[sdiy] 4-pole SVF theory

[David G Dixon]

> > So, I guess that course in "Internet Etiquette"
> > at the local Kom Vux was full, eh?  :)
> 
> Yes, you and Neil had taken the two last seats. :) 

Touche, Karl.  Touche.

> >You keep creating feedback gain matching the pole polynomial 
> you want 
> >to create.You keep creating feedforward gain matching the zero 
> >polynomial you want to create.
> 
> >The booring detail is this... for variable Q setups the 
> polynomials can 
> >get messy, unless you try to create a simple polynomical 
> matching that 
> >of a normal lowpass-filter with a feedback for Q. So adapting to 
> >"musical" filters needs a bit of care.

Might not this have more to do with what we expect filters to sound like,
then whether they actually sound "lame" or not?  I mean, really, it begs the
question, "Lame, compared to what?"

> >State-variables are neat for the theorists, as it allows for simple 
> >mapping of polynomials to filter synthesis. This does not 
> necesserilly 
> >make it neat for musical use.
> >Cheers, Magnus.
> 
> Amen to that, and that's why Neils 3pole sounds so lame Mr Dixon. ;)

This discussion is of great interest to me, because I have been thinking a
great deal about synthetic multimode filters (primarily because I am about
working on a rather unorthodox design for one), and I have also been looking
at the Schippmann multimode filter chip (the datasheet for which seems
magically to have disappeared from the interwebz, but thankfully not before
I stored a copy on my computer).  This multimode ship can purportedly create
121 distinct filter modes.  The synthetic filter modes are generated from
the unfiltered input and four outputs of a cascaded-stage four-pole filter.
Each of these five source signals can be combined with a gain of -1, 0, or
+1.  That's 3^5 = 243 combinations.  However, if you disregard the 0-0-0-0-0
combination (for which the filter is silent), and consider that half of the
remaining combinations are simply the inverse of the other half, and
therefore sonically identical, that does indeed make 121 distinct
combinations.  The only problem with this is that, given the limit of
absolute gain factors of 1, many of the resulting filter modes are going to
be very meh, based on my visual inspection of the frequency responses.

In support of my own design activities, I have tried to identify what might
be called the "canonical" filter responses which can be derived from these
five inputs, and I have come up with 30 (although I have a suspicion that
there are actually 32, and that I have missed a couple of important notch
responses -- I'm still searching).  All of these responses have transfer
functions with demonimator polynomials of the form (1 + s)^n where 0 <= n <=
4, which is the unavoidable outcome of the cascaded-stage filter outputs.
The numerators are more interesting.

To make the classic LP, HP and BP responses, the numerators are s^n where 0
<= n <= 4.

To make the true AP responses, the numerators are (s - 1)^n where 0 <= n <=
4.

To make the various true notches (by which I mean notches with one or more
zeroes on the imaginary axis, and therefore in the frequency response), the
numerator is a polynomial with coefficients corresponding to the numbers on
the rows of Pascal's triangle (as are the denominators, if expanded), but
with every other term missing.  For example, the transfer function (s^3 +
3*s)/(s + 1)^3 gives a rather interesting notch below the cutoff frequency.
The relevant Pascal row is 1-3-3-1 and only the odd terms are present.  It's
gain factors are {1,3,6,4,0}.  It's complement (3*s^2 + 1)/(s + 1)^3 gives a
frequency response which is the mirror image of the other about the cutoff
frequency.  It's gain factors are {0,3,6,4,0}.  The famous "double notch"
has the transfer function (s^4 + 6*s^2 + 1)/(s + 1)^4, with gain factors
{1,4,12,16,8}.  It's complement (4*s^3 + 4*s)/(s + 1)^4 gives a single notch
at the cutoff, with gain factors {0,4,12,16,8}.

None of these notch responses (and, indeed, very few of the classic 3- and
4-pole responses) are attainable with the Schippmann filter chip, which
makes one wonder whether the musical usefulness of the responses was ever a
consideration in the design.