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Filters

A filter alters the amplitude and/or phase of a signal dependant on the signal's frequency. Ideally, a filter does not add any new frequencies.

phase frequency response

Additionally to the magnitude frequence response, filters have also a phase frequency response.
A filter adds the value of the phase response (determined by the frequency of the filtered sound) to the filtered sound.

Order of a filter

The order of a filter is the highest power of the frequency variable in its transfer function. For example, if the amplitude of the output signal is given like this:
Amp(f) = f / (f*f + f + 1)
then, the filter is of 2nd order (f*f = f^2).
The higher the order, the steeper the roll-off rate. And the higher the resonant amplitude just prior to cut-off frequency (for low-pass and high-pass filters?).
The order of a filter is usually equal to the total number of capacitors and inductors in the circuit. (A capacitor built by combining more than one capacitor is still one capacitor).
Generally, the transfer function of an n order filter is:
/csound/n_order_filter.jpg
This equation can also be written like this:
/csound/zeroes_and_poles.jpg
z0, z1 ... zn are also known as zeroes while p0, p1 .. pn are known as poles.
zi and pi can be imaginary numbers.

The filter's Q

The filter's Q is a measure of the filter's sharpness. As the Q increases, so does the sharpness of the filter.
Low pass and high pass filters exhibit a hump for increasing Q around the cut off frequency for increasing Qs. If the Q is even larger, those humps turn into peaks. If the Q is too high, it turns the filter into an oscilator.
The Q of a band pass filter is the ratio of the center frequency to the difference between the frequencies where the signal is attenuated 3 dB (the 3 dB bandwidth):
Q = fcut / ( fh - fl)

Types of filters

Basic filter types:
  • low-pass
  • high-pass
  • band-pass
  • band-reject (notch)
  • all-pass
    Change phase only
  • comb
12dB/octave filters are sometimes called '2-pole' filters, and 24dB/octave filters are called '4-pole' filters.

Low pass filters

The cutoff frequency of a passive low-pass filter does not define the frequency at which the filter starts to work; it is itself defined as that frequency at which the signal is already attenuated by 3dB. And, since an attenuation of 3dB is easily perceived by the human ear, this means that you are already significantly affecting the signal at the cutoff frequency.

High pass filters

Band pass

Band reject (Notch Filters)

All pass (phase shift)

These filters have no direct effect on the signal's amplitude. Yet, the alter its phase (dependent on the input frequency).

n pole filters

An n pole filter attenuates n*6 dB/octave.

Common filters

The most common filter responses are the Butterworth, Chebyshev, and Bessel types.

Butterworth (maximally-flat) filter

A butterworth filter ensures a flat response in the pass band and a rollof rate of rolloff n*6 dB/octave. (n being the order of the filter)

Chebyshev

A chebyshev has a steeper rolloff than the butterworth but also has passband ripples (that usually make a chebyshev unsuitable for audio systems).

Bessel (or Thompson)

A bessel filter delays all frequencies equally.

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