Sinc filter

In signal processing, a sinc filter is an idealized 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 \left(\tfrac{\sin(\pi t)}{\pi t}\right), 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 (a.k.a. rectangular filter) is non-causal and 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: where B is an arbitrary cutoff frequency (a.k.a. bandwidth). The impulse response of such a filter is given by the inverse Fourier transform of the frequency response: where sinc is 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. Windowing and truncating a sinc filter kernel in order to use it on any practical real world data set reduces its ideal properties. 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 filter is a brick-wall low-pass filter, from which brick-wall band-pass filters and high-pass filters are easily constructed.