transfer function order (was Re: filter waveform question)

[Magnus Danielson]

>>>>> "MC" == Martin Czech <martin.czech at intermetall.de> writes:

 >> Let's look at it this way (correct me if I'm wrong):
 >> The order of a low pass filter is equal to the number of parameters
 >> which can be varied independently. A first order filter has only one

 MC> Sorry I cannot resist, forgive my the following math outbreak:

You to... ah well :)

 MC> A linear system has a transfer function that can be writen as:

 MC>         b(n)*s^(n) + b(n-1)*s^(n-1) + b(n-2)*s^(n-2) + ... + b(0)*s^0
 MC> H(s)=  ---------------------------------------------------------------
 MC>         a(m)*s^(m) + a(m-1)*s^(m-1) + a(m-2)*s^(m-2) + ... + a(0)*s^0

 MC> You have a numerator polynome NP with coefficients b(n), and a denominator
 MC> polynome DP with coefficients a(m). s is the complex frequency.

 MC> There are complex frequencys where the NP is zero. These points in the complex
 MC> plane are called zeros.

 MC> There are complex frequencys where the DP is zero. These points in the complex
 MC> plane are called poles.

All this is very true...

 MC> There exists a so called canonical form or realisation, where you find a
 MC> chain of m integrators, the b(n) and a(n) are realised by feedback
 MC> resistor values. There are many feedback loops.

 MC> Now, the order of a linear system is m, ie. the order of the
 MC> denominator polynome, or in other terms the number of integrators.

 MC> For real systems n<m, ie. all system have a lowpass characteristic (at
 MC> least at 100 GHz ;->).  But for our low frequency  stuff n=m is
 MC> possible.

In a way we should be happy that our hearing is so frequency limited
(20Hz-20kHz) since this allows many tricks that simply is not
available to designers working with GHz and above. The higher
frequency you use the more you learn about how the real aspects of
physical laws starts kicking in.

 MC> There are lp filters (Cauer or eliptical) which employ real zeros os
 MC> well, ie. there are frequencys where H(s) drops to zero (notches).
 MC> These filters are not often used, but it can be shown that they have
 MC> the steepest possible slope of all filters.

However, in musical cursuits we see a diffrent form of these filters,
they apear in EQs, phasers, flangers, choruses etc. They all use a
combination of poles and zeros to achieve their effect.

 MC> Allpass filters have the property n=m and every pole matches to a
 MC> corresponding zero, eg. p1=x+jy then z1=x-jy. There are conjugate
 MC> complex. This property ensures that |H(s)| = 1, as expected for an
 MC> allpass.

The allpass property is in reality a lowfrequency or bandwidth limited
property which we can fake efficiently good in audio. The property is
also only valid for stable sines where as sines with changine
amplitude will not experience a flat frequency responce.

 MC> Our music circuits (filters) use only very little of the possible 
 MC> pole/zero arrangements. 
 MC> On the other hand it can be shown, that any H(s) can be reproduced
 MC> by chaining or parallel use of second order filters (state variable),
 MC> and first order sections (simple RC high or lowpass).

I agree that music circuits use a very little part of the possible
solutionspace of pole/zero arrangements. One limiting factor for
making large and complex filters is it's noise property and stability
in pole/zero positions. Such analysis will be a major effort for any
larger H(s) realization, more complex structures may be used for
optimal performance (The standard Sallen-Key is not cutting it very
well).

Cheers,
Magnus