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

Neil Johnson neil.johnson97 at ntlworld.com
Wed Jul 4 22:32:50 CEST 2012


Hi,

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.

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.

Second, they are actually all rather different.

The term "state variable" comes from the method of analysis/synthesis.
 More here:

http://en.wikipedia.org/wiki/State_space_(controls)

A nice property of the state variable filter is that the Q remains
constant as the cut-off frequency, Fc, is changed.  Another nice
feature of the state variable filter is that it directly provides low
pass, band pass and high pass outputs.

The Biquad filter, while looking a little similar, is different.
Mathematically it is based on the ratio of two ("bi") quadratic
functions ("quad"), hence the name:

http://en.wikipedia.org/wiki/Biquad_filter#Biquad_filter

It has the interesting property of maintaining constant bandwidth, B,
(once set) as Fc varies.  This means that Q varies with Fc -- it is
proportional to Fc, roughly Q = Fc / B (the state variable filter
varies B to keep Q constant as Fc varies).  The classic Tow-Thomas
biquad arrangement directly only gives low pass and band pass outputs.
 It can be instantly identified by the presence of a "leaky"
integrator, as opposed to the state variable which does not.

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.

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.

Cheers,
Neil
-- 
Modules and more: http://www.cesyg.com
Homepage: http://www.njohnson.co.uk



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