In message 20160801154643.905ed816ac900a8d9a505490@kinali.ch, Attila Kinali w
rites:

Isn't that just FM modulation ?

--
Poul-Henning Kamp      | UNIX since Zilog Zeus 3.20
phk@FreeBSD.ORG        | TCP/IP since RFC 956
FreeBSD committer      | BSD since 4.3-tahoe
Never attribute to malice what can adequately be explained by incompetence.

Hi

It’s just very standard FM modulation math. The only gotcha is the
(often unknown) bandwidth of the EFC port. Even on a precision
OCXO, it might be <10 Hz, it might be over a KHz …. The trap many
fall into is the “small angle” restriction. You can get into modulation
indexes that will get the second and third order terms contributing.
It’s more common to see on vibration, but it can happen on a noisy
EFC.

Bob

On Mon, 01 Aug 2016 14:36:28 +0000
"Poul-Henning Kamp" phk@phk.freebsd.dk wrote:

Yes, it is. The problem is not the theory. The problem is to calculate
the correct values. I know i can figure it out, but if there are ready
to use formulas that are known to be correct, I rather use those.

		Attila Kinali

--
It is upon moral qualities that a society is ultimately founded. All
the prosperity and technological sophistication in the world is of no
use without that foundation.
-- Miss Matheson, The Diamond Age, Neil Stephenson

On 8/1/16 8:18 AM, Attila Kinali wrote:

Rather than deriving Bessel functions from first principles?

It's an interesting problem.. What you're really looking for is the
spectrum of the output with the FM modulation process acting on the
spectrum of the modulation. As noted by others, you need to know the
bandwidth (and then assume that it's "flat" within that bandwidth).

FM modulation isn't linear: that is, if I feed a 10 Hz and a 15 Hz
signal into a FM modulator, the spectrum I get out is not just the
superposition of the spectrum with just 10 Hz and just 15 Hz.

The spectrum of a single tone modulation is easy: it's the Bessel
function of the appropriate order with the appropriate scale factors.

Somewhere I've got a derivation of this: I was more concerned with phase
modulation (heartbeat motion and respiration motion both modulate the
reflected radar signal, so the spectrum you see is a combination of the
two): it isn't pretty in an analytical sense.  I wound up just doing
numerical simulation: you don't have to worry about whether you are
violating the small angle approximation, etc.

A couple of papers from the 60s that seem to be on point...

http://www.sciencedirect.com/science/article/pii/S0019995866800062

http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=5245193

The Medhurst paper seems to be the one you want.
"When  the  frequency  modulation  may  be  simulated  by  a  band  of
random  noise  (as  in  multiplex  telephony  carrying  large  numbers
of channels),  the  spectra  of  the  distortion  products  can,  in
principle,  be described  by  simple  algebraic  functions  of  the
characteristics  (i.e.  the minimum  and  maximum  frequencies  and  the
r.m.s.  frequency  deviaion)  of  the modulating  noise  band."

I note that "simple algebraic functions" take up the better part of a
page.  Simulation looks more and more attractive.

I don't have the answer of the top of my head, but phase noise of VCOs and
PLLs is well documented. Perhaps "loop filter noise vco" or the like may
help.

On Mon, Aug 1, 2016 at 11:18 AM, Attila Kinali attila@kinali.ch wrote:

Hi

If you are in the region that a low noise reference will apply to a low deviation precision standard, you are
deep into “small angle” territory. The higher order stuff simply does not apply. Rotate the spectrum by 1/f
(FM -> PM) and calculate the level at 1 Hz …..end of story.  If when you are done you have phase noise
that is above -60 dbc in a 1Hz bandwidth above 1 Hz, go back and look at the small angle assumption. 60 db is still
well inside the safe region so you still are likely to come back with “no problem”.

Bob