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 used before a signal sampler to restrict the bandwidth of a signal to satisfy the Nyquist–Shannon sampling theorem over the band of interest. Since the theorem states that unambiguous reconstruction of the signal from its samples is possible when the 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.}} A practical AAF makes a trade off between reduced 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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In the case of optical image sampling, as by image sensors in digital cameras, the anti-aliasing filter is also known as an optical low-pass filter (OLPF), blur filter, or AA filter. The mathematics of sampling in two spatial dimensions is similar to the mathematics of time-domain sampling, but the filter implementation technologies are different.
The typical implementation in digital cameras is two layers of birefringent material such as lithium niobate, which spreads each optical point into a cluster of four points.{{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}} 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,{{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
}}
while in color filter array (e.g. Bayer filter) cameras, an additional filter is generally needed to reduce aliasing to an acceptable level.
{{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
{{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
{{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
}}
Alternative implementations include the Pentax K-3's anti-aliasing filter, which applies small vibrations to the sensor element.{{cite web |url=http://www.dpreview.com/products/Pentax/slrs/pentax_k3 |title=Pentax K-3 |access-date=November 29, 2013}}{{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 filters 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 sharply 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 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 kHz, with a stopband Nyquist frequency of 22.05 kHz and sample rate of 44.1 kHz. The narrow 2.05 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.{{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}}{{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}}{{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.}} {{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}}{{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}}
= 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 modulated or frequency modulated signal. If one desired to sample an FM radio broadcast centered at 87.9 MHz and bandlimited to a 200 kHz band, then an appropriate anti-alias filter would be centered on 87.9 MHz with 200 kHz bandwidth (or passband of 87.8 MHz to 88.0 MHz), and the sampling rate would be no less than 400 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 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-harmonically related frequencies.{{citation |url=https://tech.ebu.ch/docs/techreview/trev_310-lund.pdf |title=Level and distortion in digital broadcasting |access-date=2021-05-11}}
Notes
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