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| {{About|aliasing in signal processing, including computer graphics|aliasing in computer programming|aliasing (computing)}}
| | http://bestwoocommercethemes.blogspot.co.uk/2014/05/the-ultimate-top-5-responsive-themes.html [http://bestwoocommercethemes.blogspot.co.uk/2014/05/the-ultimate-top-5-responsive-themes.html http://bestwoocommercethemes.blogspot.co.uk/2014/05/the-ultimate-top-5-responsive-themes.html]; Probably the best choice for e-commerce beneath a WordPress environment, WooCommerce ship all of the juice you want for a successfully e-commerce business. WooCommerce is built utilizing WordPress best practices each on the front and the again finish. This results in an efficient, strong and intuitive plugin.<br><br>With not as many choices as Magento to specify, including a product in WooCommerce works a lot quicker. There are extra choices than you see on this screenshot, because including a product is basically adding a WordPress publish with lots of variables. So you can add photos, specify one or a number of categories and add product information. But the power of WooCommerce is in these product settings. Bellissima is a responsive WooCommerce theme for WordPress. The theme is on the market in 2 different child skins to choose from. The themes listed below are best suited for vogue and clothing, bodily merchandise, digital products, magnificence & cosmetic, digital products, and so forth. About WooCommerce Plugin<br><br>The Gravity Varieties Product Add-ons extension offers you full management over how the display of the prices within the store should seem. You should use the default worth displays that WooCommerce will display, or within the case of a configurable product containing pricing fields, set the value to something akin to "As Low as $1000". I apologize upfront but I can’t assist you with your customized implementations through the comments or WordPress help forums. I'm open to contract work though. Please contact me when you’d like to hire me.<br><br>Moreover, over the previous few weeks of testing, we corresponded with the developer to make some suggestions, and several other of those suggestions have been [http://en.wiktionary.org/wiki/implemented implemented] in the present model of the plugin. These items make the plugin more usable for nonprofits, and we predict you’ll perceive why whenever you see the video. Extra Details on the Reserving and Appointments Plugin Whether or not you are working a superstore promoting hundreds of merchandise, or a just a sole trader promoting hand made arts and crafts WooCommerce has you coated. You can read all about WooCommerce' features on the WooCommerce web site<br><br>Instead of working like an unbiased, internet-hosted cart that retains its own suite of themes resembling Shopify or BigCommerce , as talked about above WooCommerce is a WordPress plugin that can theoretically be built-in into any WordPress theme. But to stop headaches like out of whack margins and formatting I counsel selecting a WooTheme on your store since many are tailored for WooCommerce and require little tweaking. Word that doing so might cost you around $100 except you opt for of three free skins Order achievement for online stores using WooCommerce is out there with this integration, launched by SkyVerge and supported by WooThemes, the developers behind WooCommerce. Features<br><br>It’s possible to make any of the WooCommerce checkout fields required or not required. The essential factor to notice is that "true" makes the field required and "false" makes it not required. As soon as the plugin was launched there were many requests to make it possible to book property, so now in version 1.1 we even have the option to guide a begin and finish date which is appropriate for the rental of property, resort reserving, lodging, and so forth. Though that is only version 1.1, it already feels prefer it’s a mature plugin and this augurs very effectively for the way forward for this add-on. |
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| [[File:Moire pattern of bricks.jpg|thumb|197px|Properly sampled image of brick wall.]]
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| [[File:Moire pattern of bricks small.jpg|thumb|197px|Spatial aliasing in the form of a [[Moiré pattern]].]]
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| In [[signal processing]] and related disciplines, '''aliasing''' refers to an effect that causes different signals to become indistinguishable (or ''aliases'' of one another) when [[sampling (signal processing)|sampled]]. It also refers to the [[distortion]] or [[artifact (error)|artifact]] that results when the signal reconstructed from samples is different from the original continuous signal.
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| Aliasing can occur in signals sampled in time, for instance [[digital audio]], and is referred to as '''temporal aliasing'''. Aliasing can also occur in spatially sampled signals, for instance [[digital image]]s. Aliasing in spatially sampled signals is called '''spatial aliasing'''.
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| == Description ==
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| [[File:Aliasing a.png|thumb|200px|Aliasing example of the A letter in Times New Roman. Left: aliased image, right: ''anti-aliased'' image.]]
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| When a digital image is viewed, a [[Signal reconstruction|reconstruction]] is performed by a display or printer device, and by the eyes and the brain. If the image data is not properly processed during sampling or reconstruction, the reconstructed image will differ from the original image, and an alias is seen.
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| 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 [[spatial anti-aliasing]]. Aliasing can be caused either by the sampling stage or the reconstruction stage; these may be distinguished by calling sampling aliasing {{anchor|prealiasing}} ''prealiasing'' and reconstruction aliasing {{anchor|postaliasing}} ''postaliasing.''<ref name="mitchell">{{cite conference
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| |title=Reconstruction filters in computer-graphics
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| |first=Don P.
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| |last=Mitchell
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| |author2=Netravali, Arun N.
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| |url=http://www.mentallandscape.com/Papers_siggraph88.pdf
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| |doi=10.1145/54852.378514
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| |conference=ACM SIGGRAPH International Conference on Computer Graphics and Interactive Techniques
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| |conferenceurl=http://portal.acm.org/toc.cfm?id=54852&type=proceeding&coll=GUIDE&dl=GUIDE,ACM&CFID=30538218&CFTOKEN=95411512
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| |pages=221–228
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| |date=August 1988
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| |volume=22
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| |number=4
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| |ISBN=0-89791-275-6
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| }}</ref>
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| 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 humans. If a piece of music is sampled at 32000 [[Sampling rate|samples per second]] (Hz), any frequency components above 16000 [[hertz|Hz]] (the [[Nyquist frequency]]) will cause aliasing when the music is reproduced by a [[digital to analog converter]] (DAC). To prevent this an [[anti-aliasing filter]] is used to remove components above the Nyquist frequency prior to sampling.
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| 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 apparent frequency of rotation. A reversal of direction can be described as a [[negative frequency]]. Temporal aliasing frequencies in video and cinematography are determined by the frame rate of the camera, but the relative intensity of the aliased frequencies is determined by the shutter timing (exposure time) or the use of a temporal aliasing reduction filter during filming.<ref name="tessive">Tessive, LLC (2010).[http://tessive.com/time-filter-technical-explanation "Time Filter Technical Explanation"]</ref>
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| 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 ([[pixel]]s) 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.
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| == Bandlimited functions ==
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| {{main|Nyquist–Shannon sampling theorem}}
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| Actual signals have finite duration and their frequency content, as defined by the Fourier transform, has no upper bound. Some amount of aliasing always occurs when such functions are sampled. Functions whose frequency content is bounded (''bandlimited'') have infinite duration. If sampled at a high enough rate, determined by the ''bandwidth'', the original function can in theory be perfectly reconstructed from the infinite set of samples.
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| == Bandpass signals ==
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| {{main|Undersampling}}
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| Sometimes aliasing is used intentionally on signals with no low-frequency content, called ''bandpass'' signals. [[Undersampling]], which creates low-frequency aliases, can produce the same result, with less effort, as frequency-shifting the signal to lower frequencies before sampling at the lower rate. Some digital channelizers<ref name="harris">{{Cite book
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| | last = harris
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| | first = frederic j.
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| | date = Aug 2006
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| | title = Multirate Signal Processing for Communication Systems
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| | publisher = Prentice Hall PTR
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| | location = Upper Saddle River, NJ
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| | isbn = 0-13-146511-2
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| }}</ref>
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| exploit aliasing in this way for computational efficiency. See [[Sampling (signal processing)#Undersampling|Sampling (signal processing)]], [[Nyquist rate#Nyquist rate relative to sampling|Nyquist rate (relative to sampling)]], and [[Filter bank]].
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| == Sampling sinusoidal functions ==
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| [[Sine wave|Sinusoid]]s 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 (with a [[Fourier series]] or [[Fourier transform|transform]]). Understanding what aliasing does to the individual sinusoids is useful in understanding what happens to their sum.
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| [[File:AliasingSines.svg|thumb|right|300px|Two different sinusoids that fit the same set of samples.]]
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| Here, a plot depicts a set of samples whose sample-interval is 1, and two (of many) different sinusoids that could have produced the samples. The sample-rate in this case is <math>f_s\,</math> = 1. For instance, if the interval is 1 second, the rate is 1 sample per second. Nine cycles of the red sinusoid and 1 cycle of the blue sinusoid span an interval of 10. The respective sinusoid frequencies are <math>f_\mathrm{red}\,</math> = 0.9 and <math>f_\mathrm{blue}\,</math> = 0.1. In general, when a sinusoid of frequency <math>f\,</math> is sampled with frequency <math>f_s,\,</math> the resulting samples are indistinguishable from those of another sinusoid of frequency <math>\scriptstyle (f - Nf_s),\,</math> for any integer N. The values corresponding to N ≠ 0 are called ''images'' or ''aliases'' of frequency <math>f.\,</math> In our example, the N=±1 aliases of <math>\scriptstyle f = f_\mathrm{red} = 0.9</math> are <math>\scriptstyle 0.9 + 1.0 = 1.9</math> and <math>\scriptstyle 0.9 - 1.0 = -0.1.</math> A [[negative frequency]] is equivalent to its absolute value, because sin(‑wt+θ)=sin(wt‑θ+π), and cos(‑wt+θ)=cos(wt‑θ).<!--‑ is a "non-breaking hyphen" character--> Therefore we can express all the image frequencies as <math>f_\mathrm{image}(N) = |f - Nf_s|,\,</math> for any integer N (with <math>\scriptstyle f_\mathrm{image}(0) = f\,</math> being the actual signal frequency). Then the N=1 alias of <math>\scriptstyle f_\mathrm{red}\,</math> is <math>\scriptstyle f_\mathrm{blue},\,</math> (and vice versa).
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| Aliasing matters when one attempts to reconstruct the original waveform from its samples. The most common reconstruction technique produces the smallest of the <math>\scriptstyle f_\mathrm{image}(N)\,</math> frequencies. So it is usually important that <math>\scriptstyle f_\mathrm{image}(0)\,</math> be the unique minimum. A necessary and sufficient condition for that is <math>\scriptstyle f_s/2\ >\ |f|,\,</math> where <math>\scriptstyle f_s/2\,</math> is commonly called the [[Nyquist frequency]] of a system that samples at rate <math>\scriptstyle f_s.\,</math> In our example, the Nyquist condition is satisfied if the original signal is the blue sinusoid (<math>\scriptstyle f = f_\mathrm{blue}</math>). But if <math>\scriptstyle f = f_\mathrm{red} = 0.9,</math> the usual reconstruction method will produce the blue sinusoid instead of the red one.
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| [[File:Aliasing-folding.png|thumb|300px|The black dots are aliases of each other. The solid red line is an <u>example</u> of adjusting amplitude vs frequency. The dashed red lines are the corresponding paths of the aliases.]]
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| === Folding ===
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| In the example above, <math>\scriptstyle f_\mathrm{red}\,</math> and <math>\scriptstyle f_\mathrm{blue}\,</math> are symmetrical around the frequency <math>\scriptstyle f_s/2.</math> And in general, as <math>\scriptstyle f</math> increases from 0 to <math>\scriptstyle f_s/2,</math> <math>\scriptstyle f_\mathrm{image}(1)</math> decreases from <math>\scriptstyle f_s</math> to <math>\scriptstyle f_s/2.</math> Similarly, as <math>\scriptstyle f</math> increases from <math>\scriptstyle f_s/2</math> to <math>\scriptstyle f_s,</math> <math>\scriptstyle f_\mathrm{image}(1)</math> continues decreasing from <math>\scriptstyle f_s/2</math> to 0.
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| A graph of amplitude vs frequency for a single sinusoid at frequency <math>\scriptstyle 0.6 f_s</math> and some of its aliases at <math>\scriptstyle 0.4 f_s,</math> <math>\scriptstyle 1.4 f_s,</math> and <math>\scriptstyle 1.6 f_s</math> would look like the 4 black dots in the adjacent figure. The red lines depict the paths ([[wikt:loci|loci]]) of the 4 dots if we were to adjust the frequency and amplitude of the sinusoid along the solid red segment (between <math>\scriptstyle f_s/2</math> and <math>\scriptstyle f_s</math>). No matter what function we choose to change the amplitude vs frequency, the graph will exhibit symmetry between 0 and <math>\scriptstyle f_s.</math> This symmetry is commonly referred to as '''folding''', and another name for <math>\scriptstyle f_s/2</math> (the Nyquist frequency) is '''folding frequency'''. Folding is most often observed in practice when viewing the [[Frequency spectrum#Spectrum analysis|frequency spectrum]] of real-valued samples using a [[discrete Fourier transform]].
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| [[File:Aliasing between a positive and a negative frequency.png|thumb|300px|right|Two complex sinusoids, colored gold and cyan, that fit the same sets of real and imaginary sample points when sampled at the rate (''f<sub>s</sub>'') indicated by the grid lines. The case shown here is: <math>\scriptstyle f_{cyan} = f_\mathrm{image}(1) = f_{gold} - 1\cdot f_s.\,</math>]]
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| === Complex sinusoids ===
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| [[Negative frequency#Complex sinusoids|Complex sinusoids]] are waveforms whose samples are [[complex numbers]], and the concept of [[negative frequency]] is necessary to distinguish them. In that case, the frequencies of the aliases are given by just''':''' <math>f_\mathrm{image}(N) = f - Nf_s.\,</math> Therefore, as <math>\scriptstyle f\,</math> increases from <math>\scriptstyle f_s/2\,</math> to <math>\scriptstyle f_s,\,</math> <math>\scriptstyle f_\mathrm{image}(1)</math> goes from <math>\scriptstyle -f_s/2\,</math> <u>up</u> to 0. Consequently, complex sinusoids do not exhibit ''folding''. Complex samples of real-valued sinusoids have zero-valued imaginary parts and do exhibit folding.
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| === Sample frequency ===
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| [[File:Aliasing.gif|thumb|300px|Illustration of 4 waveforms reconstructed from samples taken at 6 different rates. Two of the waveforms are sufficiently sampled to avoid aliasing at all 6 rates. The other two illustrate increasing distortion (aliasing) at the lower rates.]]
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| When the condition <math>\scriptstyle f_s/2\ >\ f</math> 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–Shannon sampling theorem|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]].
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| 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 actual samples of an actual signal.
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| == Historical usage ==
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| Historically the term ''aliasing'' evolved from radio engineering because of the action of [[superheterodyne receiver]]s. When the receiver shifts multiple signals down to lower frequencies, from [[Radio frequency|RF]] to [[Intermediate frequency|IF]] by [[Heterodyne|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.
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| == Angular aliasing ==
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| Aliasing occurs whenever the use of discrete elements to capture or produce a continuous signal causes frequency ambiguity.
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| Spatial aliasing, particular of angular frequency, can occur when reproducing a [[light field]]<ref>[http://lightfield.stanford.edu/lfs.html The (New) Stanford Light Field Archive]</ref> or sound field with discrete elements, as in [[3D display]]s or [[wave field synthesis]] of sound.
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| This aliasing is visible in images such as posters with [[lenticular printing]]: if they have low angular resolution, then as one moves past them, say from left-to-right, the 2D image does not initially change (so it appears to move left), then as one moves to the next angular image, the image suddenly changes (so it jumps right) – and the frequency and amplitude of this side-to-side movement corresponds to the angular resolution of the image (and, for frequency, the speed of the viewer's lateral movement), which is the angular aliasing of the 4D light field.
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| The lack of [[parallax]] on viewer movement in 2D images and in [[3-D film]] produced by [[Stereoscopy|stereoscopic]] glasses (in 3D films the effect is called "[[yaw (rotation)|yaw]]ing", as the image appears to rotate on its axis) can similarly be seen as loss of angular resolution, all angular frequencies being aliased to 0 (constant).
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| == More examples ==
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| === Online audio example ===
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| 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.
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| 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.
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| {{Listen|filename=Sawtooth-aliasingdemo.ogg|title=Sawtooth aliasing demo|description=440 Hz bandlimited, 440 Hz aliased, 880 Hz bandlimited, 880 Hz aliased, 1760 Hz bandlimited, 1760 Hz aliased|format=[[Ogg]]}}
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| === Direction finding ===
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| 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.<ref name="flangan">Flanagan J.L., ‘Beamwidth and useable bandwidth of delay- steered microphone arrays’, AT&T Tech. J., 1985, 64, pp. 983-995</ref>
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| == See also ==
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| {{Commons|Aliasing}}
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| * [[Jaggies]]
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| * [[Kell factor]]
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| * [[Sinc filter]]
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| * [[Sinc function]]
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| * [[Stroboscopic effect]]
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| * [[Wagon-wheel effect]]
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| * [[Glossary of video terms]]
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| == Further reading ==
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| "Sampling and reconstruction," [http://graphics.stanford.edu/~mmp/chapters/pbrt_chapter7.pdf Chapter 7] in.<ref name="PharrHumphreys2010">{{cite book|author1=Matt Pharr|author2=Greg Humphreys|title=Physically Based Rendering: From Theory to Implementation|url=http://books.google.com/books?id=9nJBAJhTxt8C|accessdate=3 March 2013|date=28 June 2010|publisher=Morgan Kaufmann|isbn=978-0-12-375079-2}}</ref>
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| == References ==
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| {{reflist}}
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| == External links ==
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| * [http://www.youtube.com/watch?v=g3svU5VJ8Gk&feature=plcp Aliasing by a sampling oscilloscope] by Tektronix Application Engineer
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| * [http://lavidaleica.com/content/anti-aliasing-filter-primer Anti-Aliasing Filter Primer]{{dead link|date=October 2012}} by La Vida Leica discusses its purpose and effect on the image recorded.
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| * [http://burtonmackenzie.com/2006/07/i-cant-drive-55.html Frequency Aliasing Demonstration] by Burton MacKenZie using stop frame animation and a clock.
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| {{DSP}}
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| [[Category:Digital signal processing]]
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| [[Category:Signal processing]]
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http://bestwoocommercethemes.blogspot.co.uk/2014/05/the-ultimate-top-5-responsive-themes.html http://bestwoocommercethemes.blogspot.co.uk/2014/05/the-ultimate-top-5-responsive-themes.html; Probably the best choice for e-commerce beneath a WordPress environment, WooCommerce ship all of the juice you want for a successfully e-commerce business. WooCommerce is built utilizing WordPress best practices each on the front and the again finish. This results in an efficient, strong and intuitive plugin.
With not as many choices as Magento to specify, including a product in WooCommerce works a lot quicker. There are extra choices than you see on this screenshot, because including a product is basically adding a WordPress publish with lots of variables. So you can add photos, specify one or a number of categories and add product information. But the power of WooCommerce is in these product settings. Bellissima is a responsive WooCommerce theme for WordPress. The theme is on the market in 2 different child skins to choose from. The themes listed below are best suited for vogue and clothing, bodily merchandise, digital products, magnificence & cosmetic, digital products, and so forth. About WooCommerce Plugin
The Gravity Varieties Product Add-ons extension offers you full management over how the display of the prices within the store should seem. You should use the default worth displays that WooCommerce will display, or within the case of a configurable product containing pricing fields, set the value to something akin to "As Low as $1000". I apologize upfront but I can’t assist you with your customized implementations through the comments or WordPress help forums. I'm open to contract work though. Please contact me when you’d like to hire me.
Moreover, over the previous few weeks of testing, we corresponded with the developer to make some suggestions, and several other of those suggestions have been implemented in the present model of the plugin. These items make the plugin more usable for nonprofits, and we predict you’ll perceive why whenever you see the video. Extra Details on the Reserving and Appointments Plugin Whether or not you are working a superstore promoting hundreds of merchandise, or a just a sole trader promoting hand made arts and crafts WooCommerce has you coated. You can read all about WooCommerce' features on the WooCommerce web site
Instead of working like an unbiased, internet-hosted cart that retains its own suite of themes resembling Shopify or BigCommerce , as talked about above WooCommerce is a WordPress plugin that can theoretically be built-in into any WordPress theme. But to stop headaches like out of whack margins and formatting I counsel selecting a WooTheme on your store since many are tailored for WooCommerce and require little tweaking. Word that doing so might cost you around $100 except you opt for of three free skins Order achievement for online stores using WooCommerce is out there with this integration, launched by SkyVerge and supported by WooThemes, the developers behind WooCommerce. Features
It’s possible to make any of the WooCommerce checkout fields required or not required. The essential factor to notice is that "true" makes the field required and "false" makes it not required. As soon as the plugin was launched there were many requests to make it possible to book property, so now in version 1.1 we even have the option to guide a begin and finish date which is appropriate for the rental of property, resort reserving, lodging, and so forth. Though that is only version 1.1, it already feels prefer it’s a mature plugin and this augurs very effectively for the way forward for this add-on.