[sdiy] Multimode filter topology

[David G. Dixon]

Sorry to dredge up this sorry thread again, but I've just finished a
complete spreadsheet analysis of a multimode filter based on a dedicated 4P
LPF with resonance gain distributed to the four stages for equal sine wave
oscillation amplitudes.  It's all error checked and rechecked, and it works
perfectly.  I've also analyzed a 4P LPF with HP/LP switches on each stage
(like I built a few months ago, and somewhat similar to, albeit simpler
than, Craig Anderton's "Multiple Identity Filter") to compare the two
situations.  Resonance doesn't do much on the switchable filter when one or
more of the stages are set in HP mode, which is what I wanted to confirm
with simulations.  I didn't realize just how lame it was until I compared
simulations of the two, however!

If anyone is interested in receiving a copy of the Excel spreadsheet for
their simulating pleasure, please let me know offline.


> > >  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!
> 
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