Aliasing
From Wikipedia, the free encyclopedia
- This article applies to signal processing, including computer graphics. For uses in computer programming, please refer to aliasing (computing).
In statistics, signal processing, computer graphics, and related disciplines, "signals" that are essentially continuous in space or time must be sampled, and the set of samples is never unique to the original signal. The other signals that could (or did) produce the same samples are called aliases of the original signal.
Contents |
[edit] 'Sampling' schemes
If a continuous signal is reconstructed from the samples, the result may be one of the aliases, which represents a form of distortion. For example, when we view a digital photograph, the reconstruction (interpolation) is performed by our eyes and our brain. If the original image was a lawn, we no longer see the individual blades of grass. Therefore we are seeing an alias. The term aliasing can refer to the ambiguity created by sampling or to the subsequent distortion or both.
Most sampling schemes are periodic; that is they have a characteristic sampling frequency, which we denote by the symbol 
The digital camera provides a certain number of pixels per degree or per radian, which translates into samples per meter at the distance to the subject.
An audio signal is digitized by 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.
[edit] Types of aliasing
[edit] Temporal 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. Also see sampling. Signals that contain no low frequencies are often referred to as bandpass or non-baseband.
[edit] Example of temporal aliasing
The sun moves east to west in the sky, with 24 hours between sunrises. Its frequency is 1 cycle per 24 hours = 1/24. If one were to take a picture of the sky only every 25 hours (
= 1/25), the sun would appear to move with a much slower frequency, given by 1 / 24 − 1 / 25 = 1 / 600, or 600 hours between sunrises. But note that both motions would produce the same set of photos ("samples").
Even more interesting is = 1/23, in which case the apparent frequency is 1 / 24 − 1 / 23 = − 1 / 552. The significance of the negative sign is that the sun appears to move in the opposite direction than its actual motion. The same phenomenon causes the wagon-wheel effect, spoked wheels to apparently turn at the wrong speed or in the wrong direction when filmed, or illuminated with a flashing light source — such as fluorescent lamp, a CRT, or a strobe light.
[edit] Moiré patterns
When a tweed jacket with a pronounced herringbone pattern is viewed on a TV screen with a smaller number of lines than the image of the pattern or on a computer monitor with pixels larger than the elements of the pattern, we might see large areas of darkness and lightness instead of the herringbone pattern. Another Moiré pattern is evident in the poorly pixelized image of a brick wall (see figure). Techniques that avoid such poor pixelizations are called anti-aliasing.
[edit] Sampling sinusoidal functions
Another reason to study periodic signals is that most realistic signals can be thought of as consisting of a linear combination of many sinusoids. Understanding what sampling does to the individual sinusoids can help to understand what happens to their sum. The figure below shows two sinusoids that are aliases of each other when sampled at the frequency () shown. Either sinusoid could be the original signal, and the other is an alias that fits the same set of samples.
and an alias at 
.A similarity to the sun-motion examples (above) is that the lower frequency is the difference between the higher frequency and But there are many other alias frequencies, including 
And the concept of negative frequency is not necessary, because there is always an identical sinusoid with a positive frequency: 
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 necessary and sufficient condition for that is 
where 
is commonly called the Nyquist frequency.
A case that does not satisfy that condition is 
Then the lowest image frequency is:
and a reconstruction technique that constructs the lowest possible frequency from the samples, will not reproduce the original frequency ().
Similarly, when 
the lowest image frequency is still:
which is the case depicted in the figure above, assuming the red colored sinusoid is the original signal (
).
[edit] 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 achieved by filtering the original signal to remove high frequency components before it is sampled. 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. E.g., see Whittaker–Shannon interpolation formula.
The Nyquist criterion presumes that the frequency content of the signal being sampled has an upper bound. Implicit in that assertion 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. The conclusion, that perfect reconstruction is possible, is simplistic. But like Newtonian physics, these assertions are highly useful approximations... sufficient for most purposes.
[edit] 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.
[edit] Complex signal representation
Complex signals are signals whose samples are complex numbers, and the concept of negative frequency is meaningful. 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.
[edit] 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, an unwanted signal can end up at the same frequency as the wanted one. If it is strong enough it can interfere with reception of the desired signal.
[edit] More examples
[edit] 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.
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.
- Sawtooth aliasing demo {440 Hz bandlimited, 440 Hz aliased, 880 Hz bandlimited, 880 Hz aliased, 1760 Hz bandlimited, 1760 Hz aliased}
[edit] 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.
[edit] 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
[edit] 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.
[edit] References
- ^ Harris, Frederic J. (2006). Multirate Signal Processing for Communication Systems. Upper Saddle River, NJ: Prentice Hall PTR. ISBN 0-13-146511-2.
| Digital signal processing |
|---|
| Theory — Nyquist–Shannon sampling theorem, estimation theory, detection theory |
| Sub-fields — audio signal processing | control engineering | digital image processing | speech processing | statistical signal processing |
| Techniques — Discrete Fourier transform (DFT) | Discrete-time Fourier transform (DTFT) | bilinear transform | Z-transform, advanced Z-transform |
| Sampling — oversampling | undersampling | downsampling | upsampling | aliasing | anti-aliasing filter | sampling rate | Nyquist rate/frequency |
