 Properly sampled image of brick wall. In statistics, signal processing, computer graphics and related disciplines, aliasing refers to an effect that causes different continuous signals to become indistinguishable (or aliases of one another) when sampled. It also refers to the distortion or artifact that results when a signal is sampled and reconstructed as an alias of the original signal. Image File history File links Download high resolution version (622x756, 146 KB)A brick wall demonstrating a Moiré pattern when not shown at full size {{User:Cburnett/image sig File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
Image File history File links Download high resolution version (622x756, 146 KB)A brick wall demonstrating a Moiré pattern when not shown at full size {{User:Cburnett/image sig File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
Image File history File links An example of a poorly sampled brick pattern (see Image:Moire pattern of bricks. ...
Image File history File links An example of a poorly sampled brick pattern (see Image:Moire pattern of bricks. ...
It has been suggested that Line moiré be merged into this article or section. ...
A graph of a normal bell curve showing statistics used in educational assessment and comparing various grading methods. ...
Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ...
For the journal by ACM SIGGRAPH, see Computer Graphics (Publication). ...
Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ...
A distortion is the alteration of the original shape (or other characteristic) of an object, image, sound, waveform or other form of information or representation. ...
When we view a digital photograph, the reconstruction (interpolation) is performed by a display or printer device, and by our eyes and our brain. If the reconstructed image differs from the original image, we are seeing an alias. An example of spatial aliasing is the Moiré pattern one can observe in a poorly pixelized image of a brick wall. Techniques that avoid such poor pixelizations are called anti-aliasing. It has been suggested that Line moiré be merged into this article or section. ...
In digital signal processing, anti-aliasing is the technique of minimizing the distortion artifacts known as aliasing when representing a high-resolution signal at a lower resolution. ...
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 us. If we sample it with a frequency that is too low and reconstruct the music with a digital to analog converter, we may hear the low-frequency aliases of the undersampled high frequencies. Therefore, it is common practice to remove the high frequencies with a filter before the sampling is done. Temporal aliasing is the technical term for a phenomenon also known as the stroboscopic effect or the wagon-wheel effect. ...
This article does not cite any references or sources. ...
Sound reproduction is the electrical or mechanical re-creation and/or amplification of sound, often as music. ...
In electronics, a digital-to-analog converter (DAC or D-to-A) is a device for converting a digital (usually binary) code to an analogue signal (current, voltage or charges). ...
Situations also exist where the low frequencies are removed (if necessary), and the high frequency components are intentionally undersampled and reconstructed as lower ones. Some digital channelizers [1] exploit aliasing in this way for computational efficiency; see IR/RF sampling. Signals that contain no low frequencies are often referred to as bandpass or non-baseband. In signal processing, sampling is the reduction of a continuous signal to a discrete signal. ...
The frequency axis of this symbolic diagram would be logarithmically scaled. ...
Baseband is an adjective that describes signals and systems whose range of frequencies is measured from 0 to a maximum bandwidth or highest signal frequency; it is sometimes used as a noun for a band of frequencies starting at 0. ...
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 frequency of rotation. A reversal of direction can be described as a negative frequency. The wagon-wheel effect, (alternatively, waggon-wheel effect, stagecoach-wheel effect, stroboscopic effect) is an optical illusion in which a spoked wheel appears to rotate differently from its true rotation. ...
Negative frequency is the rate of clockwise rotation in phase, where phase is defined as the arctangent of the imaginary and real parts of an electrical signal or mathematical function. ...
Like the video camera, most sampling schemes are periodic; that is they have a characteristic sampling frequency in time or in space. Digital cameras provides 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. The sampling frequency or sampling rate defines the number of samples per second taken from a continuous signal to make a discrete signal. ...
This example shows an image with a portion greatly enlarged, in which the individual pixels are rendered as little squares and can easily be seen. ...
Digitized is a method of creating sprites to games using live video footage. ...
4-channel stereo multiplexed analog-to-digital converter WM8775SEDS made by Wolfson Microelectronics placed on X-Fi Fatal1ty Pro sound card An analog-to-digital converter (abbreviated ADC, A/D or A to D) is an electronic integrated circuit (i/c) that converts continuous signals to discrete digital numbers. ...
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. Understanding what aliasing does to the individual sinusoids is a big help 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 1.0 and two (of many) different sinusoids that could have produced the samples. The sample-rate in this case is  = 1.0. For instance, if the interval is 1 second, the rate is 1 sample per second. 9 cycles of the red sinusoid and 1 cycle of the blue sinusoid span an interval of 10. The respective sinusoid frequencies are  = 0.9 and  = 0.1. Image File history File links No higher resolution available. ...
In general, when a sinusoid of frequency  is sampled with frequency  the resulting samples are indistinguishable from those of another sinusoid of frequency  for any integer  (with  being the actual signal frequency). Most reconstruction techniques produce the minimum of these frequencies, so it is often important that  be the unique minimum. A sufficient condition for that is  where  is commonly called the Nyquist frequency. The Nyquist frequency, named after Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system. ...
In our graphic example, the Nyquist condition is satisfied if the original signal is the blue sinusoid ( ). But if  the lowest image frequency is:
-
 - A reconstruction technique that constructs the lowest possible frequency from the samples will reproduce the blue sinusoid instead of the red one.
- We note that
 is also an image frequency, but since there is no way to distinguish a sinusoid of frequency  from one of frequency  all aliases can be described in terms of just positive frequencies.
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 known as the Nyquist criterion. That is typically approximated by filtering the original signal to attenuate high frequency components before it is sampled. They still generate low-frequency aliases, but at very low amplitude levels, so as not to cause a problem. A filter chosen in anticipation of a certain sample frequency is called an anti-aliasing filter. The filtered signal can subsequently be reconstructed without significant additional distortion, for example by the Whittaker–Shannon interpolation formula. The Nyquist-Shannon sampling theorem is the fundamental theorem in the field of information theory, in particular telecommunications. ...
The Whittaker–Shannon interpolation formula dates back to works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M. Whittaker in 1935 in the formulation of the Nyquist–Shannon sampling theorem by C. E. Shannon in 1949. ...
The Nyquist criterion presumes that the frequency content of the signal being sampled has an upper bound. Implicit in that assumption is that its duration has no upper bound. Similarly, the Whittaker–Shannon interpolation formula assumes instantaneous sampling and an interpolation filter with an unrealizable frequency response. These assumptions comprise a mathematical model that is only an idealized approximation, at best, to any realistic situation. The conclusion, that perfect reconstruction is possible, is mathematically correct for the model, but only an approximation for the real samples and the real signal.
Folding Based on the formula  as increases above  the image closest to 0 moves from down to 0 and then back up to  This creates a local symmetry about the frequency For example, a frequency component at  has a "mirror" image at  That effect is commonly referred to as folding. This is, of course, completely wrong, as folding would require reversal of the frequency axis & the sampling process has no mechanism for such. Explanation should always be in terms of shifted versions of the replicated spectra. The apparent folding comes from the spectrum of the negative frequencies, which under special circumstances appears as a reversal of that of the positive frequencies, leading to the widely held & very misleading concept of "folding".
Complex signal representation Complex signals are signals whose samples are complex numbers, and the concept of negative frequency is meaningful for such signals. In that case, the frequencies of the images are given by just:  So as increases above the image closest to 0 moves from  up to  and repeats that cycle. The phenomenon of "folding" does not occur. Negative frequency is the rate of clockwise rotation in phase, where phase is defined as the arctangent of the imaginary and real parts of an electrical signal or mathematical function. ...
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. The Super Heterodyne receiver (or to give it its full name, The Supersonic Heterodyne Receiver) was invented by Edwin Armstrong in 1918. ...
It has been suggested that this article or section be merged with Radio waves. ...
An intermediate frequency (IF) is a frequency to which a carrier frequency is shifted as an intermediate step in transmission or reception. ...
In telecommunications, to heterodyne is to generate new frequencies by mixing two or more signals in a nonlinear device such as a vacuum tube, transistor, or diode mixer. ...
The Superheterodyne receiver (or to give it its full name, The Supersonic Heterodyne Receiver – usually these days shortened to superhet) was invented by Edwin Armstrong in 1918. ...
More examples
Online "live" 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. This article or section does not cite its references or sources. ...
The fundamental tone, often referred to simply as the fundamental, is the lowest frequency in a harmonic series. ...
A bandlimited signal is a deterministic or stochastic signal (e. ...
The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...
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. Note that the audio file has been coded using Ogg's Vorbis codec, and as such the audio is somewhat degraded. Ogg is an open standard for a free container format for digital multimedia, unrestricted by software patents and designed for efficient streaming and manipulation. ...
Vorbis is an open source, lossy audio codec project headed by the Xiph. ...
- Sawtooth aliasing demo {440 Hz bandlimited, 440 Hz aliased, 880 Hz bandlimited, 880 Hz aliased, 1760 Hz bandlimited, 1760 Hz aliased}
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. The wavelength is the distance between repeating units of a wave pattern. ...
See also Wikimedia Commons has media related to: Aliasing Image File history File links Commons-logo. ...
In digital signal processing, anti-aliasing is the technique of minimizing the distortion artifacts known as aliasing when representing a high-resolution signal at a lower resolution. ...
The wagon-wheel effect, (alternatively, waggon-wheel effect, stagecoach-wheel effect, stroboscopic effect) is an optical illusion in which a spoked wheel appears to rotate differently from its true rotation. ...
In signal processing, a sinc filter is an idealized filter that removes all frequency components above a given bandwidth and leaves the low frequencies alone. ...
The sinc function sinc(x) from x = −8π to 8π. In mathematics, the sinc function (for sinus cardinalis), also known as the interpolation function, filtering function or the first spherical Bessel function , is the product of a sine function and a monotonically decreasing function. ...
Temporal aliasing is the technical term for a phenomenon also known as the stroboscopic effect or the wagon-wheel effect. ...
The Nyquist–Shannon sampling theorem is a fundamental result in the field of information theory, in particular telecommunications and signal processing. ...
The Whittaker–Shannon interpolation formula dates back to works of E. Borel in 1898, and E. T. Whittaker in 1915, and was cited from works of J. M. Whittaker in 1935 in the formulation of the Nyquist–Shannon sampling theorem by C. E. Shannon in 1949. ...
The sampling frequency or sampling rate defines the number of samples per second taken from a continuous signal to make a discrete signal. ...
The Nyquist frequency, named after Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system. ...
Kell factor is a number used to determine the effective resolution of a discrete display device. ...
External links - Aliasing animated gif a graph of signals sampled at different rates, showing how the character of some signals changes dramatically when the rate is too low.
- Frequency Aliasing Demonstration by Burton MacKenZie using stop frame animation and a clock.
- Your Calculator is Wrong Video from YouTube, includes some information about aliasing toward the end.
YouTube is a popular video sharing website where users can upload, view and share video clips. ...
References - ^ Harris, Frederic J. (2006). Multirate Signal Processing for Communication Systems. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-146511-2.
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