[sdiy] Fwd: FM math question

[Magnus Danielson]

> At 03:33 AM 10/16/2008, Magnus Danielson wrote:
>
>>A curious point is that a saw phase-modulated to 2*pi*k radians will
>>frequency shift (SSB) the carrier by k*fm when k is integer. For
>>non-integer k values the peak will be shifted, but with side-bands...
>
> I'm not seeing anything like that, but perhaps I don't understand what you
> mean.  Please tell me in exact detail how to set up to see that.

If you trigger your scope at the sawtooth reset, the output sine will
suddenly make a phase-jump at that point. if you adjust the phase
modulation depth correctly you force extra cycles (for positive ramp) or
less cycles (for negative ramp) to appear.

What happends is that the modulating sawtooth will make the phase
incrementally in excess of the carriers phase accumulation. When the
sawtooth resets the phase makes an abrupt jump back. When modulation is
deep enought, this allows for whole cycles to be added or removed and for
integer multiples of modulation (in cycles) this will make integer cycles
add or drop to occur.

The sawtooth PM is very peculiar in that fashion. outside of those integer
modulations the same coefficient pattern occurs with integer multiplicity
on modulation depth as the fraction has, but shifted by the integer
amount.

The most central peaks is very similar to a sin phase modulation, but the
slopes at higher k values (multiples of modulation frequency of the
carrier) is quite differently, but this is expected. You could assume a
simplified linearized model where the sine overtones of the saw contribute
to their respective multiple of the modulation frequency, but naturally
further out, but also their inheret 1/n of sawtooth overtones interact. It
is not exactly what is going on, but you can get a feel for it at least.

I think a closed form of sawtooth modulation of a sine can be expressed. I
just haven't had the time to complete it.

Cheers,
Magnus