Editing Anti-aliasing filter

{{Short description|Mathematical transformation reducing the damage caused by aliasing}}
{{Use mdy dates|date=September 2021}}
{{Lead too short|date=November 2020}}
{{Use American English|date = March 2019}}
{{more citations needed|date=June 2023}}

An '''anti-aliasing filter''' ('''AAF''') is a [[filter (signal processing)|filter]] used before a [[Sampling (signal processing)|signal sampler]] to restrict the [[Bandwidth (signal processing)|bandwidth]] of a [[signal]] to satisfy the [[Nyquist–Shannon sampling theorem]] over the [[frequency band|band of interest]]. Since the theorem states that unambiguous reconstruction of the signal from its samples is possible when the [[Spectral density|power of frequencies]] above the [[Nyquist frequency]] is zero, a [[brick wall filter]] is an idealized but impractical AAF.{{efn|Brick-wall filters that run in realtime are not physically realizable as they have infinite latency and infinite [[Order (differential equation)|order]].}} A practical AAF makes a trade off between reduced [[Bandwidth (signal processing)|bandwidth]] and increased [[aliasing]]. A practical anti-aliasing filter will typically permit some aliasing to occur or attenuate or otherwise distort some in-band frequencies close to the Nyquist limit. For this reason, many practical systems sample higher than would be theoretically required by a perfect AAF in order to ensure that all frequencies of interest can be reconstructed, a practice called [[oversampling]].

== Optical applications ==
{{See also|Spatial anti-aliasing}}

{{multiple image
| image1 = Moire_pattern_of_bricks_small.jpg
| image2 = Moire_pattern_of_bricks.jpg
| width = 189 <.-- integer fraction (1/4) of original width (756) to reduce moire -->
| footer = Simulated photographs of a brick wall without (left) and with (right) an optical low-pass filter
| total_width = 300
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[[File:Lowpassfilter - Copy.jpg|alt=Lowpassfilter|thumb|Optical low-pass filter (OLPF)|180x180px]]
In the case of [[optical]] image sampling, as by [[image sensor]]s in [[digital camera]]s, the anti-aliasing filter is also known as an '''optical low-pass filter''' ('''OLPF'''), '''blur filter''', or '''AA filter'''.  The mathematics of sampling in [[Plane (mathematics)|two spatial dimensions]] is similar to the mathematics of [[time-domain]] sampling, but the filter implementation technologies are different.

The typical implementation in [[digital camera]]s is two layers of [[birefringent]] material such as [[lithium niobate]], which spreads each optical point into a cluster of four points.<ref>{{cite book
| author = Adrian Davies and Phil Fennessy
| title = Digital imaging for photographers
| edition = Fourth
| publisher = Focal Press
| year = 2001
| url = https://books.google.com/books?id=wsxk03-gceUC&q=anti-aliasing+lithium-niobate&pg=PA30
| isbn = 0-240-51590-0}}</ref> The choice of spot separation for such a filter involves a tradeoff among sharpness, aliasing, and fill factor (the ratio of the active refracting area of a [[microlens array]] to the total contiguous area occupied by the array).  In a [[monochrome]] or [[three-CCD]] or [[Foveon X3]] camera, the microlens array alone, if near 100% effective, can provide a significant anti-aliasing function,<ref>{{cite book
 | chapter = Tradeoffs between aliasing and MTF
 | title = Proceedings of the Electro-Optical Systems Design Conference – 1974 West International Laser Exposition – San Francisco, Calif., November 5-7, 1974
 | author = S. B. Campana and D. F. Barbe
 | journal = Electro-Optical Systems Design Conference - 1974 West International Laser Exposition
 | publisher = Chicago: Industrial and Scientific Conference Management, Inc
 | year = 1974
 | pages = 1–9
 | bibcode = 1974eosd.conf....1C
 }}</ref>
while in color filter array (e.g. [[Bayer filter]]) cameras, an additional filter is generally needed to reduce aliasing to an acceptable level.<ref>
{{cite book
 | author = Brian W. Keelan
 | title = Handbook of Image Quality: Characterization and Prediction
 | publisher = Marcel–Dekker
 | year = 2004
 | url = https://books.google.com/books?id=E45MTZn17gEC&q=spot+separation+in+optical+anti-aliasing+filters&pg=RA1-PA388
 | isbn = 0-8247-0770-2
 }}</ref><ref>
{{cite book
 | title = Scientific photography and applied imaging
 | author = Sidney F. Ray
 | publisher = Focal Press
 | year = 1999
 | isbn = 978-0-240-51323-2
 | page = 61
 | url = https://books.google.com/books?id=AEFPNfghI3QC&q=aliasing+fill-factor&pg=PA61
 }}</ref><ref>
{{cite book
 | title = New Acquisition Techniques for Real Objects and Light Sources in Computer Graphics
 | author = Michael Goesele
 | publisher = Books on Demand
 | year = 2004
 | isbn = 978-3-8334-1489-3
 | page = 34
 | url = https://books.google.com/books?id=ZTJJ8QzNv1wC&q=aliasing+fill-factor+100%25+bayer&pg=PA34
 }}</ref>

Alternative implementations include the [[Pentax K-3]]'s anti-aliasing filter, which applies small [[vibration]]s to the sensor element.<ref>{{cite web |url=http://www.dpreview.com/products/Pentax/slrs/pentax_k3 |title=Pentax K-3 |access-date=November 29, 2013}}</ref>{{advert inline|date=June 2023}}

== Audio applications ==
Anti-aliasing filters are used at the input of an [[analog-to-digital converter]]. Similar filters are used as [[reconstruction filter]]s at the output of a [[digital-to-analog converter]]. In the latter case, the filter prevents imaging, the reverse process of aliasing where in-band frequencies are mirrored out of band.

=== Oversampling ===
{{main|Oversampling}}
With [[oversampling]], a higher intermediate digital sample rate is used, so that a nearly ideal [[digital filter]] can [[selectivity (radio)|sharply]] [[Cutoff frequency|cut off]] aliasing near the original low [[Nyquist frequency]] and give better [[phase response]], while a much simpler [[analog filter]] can stop frequencies above the new higher Nyquist frequency.  Because analog filters have relatively high cost and limited performance, relaxing the demands on the analog filter can greatly reduce both aliasing and cost.  Furthermore, because some [[Noise (signal processing)|noise]] is averaged out, the higher sampling rate can moderately improve [[signal-to-noise ratio]].

A signal may be intentionally sampled at a higher rate to reduce the requirements and distortion of the anti-alias filter. For example, compare [[CD audio]] with [[high-resolution audio]]. CD audio filters the signal to a passband edge of 20&nbsp;kHz, with a stopband Nyquist frequency of 22.05&nbsp;kHz and sample rate of 44.1&nbsp;kHz. The narrow 2.05&nbsp;kHz transition band requires a compromise between filter complexity and performance. High-resolution audio uses a higher sample rate, providing both a higher passband edge and larger transition band, which allows better filter performance with reduced aliasing, reduced attenuation of higher audio frequencies and reduced time and phase domain signal distortion.<ref name=AD-oversample>{{cite web|last1=Kester|first1=Walt|title=Oversampling Interpolating DACs|url=https://www.analog.com/media/en/training-seminars/tutorials/MT-017.pdf|publisher=Analog Devices|access-date=17 January 2015}}</ref><ref>{{cite magazine |magazine=[[Audioholics]] |url=http://www.audioholics.com/education/audio-formats-technology/upsampling-vs-oversampling-for-digital-audio |title=Upsampling vs. Oversampling for Digital Audio |author=Nauman Uppal |date=30 August 2004 |access-date=6 October 2012}}</ref>{{Failed verification|date=April 2023|reason=The paragraph implies that this is an intrinsic effect of storing audio at higher sampling rates ("high-resolution audio" vs. "CD audio"), but the references either describe the application of oversampling to CD audio at the DAC stage, or focus on the performance of analog filters, and therefore do not seem to support an inherent need to store the audio at that higher sampling rate.}} <ref>{{cite web|last1=Story|first1=Mike|title=A Suggested Explanation For (Some Of) The Audible Differences Between High Sample Rate And Conventional Sample Rate Audio Material |date=September 1997|url=http://sdg-master.com:80/lesestoff/aes97ny.pdf |publisher=dCS Ltd|archive-date=28 November 2009|archive-url=https://web.archive.org/web/20091128021651/http://sdg-master.com:80/lesestoff/aes97ny.pdf|url-status=live}}</ref><ref>{{cite web|last1=Lavry|first1=Dan|title=Sampling, Oversampling, Imaging and Aliasing - a basic tutorial|date=1997|url= http://lavryengineering.com/pdfs/lavry-sampling-oversampling-imaging-aliasing.pdf|publisher=Lavry Engineering|archive-date=21 June 2015|archive-url=https://web.archive.org/web/20150621202254/http://lavryengineering.com/pdfs/lavry-sampling-oversampling-imaging-aliasing.pdf|url-status=live}}</ref>

=== Bandpass signals ===
{{See also|Undersampling}}
Often, an anti-aliasing filter is a [[low-pass filter]]; this is not a requirement, however.  Generalizations of the Nyquist–Shannon sampling theorem allow sampling of other band-limited [[passband]] signals instead of [[baseband]] signals.

For signals that are bandwidth limited, but not centered at zero, a [[band-pass filter]] can be used as an anti-aliasing filter.  For example, this could be done with a [[single-sideband modulation|single-sideband modulated]] or [[frequency modulated]] signal.  If one desired to sample an [[FM radio]] broadcast centered at 87.9&nbsp;MHz and bandlimited to a 200&nbsp;kHz band, then an appropriate anti-alias filter would be centered on 87.9&nbsp;MHz with 200&nbsp;kHz bandwidth (or [[passband]] of 87.8&nbsp;MHz to 88.0&nbsp;MHz), and the sampling rate would be no less than 400&nbsp;kHz, but should also satisfy other constraints to prevent [[aliasing]].{{specify|date=April 2021}}

=== Signal overload ===
It is very important to avoid input signal overload when using an anti-aliasing filter. If the signal is strong enough, it can cause [[Clipping (signal processing)|clipping]] at the [[analog-to-digital converter]], even after filtering. When [[distortion]] due to clipping occurs after the anti-aliasing filter, it can create components outside the [[passband]] of the anti-aliasing filter; these components can then alias, causing the reproduction of other non-[[harmonic]]ally related frequencies.<ref>{{citation |url=https://tech.ebu.ch/docs/techreview/trev_310-lund.pdf |title=Level and distortion in digital broadcasting |access-date=2021-05-11}}</ref>

== Notes ==
{{Notelist}}

== References ==
{{Reflist}}

{{DSP}}

[[Category:Digital signal processing]]
[[Category:Linear filters]]
[[Category:Electronic filter applications]]
[[Category:Anti-aliasing]]