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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 ...
and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or ''aliases'' of one another) when
sampled Sample or samples may refer to: Base meaning * Sample (statistics), a subset of a population – complete data set * Sample (signal), a digital discrete sample of a continuous analog signal * Sample (material), a specimen or small quantity of so ...
. It also often refers to the
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 signal ...
or artifact that results when a signal reconstructed from samples is different from the original continuous signal. Aliasing can occur in signals sampled in time, for instance
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 ...
, or the
stroboscopic effect The stroboscopic effect is a visual phenomenon caused by aliasing that occurs when continuous rotational or other cyclic motion is represented by a series of short or instantaneous samples (as opposed to a continuous view) at a sampling rate clos ...
, and is referred to as temporal aliasing. It can also occur in spatially sampled signals (e.g.
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 ...
s in
digital image A digital image is an image composed of picture elements, also known as ''pixels'', each with ''finite'', '' discrete quantities'' of numeric representation for its intensity or gray level that is an output from its two-dimensional functions ...
s); this type of aliasing is called spatial aliasing. Aliasing is generally avoided by applying
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 ...
s or
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 reconstruction ...
s (AAF) to the input signal before sampling and when converting a signal from a higher to a lower sampling rate. Suitable
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 ...
ing should then be used when restoring the sampled signal to the continuous domain or converting a signal from a lower to a higher sampling rate. For
spatial anti-aliasing In digital signal processing, spatial anti-aliasing is a technique for minimizing the distortion artifacts (aliasing) when representing a high-resolution image at a lower resolution. Anti-aliasing is used in digital photography, computer graphics ...
, the types of anti-aliasing include
fast approximate anti-aliasing Fast approximate anti-aliasing (FXAA) is a screen-space anti-aliasing algorithm created by Timothy Lottes at Nvidia. FXAA 3 is released under a public domain license. A later version, FXAA 3.11, is released under a 3-clause BSD license. Algori ...
(FXAA),
multisample anti-aliasing Multisample anti-aliasing (MSAA) is a type of spatial anti-aliasing, a technique used in computer graphics to remove jaggies. Definition The term generally refers to a special case of supersampling. Initial implementations of full-scene anti-al ...
, and
supersampling Supersampling or supersampling anti-aliasing (SSAA) is a spatial anti-aliasing method, i.e. a method used to remove aliasing (jagged and pixelated edges, colloquially known as "jaggies") from images rendered in computer games or other computer p ...
.


Description

When a digital image is viewed, a
reconstruction 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'' (Russian for "reconstruction"), a late 20th century Soviet Unio ...
is performed by a display or printer device, and by the eyes and the brain. If the image data is processed in some way during sampling or reconstruction, the reconstructed image will differ from the original image, and an alias is seen. An example of spatial aliasing is the
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 ...
observed in a poorly pixelized image of a brick wall.
Spatial anti-aliasing In digital signal processing, spatial anti-aliasing is a technique for minimizing the distortion artifacts (aliasing) when representing a high-resolution image at a lower resolution. Anti-aliasing is used in digital photography, computer graphics ...
techniques avoid such poor pixelizations. Aliasing can be caused either by the sampling stage or the reconstruction stage; these may be distinguished by calling sampling aliasing ''prealiasing'' and reconstruction aliasing ''postaliasing.'' Temporal aliasing is a major concern in the sampling of video and audio signals. Music, for instance, may contain high-frequency components that are inaudible to humans. If a piece of music is sampled at 32,000
samples per second 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 s ...
(Hz), any frequency components at or above 16,000 Hz (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 ...
for this sampling rate) will cause aliasing when the music is reproduced by a
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 architec ...
(DAC). The high frequencies in the analog signal will appear as lower frequencies (wrong alias) in the recorded digital sample and, hence, cannot be reproduced by the DAC. To prevent this, an
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 reconstruction ...
is used to remove components above the Nyquist frequency prior to sampling. In video or cinematography, temporal aliasing results from the limited frame rate, and causes the
wagon-wheel effect The wagon-wheel effect (alternatively called stagecoach-wheel effect or stroboscopic effect) is an optical illusion in which a spoked wheel appears to rotate differently from its true rotation. The wheel can appear to rotate more slowly than the ...
, whereby a spoked wheel appears to rotate too slowly or even backwards. Aliasing has changed its apparent frequency of rotation. A reversal of direction can be described as a
negative frequency The concept of signed frequency (negative and positive frequency) can indicate both the rate and sense of rotation; it can be as simple as a wheel rotating clockwise or counterclockwise. The rate is expressed in units such as revolutions (a.k.a. ''c ...
. Temporal aliasing frequencies in video and cinematography are determined by the frame rate of the camera, but the relative intensity of the aliased frequencies is determined by the shutter timing (exposure time) or the use of a temporal aliasing reduction filter during filming. Like the video camera, most sampling schemes are periodic; that is, they have a characteristic
sampling frequency 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 sp ...
in time or in space. Digital cameras provide a certain number of samples (
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 smal ...
s) per degree or per radian, or samples per mm in the focal plane of the camera. Audio signals are sampled (
digitized 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- ...
) with 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 ...
, which produces a constant number of samples per second. Some of the most dramatic and subtle examples of aliasing occur when the signal being sampled also has periodic content.


Bandlimited functions

Actual signals have a finite duration and their frequency content, as defined by the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
, has no upper bound. Some amount of aliasing always occurs when such functions are sampled. Functions whose frequency content is bounded (''bandlimited'') have an infinite duration in the time domain. If sampled at a high enough rate, determined by the ''bandwidth'', the original function can, in theory, be perfectly reconstructed from the infinite set of samples.


Bandpass signals

Sometimes aliasing is used intentionally on signals with no low-frequency content, called ''bandpass'' signals.
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 ...
, which creates low-frequency aliases, can produce the same result, with less effort, as frequency-shifting the signal to lower frequencies before sampling at the lower rate. Some digital channelizers exploit aliasing in this way for computational efficiency.  (See
Sampling (signal processing) 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 sp ...
, Nyquist rate (relative to sampling), and
Filter bank In signal processing, a filter bank (or filterbank) is an array of bandpass filters that separates the input signal into multiple components, each one carrying a single frequency Sub-band coding, sub-band of the original signal. One application of ...
.)


Sampling sinusoidal functions

Sinusoid A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...
s are an important type of periodic function, because realistic signals are often modeled as the summation of many sinusoids of different frequencies and different amplitudes (for example, with a
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
or
transform Transform may refer to: Arts and entertainment * Transform (scratch), a type of scratch used by turntablists * ''Transform'' (Alva Noto album), 2001 * ''Transform'' (Howard Jones album) or the title song, 2019 * ''Transform'' (Powerman 5000 album ...
). Understanding what aliasing does to the individual sinusoids is useful in understanding what happens to their sum. When sampling a function at frequency (intervals ), the following functions of time yield identical sets of samples: . A
frequency spectrum The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
of the samples produces equally strong responses at all those frequencies. Without collateral information, the frequency of the original function is ambiguous. So the functions and their frequencies are said to be ''aliases'' of each other. Noting the trigonometric identity: : \sin(2\pi (f+Nf_)t + \phi) = \left\{ \begin{array}{ll} +\sin(2\pi (f+Nf_{\rm s})t + \phi), & f+Nf_{\rm s} \ge 0 \\ -\sin(2\pi , f+Nf_{\rm s}, t - \phi), & f+Nf_{\rm s} < 0 \\ \end{array} \right. we can write all the alias frequencies as positive values:  f_{_N}(f) \triangleq \left, f+Nf_{\rm s}\. For example, a snapshot of the lower right frame of Fig.2 shows a component at the actual frequency f and another component at alias f_{_{-1(f). As f increases during the animation, f_{_{-1(f) decreases. The point at which they are equal (f=f_s/2) is an axis of symmetry called the ''folding frequency'', also known as ''
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 ...
''. Aliasing matters when one attempts to reconstruct the original waveform from its samples. The most common reconstruction technique produces the smallest of the f_{_N}(f) frequencies. So it is usually important that f_0(f) be the unique minimum.  A necessary and sufficient condition for that is f_s/2 > , f, , called the ''Nyquist condition''. The lower left frame of Fig.2 depicts the typical reconstruction result of the available samples. Until f exceeds the Nyquist frequency, the reconstruction matches the actual waveform (upper left frame). After that, it is the low frequency alias of the upper frame.


Folding

The figures below offer additional depictions of aliasing, due to sampling. A graph of amplitude vs frequency (not time) for a single sinusoid at frequency    and some of its aliases at      and    would look like the 4 black dots in Fig.3. The red lines depict the paths ( loci) of the 4 dots if we were to adjust the frequency and amplitude of the sinusoid along the solid red segment (between    and  ).  No matter what function we choose to change the amplitude vs frequency, the graph will exhibit symmetry between 0 and    Folding is often observed in practice when viewing the
frequency spectrum The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
of real-valued samples, such as Fig.4.. {, , , ,


Complex sinusoids

Complex sinusoids are waveforms whose samples are
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 ...
, and the concept of
negative frequency The concept of signed frequency (negative and positive frequency) can indicate both the rate and sense of rotation; it can be as simple as a wheel rotating clockwise or counterclockwise. The rate is expressed in units such as revolutions (a.k.a. ''c ...
is necessary to distinguish them. In that case, the frequencies of the aliases are given by just:    Therefore, as    increases from    to       also increases (from    to 0).  Consequently, complex sinusoids do not exhibit ''folding''.


Sample frequency

When the condition    is met for the highest frequency component of the original signal, then it is met for all the frequency components, a condition called the Nyquist criterion. That is typically approximated by filtering the original signal to attenuate high frequency components before it is sampled. These attenuated high frequency components still generate low-frequency aliases, but typically at low enough amplitudes that they do not cause problems. A filter chosen in anticipation of a certain sample frequency is called an
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 reconstruction ...
. The filtered signal can subsequently be reconstructed, by interpolation algorithms, without significant additional distortion. Most sampled signals are not simply stored and reconstructed. But the fidelity of a theoretical reconstruction (via 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 in ...
) is a customary measure of the effectiveness of sampling.


Historical usage

Historically the term ''aliasing'' evolved from radio engineering because of the action of
superheterodyne receiver A superheterodyne receiver, often shortened to superhet, is a type of radio receiver that uses frequency mixing to convert a received signal to a fixed intermediate frequency (IF) which can be more conveniently processed than the original carr ...
s. When the receiver shifts multiple signals down to lower frequencies, from RF to IF by
heterodyning A heterodyne is a signal frequency that is created by combining or mixing two other frequencies using a signal processing technique called ''heterodyning'', which was invented by Canadian inventor-engineer Reginald Fessenden. Heterodyning is u ...
, an unwanted signal, from an RF frequency equally far from the
local oscillator In electronics, a local oscillator (LO) is an electronic oscillator used with a mixer to change the frequency of a signal. This frequency conversion process, also called heterodyning, produces the sum and difference frequencies from the frequenc ...
(LO) frequency as the desired signal, but on the wrong side of the LO, can end up at the same IF frequency as the wanted one. If it is strong enough it can interfere with reception of the desired signal. This unwanted signal is known as an ''image'' or ''alias'' of the desired signal. The first use of the term ``aliasing" in this context is due to Blackman and
Tukey John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the Cooley–Tukey FFT algorithm, fast Fourier Transform (FFT) algorithm and box plot. The Tukey's range test ...
in 1958. In their preface to the Dover reprint of this paper, they point out that the idea of aliasing had been illustrated graphically by Stumpf ten years earlier.


Angular aliasing

Aliasing occurs whenever the use of discrete elements to capture or produce a continuous signal causes frequency ambiguity. Spatial aliasing, particular of angular frequency, can occur when reproducing a
light field The light field is a vector function that describes the amount of light flowing in every direction through every point in space. The space of all possible '' light rays'' is given by the five-dimensional plenoptic function, and the magnitude of ea ...
or sound field with discrete elements, as in
3D display A 3D display is a display device capable of conveying depth perception, depth to the viewer. Many 3D displays are stereoscopic displays, which produce a basic 3D effect by means of stereopsis, but can cause eye strain and visual fatigue. Newer ...
s or
wave field synthesis Wave field synthesis (WFS) is a spatial audio rendering technique, characterized by creation of virtual acoustic space, acoustic environments. It produces ''artificial'' wavefronts synthesized by a large number of individually driven loudspeake ...
of sound. This aliasing is visible in images such as posters with
lenticular printing Lenticular printing is a technology in which lenticular lenses (a technology also used for 3D displays) are used to produce printed images with an illusion of depth, or the ability to change or move as they are viewed from different angles. Ex ...
: if they have low angular resolution, then as one moves past them, say from left-to-right, the 2D image does not initially change (so it appears to move left), then as one moves to the next angular image, the image suddenly changes (so it jumps right) – and the frequency and amplitude of this side-to-side movement corresponds to the angular resolution of the image (and, for frequency, the speed of the viewer's lateral movement), which is the angular aliasing of the 4D light field. The lack of
parallax Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight and is measured by the angle or semi-angle of inclination between those two lines. Due to foreshortening, nearby objects ...
on viewer movement in 2D images and in
3-D film 3D films are motion pictures made to give an illusion of three-dimensional solidity, usually with the help of special glasses worn by viewers. They have existed in some form since 1915, but had been largely relegated to a niche in the motion pict ...
produced by
stereoscopic Stereoscopy (also called stereoscopics, or stereo imaging) is a technique for creating or enhancing the depth perception, illusion of depth in an image by means of stereopsis for binocular vision. The word ''stereoscopy'' derives . Any stere ...
glasses (in 3D films the effect is called " yawing", as the image appears to rotate on its axis) can similarly be seen as loss of angular resolution, all angular frequencies being aliased to 0 (constant).


More examples


Audio example

The qualitative effects of aliasing can be heard in the following audio demonstration. Six
sawtooth wave The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an intermittently triggered sawtooth, is called a ...
s are played in succession, with the first two sawtooths having a
fundamental frequency The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In ...
of 440 Hz (A4), the second two having fundamental frequency of 880 Hz (A5), and the final two at 1,760 Hz (A6). The sawtooths alternate between
bandlimited Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency. A band-limited signal is one whose Fourier transform or spectral density has bounded support. A bandlimit ...
(non-aliased) sawtooths and aliased sawtooths and the sampling rate is 22,05 kHz. The bandlimited sawtooths are synthesized from the sawtooth waveform's
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
such that no harmonics above 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 ...
are present. The aliasing distortion in the lower frequencies is increasingly obvious with higher fundamental frequencies, and while the bandlimited sawtooth is still clear at 1,760 Hz, the aliased sawtooth is degraded and harsh with a buzzing audible at frequencies lower than the fundamental.


Direction finding

A form of spatial aliasing can also occur in antenna arrays or microphone arrays used to estimate the direction of arrival of a wave signal, as in geophysical exploration by seismic waves. Waves must be sampled more densely than two points per
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
, or the wave arrival direction becomes ambiguous.


See also

*
Brillouin zone In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice i ...
*
Glossary of video terms This glossary defines terms that are used in the documen"Defining Video Quality Requirements: A Guide for Public Safety" developed by thVideo Quality in Public Safety (VQIPS) Working Group It contains terminology and explanations of concepts relev ...
*
Jaggies "Jaggies" is the informal name for artifacts in raster images, most frequently from aliasing, which in turn is often caused by non-linear mixing effects producing high-frequency components, or missing or poor anti-aliasing filtering prior to samp ...
*
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 ...
*
Sinc filter In signal processing, a sinc filter is an idealized filter that removes all frequency components above a given cutoff frequency, without affecting lower frequencies, and has linear phase response. The filter's impulse response is a sinc functio ...
*
Sinc function In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the u ...
*
Spectral leakage The Fourier transform of a function of time, s(t), is a complex-valued function of frequency, S(f), often referred to as a frequency spectrum. Any linear time-invariant operation on s(t) produces a new spectrum of the form H(f)•S(f), which chang ...
*
Stroboscopic effect The stroboscopic effect is a visual phenomenon caused by aliasing that occurs when continuous rotational or other cyclic motion is represented by a series of short or instantaneous samples (as opposed to a continuous view) at a sampling rate clos ...
*
Wagon-wheel effect The wagon-wheel effect (alternatively called stagecoach-wheel effect or stroboscopic effect) is an optical illusion in which a spoked wheel appears to rotate differently from its true rotation. The wheel can appear to rotate more slowly than the ...
*


Notes


References


Further reading

* Pharr, Matt; Humphreys, Greg. (28 June 2010)
''Physically Based Rendering: From Theory to Implementation''.
Morgan Kaufmann Morgan Kaufmann Publishers is a Burlington, Massachusetts (San Francisco, California until 2008) based publisher specializing in computer science and engineering content. Since 1984, Morgan Kaufmann has published content on information technology ...
.
Chapter 7 (''Sampling and reconstruction'')
Retrieved 3 March 2013.


External links

* by Tektronix Application Engineer
Anti-Aliasing Filter Primer
by La Vida Leica, discusses its purpose and effect on recorded images
Interactive examples demonstrating the aliasing effect
{{Authority control Digital signal processing Signal processing