[sdiy] Multimode filter topology

[David G. Dixon]

> >  By permanently wiring the filter as a 4-pole LP (which resonates
> > easily), the stagewise outputs can be added and subtracted in various
> > ways to arrive at all possible LP, HP and BP filter modes, and I'm
> > assuming they will all resonate (although I haven't tested this).
> 
> Congrats, you essentially rediscovered what Oberheim did in Matrix 12 ;)

Yes, I'm aware of that.  Those Oberheim dudes were very clever!

> Tap each the filter input and each stage output and mix with varying
> factors in the final output while always taking resonance input directly
> from last stage output.

Yes, that was the idea.  Happily, in my filter, I distribute the resonance
gain equally across the stages (except for the first stage, which is higher)
to ensure equal amplitude from every stage.  This would not change in a
multimode filter.  Indeed, the output of stage 4 would always be a -24dB LPF
with resonance no matter what else was being tapped from the other stages.

What I'm really asking, Antti, is this:  What is the "meaning" of all the
various combinations?  I know about the basic LP/BP/HP combos, and also the
simple n-pole notch combos; i.e., (s^n + 1)/{s+1)^n.  I'm curious about all
of the other combinations which give various polynomials in the numerator.

Someone could start by helping me understand why Grant's configuration

s^3 + 3s
--------
(s+1)^3

is an allpass filter!  This is a 3P HP mixed with a 1P/2P BP at triple the
gain.  Why is this AP?  From what I can tell, the transfer function for a
3-pole allpass filter is defined as follows:

       c(3)s^3 + c(2)s^2 + c(1)s + 1
H(s) = -----------------------------
       c(3) + c(2)s + c(1)s^2 + s^3

Grant's configuration does not satisfy this relationship.  Of course, if one
uses the Pascal's triangle entries {1,3,3,1} for the coefficients in this
transfer function, one simply obtains the trivial result of 1, which is just
the input (with no phase shift), so it could be argued that (in the absense
of resonance) one could not achieve any meaningful allpass configurations by
adding up cascaded 4-pole outputs.  (Is this correct?)

So, is Grant's AP actually an AP, or just an approximation?  If so, why?  If
not, then what is it, and how do I classify other, similar transfer
functions?  That's what I'm asking.  If someone could just point me in the
direction of a decent reference, that would be great!