[time-nuts] Crosscorrelation phase noise measurements

[John Miles]

> 
> I've heard and read some documents about using cross-correlation using
> two distinct reference oscillators when trying to measure the phase
> noise from a source to reduce the influence of the reference
> oscillators phase noise.
> 
> Unfortunately, it's still not exactly clear to me how it works ...
> does anyone have a concrete example maybe with data and the exact math
> that was done on them to get the result ?

Cross correlation is a fancy term for taking the dot product of two spectra,
in this case two FFTs measured simultaneously by two different instruments.
Recall that the dot product of two 2D vectors is a measure of how close the
vectors are to each other in 2D space.   The I,Q vectors from two
hypothetical noiseless instruments making the same measurement will be
identical, with no distance between them, so the magnitude of their dot
product would simply be the same 'correct' value.

The key idea is that if the instrument noise is randomly distributed in 2D
space, the I and Q components measured by each instrument at any given time
will be perturbed by a random amount in a random direction.  If averaged
over time, the individual I and Q components of the dot product in each bin
should approach the correct values for that bin, assuming each instrument's
additive noise is truly random and uncorrelated with respect to the other
instrument.  The magnitude of that average value can be plotted on a dB
scale.

Cross correlation is one of life's few free lunches, but you do have to wait
for it to work.  Magnus is correct in that you can get a 3 dB advantage in
noise reduction per sweep, but that only applies to the first sweep.  The
actual improvement in the instrument noise is related to the square root of
the number of averages taken.  This means that a cross-correlating phase
noise analyzer is very effective for broadband noise, where minimal
decimation has to be done prior to the FFT.  The input buffers fill up very
rapidly in that case, and you can get hundreds of thousands of averages in
just a few minutes.  But measurement of low levels of 1/f^n noise close to
the carrier can take a lot longer.  

As an example, one customer just sent me a plot of the residual noise of a
high-quality distribution amplifier.  The indicated PN was below -160 dBc/Hz
at 10 Hz, but the instrument floor at 10 Hz was still -165 dBc/Hz after a
50-hour run!  Over that time, the 10 kHz-100 kHz FFT segment had undergone
almost 30 million averages, improving the instrument floor in that region by
over 35 dB.  But the segment containing 10 Hz had been averaged only 17000
times for a 21 dB improvement.  

As far as concrete examples go, in addition to Magnus's suggestion of the
NIST documents, I'd recommend this one:

http://www.congrex.nl/EFTF_Proceedings/Papers/Session_14_Oscillators_and_Noi
se/14_04_Bale.pdf
Here, the authors use a pair of 1980s-vintage HP 3048A phase noise
measurement systems with a newer signal analyzer that performs
cross-correlation.  They are measuring additive noise from two-port devices,
rather than oscillator noise, but the same principles apply to both.  

http://cp.literature.agilent.com/litweb/pdf/5989-1617EN.pdf
The modern E5052A/B signal analyzers do the same thing in one box,
basically. 

http://arxiv.org/pdf/1003.0113v1.pdf
A good introductory article by Enrico Rubiola, with more specific math than
the other two links.

-- john, KE5FX
www.miles.io