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| [[File:Sinc function (normalized).svg|thumb|The normalized [[sinc function]], the [[impulse response]] of the sinc filter.]]
| | My name is Vicky and I am studying Biochemistry and Athletics and Physical Education at Waanrode / Belgium.<br><br>Also visit my web blog :: [http://zmyslowoinamietnie29.rus.tl/blog_455000_Kupowanie-w-sklepie-erotycznym.html 5 relations of confucianism] |
| [[File:Rectangular function.svg|thumb|The [[rectangular function]], the [[frequency response]] of the sinc filter.]]
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| In [[signal processing]], a '''sinc filter''' is an idealized [[Filter (signal processing)|filter]] that removes all frequency components above a given [[cutoff frequency]], without affecting lower frequencies, and has [[linear phase]] response. The filter's [[impulse response]] is a [[sinc function]] in the time domain, and its [[frequency response]] is a [[rectangular function]].
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| It is an "ideal" [[low-pass filter]] in the frequency sense, perfectly passing low frequencies, perfectly cutting high frequencies; and thus may be considered to be a brick-wall filter.
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| Real-time filters can only approximate this ideal, since an ideal sinc filter (aka ''rectangular filter'') is [[causal filter|non-causal]] and has an infinite delay, but it is commonly found in conceptual demonstrations or proofs, such as the [[Nyquist–Shannon sampling theorem|sampling theorem]] and the [[Whittaker–Shannon interpolation formula]].
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| In mathematical terms, the desired frequency response is the [[rectangular function]]:
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| :<math>H(f) = \mathrm{rect} \left( \frac{f}{2B} \right)</math>
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| where <math>B\,</math> is an arbitrary cutoff frequency (aka ''bandwidth''). The impulse response of such a filter is given by the [[Continuous Fourier transform#Table of important Fourier transforms|inverse Fourier transform]] of the frequency response:
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| : <math>
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| \begin{align}
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| h(t) = \mathcal{F}^{-1} \{ H (f)\} & = 2B \frac{\sin(2\pi Bt)}{2\pi Bt} \\
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| & = 2B \, \mathrm{sinc}(2 B t)
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| \end{align}
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| </math>
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| the normalized [[sinc function]].
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| As the sinc filter has infinite impulse response in both positive and negative time directions, it must be approximated for real-world (non-abstract) applications; a [[window function|windowed]] sinc filter is often used instead. Windowing and truncating a sinc filter [[Convolution kernel|kernel]] in order to use it on any practical real world data set destroys its ideal properties.
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| ==Brick-wall filters==
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| An idealized [[electronic filter]], one that has full transmission in the pass band, and complete attenuation in the stop band, with abrupt transitions, is known colloquially as a "brick-wall filter", in reference to the shape of the [[transfer function]]. The sinc filter is a brick-wall [[low-pass filter]], from which brick-wall [[band-pass filter]]s and [[high-pass filter]]s are easily constructed.
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| The lowpass filter with brick-wall cutoff at frequency ''B''<sub>''L''</sub> has impulse response and transfer function given by:
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| :<math> h_{LPF}(t) = 2B_L \, \mathrm{sinc}\left(2B_L t\right)</math>
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| :<math> H_{LPF}(f) = \mathrm{rect}\left( \frac{f}{2B_L} \right).</math>
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| The band-pass filter with lower band edge ''B''<sub>''L''</sub> and upper band edge ''B''<sub>''H''</sub> is just the difference of two such sinc filters (since the filters are zero phase, their magnitude responses subtract directly):<ref>{{cite book
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| | title = Practical signal processing
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| | author = Mark Owen
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| | publisher = Cambridge University Press
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| | year = 2007
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| | isbn = 978-0-521-85478-8
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| | page = 81
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| | url = http://books.google.com/?id=lx-tqq-MkK0C&pg=RA1-PA81&dq=sinc-function+high-pass+band-pass+difference&q=sinc-function%20high-pass%20band-pass%20difference
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| }}</ref>
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| :<math> h_{BPF}(t) = 2B_H \, \mathrm{sinc}\left(2B_H t\right) - 2B_L \, \mathrm{sinc}\left(2B_L t\right)</math>
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| :<math> H_{BPF}(f) = \mathrm{rect}\left( \frac{f}{2B_H} \right) - \mathrm{rect}\left( \frac{f}{2B_L} \right).</math>
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| The high-pass filter with lower band edge ''B''<sub>''H''</sub> is just a transparent filter minus a sinc filter, which makes it clear that the [[Dirac delta function]] is the limit of a narrow-in-time sinc filter:
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| :<math> h_{HPF}(t) = \delta(t) - 2B_H \, \mathrm{sinc}\left(2B_H t\right)</math>
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| :<math> H_{HPF}(f) = 1 - \mathrm{rect}\left( \frac{f}{2B_H} \right).</math>
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| Brick-wall filters that run in realtime are not physically realizable as they have infinite latency (i.e., its [[compact support]] in the [[frequency domain]] forces its time response not to have compact support meaning that it is ever-lasting) and infinite order (i.e., the response cannot be expressed as a [[linear differential equation]] with a finite sum), but approximate implementations are sometimes used and they are frequently called brick-wall filters.{{Citation needed|date=May 2009}}
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| ==Frequency-domain sinc==
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| The name "sinc filter" is applied also to the filter shape that is rectangular in time and a sinc function in frequency, as opposed to the ideal low-pass sinc filter, which is sinc in time and rectangular in frequency. In case of confusion, one may refer to these as '''sinc-in-frequency''' and '''sinc-in-time,''' according to which domain the filter is sinc in.
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| Sinc-in-frequency [[Cascaded integrator-comb|CIC]] filters, among many other applications, are almost universally used for [[Decimation (signal processing)|decimating]] [[Delta-sigma modulation|delta-sigma]] [[Analog-to-digital converter|ADCs]], as they are easy to implement and nearly optimal for this use.<ref>{{cite journal | title=Time domain analysis of sigma delta modulation | author=Chou, W.; Meng, T.H.; Gray, R.M. | journal=Acoustics, Speech, and Signal Processing | year=1990 |pages=1751–1754 | volume=3 | doi=10.1109/ICASSP.1990.115820 }}</ref>
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| ==Stability==
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| The sinc filter is not [[BIBO stability|bounded-input–bounded-output (BIBO) stable]]. That is, a bounded input can produce an unbounded output, because the integral of the absolute value of the sinc function is infinite. A bounded input that produces an unbounded output is sgn(sinc(''t'')). Another is sin(2{{pi}}''Bt'')u(''t''), a sine wave starting at time 0, at the cutoff frequency.
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| ==See also==
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| * [[Lanczos resampling]]
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| * [[Aliasing]]
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| * [[Anti-aliasing filter]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://www.audioholics.com/education/audio-formats-technology/brick-wall-digital-filters-and-phase-deviations Brick Wall Digital Filters and Phase Deviations]
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| * [http://www.sweetwater.com/expert-center/glossary/t--BrickwallFilter Brick-wall filters]
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| {{DEFAULTSORT:Sinc Filter}}
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| [[Category:Signal processing]]
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| [[Category:Digital signal processing]]
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| [[Category:Filter theory]]
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| [[Category:Filter frequency response]]
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My name is Vicky and I am studying Biochemistry and Athletics and Physical Education at Waanrode / Belgium.
Also visit my web blog :: 5 relations of confucianism