Filter types

Summary: Describes different filter types. Lowpass, highpass, bandpass, bandstop, allpass and notch filter.

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( f_c) through, see Figure 1.

**Figure 1:** An ideal lowpass filter.

Highpass

Highpass filters stops low frequencies, letting higher frequencies through, see Figure 2.

**Figure 2:** An ideal highpass filter.

Bandpass

Letting through only frequencies in a certain range, see Figure 3.

**Figure 3:** An ideal bandpass filter.

Bandstop

Stopping frequencies in a certain range, see Figure 4.

**Figure 4:** An ideal bandstop filter.

Allpass

Letting all frequencies through, see see Figure 5.

**Figure 5:** 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(ⅇ^ⅈ2πf)|=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 Figure 6.

**Figure 6:** 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.