# [music-dsp] Non-linear processing/Volterra series?

Dave Gamble signalzerodb at yahoo.co.uk
Wed Mar 1 11:12:51 EST 2006

```On 28 Feb 2006, at 14:37, Knut Hvidsten wrote:

> Anyone here with any experience, online links or book references on
> non-linear processing/analys for dsp and audio/music?
>
Martin Schetzen's book on nonlinear systems is a bible
So is Wilson Rugh's book.
They both have Volterra and Wiener in the titles.

> I should probably get a book on nonlinear system theory, but I imagine
> that it would be a lot more motivating if it was put into a practical
> context where I could see immidiate use :-)
>
> Theese are interesting:
> http://pcfarina.eng.unipr.it/Public/papers/147-AES00.PDF
Someone sent me a masters thesis the other day.
That paper is based on Farina's 'nonlinear convolution' technique,
and the thesis
showed the maths in the paper to be wrong. So tread carefully.

> Basically I want to know if similar methods exist for nonlinear system
> as those for linear ones. Is there a way to measure for different
> amplitude levels that characterises a loudspeaker or tube amplfier as
> accurately as one wishes, without taking it apart and modelling each
> physical component, similar to IR analysis for linear systems?
>
Yes. That would be the Volterra/Wiener strategy.

> Can any time-invariant non-linear system be described as a set of FIR
> filters (of potentially infinite length N), for each x^1, x^2, ...,
> x^M (where M can be infinite) so that the total number of
> multiplications per sample is N*M, and is this the essence of
> Volterra?
>
Worse than that; sorry.

Take a constant. Add an FIR.
Now generalise; muiltiply every pair of inputs, weight and sum them.
Now triplets
Keep going.

That's the Volterra system.

> What kinds of tools exists for analyzing and inverting such systems?
It all starts in the 60s. Probably the first 'easily obtainable'
publication is
Wiener's "Nonlinear Problems in Random Theory",
which is aimed at stochastics, to be honest, but is the first point.
Credit needs to be given to the MIT team; Brilliant, Schetzen and
others...
but I'll be amazed if you can ever get hold of a copy of those
technical reports.

>
Not really. Search for Wiener, Wiener-Hammerstein, Volterra.
There's /some/ stuff in control theory.