[sdiy] 4-pole SVF theory

[Magnus Danielson]

Tom,

On 05/08/11 00:18, Tom Wiltshire wrote:
> Hi All,
>
> Can anyone point me at some good resources for state variable filter theory? I've spent a long time playing with the classic 2-pole, 12dB oct SVF filter, and I think I get that now. So I want to move along. How do I convert it into a 4-pole filter? (without just sticking a second one on the end of the first one!)
> Presumably there are good reasons why you don't see lots of 4-pole SVF filters around, otherwise I'd have thought it'd be the killer synth filter  - LP, HP, BP outputs, independent Q control - cool, no? So what's wrong with it?
>
> Pleez, help me edukate meself!

You keep stacking integrator stages to the desired length.
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.

With these hints, get back to your second degree state variable, to see 
what you actually did there, then try applying it to a fourth-pole.

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. Not unreachable tought.

Look at the Xpander/Matrix-12 for inspiration on the output zero 
polynomial side.

This is my quick hint for you, but I can (as you very well know) dwell 
into details if needed.

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