a high-resolution image at a lower resolution...

filter (usually a low-pass filter
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...

with cutoff near the Nyquist frequency) before conversion to the sampled discrete representation.

### Observation period

The observation period is the span of time during which a series of data samples are collected at regular intervals. More broadly, it can refer to any specific period during which a set of data points is gathered, regardless of whether or not the data is periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

in nature. Thus a researcher might study the incidence of earthquake
Earthquake
An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The seismicity, seismism or seismic activity of an area refers to the frequency, type and size of earthquakes experienced over a period of time...

s and tsunami
Tsunami
A tsunami is a series of water waves caused by the displacement of a large volume of a body of water, typically an ocean or a large lake...

s over a particular time period, such as a year or a century.

The observation period is simply the span of time during which the data is studied, regardless of whether data so gathered represents a set of discrete events having arbitrary timing within the interval, or whether the samples are explicitly bound to specified sub-intervals.

## Practical implications

In practice, the continuous signal is sampled using an analog-to-digital converter
Analog-to-digital converter
An analog-to-digital converter is a device that converts a continuous quantity to a discrete time digital representation. An ADC may also provide an isolated measurement...

(ADC), a non-ideal device with various physical limitations. This results in deviations from the theoretically perfect reconstruction capabilities, collectively referred to as distortion.

Various types of distortion can occur, including:
• Aliasing
Aliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...

. A precondition of the sampling theorem is that the signal be bandlimited. However, in practice, no time-limited signal can be bandlimited. Since signals of interest are almost always time-limited (e.g., at most spanning the lifetime of the sampling device in question), it follows that they are not bandlimited. However, by designing a sampler with an appropriate guard band
Guard band
-Radio and electronic signalling:In radio, a guard band is an unused part of the radio spectrum between radio bands, for the purpose of preventing interference....

, it is possible to obtain output that is as accurate as necessary.
• Integration effect or aperture effect. This 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. The integration effect is readily noticeable in photography
Photography
Photography is the art, science and practice of creating durable images by recording light or other electromagnetic radiation, either electronically by means of an image sensor or chemically by means of a light-sensitive material such as photographic film...

when the exposure
Exposure (photography)
In photography, exposure is the total amount of light allowed to fall on the photographic medium during the process of taking a photograph. Exposure is measured in lux seconds, and can be computed from exposure value and scene luminance over a specified area.In photographic jargon, an exposure...

is too long and creates a blur in the image. An ideal camera would have an exposure time of zero. In a capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...

-based sample and hold
Sample and hold
In electronics, a sample and hold circuit is an analog device that samples the voltage of a continuously varying analog signal and holds its value at a constant level for a specified minimal period of time. Sample and hold circuits and related peak detectors are the elementary analog memory...

circuit, the integration effect is introduced because the capacitor cannot instantly change voltage thus requiring the sample to have non-zero width.
• Jitter
Jitter
Jitter is the undesired deviation from true periodicity of an assumed periodic signal in electronics and telecommunications, often in relation to a reference clock source. Jitter may be observed in characteristics such as the frequency of successive pulses, the signal amplitude, or phase of...

or deviation from the precise sample timing intervals.
• Noise, including thermal sensor noise, analog circuit noise, etc.
• Slew rate
Slew rate
In electronics, the slew rate represents the maximum rate of change of a signal at any point in a circuit.Limitations in slew rate capability can give rise to non linear effects in electronic amplifiers...

limit error, caused by an inability for an ADC output value to change sufficiently rapidly.
• Quantization
Quantization (signal processing)
Quantization, in mathematics and digital signal processing, is the process of mapping a large set of input values to a smaller set – such as rounding values to some unit of precision. A device or algorithmic function that performs quantization is called a quantizer. The error introduced by...

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).

The conventional, practical digital-to-analog converter
Digital-to-analog converter
In electronics, a digital-to-analog converter is a device that converts a digital code to an analog signal . An analog-to-digital converter performs the reverse operation...

(DAC) does not output a sequence of dirac impulses (such that, if ideally low-pass filtered, result in the original signal before sampling) but instead output a sequence of piecewise constant values or rectangular pulses. This means that there is an inherent effect of the zero-order hold
Zero-order hold
The zero-order hold is a mathematical model of the practical signal reconstruction done by a conventional digital-to-analog converter . That is, it describes the effect of converting a discrete-time signal to a continuous-time signal by holding each sample value for one sample interval...

on the effective frequency response of the DAC resulting in a mild roll-off
Roll-off
Roll-off is a term commonly used to describe the steepness of a transmission function with frequency, particularly in electrical network analysis, and most especially in connection with filter circuits in the transition between a passband and a stopband...

of gain at the higher frequencies (a 3.9224 dB loss at the Nyquist frequency
Nyquist frequency
The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system...

). This zero-order hold effect is a consequence of the hold action of the DAC and is not due to the sample and hold that might precede a conventional ADC as is often misunderstood. The DAC can also suffer errors from jitter, noise, slewing, and non-linear mapping of input value to output voltage.

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
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...

ing. The non-linearities of either ADC or DAC are analyzed by replacing the ideal linear function
Linear function
In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....

mapping with a proposed nonlinear function.

### Audio sampling

Digital audio
Digital audio
Digital audio is sound reproduction using pulse-code modulation and digital signals. Digital audio systems include analog-to-digital conversion , digital-to-analog conversion , digital storage, processing and transmission components...

uses pulse-code modulation
Pulse-code modulation
Pulse-code modulation is a method used to digitally represent sampled analog signals. It is the standard form for digital audio in computers and various Blu-ray, Compact Disc and DVD formats, as well as other uses such as digital telephone systems...

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.

#### Sampling rate

When it is necessary to capture audio covering the entire 20–20,000 Hz range of human hearing
Auditory system
The auditory system is the sensory system for the sense of hearing.- Outer ear :The folds of cartilage surrounding the ear canal are called the pinna...

, such as when recording music or many types of acoustic events, audio waveforms are typically sampled at 44.1 kHz (CD
Compact Disc
The Compact Disc is an optical disc used to store digital data. It was originally developed to store and playback sound recordings exclusively, but later expanded to encompass data storage , write-once audio and data storage , rewritable media , Video Compact Discs , Super Video Compact Discs ,...

), 48 kHz (professional audio
Professional audio
Professional audio, also 'pro audio', refers to both an activity and a type of audio equipment. Typically it encompasses the production or reproduction of sound for an audience, by individuals who do such work as an occupation like live event support, using sound reinforcement systems designed for...

), or 96 kHz. The approximately double-rate requirement is a consequence of the Nyquist theorem.

There has been an industry trend towards sampling rates well beyond the basic requirements; 96 kHz and even 192 kHz are available. This is in contrast with laboratory experiments, which have failed to show that ultrasonic
Ultrasound
Ultrasound is cyclic sound pressure with a frequency greater than the upper limit of human hearing. Ultrasound is thus not separated from "normal" sound based on differences in physical properties, only the fact that humans cannot hear it. Although this limit varies from person to person, it is...

frequencies are audible to human observers; however in some cases ultrasonic sounds do interact with and modulate the audible part of the frequency spectrum (intermodulation distortion). It is noteworthy that intermodulation distortion is not present in the live audio and so it represents an artificial coloration to the live sound.

One advantage of higher sampling rates is that they can relax the low-pass filter design requirements for ADCs
Analog-to-digital converter
An analog-to-digital converter is a device that converts a continuous quantity to a discrete time digital representation. An ADC may also provide an isolated measurement...

and DACs
Digital-to-analog converter
In electronics, a digital-to-analog converter is a device that converts a digital code to an analog signal . An analog-to-digital converter performs the reverse operation...

, but with modern oversampling sigma-delta converters this advantage is less important.

#### Bit depth (quantization)

Audio is typically recorded at 8-, 16-, and 20-bit depth, which yield a theoretical maximum signal to quantization noise ratio (SQNR) for a pure sine wave
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

of, approximately, 49.93 dB
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

, 98.09 dB and 122.17 dB. Eight-bit audio is generally not used due to prominent and inherent quantization noise (low maximum SQNR), although the A-law and u-law 8-bit encodings pack more resolution into 8 bits while increase total harmonic distortion
Total harmonic distortion
The total harmonic distortion, or THD, of a signal is a measurement of the harmonic distortion present and is defined as the ratio of the sum of the powers of all harmonic components to the power of the fundamental frequency...

. CD quality audio is recorded at 16-bit. In practice, not many consumer stereos can produce more than about 90 dB of dynamic range, although some can exceed 100 dB. 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; consequently, few situations will require more than 20-bit quantization.

For playback and not recording purposes, a proper analysis of typical programme levels throughout an audio system reveals that the capabilities of well-engineered 16-bit material far exceed those of the very best hi-fi systems, with the microphone noise and loudspeaker headroom being the real limiting factors.

#### 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
Phoneme
In a language or dialect, a phoneme is the smallest segmental unit of sound employed to form meaningful contrasts between utterances....

s, almost all of the energy is contained in the 5Hz-4 kHz range, allowing a sampling rate of 8 kHz. This is the sampling rate used by nearly all telephony
Telephony
In telecommunications, telephony encompasses the general use of equipment to provide communication over distances, specifically by connecting telephones to each other....

systems, which use the G.711
G.711
G.711 is an ITU-T standard for audio companding. It is primarily used in telephony. The standard was released for usage in 1972. Its formal name is Pulse code modulation of voice frequencies. It is required standard in many technologies, for example in H.320 and H.323 specifications. It can also...

sampling and quantization specifications.

### Video sampling

Standard-definition television
Standard-definition television
Sorete-definition television is a television system that uses a resolution that is not considered to be either enhanced-definition television or high-definition television . The term is usually used in reference to digital television, in particular when broadcasting at the same resolution as...

(SDTV) uses either 720 by 480 pixels (US NTSC
NTSC
NTSC, named for the National Television System Committee, is the analog television system that is used in most of North America, most of South America , Burma, South Korea, Taiwan, Japan, the Philippines, and some Pacific island nations and territories .Most countries using the NTSC standard, as...

525-line) or 704 by 576 pixels (UK PAL
PAL
PAL, short for Phase Alternating Line, is an analogue television colour encoding system used in broadcast television systems in many countries. Other common analogue television systems are NTSC and SECAM. This page primarily discusses the PAL colour encoding system...

625-line) for the visible picture area.

High-definition television
High-definition television
High-definition television is video that has resolution substantially higher than that of traditional television systems . HDTV has one or two million pixels per frame, roughly five times that of SD...

(HDTV) is currently moving towards three standards referred to as 720p
720p
720p is the shorthand name for 1280x720, a category of High-definition television video modes having a resolution of 1080 or 720p and a progressive scan...

(progressive), 1080i
1080i
1080i is the shorthand name for a high-definition television mode. The i means interlaced video; 1080i differs from 1080p, in which the p stands for progressive scan. The term 1080i assumes a widescreen aspect ratio of 16:9, implying a frame size of 1920×1080 pixels...

(interlaced) and 1080p
1080p
1080p is the shorthand identification for a set of HDTV high-definition video modes that are characterized by 1080 horizontal lines of resolution and progressive scan, meaning the image is not interlaced as is the case with the 1080i display standard....

(progressive, also known as Full-HD) which all 'HD-Ready' sets will be able to display.

## Undersampling

When one samples a bandpass signal at a rate lower than the Nyquist rate, the samples are equal to samples of a low-frequency alias
Aliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...

of the high-frequency signal; the original signal will still be uniquely represented and recoverable if the spectrum of its alias does not cross over half the sampling rate. Such undersampling
Undersampling
In signal processing, undersampling or bandpass sampling is a technique where one samples a bandpass filtered signal at a sample rate below the usual Nyquist rate In signal processing, undersampling or bandpass sampling is a technique where one samples a bandpass filtered signal at a sample rate...

is also known as bandpass sampling, harmonic sampling, IF sampling, and direct IF to digital conversion.

## Oversampling

Oversampling
Oversampling
In signal processing, oversampling is the process of sampling a signal with a sampling frequency significantly higher than twice the bandwidth or highest frequency of the signal being sampled...

is used in most modern analog-to-digital converters to reduce the distortion introduced by practical digital-to-analog converter
Digital-to-analog converter
In electronics, a digital-to-analog converter is a device that converts a digital code to an analog signal . An analog-to-digital converter performs the reverse operation...

s, such as a zero-order hold
Zero-order hold
The zero-order hold is a mathematical model of the practical signal reconstruction done by a conventional digital-to-analog converter . That is, it describes the effect of converting a discrete-time signal to a continuous-time signal by holding each sample value for one sample interval...

instead of idealizations like the Whittaker–Shannon interpolation formula
Whittaker–Shannon interpolation formula
The Whittaker–Shannon interpolation formula or sinc interpolation is a method to reconstruct a continuous-time bandlimited signal from a set of equally spaced samples.-Definition:...

.

## Complex sampling

Complex sampling refers to the simultaneous sampling of two different, but related, waveforms, resulting in pairs of samples that are subsequently treated as complex numbers. Usually one waveform  is the Hilbert transform
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...

of the other waveform  and the complex-valued function,    is called an analytic signal
Analytic signal
In mathematics and signal processing, the analytic representation of a real-valued function or signal facilitates many mathematical manipulations of the signal. The basic idea is that the negative frequency components of the Fourier transform of a real-valued function are superfluous, due to the...

,  whose Fourier transform is zero for all negative values of frequency. In that case, the Nyquist rate
Nyquist rate
In signal processing, the Nyquist rate, named after Harry Nyquist, is two times the bandwidth of a bandlimited signal or a bandlimited channel...

for a waveform with no frequencies ≥ B can be reduced to just B (complex samples/sec), instead of 2B (real samples/sec).When the complex sample-rate is B, a frequency component at 0.6 B, for instance, will have an alias at -0.4 B, which is unambiguous because of the constraint that the pre-sampled signal was analytic. Also see Aliasing#Complex signal representation 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 much more commonly 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 sequenceWhen s(t) is sampled at the Nyquist frequency (1/T = 2B), the product sequence simplifies to  through a digital lowpass filter whose cutoff frequency is B/2.The sequence of complex numbers is convolved with the impulse response of a filter with real-valued coefficients. That is equivalent to separately filtering the sequences of real parts and imaginary parts and reforming complex pairs at the outputs. 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.

• Beta encoder
Beta encoder
A beta encoder is an analog to digital conversion 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...

• Digitizing
Digitizing
Digitizing or digitization is the representation of an object, image, sound, document or a signal by a discrete set of its points or samples. The result is called digital representation or, more specifically, a digital image, for the object, and digital form, for the signal...

• Kell factor
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 devices, usually taken to be 0.7. The number was first measured in 1934...

• Digital signal processing
Digital signal processing
Digital signal processing is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing...

• Downsampling
Downsampling
In signal processing, downsampling is the process of reducing the sampling rate of a signal. This is usually done to reduce the data rate or the size of the data....

• Upsampling
Upsampling
Upsampling is the process of increasing the sampling rate of a signal. For instance, upsampling raster images such as photographs means increasing the resolution of the image....

• Oversampling
Oversampling
In signal processing, oversampling is the process of sampling a signal with a sampling frequency significantly higher than twice the bandwidth or highest frequency of the signal being sampled...