This article applies to signal processing, including computer graphics. For uses in computer programming, please refer to aliasing (computing). In Computing, aliasing describes a situation in which a data location in memory can be accessed through different symbolic names in the program

Properly sampled image of brick wall.

Spatial aliasing in the form of a Moiré pattern.

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. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data In Signal processing, sampling is the reduction of a Continuous signal to a Discrete signal. It also refers to the distortion or artifact that results when a signal is sampled and reconstructed as an alias of the original signal. 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 In Natural science and Signal processing, an artifact is any perceived Distortion or other Data error caused by the instrument of observation

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. In Digital signal processing, anti-aliasing is the technique of minimizing the distortion artifacts known as Aliasing when representing a high-resolution signal

Temporal aliasing is a major concern in the sampling of video and audio signals. Temporal aliasing is the term applied to a visual Phenomenon also known as the stroboscopic effect. Video is the technology of electronically capturing, Recording, processing storing transmitting and reconstructing a sequence of Still images 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. In Electronics, a digital-to-analog converter ( DAC or D-to-A) is a device for converting a digital (usually binary code to an Analog signal Therefore, it is common practice to remove the high frequencies with a filter before the sampling is done.

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. In Signal processing, sampling is the reduction of a Continuous signal to a Discrete signal. Signals that contain no low frequencies are often referred to as bandpass or non-baseband. A band-pass filter is a device that passes frequencies within a certain range and rejects ( Attenuates frequencies outside that range In Signal processing, baseband is an adjective that describes signals and systems whose range of Frequencies is measured from zero to a maximum bandwidth

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. The wagon-wheel effect (alternatively or stagecoach-wheel effect, stroboscopic effect) is an Optical illusion in which a Spoked Wheel Aliasing has changed its frequency of rotation. A reversal of direction can be described as a negative frequency. The concept of negative and positive frequency can be as simple as a wheel rotating one way or the other way

Like the video camera, most sampling schemes are periodic; that is they have a characteristic sampling frequency in time or in space. Sampling theorem The Nyquist–Shannon sampling theorem states that perfect reconstruction 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. In Digital imaging, a pixel ( pict ure el ement is the smallest piece of information in an image Audio signals are sampled (digitized) with an analog-to-digital converter, which produces a constant number of samples per second. An analog-to-digital converter (abbreviated ADC, A/D or A to D) is an electronic integrated circuit which converts continuous signals to Some of the most dramatic and subtle examples of aliasing occur when the signal being sampled also has periodic content.

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.

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 the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the Sampling frequency

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

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.

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. The Nyquist–Shannon sampling theorem is a fundamental result in the field of Information theory, in particular Telecommunications and Signal processing 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. An anti-aliasing filter is a filter used before a signal sampler to restrict the bandwidth of a signal to approximately satisfy the sampling theorem. The filtered signal can subsequently be reconstructed without significant additional distortion, for example by the Whittaker–Shannon interpolation formula. The Whittaker–Shannon interpolation formula is a method to reconstruct a Continuous-time Bandlimited signal from a set of equally spaced samples

The Nyquist criterion presumes that the frequency content of the signal being sampled has an upper bound. Implicit in that assumption is that the signal's duration has no upper bound. Similarly, the Whittaker–Shannon interpolation formula represents an interpolation filter with an unrealizable frequency response. These assumptions make up a mathematical model that is 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 real samples of a real signal.

Complex signal representation

Complex signals are signals whose samples are complex numbers, and the concept of negative frequency is necessary for such signals. The concept of negative and positive frequency can be as simple as a wheel rotating one way or the other way  In that case, the frequencies of the aliases are given by just:    Therefore, as increases from   to    the image closest to 0 moves from     up to 0.

Folding

Real-valued sinusoids have the same negative-frequency aliases as complex ones. The absolute value operator,    is possible because there is always an equivalent sinusoid with a positive frequency. Therefore, as increases from   to    an image moves from down to 0.  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.  And another name for   (the Nyquist frequency)  is  folding frequency.

Historical usage

Historically the term aliasing evolved from radio engineering because of the action of superheterodyne receivers. In Electronics, the superheterodyne receiver (also known by its full name the supersonic heterodyne receiver, or by the abbreviated form superhet) is a 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. Radio frequency ( RF) is a Frequency or rate of Oscillation within the range of about 3 Hz to 300 GHz In communications and Electronic engineering, an intermediate frequency ( IF) is a Frequency to which a carrier frequency is shifted as an In Radio and Signal processing, heterodyning is the generation of new frequencies by mixing or multiplying two Oscillating waveforms A local oscillator is an electronic device used to generate a signal normally for the purpose of converting a signal of interest to a different frequency using a mixer 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.

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 sawtooth wave (or saw wave) is a kind of Non-sinusoidal waveform. The fundamental tone, often referred to simply as the fundamental and abbreviated fo, is the lowest frequency in a harmonic series. The sawtooths alternate between bandlimited (non-aliased) sawtooths and aliased sawtooths and the sampling rate is 22. A bandlimited signal is a Deterministic or Stochastic signal whose Fourier transform or Power spectral density is zero above a certain finite 05 kHz. The bandlimited sawtooths are synthesized from the sawtooth waveform's Fourier series such that no harmonics above the Nyquist frequency are present. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions

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. To learn how to create video or audio files for Wikipedia and its sister projects check WikipediaCreation and usage of media files. Vorbis is a free and 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. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency.

See also

• Anti-aliasing
• Wagon-wheel effect
• Sinc filter
• Sinc function
• Temporal aliasing
• Nyquist–Shannon sampling theorem
• Whittaker–Shannon interpolation formula
• Nyquist rate
• Nyquist frequency
• Kell factor

References

External links

• 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 video sharing website where users can upload view and share Video clips YouTube was created in February 2005 by three former PayPal employees