# [time-nuts] Effect of EFC noise on phase noise

Mon Aug 1 15:01:19 EDT 2016

```On 8/1/16 8:18 AM, Attila Kinali wrote:
> On Mon, 01 Aug 2016 14:36:28 +0000
> "Poul-Henning Kamp" <phk at phk.freebsd.dk> wrote:
>
>>> I need some formulas that relate EFC noise to the (added) phase noise of
>>> an OCXO. It shouldn't be too difficult to come up with something. But
>>> before I make some stupid mistakes, i wanted to ask whether someone
>>> has already done this or has any references to papers? My google-foo
>>> was not strong enough to find something.
>>
>> Isn't that just FM modulation ?
>
> 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.

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.

>
> 			Attila Kinali
>

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