So what is a filter? In general a filter is a device that discriminates, according to one or more attributes at its input, what passes through it. One example is the colour filter which absorbs light at certain wavelengths. Here we shall describe frequency-selective filters. It is called freqency-selective because it discriminates among the various frequency compononents of its input. By filter design we can create filters that pass signals with frequency components in some bands, and attenuates signals with content in other frequency bands. It is customary to classify filters according to their frequency domain charachteristics. In the following we will take a look at: lowpass, highpass, bandpass, bandstop, allpass and notch filters. (All of the filters shown are discrete-time)

Ideal filter types

Lowpass Attenuates frequencies above cutoff frequency, letting frequencies below cutoff( fc ) through, see .

An ideal lowpass filter.

Highpass Highpass filters stops low frequencies, letting higher frequencies through, see .

An ideal highpass filter.

Bandpass Letting through only frequencies in a certain range, see .

An ideal bandpass filter.

Bandstop Stopping frequencies in a certain range, see .

An ideal bandstop filter.

Allpass Letting all frequencies through, see see .

An ideal allpass filter. Does this imply that the allpass filter is useless? The answer is no, because it may have effect on the signals phase. A filter is allpass if H 2 f 1 , f . The allpass filter finds further applications as building blocks for many higher order filters.

Other filter types

Notch filter The notch filter recognized by its perfect nulls in the frequency response, see .

Notch filter. Notch filters have many applications. One of them is in recording systems, where the notch filter serve to remove the power-line frequency 50 Hz and its harmonics(100 Hz, 150 Hz,...). Some audio equalisers include a notch filter.

Matlab files idealFilters.m, notchFilter.m