Sinc filter: Information from Answers.com

The normalized sinc function, the impulse response of the sinc filter.

The rectangular function, the frequency response of the sinc filter.

In signal processing, a sinc filter is an idealized filter that removes all frequency components above a given bandwidth, leaves the low frequencies alone, and has linear phase. 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) has an infinite delay, but it is commonly found in conceptual demonstrations or proofs, such as the sampling theorem and the Whittaker–Shannon interpolation formula.

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

$H(f) = \mathrm{rect} \left( \frac{f}{2B} \right)$

where $B\,$ is an arbitrary cutoff frequency (aka bandwidth) (in Hz). The impulse response of such a filter is given by the inverse Fourier transform:

$h(t) = \mathcal{F}^{-1} \{ H \}(t)$	$= 2B \frac{\sin(2\pi Bt)}{2\pi Bt}$
	$= 2B \mathrm{sinc}(2 B t)\,$ , in terms of 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 windowed sinc filter is often used instead.

1 Brick-wall filters
2 Frequency-domain sinc
3 References
4 See also
5 External links

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 filters and high-pass filters are easily constructed.

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

$h_{LPF}(t) = 2B_L \mathrm{sinc}\left(2B_L t\right)$

$H_{LPF}(f) = \mathrm{rect}\left( \frac{f}{2B_L} \right).$

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):^[1]

$h_{BPF}(t) = 2B_H \mathrm{sinc}\left(2B_H t\right) - 2B_L \mathrm{sinc}\left(2B_L t\right)$

$H_{BPF}(f) = \mathrm{rect}\left( \frac{f}{2B_H} \right) - \mathrm{rect}\left( \frac{f}{2B_L} \right).$

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:

$h_{LPF}(t) = \delta(t) - 2B_H \mathrm{sinc}\left(2B_H t\right)$

$H_{LPF}(f) = 1 - \mathrm{rect}\left( \frac{f}{2B_H} \right).$

Brick-wall filters that run in realtime are not physically realizable as they have infinite latency (i.e., its compact support in the Fourier transform forces its time response to be 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]}

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 filters, among many other applications, are almost universally used for decimating sigma–delta ADCs, as they are easy to implement and nearly optimum for this use.^[2]

References

^ Mark Owen (2007). Practical signal processing. Cambridge University Press. p. 81. ISBN 9780521854788. http://books.google.com/books?id=lx-tqq-MkK0C&pg=RA1-PA81&dq=sinc-function+high-pass+band-pass+difference&lr=&as_brr=3&ei=5GxySsOSK4zSkATn_4Vx#v=onepage&q=sinc-function%20high-pass%20band-pass%20difference&f=false.
^ Chou, W.; Meng, T.H.; Gray, R.M. (1990). "Time domain analysis of sigma delta modulation". Acoustics, Speech, and Signal Processing 3: 1751–1754. doi:10.1109/ICASSP.1990.115820.

External links

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Sinc filter

Contents

Brick-wall filters

Frequency-domain sinc

References

See also

External links