[sdiy] FM math question

[Magnus Danielson]

Hi Ian,

> 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?

As you waveshape to sine it is very simple and infact maps very neatly
into the theory. Your sawtooth oscillator/phase accumulator generates the
increasing phase. You could express total phase by accumulating the number
of cycles since some T0 time. The waveshaping to sine is really just
converting the oscillator phase into a sine curve. You can express this as

x(t) = A*sin(phi_0 + omega_c t)

That you had an intermediary waveform of

w(t) = omega_c t mod 2 pi

does not change anything since sin(phi) = sin(phi mod 2 pi).

That your sin(phi) representation also uses the fact that cos(phi) =
cos(-phi) for the triangular intermediary for does not change anything.

So theory wise it is simple, for once.

>  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.

This is expected.

> 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?

This is exactly what is expected. The power of these depends on the Bessel
coefficients, which depends on the modulation index. Increase the
modulation and the central carrier disappears!

Welcome to the wonderful world of FM/PM modulation theory.

Cheers,
Magnus