[sdiy] FM math question

[Ian Fritz]

This is a question concerning my new FM (technically PM, of course) 
VCO.  It uses a sawtooth core waveform, modulated by anything, but we can 
assume a Sin modulation for now.

It's easy enough to calculate the output spectrum -- just Fourier transform 
the Saw and apply the standard FM equation to each component.  But my 
question is about what happens after that signal is passed through two 
waveshapers, first a Saw -> Tri shaper, then a distortion circuit for Tri 
-> Sin.

Has anyone looked at how to do the math for this 
waveshaping?  Conceptually, I suppose you might be able to express the 
output of the VCO core as a sum of sawtooths, look at the effect of the 
first shaper on each and then express that result as a sum of triangles and 
then look at the effect of the second shaper to those.  The problem with 
that approach is that the various input signals to each shaper will have 
differing amplitudes, so the effects of the shapers are not simple.

The reason I am asking this is that I seem to be getting a surprising 
result: the final output looks as if it has (at least roughly) the same 
harmonic components as would a modulated Sin wave having the same frequency 
as the Saw!  In other words, the strongest frequency components of the 
output are at the fundamental frequency of the VCO plus or minus multiples 
of the modulation frequency.

How can this come about?  It seems to imply some sort of commutivity 
between the the nonlinear modulation process and the nonlinear waveshaping 
process.  Am I missing some simple way to look at this result?

I posted a couple of clips using the Sin shaper output here:
http://electro-music.com/forum/topic-29149-25.html
You should be able to hear that the harmonic content is much more subdued 
than that of the raw sawtooth output (clip on previous page).

   Ian