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:
sin ( π t ) π t {\displaystyle {\frac {\sin(\pi t)}{\pi t}}}
while its frequency response is a rectangular function:
H ( f ) = rect ( f 2 B ) = { 0 , if | f | > B , 1 2 , if | f | = B , 1 , if | f | < B , {\displaystyle H(f)=\operatorname {rect} \left({\frac {f}{2B}}\right)={\begin{cases}0,&{\text{if }}|f|>B,\\{\frac {1}{2}},&{\text{if }}|f|=B,\\1,&{\text{if }}|f|<B,\end{cases}}}
where B {\displaystyle B} (representing its bandwidth) is an arbitrary cutoff frequency. Its impulse response is given by the inverse Fourier transform of its frequency response:
h ( t ) = F − 1 { H ( f ) } = ∫ − B B exp ( 2 π i f t ) d f = 2 B sinc ( 2 B t ) {\displaystyle {\begin{aligned}h(t)={\mathcal {F}}^{-1}\{H(f)\}&=\int _{-B}^{B}\exp(2\pi ift)\,df\\&=2B\operatorname {sinc} (2Bt)\end{aligned}}}
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:
h L P F ( t ) = 2 B L sinc ( 2 B L t ) {\displaystyle h_{LPF}(t)=2B_{L}\operatorname {sinc} \left(2B_{L}t\right)}
H L P F ( f ) = rect ( f 2 B L ) . {\displaystyle H_{LPF}(f)=\operatorname {rect} \left({\frac {f}{2B_{L}}}\right).}
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