Jordan curve theorem

2013-10-11T14:27:04Z

86.59.154.11: /* History and further proofs */ link to Béla Kerékjártó

[[File:Sinc function (normalized).svg|thumb|The normalized [[sinc function]], the [[impulse response]] of the sinc filter.]]
[[File:Rectangular function.svg|thumb|The [[rectangular function]], the [[frequency response]] of the sinc filter.]]

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]].

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.

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]].

In mathematical terms, the desired frequency response is the [[rectangular function]]:

:<math>H(f) = \mathrm{rect} \left( \frac{f}{2B} \right)</math>

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:

: <math>
\begin{align}
h(t) = \mathcal{F}^{-1} \{ H (f)\} & = 2B \frac{\sin(2\pi Bt)}{2\pi Bt} \\
& = 2B \, \mathrm{sinc}(2 B t)
\end{align}
</math>

the normalized [[sinc function]].

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.

==Brick-wall filters==
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.

The lowpass filter with brick-wall cutoff at frequency ''B''''L'' has impulse response and transfer function given by:

:<math> h_{LPF}(t) = 2B_L \, \mathrm{sinc}\left(2B_L t\right)</math>

:<math> H_{LPF}(f) = \mathrm{rect}\left( \frac{f}{2B_L} \right).</math>

The band-pass filter with lower band edge ''B''''L'' and upper band edge ''B''''H'' is just the difference of two such sinc filters (since the filters are zero phase, their magnitude responses subtract directly):<ref>{{cite book
| title = Practical signal processing
| author = Mark Owen
| publisher = Cambridge University Press
| year = 2007
| isbn = 978-0-521-85478-8
| page = 81
| 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
}}</ref>

:<math> h_{BPF}(t) = 2B_H \, \mathrm{sinc}\left(2B_H t\right) - 2B_L \, \mathrm{sinc}\left(2B_L t\right)</math>

:<math> H_{BPF}(f) = \mathrm{rect}\left( \frac{f}{2B_H} \right) - \mathrm{rect}\left( \frac{f}{2B_L} \right).</math>

The high-pass filter with lower band edge ''B''''H'' 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:

:<math> h_{HPF}(t) = \delta(t) - 2B_H \, \mathrm{sinc}\left(2B_H t\right)</math>

:<math> H_{HPF}(f) = 1 - \mathrm{rect}\left( \frac{f}{2B_H} \right).</math>

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}}

==Frequency-domain sinc==
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.

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>

==Stability==

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.

==See also==
* [[Lanczos resampling]]
* [[Aliasing]]
* [[Anti-aliasing filter]]

==References==
{{reflist}}

==External links==
* [http://www.audioholics.com/education/audio-formats-technology/brick-wall-digital-filters-and-phase-deviations Brick Wall Digital Filters and Phase Deviations]
* [http://www.sweetwater.com/expert-center/glossary/t--BrickwallFilter Brick-wall filters]

{{DEFAULTSORT:Sinc Filter}}
[[Category:Signal processing]]
[[Category:Digital signal processing]]
[[Category:Filter theory]]
[[Category:Filter frequency response]]

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Jordan curve theorem