Aliasing
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Aliasing refers to an effect that causes different continuous signals to become indistinguishable (or aliases of one another) when sampled in statistics, signal processing, computer graphics and related disciplines. Aliasing also refers to the distortion or artifact that results when a signal is sampled and reconstructed as an alias of the original signal. 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. 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. 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 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. 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.
Sampling sinusoidal functionsSinusoids 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. Image:AliasingSines.png
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 Failed to parse (Missing texvc executable; please see math/README to configure.): f_s\,
= 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 Failed to parse (Missing texvc executable; please see math/README to configure.): f_\mathrm{red}\,
= 0.9 and Failed to parse (Missing texvc executable; please see math/README to configure.): f_\mathrm{blue}\,
= 0.1.
In general, when a sinusoid of frequency Failed to parse (Missing texvc executable; please see math/README to configure.): f\, is sampled with frequency Failed to parse (Missing texvc executable; please see math/README to configure.): f_s,\, the resulting samples are indistinguishable from those of another sinusoid of frequency Failed to parse (Missing texvc executable; please see math/README to configure.): f_\mathrm{image}(N) = |f - Nf_s|\,
for any integer Failed to parse (Missing texvc executable; please see math/README to configure.): N\,
(with Failed to parse (Missing texvc executable; please see math/README to configure.): f_\mathrm{image}(0) = f\,
being the actual signal frequency). Most reconstruction techniques produce the minimum of these frequencies, so it is often important that Failed to parse (Missing texvc executable; please see math/README to configure.): f_\mathrm{image}(0)\,
be the unique minimum. A sufficient condition for that is Failed to parse (Missing texvc executable; please see math/README to configure.): f_s/2 > f,\,
where Failed to parse (Missing texvc executable; please see math/README to configure.): f_s/2\,
is commonly called the Nyquist frequency.
In our graphic example, the Nyquist condition is satisfied if the original signal is the blue sinusoid (Failed to parse (Missing texvc executable; please see math/README to configure.): f = f_\mathrm{blue}\, ). But if Failed to parse (Missing texvc executable; please see math/README to configure.): f = f_\mathrm{red},\, the lowest image frequency is:
is also an image frequency, but since there is no way to distinguish a sinusoid of frequency Failed to parse (Missing texvc executable; please see math/README to configure.): -f\, from one of frequency Failed to parse (Missing texvc executable; please see math/README to configure.): f,\, all aliases can be described in terms of just positive frequencies. Sample frequencyImage:Aliasing.gif
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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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. 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 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. Complex signal representationComplex signals are signals whose samples are complex numbers, and the concept of negative frequency is necessary for such signals. In that case, the frequencies of the aliases are given by just: Failed to parse (Missing texvc executable; please see math/README to configure.): f_\mathrm{image}(N) = f - Nf_s.\, Therefore, as Failed to parse (Missing texvc executable; please see math/README to configure.): f\, increases from Failed to parse (Missing texvc executable; please see math/README to configure.): f_s/2\, to Failed to parse (Missing texvc executable; please see math/README to configure.): f_s,\, the image closest to 0 moves from Failed to parse (Missing texvc executable; please see math/README to configure.): -f_s/2\, up to 0. FoldingReal-valued sinusoids have the same negative-frequency aliases as complex ones. The absolute value operator, Failed to parse (Missing texvc executable; please see math/README to configure.): |f - Nf_s|,\, is possible because there is always an equivalent sinusoid with a positive frequency. Therefore, as Failed to parse (Missing texvc executable; please see math/README to configure.): f\, increases from Failed to parse (Missing texvc executable; please see math/README to configure.): f_s/2\, to Failed to parse (Missing texvc executable; please see math/README to configure.): f_s,\, an image moves from Failed to parse (Missing texvc executable; please see math/README to configure.): f_s/2\, down to 0. This creates a local symmetry about the frequency Failed to parse (Missing texvc executable; please see math/README to configure.): f_s/2.\, For example, a frequency component at Failed to parse (Missing texvc executable; please see math/README to configure.): 0.6 f_s\, has a "mirror" image at Failed to parse (Missing texvc executable; please see math/README to configure.): 0.4 f_s.\, That effect is commonly referred to as folding. And another name for Failed to parse (Missing texvc executable; please see math/README to configure.): f_s/2\, (the Nyquist frequency) is folding frequency. Historical usageHistorically 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. More examplesOnline "live" exampleThe 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. Note that the audio file has been coded using Ogg's Vorbis codec, and as such the audio is somewhat degraded.
Direction findingA 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. See also Wikimedia Commons has media related to:
References
External links
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