Filters :

The simplest ideal filters aim at retaining a portion of the spectrum of the input in some pre-defined region of the frequency axis and removing the rest.

A LOWPASS FILTER is a filter that passes low frequencies – i.e. around f = 0 and rejects the higher ones, i.e: it multiplies the input spectrum with the following:

A High pass filter passes high frequencies and rejects low ones by multiplying the input spectrum by:

A BANDPASS FILTER passes a band of frequencies and rejects both higher and lower than those in the band that is passed, thus multiplying the input spectrum by:

A BANDSTOP FILTER stops or rejects a band of frequencies and passes the rest of the spectrum, thus multiplying the input spectrum by: