[sdiy] Multimode filter topology

[Tim Ressel]

You could use the patented T.R. approach and build up the circuit and play with it. Use digital pots to adjust things or whatever. Take notes on what patterns work and what don't.

Alternately CSound is a way to breadboard up stuff for testing without all that mucking about with copper and solder.

--Tim (lazy pragmatist) Ressel


--- On Mon, 8/17/09, David G. Dixon <dixon at interchange.ubc.ca> wrote:

> From: David G. Dixon <dixon at interchange.ubc.ca>
> Subject: [sdiy] Multimode filter topology
> To: "'synthdiy DIY'" <synth-diy at dropmix.xs4all.nl>
> Date: Monday, August 17, 2009, 4:22 PM
> (Apologies in advance: very long post
> with lame questions at the end):
> 
> Lately I've been thinking about multimode filters, and I'm
> very intrigued by
> Grant Richter's scheme of adding or subtracting the outputs
> from individual
> single poles in a cascaded 4-pole filter to achieve various
> modes:
> 
> http://www.musicsynthesizer.com/Circuitry/Multi-Function%20VCF.htm
> 
> In my own cascaded 4-pole filter, I simply put a switch on
> each filter stage
> to swap the positions of the resistor (OTA) and the
> capacitor to switch
> between LP and HP.  However, the problem with this
> scheme is that as soon as
> the modes are mixed, the filter will no longer resonate
> because the
> resonance feedback is insufficient.  Also, the output
> level diminishes
> sharply.  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).
> 
> I personally would alter Grant's technique slightly: I
> don't like the fact
> that he must disengage the first filter stage for half of
> his functions.  If
> one instead takes the incoming signal as one of five
> additive terms, then
> one can achieve 15 distinct modes (and their inverses) by
> applying Pascal's
> triangle to the coefficients (while alternating their
> signs).
> 
> Here is Pascal's triangle (up to five terms):
> 
>         1
>       1   1
>     1   2   1
>  
> 1   3   3   1
> 1   4   6   4   1
> 
> Here are the five terms corresponding to the outputs of
> each stage (and the
> input), multiplied by factors from the triangle and
> alternatively added and
> subtracted:
> 
>    A     
>    B     
>    C     
>    D         E
> ------- - ------- + ------- - ------- + -------
> (s+1)^0   (s+1)^1   (s+1)^2   (s+1)^3   (s+1)^4
> 
> (The signs may all be switched as well.)  Here are the
> various modal
> possibilities:
> 
> {A,B,C,D,E} = {1,0,0,0,0} = no filtering
>             = {0,1,0,0,0}
> =  -6dB LP
>             = {0,0,1,0,0} =
> -12dB LP
>             = {0,0,0,1,0} =
> -18dB LP
>             = {0,0,0,0,1} =
> -24dB LP
> 
>             = {1,1,0,0,0}
> =  +6dB HP
>             = {0,1,1,0,0}
> =  +6/ -6dB BP
>             = {0,0,1,1,0}
> =  +6/-12dB BP
>             = {0,0,0,1,1}
> =  +6/-18dB BP
> 
>             = {1,2,1,0,0} =
> +12dB HP
>             = {0,1,2,1,0} =
> +12/ -6dB BP
>             = {0,0,1,2,1} =
> +12/-12dB BP
> 
>             = {1,3,3,1,0} =
> +18dB HP
>             = {0,1,3,3,1} =
> +18/ -6dB BP
> 
>             = {1,4,6,4,1} =
> +24dB HP
> 
> Two of these entries are not available in Grant's scheme.
> 
> Now, one can also achieve various notch and allpass modes
> by using other
> additive schemes, or combinations of the Pascal's triangle
> schemes.  For
> example, a 2-pole notch filter with transfer function:
> 
> s^2 + 1
> -------
> (s+1)^2
> 
> is achieved with:
> 
> {A,B,C,D,E} = {1,2,2,0,0}
> 
>             = {1,2,1,0,0} +
> {0,0,1,0,0}
> 
> and a 3-pole allpass filter with the transfer function:
> 
> s^3 + 3s
> --------
> (s+1)^3
> 
> is achieved with:
> 
> {A,B,C,D,E} = {1,3,6,4,0}
> 
>             = {1,4,6,4,1} -
> {0,1,0,0,0} - {0,0,0,0,1}
> 
> These are both options in Grant's scheme, along with two
> others with one
> additional LP pole.
> 
> So, at long last, here is my lame-o question:
> 
> Where can I go (decent book or website) to learn which of
> the plethora of
> combinations are meaningful, so I can begin to put together
> an algorithm for
> combining terms to achieve meaningful filter modes beyond
> Pascal's triangle?
> For instance, I can find no reference to the aforementioned
> allpass mode on
> the web or in any of the books on my shelf; the only
> reference to it is in
> Grant's webpage given above.  One thing I can do,
> given that my 4-pole VCF
> has + and - outputs available from every stage, is to test
> every combination
> I can think of to see if they are any good.  However,
> there must be some
> systematic approach to this question beyond just the
> LP/BP/HP combos.  What
> about various polynomials in the numerator?  For
> example, how would one
> classify this mode:
> 
> s^3 + 3s + 1
> ------------
>   (s+1)^3
> 
> Given by {A,B,C,D,E} = {1,3,6,3,0} 
> 
>                
>      = {1,4,6,4,1} - {0,1,0,0,0} -
> {0,0,0,1,0}?
> 
> Also, what if some of the signs in the numerator are
> changed?
> 
> Any help appreciated!
> 
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