# [time-nuts] simple explanation of noise spectra with mixing

Joseph Gwinn joegwinn at comcast.net
Sun Mar 1 13:23:03 EST 2015

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> time-nuts Digest, Vol 128, Issue 1, Message: 8
>
> Date: Sat, 28 Feb 2015 17:46:18 -0800
> From: Jim Lux <jimlux at earthlink.net>
> To: Discussion of precise time and frequency measurement
> 	<time-nuts at febo.com>
> Subject: [time-nuts] simple explanation of noise spectra with mixing,
> 	etc.
> Content-Type: text/plain; charset=utf-8; format=flowed
>
> Is there a handy "one pager" kind of explanation of noise spectra after
> some forms of signal processing..

The best source for the math is probably Fred Walls:

F. L. Walls, “Correlation between upper and lower sidebands” IEEE
Trans. UFFC, Vol. 47, pp 407-410, 2000.

"PM and AM Noise of Combined Signal Sources", Fred L. Walls, Total
Frequency, fredlwalls at cs.com, Proceedings of the 2003 IEEE
International Frequency Control Symposium and PDA Exhibition Jointly
with the 17th European Frequency and Time Forum,
0-7803-7688-9/03/\$17.00 © 2003 IEEE, pages 532-540.

> For instance, if you have a oscillator which has a 1/f characteristic,
> and you mix it with itself, what is the spectra of the output of the mixer.

Mixing is a multiplicative process, so this is equivalent to squatting
the signal, which doubles its frequency, so the effect will be 20
Log10(2)= 6 db increase of phase noise on the double-frequency terms.

Your bottom-line question will be if there is any cancellation of phase
noise; this will involve the time delay for the rata signal to get to
the target and return.  My guess is that there will be no
cancellation.

> Or if you have a 1/f^3 characteristic (e.g. from a crystal oscillator,
> very close in) or a 1/f^2 (from a VCO).

The double rule will apply to each point in the phase noise  as if it
were alone.

> Specifically, I've got some folks working with homodyne radars (where
> you demodulate the received signal with a sample of the transmitted
> signal, but sometimes with an offset mixed in, etc.) and I'm looking for
> a quick intro to this kind of thing at a sort of
> empirical/phenomenological as opposed to analytical..
>
> "If you see X on a spectrum analyzer or FFT, it is because of Y"...
>
> Similarly, they're building PLLs and know about the 20log10(N) thing,
> but what should the shape of things underneath be.

The 20Loh10(N) applies regardless of offset frequency, so the whole
phase noise spectrum will move up and down as a unit.

Joe Gwinn
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