Fwd: [sdiy] FM math question

[ASSI]

On Donnerstag 16 Oktober 2008, Mattias Rickardsson wrote:
> I don't understand why a  sawtooth core -> tri shaper -> sine
> shaper would not be a phase-to-sine mapping.

Good call, that's the core of the question and exactly where my addled 
brain requested some sleep yesterday evening. :-)

If we consider the sawtooth waveform as a variable [-1, +1], then a 
direct phase-to-sine mapping is possible by multiplying with pi to 
get at the (wrapped) phase.  The tri-shaper needs to do something 
peculiar: rectification, multiplication by -2 and shift by +1.  The 
result is not an expression of a phase variable anymore, but again 
within [-1, +1] and can be shaped into a cosine.  In this case it 
seems that there still is a mapping of phase (sawtooth wave) to 
amplitude all the way through the shapers, perhaps with some constant 
phase shift.  If we could show the unconditional existence of such a 
mapping, we can declare that PM on the sawtooth and then shaping to 
sine is equivalent to first shaping to sine and then do the PM as 
constant phase shifts nullify in PM.  

Now all my attempts to come up with such a mapping have always 
produced some very similar waveforms (which corroborates what Ians 
sound examples let us _hear_), but nothing that can be called 
identical.  The crux is that pesky multiplication by 2 after 
rectification; however if you don't do this, waveshaping will produce 
rectified waveforms.  Do a few plots of the whole signal path with a 
sawtooth waveform with some moderate noise on it to see for yourself.


Achim.
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