[music-dsp] Name for the integrator replacement technique

Vadim Zavalishin vadim.zavalishin at native-instruments.de
Mon Jun 8 10:25:57 EDT 2009

>> I'd suggest "topology-preserving transform" or TPT. The reason
>> is that the transfer function of the resulting system is
>> exactly what it would have been if one applied an s- to
>> z-plane transform (most commonly BLT) to the analog prototype
>> transfer function, but simultaneously also the topology of the
>> system is preserved.
> You're assuming that the analog circuit is based on cascaded
> integrators in the first place,

Not necessarily cascaded (as I understand that term implies serial 
connection), but simply somehow interconnected. I was assuming that any 
real-world analog circuit is expressible using integrators, directly 
corresponding to the state-space equations describing the circuit. At any 
rate, the analog reactive elements (capacitors and inductors) essentially 
translate as integrators in block diagrams.

> so "topology-preserving" is
> underspecific. How about "state variable preserving"?

Considering the above, the state variables should be identical to the 
integrator states. Which other state variables do you have in mind?

Also "state-variable preserving" is less specific, as you can have different 
block diagrams having the same state variables. The time-variant behavior 
will probably be the same though, but only if there are no nonlinearities 
and if we ignore the computation precision issues. So IMHO the preservation 
of the state variables is a side effect, and the relation between those is 
logical implication, not the equivalence:

preserved topology ==> preserved state variables


Vadim Zavalishin
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