Insignal processing and related disciplines,aliasing is a phenomenon that a reconstructed signal from samples of the original signal contains low frequency components that are not present in the original one. This is caused when, in the original signal, there are components at frequency exceeding a certain frequency calledNyquist frequency,, where is the sampling frequency (undersampling). This is because typical reconstruction methods use low frequency components while there are a number of frequency components, called aliases, which sampling result in the identical sample. It also often refers to thedistortion orartifact 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 indigital audio or thestroboscopic effect, and is referred to astemporal aliasing. Aliasing in spatially sampled signals (e.g.,moiré patterns indigital images) is referred to asspatial aliasing.
Aliasing is generally avoided by applyinglow-pass filters oranti-aliasing filters (AAF) to the input signal before sampling and when converting a signal from a higher to a lower sampling rate. Suitablereconstruction filtering 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. Forspatial anti-aliasing, the types of anti-aliasing includefast approximate anti-aliasing (FXAA),multisample anti-aliasing, andsupersampling.
When a digital image is viewed, areconstruction is performed by a display or printer device, and by the eyes and the brain. If the image data is processed incorrectly during sampling or reconstruction, the reconstructed image will differ from the original image, and an alias is seen.
An example of spatial aliasing is themoiré pattern observed in a poorly pixelized image of a brick wall.Spatial anti-aliasing 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 aliasingprealiasing and reconstruction aliasingpostaliasing.[1]
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 32,000samples per second (Hz), any frequency components at or above 16,000Hz (theNyquist frequency for this sampling rate) will cause aliasing when the music is reproduced by adigital-to-analog converter (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, ananti-aliasing filter 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 thewagon-wheel effect, 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 anegative frequency. 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.[2][unreliable source?]
Like the video camera, most sampling schemes are periodic; that is, they have a characteristicsampling frequency in time or in space. Digital cameras provide a certain number of samples (pixels) per degree or per radian, or samples per mm in the focal plane of the camera. Audio signals are sampled (digitized) with ananalog-to-digital converter, 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.
Actual signals have a finite duration and their frequency content, as defined by theFourier transform, has no upper bound. Some amount of aliasing always occurs when such continuous functions over time 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 thebandwidth, the original function can, in theory, be perfectly reconstructed from the infinite set of samples.
Sometimes aliasing is used intentionally on signals with no low-frequency content, calledbandpass signals.Undersampling, 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.[3] (SeeSampling (signal processing),Nyquist rate (relative to sampling), andFilter bank.)
Sinusoids 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 aFourier series ortransform). Understanding what aliasing does to the individual sinusoids is useful in understanding what happens to their sum.
When sampling a function at frequencyfs (i.e., the sampling interval is1/fs), the following functions of time(t) yield identical sets of samples if the sampling starts from such that where, and so on:
Afrequency spectrum 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 bealiases of each other. Noting the sine functions as odd functions:
thus, we can write all the alias frequencies as positive values:. For example, a snapshot of the lower right frame of Fig.2 shows a component at the actual frequency and another component at alias. As increases during the animation, decreases. The point at which they are equal is an axis of symmetry called thefolding frequency, also known asNyquist frequency.
Aliasing matters when one attempts to reconstruct the original waveform from its samples. The most common reconstruction technique produces the smallest of the frequencies. So, it is usually important that be the unique minimum. A necessary and sufficient condition for that is called theNyquist condition. The lower left frame of Fig.2 depicts the typical reconstruction result of the available samples. Until 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.
The figures below offer additional depictions of aliasing, due to sampling. A graph of amplitude vs frequency (not time) for a single sinusoid at frequency0.6fs and some of its aliases at0.4fs,1.4fs, and1.6fs 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 (betweenfs/2 andfs). No matter what function we choose to change the amplitude vs frequency, the graph will exhibit symmetry between 0 andfs. Folding is often observed in practice when viewing thefrequency spectrum of real-valued samples, such as Fig.4.
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Complex sinusoids are waveforms whose samples arecomplex numbers (), and the concept ofnegative frequency is necessary to distinguish them. In that case, the frequencies of the aliases are given by just:fN( f) =f +N fs. (In real sinusoids, as shown in the above, all alias frequencies can be written as positive frequencies because of sine functions as odd functions.) Therefore, asf increases from0 tofs,f−1( f) also increases (from–fs to 0). Consequently, complex sinusoids do not exhibitfolding.
When the conditionfs/2 > f is met for the highest frequency component of the original signal, then it is met for all the frequency components, a condition called theNyquist 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 ananti-aliasing filter.
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 theWhittaker–Shannon interpolation formula) is a customary measure of the effectiveness of sampling.
Historically the termaliasing evolved from radio engineering because of the action ofsuperheterodyne receivers. When the receiver shifts multiple signals down to lower frequencies, fromRF toIF byheterodyning, an unwanted signal, from an RF frequency equally far from thelocal oscillator (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 animage oralias of the desired signal.
The first written use of the terms "alias" and "aliasing" in signal processing appears to be in a 1949 unpublished Bell Laboratories technical memorandum[4] byJohn Tukey andRichard Hamming. That paper includes an example of frequency aliasing dating back to 1922. The firstpublished use of the term "aliasing" in this context is due toBlackman and Tukey in 1958.[5] In their preface to the Dover reprint[6] of this paper, they point out that the idea of aliasing had been illustrated graphically by Stumpf[7] ten years prior.
The 1949 Bell technical report refers to aliasing as though it is a well-known concept, but does not offer a source for the term.Gwilym Jenkins andMaurice Priestley credit Tukey with introducing it in this context,[8] though ananalogous concept of aliasing had been introduced a few years earlier[9] infractional factorial designs. While Tukey did significant work in factorial experiments[10] and was certainly aware of aliasing in fractional designs,[11] it cannot be determined whether his use of "aliasing" in signal processing was consciously inspired by such designs.
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 alight field or sound field with discrete elements, as in3D displays orwave field synthesis of sound.[12]
This aliasing is visible in images such as posters withlenticular printing: 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 ofparallax on viewer movement in 2D images and in3-D film produced bystereoscopic 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).
The qualitative effects of aliasing can be heard in the following audio demonstration. Sixsawtooth waves are played in succession, with the first two sawtooths having afundamental frequency of 440 Hz (A4), the second two having fundamental frequency of 880 Hz (A5), and the final two at 1760 Hz (A6). The sawtooths alternate betweenbandlimited (non-aliased) sawtooths and aliased sawtooths and the sampling rate is 22050 Hz. The bandlimited sawtooths are synthesized from the sawtooth waveform'sFourier series such that no harmonics above theNyquist frequency (11025 Hz = 22050 Hz / 2 here) 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 1760 Hz, the aliased sawtooth is degraded and harsh with a buzzing audible at frequencies lower than the fundamental.
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 perwavelength, or the wave arrival direction becomes ambiguous.[13]
