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In Statistics , Signal Processing , and related disciplines, aliasing is an effect that causes different continuous signals to become indistinguishable (or ''aliases'' of one another) when Sampled . When this happens, the original signal cannot be uniquely reconstructed from the sampled signal. Aliasing can take place either in time, Temporal Aliasing , or in space, Spatial Aliasing . Aliasing is a major concern in the will cause high- Frequency components to be aliased with genuine low-frequency ones, and be incorrectly Reconstructed as such during the subsequent Digital-to-analog Conversion . To prevent this problem, the signals must be appropriately Filtered before sampling. It is also a major concern in Digital Imaging and Computer Graphics , where it may give rise to Moiré Pattern s (when the original Image is finely textured) or jagged outlines (when the original has sharp contrasting edges, e.g. Screen Fonts ). Anti-aliasing techniques are used to reduce such Artifact s. OVERVIEW Aliasing in periodic phenomena The Sun moves east to west in the sky, with 24 hours between sunrises. If one were to take a picture of the sky every 23 hours, the sun would appear to move west to east, with 24 × 23 = 552 hours between sunrises. Note that both motions would result in the same pictures. The same phenomenon causes 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 . These are examples of Temporal Aliasing . If someone wearing a Tweed jacket with a pronounced Herringbone pattern was videoed, and the video played 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, then one would see large areas of darkness and lightness over the image of the jacket and not the herringbone pattern. This is an example of Spatial Aliasing , also known as a Moiré Pattern ; how it is produced is illustrated next. Sampling a sinusoidal signal In the same way, when one measures a Sinusoid al Signal at regular intervals, one may obtain the same Sequence of samples that one would get from a sinusoid with lower Frequency . Specifically, if a sinusoid of frequency f (in cycles per second for a time-varying signal, or in cycles per centimeter for space-varying signal) is sampled s times per second or s intervals per centimeter, with s ≤ 2 f, the resulting samples will also be compatible with a sinusoid of frequency 2 f - s. In the area's jargon, each sinusoid gets aliased to (becomes an ''alias'' for) the other. Therefore, if we sample at frequency a continuous signal that may contain both sinusoids, we will not be able to Reconstruct the original signal from the samples, because it is mathematically impossible to tell how much of each component we should take. The Nyquist criterion One way to avoid such aliasing is to make sure that the signal does not contain any Sinusoidal Component with a frequency greater than s/2. More generally, it suffices to ensure that the signal is appropriately ''band-limited'', namely that the difference between the frequencies of any two of its sinusoidal components must be strictly less than s/2. This condition is called the Nyquist Criterion , and is equivalent to saying that the sampling frequency (''s'') must be strictly greater than twice the signal's '' Bandwidth '', the difference between the maximum and minimum frequencies of its sinusoidal components. Origin of the term The term "aliasing" derives from the usage in radio engineering, where a radio signal could be picked up at two different positions on the radio dial in a Superheterodyne radio: one where the local oscillator was above the radio frequency, and one where it was below. This is analogous to the frequency-space "wrapround" that is one way of understanding aliasing. An audio example The qualitative effects of aliasing can be heard in the following audio demonstration. Six Sawtooth Wave s 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.
MATHEMATICAL EXPLANATION OF ALIASING The preceding explanation and the Nyquist criterion are somewhat idealised, because they assume instantaneous sampling and other slightly unrealistic hypotheses, although useful approximations to these things do exist. The following is a more detailed explanation of the phenomenon in terms of Function Approximation theory. Continuous signals For the purposes of this analysis, we define (''continuous'') ''signal'' as a Real or Complex valued Function whose domain is the Interval {Link without Title} . To quantify the "magnitude" of a signal (and, in particular, to measure the difference between two signals), we will use the Root Mean Square norm (see Lp Space s for some details), namely | ||
| :<math>L^2 | L^2( f:[0,1
ightarrow \Bbb C : f<\infin
ight\}\,</math> |
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| To Be Precise, We Do Not Distinguish Between Functions That Differ Only On Sets Of | "http://wwwinformationdelightinfo/encyclopedia/entry/measure_zero" class="copylinks">Zero Measure This technicality turns into a Norm , and explains some of the difficulties (see the ''S''<sub>0</sub> sampling method, below) For details, see Lp Space s |
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| For Instance, When Observing A Signal Which Is Known To Be Of The Form F(t) | (exp(it)-a)/exp(it)-a (see Figure 1 where the trajectory of f(t) is plotted, with a=exp(i)) where a is some unit complex number, then in fact the set of possible signals can be viewed as a one-dimensional Manifold Since a is unit, f will have a singularity somewhere and therefore will contain very high frequency information Yet, the only necessary information is a from this we are able to reconstruct the entirety of the signal with arbitrary precision To recover a, it is sufficient to have any single Fourier coefficient of f, except the zeroth one In other words, while f has extremely wide bandwidth (and the Nyquist theorem would seem to suggest that we need many samples) a single sample of f suffices with the <math>S_0</math> method to recover f fully, to arbitrary precision The reconstruction filter to recover f from <math>S_0f</math> in this case is not any of the discussed reconstruction formulae |
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| :<math>(w,z) R | \sqrt{w^2+z^2} r = Z_0</math>, |
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