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1648:: 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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1528:. That is typically approximated by filtering the original signal to attenuate high frequency components before it is sampled. These attenuated high frequency components still generate low-frequency aliases, but typically at low enough amplitudes that they do not cause problems. A filter chosen in anticipation of a certain sample frequency is called an
63:
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1748:
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.
258:. 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.
325:, 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 exploit aliasing in this way for computational efficiency. (See
90:. The speed of the "camera", moving towards the right, constantly increases at the same rate with the objects sliding to the left. Halfway through the 24-second loop, the objects appear to suddenly shift and head in the reverse direction, towards the right.
1535:
The filtered signal can subsequently be reconstructed, by interpolation algorithms, without significant additional distortion. Most sampled signals are not simply stored and reconstructed. But the fidelity of a theoretical reconstruction (via the
1757:
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 more densely than two points per
1572:(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
1506:
Illustration of 4 waveforms reconstructed from samples taken at six different rates. Two of the waveforms are sufficiently sampled to avoid aliasing at all six rates. The other two illustrate increasing distortion (aliasing) at the lower
1618:. While Tukey did significant work in factorial experiments and was certainly aware of aliasing in fractional designs, it cannot be determined whether his use of "aliasing" in signal processing was consciously inspired by such designs.
204:
is performed by a display or printer device, and by the eyes and the brain. If the image data is processed incorrectly during sampling or reconstruction, the reconstructed image will differ from the original image, and an alias is seen.
752:{\displaystyle \sin(2\pi (f+Nf_{\rm {s}})t+\phi )=\left\{{\begin{array}{ll}+\sin(2\pi (f+Nf_{\rm {s}})t+\phi ),&f+Nf_{\rm {s}}\geq 0\\-\sin(2\pi |f+Nf_{\rm {s}}|t-\phi ),&f+Nf_{\rm {s}}<0\\\end{array}}\right.}
473:
of the samples produces equally strong responses at all those frequencies. Without collateral information, the frequency of the original function is ambiguous. So the functions and their frequencies are said to be
1702:
226:
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 32,000
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412:
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 (for example, with a
243:(DAC). The high frequencies in the analog signal will appear as lower frequencies (wrong alias) in the recorded digital sample and, hence, cannot be reproduced by the DAC. To prevent this, an
81:
400:
Fourier transform of just the available samples. The presence of two components means the samples can fit at least two different sinusoids, one of which is the true frequency (upper-right).
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189:
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Animation depicts a sequence of sinusoids, each with a higher frequency than the previous ones. These "true" signals are also being sampled (blue dots) at a constant frequency/rate,
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techniques avoid such poor pixelizations. Aliasing can be caused either by the sampling stage or the reconstruction stage; these may be distinguished by calling sampling aliasing
277:, 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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and Tukey in 1958. In their preface to the Dover reprint of this paper, they point out that the idea of aliasing had been illustrated graphically by Stumpf ten years prior.
254:, 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
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The figures below offer additional depictions of aliasing, due to sampling. A graph of amplitude vs frequency (not time) for a single sinusoid at frequency
125:
from samples which causes the reconstructed signal to differ from the original continuous signal. Aliasing that occurs in signals sampled in time, for instance in
1190:
exceeds the
Nyquist frequency, the reconstruction matches the actual waveform (upper left frame). After that, it is the low frequency alias of the upper frame.
1032:
Aliasing matters when one attempts to reconstruct the original waveform from its samples. The most common reconstruction technique produces the smallest of the
19:
This article is about aliasing in signal processing, including computer graphics. For accessing the same data using different names in computer programming, see
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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
1737:(non-aliased) sawtooths and aliased sawtooths and the sampling rate is 22050 Hz. The bandlimited sawtooths are synthesized from the sawtooth waveform's
1583:
The first written use of the terms "alias" and "aliasing" in signal processing appears to be in a 1949 unpublished Bell
Laboratories technical memorandum by
1667:", 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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2030:
1302:
Fig.4: The
Fourier transform of music sampled at 44,100 samples/sec exhibits symmetry (called "folding") around the Nyquist frequency (22,050 Hz).
2101:
Tukey, John W.; Hamming, R. W. (1984) . "Mathematics 596: An introduction to the frequency analysis of time series". In
Brillinger, David R. (ed.).
1524: is met for the highest frequency component of the original signal, then it is met for all the frequency components, a condition called the
164:
should then be used when restoring the sampled signal to the continuous domain or converting a signal from a lower to a higher sampling rate. For
391:
Fourier transform of the sinusoid (not the samples). The single non-zero component, depicting the actual frequency, means there is no ambiguity.
1821:
295:, has no upper bound. Some amount of aliasing always occurs when such functions are sampled. Functions whose frequency content is bounded (
2180:
1708:
440 Hz bandlimited, 440 Hz aliased, 880 Hz bandlimited, 880 Hz aliased, 1760 Hz bandlimited, 1760 Hz aliased
1537:
765:
1263:). No matter what function we choose to change the amplitude vs frequency, the graph will exhibit symmetry between 0 and
1602:
The 1949 Bell technical report refers to aliasing as though it is a well-known concept, but does not offer a source for the term.
1348:
Two complex sinusoids, colored gold and cyan, that fit the same sets of real and imaginary sample points when sampled at the rate (
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1298:
2176:
1914:
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Aliasing occurs whenever the use of discrete elements to capture or produce a continuous signal causes frequency ambiguity.
1244:) of the 4 dots if we were to adjust the frequency and amplitude of the sinusoid along the solid red segment (between
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1525:
286:
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160:(AAF) to the input signal before sampling and when converting a signal from a higher to a lower sampling rate. Suitable
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420:). Understanding what aliasing does to the individual sinusoids is useful in understanding what happens to their sum.
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Using the same samples (now in orange), the default reconstruction algorithm produces the lower-frequency sinusoid.
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1964:
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1170:. The lower left frame of Fig.2 depicts the typical reconstruction result of the available samples. Until
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269:) per degree or per radian, or samples per mm in the focal plane of the camera. Audio signals are sampled (
1962:(1958). "The measurement of power spectra from the point of view of communications engineering - Part I".
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240:
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Jenkins, G. M.; Priestley, M. B. (1957). "Discussion (Symposium on
Spectral Approach to Time Series)".
1344:
838:. For example, a snapshot of the lower right frame of Fig.2 shows a component at the actual frequency
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2252:
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of amplitude varying with frequency. The dashed red lines are the corresponding paths of the aliases.
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303:, the original function can, in theory, be perfectly reconstructed from the infinite set of samples.
173:
299:) have an infinite duration in the time domain. If sampled at a high enough rate, determined by the
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Fig.5: Graph of frequency aliasing, showing folding frequency and periodicity. Frequencies above
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861:
250:
In video or cinematography, temporal aliasing results from the limited frame rate, and causes the
48:
This full-sized image shows what a properly sampled image of a brick wall should look like with a
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1776:
1035:
971:
1930:
Tukey, John W.; Hamming, R. W. (1984) . "Measuring noise color". In
Brillinger, David R. (ed.).
1415:
is necessary to distinguish them. In that case, the frequencies of the aliases are given by just
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261:
Like the video camera, most sampling schemes are periodic; that is, they have a characteristic
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86:
The motion of the 'camera' at a constant shutter speed creates temporal aliasing known as the
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The qualitative effects of aliasing can be heard in the following audio demonstration. Six
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Sometimes aliasing is used intentionally on signals with no low-frequency content, called
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1591:. That paper includes an example of frequency aliasing dating back to 1922. The first
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1992:
The
Measurement of Power Spectra from the Point of View of Communications Engineering
1935:
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1866:
1855:
ACM SIGGRAPH International
Conference on Computer Graphics and Interactive Techniques
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Actual signals have a finite duration and their frequency content, as defined by the
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1240: would look like the 4 black dots in Fig.3. The red lines depict the paths (
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Spatial aliasing, particular of angular frequency, can occur when reproducing a
62:
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1771:
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1603:
413:
2198:
2049:
Finney, D. J. (1945). "The fractional replication of factorial arrangements".
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be the unique minimum. A necessary and sufficient condition for that is
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for this sampling rate) will cause aliasing when the music is reproduced by a
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1556:. When the receiver shifts multiple signals down to lower frequencies, from
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is used to remove components above the
Nyquist frequency prior to sampling.
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110:
2125:
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Fig.3: The black dots are aliases of each other. The solid red line is an
265:
in time or in space. Digital cameras provide a certain number of samples (
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2139:, "Beamwidth and useable bandwidth of delay-steered microphone arrays",
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1987:
1959:
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by La Vida Leica, discusses its purpose and effect on recorded images
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1490: to 0). Consequently, complex sinusoids do not exhibit
1308:
345:
193:
80:
68:
When the resolution is reduced, aliasing appears in the form of a
1781:
831:{\displaystyle f_{_{N}}(f)\triangleq \left|f+Nf_{\rm {s}}\right|}
762:
we can write all the alias frequencies as positive values:
1729:
are played in succession, with the first two sawtooths having a
1568:, an unwanted signal, from an RF frequency equally far from the
1273: Folding is often observed in practice when viewing the
1995:
266:
232:
1610:
credit Tukey with introducing it in this context, though an
105:
is the overlapping of frequency components resulting from a
1540:) is a customary measure of the effectiveness of sampling.
1502:
746:
1839:
1822:
Nyquist–Shannon sampling theorem § Critical frequency
2164:
Physically Based
Rendering: From Theory to Implementation
1357:) indicated by the grid lines. The case shown here is:
1644:
This aliasing is visible in images such as posters with
1552:
evolved from radio engineering because of the action of
1976:
2217:
Interactive examples demonstrating the aliasing effect
1948:
212:
observed in a poorly pixelized image of a brick wall.
1903:
Multirate Signal Processing for Communication Systems
1595:
use of the term "aliasing" in this context is due to
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1038:
974:
929:
909:
864:
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1982:
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1762:, or the wave arrival direction becomes ambiguous.
2031:Journal of the Royal Statistical Society, Series B
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231:(Hz), any frequency components at or above 16,000
192:Dots in the sky due to spatial aliasing caused by
478:of each other. Noting the trigonometric identity
2473:
2027:
340:
2069:
137:. Aliasing in spatially sampled signals (e.g.,
2094:
1074:frequencies. So it is usually important that
968:decreases. The point at which they are equal
2251:
2237:
2011:Grundlagen und Methoden der Periodenforschung
1633:or sound field with discrete elements, as in
2100:
2076:Tukey, John W. (1992). Cox, David R. (ed.).
1929:
1923:
1847:Reconstruction filters in computer-graphics
1663:glasses (in 3D films the effect is called "
1614:had been introduced a few years earlier in
2244:
2230:
2119:
152:Aliasing is generally avoided by applying
23:. For aliasing in statistical design, see
1501:
1343:
1307:
1297:
1283:
1277:of real-valued samples, such as Fig.4..
1208: and some of its aliases at
344:
280:
187:
2130:
2105:. Vol. 1. Wadsworth. p. 571.
1833:
1655:on viewer movement in 2D images and in
1538:Whittaker–Shannon interpolation formula
2474:
2126:The (New) Stanford Light Field Archive
2048:
2008:
1897:
1879:
423:When sampling a function at frequency
208:An example of spatial aliasing is the
2225:
2075:
2042:
2002:
1934:. Vol. 1. Wadsworth. p. 5.
1891:
168:, the types of anti-aliasing include
2103:The Collected Works of John W. Tukey
2078:The Collected Works of John W. Tukey
1932:The Collected Works of John W. Tukey
1752:
1399:
2200:Aliasing by a sampling oscilloscope
2161:; Humphreys, Greg. (28 June 2010).
1887:"Time Filter Technical Explanation"
1621:
1543:
1497:
1336:whose value is given by this graph.
442:), the following functions of time
331:Nyquist rate (relative to sampling)
306:
13:
2151:
2063:10.1111/j.1469-1809.1943.tb02333.x
1857:. Vol. 22. pp. 221–228.
1682:
1480: also increases (from
1015:is an axis of symmetry called the
817:
730:
687:
633:
592:
524:
200:When a digital image is viewed, a
14:
2498:
2207:by Tektronix Application Engineer
2192:
1741:such that no harmonics above the
923:increases during the animation,
450:yield identical sets of samples:
2282:Nyquist–Shannon sampling theorem
1716:Problems playing this file? See
1698:
1675:
1670:
1407:are waveforms whose samples are
287:Nyquist–Shannon sampling theorem
79:
61:
41:
25:Aliasing (factorial experiments)
2368:Discrete-time Fourier transform
1157:{\displaystyle f_{s}/2>|f|,}
858:and another component at alias
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183:
170:fast approximate anti-aliasing
1:
2313:Statistical signal processing
1965:Bell System Technical Journal
1827:
1612:analogous concept of aliasing
341:Sampling sinusoidal functions
220:and reconstruction aliasing
196:resized to a lower resolution
1616:fractional factorial designs
1450: increases from
961:{\displaystyle f_{_{-1}}(f)}
896:{\displaystyle f_{_{-1}}(f)}
327:Sampling (signal processing)
7:
2211:Anti-Aliasing Filter Primer
2183:Sampling and reconstruction
1765:
1446: Therefore, as
1067:{\displaystyle f_{_{N}}(f)}
1008:{\displaystyle (f=f_{s}/2)}
275:analog-to-digital converter
241:digital-to-analog converter
10:
2503:
2362:Discrete Fourier transform
2339:Matched Z-transform method
2080:. Vol. 7. Wadsworth.
1905:. Upper Saddle River, NJ:
1193:
310:
284:
113:. This overlap results in
18:
2482:Digital signal processing
2407:
2356:Discrete cosine transform
2321:
2290:
2259:
2253:Digital signal processing
2187:. Retrieved 3 March 2013.
1554:superheterodyne receivers
1511:When the condition
174:multisample anti-aliasing
101:and related disciplines,
2389:Post's inversion formula
2303:Digital image processing
1103:{\displaystyle f_{0}(f)}
162:reconstruction filtering
16:Signal processing effect
2298:Audio signal processing
2145:, 1985, 64, pp. 983–995
1777:Glossary of video terms
1580:of the desired signal.
123:signal is reconstructed
1694:Sawtooth aliasing demo
1687:
1548:Historically the term
1508:
1396:
1337:
1303:
1293:
1184:
1158:
1104:
1068:
1009:
962:
917:
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832:
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406:
377:
376:{\displaystyle f_{s}.}
197:
35:An example of aliasing
2009:Stumpf, Karl (1937).
1731:fundamental frequency
1686:
1505:
1411:, and the concept of
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1159:
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1010:
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348:
281:Bandlimited functions
214:Spatial anti-aliasing
191:
166:spatial anti-aliasing
158:anti-aliasing filters
2422:Anti-aliasing filter
2351:Constant-Q transform
2334:Advanced z-transform
1885:Tessive, LLC (2010).
1863:10.1145/54852.378514
1639:wave field synthesis
1530:anti-aliasing filter
1174:
1114:
1078:
1036:
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842:
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245:anti-aliasing filter
145:) is referred to as
133:, is referred to as
21:Aliasing (computing)
1899:Harris, Frederic J.
1812:Stroboscopic effect
1646:lenticular printing
467:= 0, ±1, ±2, ±3,...
131:stroboscopic effect
2379:Integral transform
2374:Impulse invariance
2346:Bilinear transform
2137:Flanagan, James L.
2051:Annals of Eugenics
1842:Netravali, Arun N.
1840:Mitchell, Don P.;
1817:Wagon-wheel effect
1688:
1509:
1413:negative frequency
1397:
1338:
1304:
1294:
1275:frequency spectrum
1180:
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471:frequency spectrum
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263:sampling frequency
256:negative frequency
252:wagon-wheel effect
229:samples per second
198:
88:wagon wheel effect
2487:Signal processing
2469:
2468:
2394:Starred transform
2384:Laplace transform
2308:Speech processing
2277:Estimation theory
2177:978-0-12-375079-2
2142:AT&T Tech. J.
1916:978-0-13-146511-4
1907:Prentice Hall PTR
1753:Direction finding
1743:Nyquist frequency
1703:
1608:Maurice Priestley
1526:Nyquist criterion
1405:Complex sinusoids
1400:Complex sinusoids
1342:
1341:
1254: and
1230: and
1183:{\displaystyle f}
1167:Nyquist condition
1025:Nyquist frequency
1018:folding frequency
916:{\displaystyle f}
851:{\displaystyle f}
293:Fourier transform
237:Nyquist frequency
135:temporal aliasing
99:signal processing
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2267:Detection theory
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1807:Spectral leakage
1802:Spectral density
1705:
1704:
1685:
1622:Angular aliasing
1570:local oscillator
1544:Historical usage
1523:
1498:Sample frequency
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1454: to
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1021:, also known as
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307:Bandpass signals
154:low-pass filters
147:spatial aliasing
83:
65:
45:
2502:
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2272:Discrete signal
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2169:Morgan Kaufmann
2154:
2152:Further reading
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2003:
1984:Blackman, R. B.
1981:
1977:
1956:Blackman, R. B.
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1949:
1942:
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1917:
1901:(August 2006).
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1880:
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1850:
1844:(August 1988).
1838:
1834:
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1768:
1755:
1723:
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1714:
1712:
1711:
1710:
1709:
1706:
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1683:
1678:
1673:
1624:
1589:Richard Hamming
1546:
1518:
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1500:
1488:
1481:
1473:
1472:
1465:
1461:
1455:
1451:
1447:
1443:
1428:
1427:
1420:
1409:complex numbers
1402:
1394:
1387:
1380:
1373:
1372:
1364:
1358:
1355:
1349:
1333:
1327:
1319:
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1255:
1251:
1245:
1238:
1231:
1227:
1220:
1216:
1209:
1206:
1199:
1196:
1175:
1172:
1171:
1146:
1138:
1127:
1121:
1117:
1115:
1112:
1111:
1085:
1081:
1079:
1076:
1075:
1047:
1044:
1043:
1039:
1037:
1034:
1033:
994:
988:
984:
973:
970:
969:
938:
935:
934:
930:
928:
925:
924:
908:
905:
904:
873:
870:
869:
865:
863:
860:
859:
843:
840:
839:
816:
815:
811:
801:
797:
777:
774:
773:
769:
767:
764:
763:
743:
742:
729:
728:
724:
713:
693:
686:
685:
681:
667:
646:
645:
632:
631:
627:
616:
591:
590:
586:
552:
548:
523:
522:
518:
489:
486:
485:
458:
451:
443:
440:
433:
430:
424:
401:
392:
383:
364:
360:
358:
355:
354:
343:
315:
309:
289:
283:
186:
95:
94:
93:
92:
91:
84:
75:
74:
73:
66:
58:
57:
46:
37:
36:
28:
17:
12:
11:
5:
2500:
2490:
2489:
2484:
2467:
2466:
2464:
2463:
2458:
2453:
2448:
2443:
2438:
2429:
2424:
2419:
2413:
2411:
2405:
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2336:
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2319:
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2300:
2294:
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2287:
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2284:
2279:
2274:
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2257:
2256:
2249:
2248:
2241:
2234:
2226:
2220:
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2214:
2208:
2194:
2193:External links
2191:
2189:
2188:
2155:
2153:
2150:
2148:
2147:
2129:
2118:
2111:
2093:
2086:
2068:
2041:
2020:
2001:
1998:. p. vii.
1975:
1947:
1940:
1922:
1915:
1890:
1878:
1871:
1831:
1829:
1826:
1825:
1824:
1819:
1814:
1809:
1804:
1799:
1794:
1789:
1784:
1779:
1774:
1772:Brillouin zone
1767:
1764:
1754:
1751:
1739:Fourier series
1727:sawtooth waves
1713:
1707:
1697:
1692:
1691:
1690:
1681:
1680:
1679:
1677:
1674:
1672:
1669:
1623:
1620:
1604:Gwilym Jenkins
1545:
1542:
1516:
1499:
1496:
1486:
1470:
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1425:
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826:
819:
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772:
760:
759:
747:
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547:
544:
541:
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532:
526:
521:
517:
514:
511:
508:
505:
502:
499:
496:
493:
456:
438:
428:
414:Fourier series
372:
367:
363:
342:
339:
311:Main article:
308:
305:
285:Main article:
282:
279:
202:reconstruction
185:
182:
143:digital images
139:moiré patterns
85:
78:
77:
76:
67:
60:
59:
52:of sufficient
47:
40:
39:
38:
34:
33:
32:
31:
15:
9:
6:
4:
3:
2:
2499:
2488:
2485:
2483:
2480:
2479:
2477:
2462:
2459:
2457:
2456:Undersampling
2454:
2452:
2451:Sampling rate
2449:
2447:
2444:
2442:
2439:
2437:
2433:
2430:
2428:
2425:
2423:
2420:
2418:
2415:
2414:
2412:
2410:
2406:
2400:
2399:Zak transform
2397:
2395:
2392:
2390:
2387:
2385:
2382:
2380:
2377:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2352:
2349:
2347:
2344:
2340:
2337:
2335:
2332:
2331:
2330:
2327:
2326:
2324:
2320:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2295:
2293:
2289:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2264:
2262:
2258:
2254:
2247:
2242:
2240:
2235:
2233:
2228:
2227:
2224:
2218:
2215:
2212:
2209:
2206:
2202:
2197:
2196:
2186:
2184:
2178:
2174:
2170:
2167:
2165:
2160:
2157:
2156:
2144:
2143:
2138:
2133:
2127:
2122:
2114:
2112:0-534-03303-2
2108:
2104:
2097:
2089:
2087:0-534-05104-9
2083:
2079:
2072:
2064:
2060:
2056:
2052:
2045:
2037:
2033:
2032:
2024:
2017:. p. 45.
2016:
2012:
2005:
1997:
1993:
1989:
1985:
1979:
1971:
1967:
1966:
1961:
1957:
1951:
1943:
1941:0-534-03303-2
1937:
1933:
1926:
1918:
1912:
1908:
1904:
1900:
1894:
1888:
1882:
1874:
1872:0-89791-275-6
1868:
1864:
1860:
1856:
1849:
1848:
1843:
1836:
1832:
1823:
1820:
1818:
1815:
1813:
1810:
1808:
1805:
1803:
1800:
1798:
1797:Sinc function
1795:
1793:
1790:
1788:
1785:
1783:
1780:
1778:
1775:
1773:
1770:
1769:
1763:
1761:
1750:
1746:
1745:are present.
1744:
1740:
1736:
1732:
1728:
1721:
1719:
1695:
1676:Audio example
1671:More examples
1668:
1666:
1662:
1658:
1654:
1649:
1647:
1642:
1640:
1636:
1632:
1627:
1619:
1617:
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1605:
1600:
1598:
1594:
1590:
1586:
1581:
1579:
1575:
1571:
1567:
1563:
1559:
1555:
1551:
1541:
1539:
1533:
1531:
1527:
1522:
1515:
1504:
1495:
1493:
1485:
1477:
1468:
1458:
1440:
1436:
1432:
1423:
1418:
1414:
1410:
1406:
1391:
1384:
1377:
1368:
1361:
1352:
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1330:
1325:
1316:
1310:
1306:
1300:
1296:
1291:
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1282:
1281:
1278:
1276:
1267:
1258:
1248:
1243:
1235:
1224:
1213:
1203:
1191:
1177:
1169:
1168:
1151:
1143:
1135:
1132:
1128:
1122:
1118:
1094:
1086:
1082:
1058:
1048:
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1040:
1030:
1028:
1027:
1026:
1020:
1019:
999:
995:
989:
985:
981:
978:
952:
942:
939:
936:
931:
910:
887:
877:
874:
871:
866:
845:
824:
812:
808:
805:
802:
798:
794:
788:
778:
775:
770:
739:
736:
725:
721:
718:
715:
710:
704:
701:
698:
682:
678:
675:
672:
664:
661:
655:
652:
649:
642:
639:
628:
624:
621:
618:
613:
607:
604:
601:
587:
583:
580:
577:
571:
568:
562:
559:
556:
549:
545:
539:
536:
533:
519:
515:
512:
509:
503:
500:
494:
491:
484:
483:
482:
481:
477:
472:
466:
462:
455:
447:
437:
427:
421:
419:
415:
411:
404:
399:
395:
390:
386:
370:
365:
361:
352:
347:
338:
336:
332:
328:
324:
323:Undersampling
320:
314:
313:Undersampling
304:
302:
298:
294:
288:
278:
276:
272:
268:
264:
259:
257:
253:
248:
246:
242:
238:
234:
230:
224:
223:
222:postaliasing.
219:
215:
211:
210:moiré pattern
206:
203:
195:
190:
181:
179:
178:supersampling
175:
171:
167:
163:
159:
155:
150:
148:
144:
140:
136:
132:
128:
127:digital audio
124:
120:
116:
112:
108:
104:
100:
89:
82:
71:
70:moiré pattern
64:
55:
51:
44:
30:
26:
22:
2446:Quantization
2441:Oversampling
2432:Nyquist rate
2427:Downsampling
2416:
2182:
2163:
2140:
2132:
2121:
2102:
2096:
2077:
2071:
2054:
2050:
2044:
2035:
2029:
2023:
2010:
2004:
1994:. New York:
1991:
1978:
1969:
1963:
1950:
1931:
1925:
1902:
1893:
1881:
1846:
1835:
1756:
1747:
1724:
1715:
1661:stereoscopic
1659:produced by
1651:The lack of
1650:
1643:
1628:
1625:
1601:
1592:
1582:
1577:
1573:
1566:heterodyning
1549:
1547:
1534:
1520:
1513:
1510:
1491:
1483:
1475:
1466:
1456:
1438:
1434:
1430:
1421:
1416:
1403:
1389:
1382:
1375:
1366:
1359:
1350:
1328:
1323:
1314:
1289:
1265:
1256:
1246:
1233:
1222:
1211:
1201:
1197:
1166:
1165:
1031:
1023:
1022:
1017:
1016:
761:
479:
475:
464:
460:
453:
445:
435:
425:
422:
408:
402:
397:
394:Lower right:
393:
388:
385:Upper right:
384:
350:
318:
316:
300:
296:
290:
260:
249:
225:
221:
217:
207:
199:
151:
146:
134:
111:Nyquist rate
102:
96:
29:
2329:Z-transform
2181:Chapter 7 (
2159:Pharr, Matt
2057:: 291–301.
1988:J. W. Tukey
1960:J. W. Tukey
1792:Sinc filter
1787:Kell factor
1735:bandlimited
1635:3D displays
1631:light field
1164:called the
432:(intervals
403:Lower left:
351:Upper left:
335:Filter bank
297:bandlimited
218:prealiasing
184:Description
107:sample rate
2476:Categories
2461:Upsampling
2322:Techniques
2291:Sub-fields
2013:. Berlin:
1828:References
1760:wavelength
1718:media help
1641:of sound.
1585:John Tukey
389:continuous
273:) with an
115:distortion
109:below the
54:resolution
2436:frequency
1972:(1): 216.
1593:published
940:−
875:−
795:≜
705:ϕ
702:−
665:π
656:
650:−
640:≥
608:ϕ
572:π
563:
540:ϕ
504:π
495:
452:{sin(2π(
418:transform
410:Sinusoids
321:signals.
301:bandwidth
271:digitized
121:when the
119:artifacts
2417:Aliasing
2409:Sampling
2038:(1): 59.
2015:Springer
1990:(1959).
1766:See also
1657:3-D film
1653:parallax
1597:Blackman
1550:aliasing
1519:/2 >
1322:have an
398:discrete
319:bandpass
194:halftone
172:(FXAA),
103:aliasing
2205:YouTube
1782:Jaggies
1492:folding
1464:
1290:example
1219:
1194:Folding
476:aliases
129:or the
2370:(DTFT)
2260:Theory
2175:
2109:
2084:
1938:
1913:
1869:
1665:yawing
1507:rates.
1419:
1326:below
903:. As
469:}. A
463:+ φ),
349:Fig.2
333:, and
267:pixels
176:, and
50:screen
2364:(DFT)
2358:(DCT)
1996:Dover
1851:(PDF)
1578:alias
1574:image
1324:alias
235:(the
2173:ISBN
2107:ISBN
2082:ISBN
1936:ISBN
1911:ISBN
1867:ISBN
1606:and
1587:and
1433:) =
1386:gold
1381:) =
1379:gold
1363:cyan
1242:loci
1232:1.6
1221:1.4
1210:0.4
1200:0.6
1136:>
737:<
454:f+Nf
396:The
387:The
2203:on
2059:doi
1859:doi
1637:or
1576:or
1564:by
1560:to
1439:N f
1334:/2,
653:sin
560:sin
492:sin
416:or
337:.)
156:or
141:in
117:or
97:In
2478::
2434:/
2179:.
2171:.
2055:12
2053:.
2036:19
2034:.
1986:;
1970:37
1968:.
1958:;
1909:.
1865:.
1853:.
1562:IF
1558:RF
1532:.
1521:f
1494:.
1476:f
1471:−1
1437:+
1431:f
1388:–
1371:−1
1365:=
1320:/2
1252:/2
1029:.
459:)
434:1/
329:,
233:Hz
180:.
149:.
2245:e
2238:t
2231:v
2185:)
2166:.
2115:.
2090:.
2065:.
2061::
1944:.
1919:.
1875:.
1861::
1720:.
1517:s
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1484:f
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1478:)
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1467:f
1462:,
1460:s
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1442:s
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1429:(
1426:N
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1417::
1393:s
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1000:2
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990:s
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979:f
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611:)
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593:s
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566:(
557:+
550:{
546:=
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429:s
426:f
371:.
366:s
362:f
72:.
56:.
27:.
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