[sdiy] FM math question

[David Moylan]

Ian Fritz wrote:
> 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.
> 
I don't think analyzing as a sum is valid since the waveshapers are 
non-linear therefore they don't necessarily obey superposition, meaning 
f(x) + f(y) does not necessarily equal f(x + y), nor necessarily 
commutation, f1(f2(x)) does not necessarily equal f2(f1(x)).  But that 
doesn't rule out the possibility that commutation will hold.  And as 
you've stated commutation seems to work here.

> 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 think the waveshapers are sensitive to amplitude changes but not 
frequency changes.  The saw->tri has no memory so it's really only 
working on the instantaneous saw voltage and converting that to the 
proper instantaneous tri voltage, sort of like an analog lookup table. 
Since the FM of the saw is assumably giving you the proper instantaneous 
voltages for an FM'd sawtooth, and the lookup is static, the output will 
also be correct (although it might take some serious math to prove it).

Again, I think we can say that the waveshapers are not frequency 
dependent; they work the same across the entire VCO range (a safe 
assumption).  So if we think about a slow FM modulation the frequency is 
changing but we don't expect the waveforms to be different as the 
frequency increases or decreases.  Intuitively, we might suspect that if 
it works at slow modulation it will work at fast modulations.  Although, 
I do have a mental block trying to imagine what's happening as the 
modulation reaches and passes the base frequency.

You can also think about this for the tri-sin case.  As long as you can 
assume that the incoming voltage is correct in the time domain for a 
triangle of instantaneous frequency X than the output should be correct 
for a sin with instantaneous frequency X.

At least, that's how I imagine it....

> 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
> 
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