[sdiy] semi OT: nonlinear idiot needs help

[Magnus Danielson]

From: Czech Martin <Martin.Czech at Micronas.com>
Subject: [sdiy] semi OT: nonlinear idiot needs help
Date: Wed, 5 Sep 2001 14:31:04 +0200 

Hi Martin,

> I think by now I have a good position in understanding linear
> time invariant systems. I mean convolution, transforms,
> transfer function etc. (thanx Magnus, Don, Juergen and all
> the others who teached me).

You are wellcome!

> This linear stuff seems pretty much complete and all worked out.

Indeed. By know it is pretty much a mature science even if some work
is still in the cooking. Most attention has moved to nonlinear aspects
of all possible kinds.

> But: we all know (not only guitar players) that the
> real fun starts with nonlinearity at least.

Indeed.

> I only know little about nonlinear or time varying systems.
> I knew basic control theory about that, and some chaos
> stuff like Lorentz attractors and and Runge-Kutta
> integration of first order non linear differential
> equations and stuff like that.
> 
> Is there anything like a complete theory of said systems?
> I guess it's far away from completeness and still growing.
> Any hints?

Complete theory - naw!

There are bits and pieces thought. Many of them actually. Many of them
is common to everyday electronics.

The static non-linearity will not change properties over time.

The dynamic non-linearity will however change properties over
time. Just think of a tweakable filter, a VCA or a compressor. The
former will change properties with time as a function of some control
signal which varies with time. The compressor will however change
properties with time as a result of previous sent signals.

If you wire 230V to your mic-input you also see non-linearity, since
you will probably not get the same level of input signal from a mic
after you gave the mic-input the 230V treatment as compared to the
level you had before the said treatment. That also counts as
non-linearity - the BREAKING of things. Even the most linear pre-amp
we can dig up has this form of non-linearness (they are all a fake!) ;O)

You find it all over the place, and depending on the application and
signals the result may be more or less predictable. It is very common
to make linearized approximations or at least less computationally
heavy non-linear approximations. Piece-wise linear models is also a
way to handle non-linearity.

Some of the nice math-tricks we are used to in linear systems is just
to forget for an exact solution. It is often that one finds that the
super-position principle (could almost define a linear systen) breaks
in the non-linear systems.

If you are badly out for a research area, I'm sure you could find your
own nice corner of non-linear physics or something and do a thesis on
it. God forbid that the applications are practiacally usefull! ;O)

Actually, PLLs (which I like to fiddle with) is also heavilly
non-linear. This certainly doesn't help when trying to design with
them. Linear approximations are commonly used for the steady-state
case. The locking-process is... eh... caotic!

Cheers,
Magnus