Sinc Filter - Brick-wall Filters

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:

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):

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:

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.