Path: utzoo!utgpu!jarvis.csri.toronto.edu!eecg.toronto.edu!johns
Newsgroups: sci.electronics
From: johns@eecg.toronto.edu (David Johns)
Subject: Re: State Variable Filters
Message-ID: <1989Sep8.163246.2233@jarvis.csri.toronto.edu>
Organization: EECG, University of Toronto
References: <2376@radio.oakhill.UUCP> <17660018@hpfcdj.HP.COM>
Date: 8 Sep 89 20:32:47 GMT

In article <17660018@hpfcdj.HP.COM> myers@hpfcdj.HP.COM (Bob Myers) writes:
>>For years I have seen a 3 op-amp filter structure known as a
>>'state variable' or 'biquad' filter.  The filter provides simultaneous
>>highpass, lowpass, and bandpass outputs.  I understand how the filter
>>works but WHY is it called a 'state variable' / 'biquad' filter???
> 
(Part about biquad deleted) 
>The "state-variable biquad" is one name for this three-amplifier realization of
>a biquadratic transfer function, but I'm not certain how "state variable"
>got hung on this particular circuit.  (There is at least one other three-amp
>biquad circuit to which I have found reference: the "feedforward three-amp
>biquad".)
>
> 
>Bob Myers  KC0EW   HP Graphics Tech. Div.|  Opinions expressed here are not
>                   Ft. Collins, Colorado |  those of my employer or any other
>myers%hpfcla@hplabs.hp.com               |  sentient life-form on this planet.

I believe that the term "state-variable" is used since this particular
configuration realizes the "direct-form" implementation of a state-variable
filter.  In other words, at the output of the two integrators one has
1/e(s) and s/e(s), where e(s) is the pole polynomial.  Since the numerators
of these two integrator output are quite simple, it is an easy matter to
sum them together and create an arbitrary zero polynomial and therefore
an arbitrary biquadratic function.  (To create a filter with a non-zero 
gain at infinity, one has to also sum up the input signal.)  

It is interesting that this direct-form (direct-form is used in the 
control literature) filter structure is quite poor for higher order filters 
but is almost optimum for the second order case.  (For active filter designers,
the high order case of the direct-form structure is often called 
companion-form filters.)

David Johns