[sdiy] Voltera-Wiener nonlinear system analysis was

[Czech Martin]

Ok, during the weekend I found severall links covering this subject.
I think it is slightly ON TOPIC here.

Surprisingly, I found a lot of German papers, thesis' and the like.
They seem to be written arround '90, so I'm pretty sure that this
subject is not so well known in the eng. community.
At least I can see that it wasn't even mentioned in my university
courses.

What is it all about:
Linear systems with or without memory are relatively easy to
analyse. One can proove that the so called impulse response
will completely specify the system, regarding it's input/output
characteristic. For this you do not need to know what's
inside, i.e. black box approach.
E.g. a speaker and a microphone in a cathedral (let's assume
they are linear) make up a very complicated system (think
of all those reflexions). But nevertheless, the inpulse
response of that system can be obtained, thus giving a 
very good model without dealing with all this raytracing,
reflexion path computing. Or turning it the other way:
once you have the impulse response, you can compute 
things like you and your Mini Moog playing in that cathedral.

Some systems we all know are clearly nonlinear (overdriven amps
and filters, etc.). It is very difficult at times to separate
the internal components from each other. Therefore
my question always was if a nonlinear system with memory could
be analysed in a black box method, too.

This seems to be theoretically possible using the Voltera/Wiener method,
at least for systems that are unique (no hysteresis).
The starting point seems to be taylor series expansion.
This makes the first difficulty clear: the sampling theorem
must be observed (assuming you run that on a digital computer),
and the higher order parts of the taylor series will 
widen the neccisary bandwidth.

And then convoluted convolutions, which I have not understood so
far.

There seems to be even something like a spectrum response
characteristic to the system response, but since the linear
properties do not apply, it's usefullness is very limited
in comparison to the linear case. Problems seem to get worse
when transforming them into frequency domain.

Though the computational effort seems to be, well, absurd high,
it is still exciting that there is a method to analyse a black box
nonlinear system with memory.

To my knowledge there is no product on the market which takes
advantage out of these developments (but of course I could be
totally wrong).

Perhaps this will sheed some new light on the question if
"digital" emulation of "analog" effects and synths will be possible
or not.

Cheers

m.c.