Aliasing

This article is about aliasing in signal processing, including computer graphics. For aliasing in computer programming, see aliasing (computing).

Properly sampled image of a brick wall

Spatial aliasing in the form of a moiré pattern

In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or aliases of one another) when sampled. It also refers to the distortion or artifact that results when the signal reconstructed from samples is different from the original continuous signal.

Aliasing can occur in signals sampled in time, for instance digital audio, and is referred to as temporal aliasing. Aliasing can also occur in spatially sampled signals, for instance digital images. Aliasing in spatially sampled signals is called spatial aliasing.

Description

Left: An aliased image of the letter A in Times New Roman. Right: An anti-aliased image. (See also: Font rasterization)

When a digital image is viewed, a reconstruction is performed by a display or printer device, and by the eyes and the brain. If the image data is not properly processed 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 one can observe 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 aliasing prealiasing and reconstruction aliasing postaliasing.^[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 32000 samples per second (Hz), any frequency components above 16000 Hz (the Nyquist frequency for this sampling rate) will cause aliasing when the music is reproduced by a digital to analog converter (DAC). To prevent this an anti-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 the wagon-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 a negative 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]

Like the video camera, most sampling schemes are periodic; that is, they have a characteristic sampling 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 an analog-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.

Bandlimited functions

Main article: Nyquist–Shannon sampling theorem

Actual signals have finite duration and their frequency content, as defined by the Fourier transform, has no upper bound. Some amount of aliasing always occurs when such functions are sampled. Functions whose frequency content is bounded (bandlimited) have infinite duration. 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

Main article: Undersampling

Sometimes aliasing is used intentionally on signals with no low-frequency content, called bandpass 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^[3] exploit aliasing in this way for computational efficiency. See Sampling (signal processing), Nyquist rate (relative to sampling), and Filter bank.

Sampling sinusoidal functions

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 (as, for example, with a Fourier series or transform). Understanding what aliasing does to the individual sinusoids is useful in understanding what happens to their sum.

Two different sinusoids that fit the same set of samples.

Here, a plot depicts a set of samples whose sample-interval is 1, and two (of many) different sinusoids that could have produced the samples. The sample-rate in this case is $f_s=1\,$ . For instance, if the interval is 1 second, the rate is 1 sample per second. Nine cycles of the red sinusoid and 1 cycle of the blue sinusoid span an interval of 10 samples. The corresponding number of cycles per sample are $f_\mathrm{red}=0.9\,$ and $f_\mathrm{blue}=0.1\,$ . If these samples were produced by sampling functions cos(2π(0.9)x−θ) and cos(2π(0.1)x−φ), they could also have been produced by the trigonometrically identical functions, cos(2π(−0.9)x +θ) and cos(2π(−0.1)x +φ), which introduces the useful concept of Negative frequency.

In general, when a sinusoid of frequency $f\,$ is sampled with frequency $f_s,\,$ the resulting number of cycles per sample is $f/f_s$ (known as normalized frequency), and the samples are indistinguishable from those of another sinusoid (called an alias) whose normalized frequency differs from $f/f_s$ by any integer (positive or negative).^{[note 1]} Replacing negative frequency sinusoids by their equivalent positive frequency representations, we can express all the aliases of frequency $f$ as $f_\mathrm{alias}(N)\ \stackrel{\mathrm{def}}{=}\ |f - N f_s|,\,$ for any integer N, with $\scriptstyle f_\mathrm{alias}(0) = f\,$ being the true value, and N has units of cycles per sample. Then the N = 1 alias of $\scriptstyle f_\mathrm{red}\,$ is $\scriptstyle f_\mathrm{blue},\,$ (and vice versa).

Aliasing matters when one attempts to reconstruct the original waveform from its samples. The most common reconstruction technique produces the smallest of the $\scriptstyle f_\mathrm{alias}(N)\,$ frequencies. So it is usually important that $\scriptstyle f_\mathrm{alias}(0)\,$ be the unique minimum. A necessary and sufficient condition for that is $\scriptstyle f_s/2\ >\ |f|,\,$ where $\scriptstyle f_s/2\,$ is commonly called the Nyquist frequency of a system that samples at rate $\scriptstyle f_s.\,$ In our example, the Nyquist condition is satisfied if the original signal is the blue sinusoid ( $\scriptstyle f = f_\mathrm{blue}$ ). But if $\scriptstyle f = f_\mathrm{red} = 0.9,$ the usual reconstruction method will produce the blue sinusoid instead of the red one.

The black dots are aliases of each other. The solid red line is an example of amplitude varying with frequency. The dashed red lines are the corresponding paths of the aliases.

Folding

In the example above, $\scriptstyle f_\mathrm{red}\,$ and $\scriptstyle f_\mathrm{blue}\,$ are symmetrical around the frequency $\scriptstyle f_s/2.$ And in general, as $\scriptstyle f$ increases from 0 to $\scriptstyle f_s/2,$ $\scriptstyle f_\mathrm{alias}(1)$ decreases from $\scriptstyle f_s$ to $\scriptstyle f_s/2.$ Similarly, as $\scriptstyle f$ increases from $\scriptstyle f_s/2$ to $\scriptstyle f_s,$ $\scriptstyle f_\mathrm{alias}(1)$ continues decreasing from $\scriptstyle f_s/2$ to 0.

A graph of amplitude vs frequency for a single sinusoid at frequency $\scriptstyle 0.6 f_s$ and some of its aliases at $\scriptstyle 0.4 f_s,$ $\scriptstyle 1.4 f_s,$ and $\scriptstyle 1.6 f_s$ would look like the 4 black dots in the adjacent figure. 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 $\scriptstyle f_s/2$ and $\scriptstyle f_s$ ). No matter what function we choose to change the amplitude vs frequency, the graph will exhibit symmetry between 0 and $\scriptstyle f_s.$ This symmetry is commonly referred to as folding, and another name for $\scriptstyle f_s/2$ (the Nyquist frequency) is folding frequency. Folding is most often observed in practice when viewing the frequency spectrum of real-valued samples using a discrete Fourier transform.

Two complex sinusoids, colored gold and cyan, that fit the same sets of real and imaginary sample points when sampled at the rate (f_s) indicated by the grid lines. The case shown here is:

\scriptstyle f_\mathrm{cyan} = f_\mathrm{alias}(1) = f_\mathrm{gold} - 1\cdot f_s.\,

Complex sinusoids

Complex sinusoids are waveforms whose samples are complex numbers, and the concept of negative frequency is necessary to distinguish them. In that case, the frequencies of the aliases are given by just: $f_\mathrm{alias}(N) = f - Nf_s.\,$ Therefore, as $\scriptstyle f\,$ increases from $\scriptstyle f_s/2\,$ to $\scriptstyle f_s,\,$ $\scriptstyle f_\mathrm{alias}(1)$ goes from $\scriptstyle -f_s/2\,$ up to 0. Consequently, complex sinusoids do not exhibit folding. Complex samples of real-valued sinusoids have zero-valued imaginary parts and do exhibit folding.

Sample frequency

Illustration of 4 waveforms reconstructed from samples taken at six different rates. Two of the waveforms are sufficiently sampled to avoid aliasing at all six rates. The other two illustrate increasing distortion (aliasing) at the lower rates.

When the condition $\scriptstyle f_s/2\ >\ f$ is met for the highest frequency component of the original signal, then it is met for all the frequency components, a condition known as 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.

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) 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 receivers. When the receiver shifts multiple signals down to lower frequencies, from RF to IF by heterodyning, an unwanted signal, from an RF frequency equally far from the local 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 an image or alias of the desired signal.

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^[4] or sound field with discrete elements, as in 3D displays or wave field synthesis of sound.

This aliasing is visible in images such as posters with lenticular 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 of parallax on viewer movement in 2D images and in 3-D film produced by stereoscopic 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

Online audio example

The qualitative effects of aliasing can be heard in the following audio demonstration. Six sawtooth waves are played in succession, with the first two sawtooths having a fundamental 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 between bandlimited (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 such that no harmonics above the Nyquist frequency 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.

	Sawtooth aliasing demo 440 Hz bandlimited, 440 Hz aliased, 880 Hz bandlimited, 880 Hz aliased, 1760 Hz bandlimited, 1760 Hz aliased
Problems playing this file? See media help.

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 at more than two points per wavelength, or the wave arrival direction becomes ambiguous.^[5]

Notes

↑ Adding an integer number of cycles between the samples of a sinusoid has no effect on the values at the sample points. That is the essence of aliasing.

Citations

↑ Mitchell, Don P.; Netravali, Arun N. (August 1988). Reconstruction filters in computer-graphics (PDF). ACM SIGGRAPH International Conference on Computer Graphics and Interactive Techniques. pp. 221–228. doi:10.1145/54852.378514. ISBN 0-89791-275-6.
↑ Tessive, LLC (2010)."Time Filter Technical Explanation"
↑ harris, frederic j. (Aug 2006). Multirate Signal Processing for Communication Systems. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-146511-2.
↑ The (New) Stanford Light Field Archive
↑ Flanagan J.L., ‘Beamwidth and useable bandwidth of delay- steered microphone arrays’, AT&T Tech. J., 1985, 64, pp. 983–995