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

jimlux jimlux at earthlink.net
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...



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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