transfer function order (was Re: filter waveform question)

[Martin Czech]

> 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

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

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

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

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

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

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

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

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

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

Common lowpass filters (bessel,butterworth, chebychev) have a constant
numerator, ie. n=0.  The only difference between them for a given order
and frequency is different pole location.

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

Highpass filters have n=m, the common types are derived from above
common lp types by transformation, this gives b(n) != 0 only for n=m,
the other b(n) are zero. This means you have n=m real zeros at s=0,
which leads to the fact that hp filters are closed for low frequency.

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

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

m.c.