Sinc filter
In signal processing, a sinc filter can refer to either a sinc-in-time filter whose impulse response is a sinc function and whose frequency response is rectangular, or to a sinc-in-frequency filter whose impulse response is rectangular and whose frequency response is a sinc function. Calling them according to which domain the filter resembles a sinc avoids confusion. If the domain is unspecified, sinc-in-time is often assumed, or context hopefully can infer the correct domain.
Sinc-in-time
Sinc-in-time is an ideal filter that removes all frequency components above a given cutoff frequency, without attenuating lower frequencies, and has linear phase response. It may thus be considered a brick-wall filter or rectangular filter.
Its impulse response is a sinc function in the time domain:
while its frequency response is a rectangular function:
where (representing its bandwidth) is an arbitrary cutoff frequency.
Its impulse response is given by the
where sinc is the normalized sinc function.
Brick-wall filters
An idealized electronic filter with full transmission in the pass band, complete attenuation in the stop band, and abrupt transitions is known colloquially as a "brick-wall filter" (in reference to the shape of the transfer function). The sinc-in-time 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 BL has impulse response and transfer function given by:
The band-pass filter with lower band edge BL and upper band edge BH is just the difference of two such sinc-in-time filters (since the filters are zero phase, their magnitude responses subtract directly):[1]
The high-pass filter with lower band edge BH is just a transparent filter minus a sinc-in-time filter, which makes it clear that the Dirac delta function is the limit of a narrow-in-time sinc-in-time filter:
Unrealizable
As the sinc-in-time filter has infinite impulse response in both positive and negative time directions, it is
Sinc-in-time filters must be approximated for real-world (non-abstract) applications, typically by
Stability
The sinc filter is not 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(2πBt)u(t), a sine wave starting at time 0, at the cutoff frequency.
Frequency-domain sinc
The simplest implementation of a sinc-in-frequency filter uses a
This filter can be used for crude but fast and easy
A group averaging filter processing samples has transmission zeroes evenly-spaced by with the lowest zero at and the highest zero at (the Nyquist frequency). Above the Nyquist frequency, the frequency response is mirrored and then is repeated periodically above forever.
The
An inverse sinc filter may be used for
See Window function § Rectangular window for application of the sinc kernel as the simplest windowing function.
See also
References
- ISBN 978-0-521-85478-8.
- ^ Verbeure, Tom (2020-09-30). "An Intuitive Look at Moving Average and CIC Filters". Electronics etc…. Archived from the original on 2023-04-02. Retrieved 2023-08-24.
- ^ "APPLICATION NOTE 3853: Equalizing Techniques Flatten DAC Frequency Response". Analog Devices. 2012-08-20. Archived from the original on 2023-09-18. Retrieved 2024-01-02.