[sdiy] FM math question

[Magnus Danielson]

Dear Ian,

Ian Fritz wrote:
> At 01:43 PM 10/16/2008, harrybissell at wowway.com wrote:
> 
>> I have some simulations and the end result does look like a modulated 
>> sine
>> wave.  Now if you make each cycle REALLY non-linear... that will not 
>> be the
>> case anymore.
> 
> Thanks for your results.
> 
> I have been trying to be too fancy about this -- imagining the spectrum 
> of the modulated Saw and then how the shaper would effect each component.
> 
> Instead -- as everyone keeps saying -- since the phase angle is the same 
> in the two cases the non-linearities are irrelevant.  In other words, 
> for a Sin core the signal is
> 
> Sin(Wo*t + m*Cos(Wm*t)  = Sin(phase),
> 
> and for the Saw core plus shaper it is
> 
> Shaperfunction(Sawwaveform(phase)) = Sin(phase).
> 
> Thanks everyone for beating this into me.  :-)

I have been doing theoretical analysis of a phase modulated sine using 
either sine or sawtooth modulation.

The sine modulation is the toughest one until you realize that it is the 
Bessel function that just popped up and the messy integration can be 
mapped into the Bessel function J_n(x). For a phase modulation of 
2*pi*a*sin(2*pi*f_m*t) the f_c+k*f_m tones (for integer k values) 
becomes J_k(2*pi*a).

For a sawtooth modulation of 2*pi*a*saw(2*pi*f_m*t) the f_c+k*f_m tones 
(for integer k values) becomes

sin(pi*(a-k))/(pi*(a-k))

I have run this next to the simulation and it matches very well, with 
only simulation shortcommings inducing errors. This formula also shows 
that modulation index a shifts the responce, so when a is an integer, 
then for k=a there is a value of 1 and zero for other k values.

Cheers,
Magnus