[sdiy] 'State-Variable' vs. 'Multi-Mode' filters - differences?

[Donald Tillman]

On Jul 4, 2012, at 1:32 PM, Neil Johnson wrote:

> Donald wrote:
>> "State Variable", "Multi Mode" and "BiQuad" are almost always used interchangeably referring to a second order filter built with two integrators.  But they are different.
> 
> Ummm.... no, if I may politely disagree.


I don't see that you're actually disagreeing.  I did differentiate between common use of the phrases and the technical issues.


> First, there is nothing that says they have to be second order.  It is
> perfectly possible to build 3rd order, 4th order, and higher order
> state variable filters.  It gets tricky beyond two, but is
> none-the-less achievable.  My own 3rd order state variable filter
> being one example, the Linkwitz-Riley 4th order being another example.

Right; I did say that a state variable filter can have "one or more" state variables.  Yours is a third order state variable filter.  (Interesting design, by the way.)  Technically it's not a BiQuad, as the "quad" in BiQuad refers to a second order quadratic equation, but it's certainly built the same way.

The Linkwitz-Riley is just two cascaded filters with some interesting mathematical results.


> Second, they are actually all rather different.

And that was exactly my second sentence.  :-)


> The term "state variable" comes from the method of analysis/synthesis.

Yep, as I said.


> http://en.wikipedia.org/wiki/Biquad_filter#Biquad_filter
> 
> It has the interesting property of maintaining constant bandwidth, B,
> (once set) as Fc varies.


Please, let's not reference bogus Wikipedia entries...  :-)

Ah, *this* is where I'll disagree.

The only difference between a BiQuad filter that maintains a constant bandwidth and one that maintains constant Q is whether there's a frequency multiplier element in series with the damping factor feedback path or not.  Many State Variable / BiQuad variations allow either way, so I'll claim it's a trivial difference.


> With a bit of hand-waving you can say that a biquad filter is a (form
> of) state variable filter, in the same way that you can say a Minimoog
> is the same as a Fairlight.

'Lost me.


> Multi-mode filters tend to consist of a number of integrators and
> summers, and analogue switches to connect them in various ways.  The
> standard example being the Oberheim Matrix 12.  Or there is David's
> more recent take on the theme.

Hmm, when you have analog switches flipping around the topology all bets are off, right?  I mean, I could have analog switch changing a filter into a ring modulator.

  -- Don

--
Don Tillman
Palo Alto, California
don at till.com
http://www.till.com