Hi

This process is typically referred to as a “three corner hat” approach. There are
a lot of papers about doing it and the often surprising results. These days it is
commonly done with simultaneous sampling ADC based setups.

Back when they tried it with “analog” sorts of approaches ( take three data sets
and then subtract this from that) the results often came out a bit crazy. That improved
with the ADC setups. You still can easily get nutty results. This is often true if the
noise levels of this or that DUT are much better / much worse than the others.

Sometimes “nutty” is very obvious. You can’t have noise that low ( or negative
results). The more pesky case is when the result might be correct. Fortunately
the really crazy outcome is the more likely failure mode.

By far the most simple approach ( and one with very few gotchas) is to simply
compare A to B, B to C, and A to C. One of those pairs will be better than the
others. Put the oscillator that is not part of that pair back on the shelf and start
with another. It’s not fast, but neither are the sorts of runs involved with the ADC
setups.

Bob

Also using one as the external refrence of the counter will super impose
its errors on to channel 1 and 2. So the errors of channel 1 will be
channel 1 error + ref errors.

Eric

On Tue, Jul 19, 2022, 4:06 PM Bob kb8tq via time-nuts <
time-nuts@lists.febo.com> wrote:

Hi Erik,

On 7/19/22 19:25, Erik Kaashoek via time-nuts wrote:

Yes, you can do that, if you have the raw-information so you can get the
A-REF, B-REF and A-B time-interval measures. Letting REF be C in a
classic "three corner hat" works.

You can do that. Considering you get A-REF and B-REF you have two of the
measures. Then subtracting these yields (A-REF)-(B-REF) = A-REF-B+REG =
A-B. Then for A-B the REF is only a transfer oscillator.

The three-cornered hat uses the ADEV stability sigma^2_AB, sigma^2_AC
and sigma^2_BC. Considering that these are power-sums from sigma^2_A,
sigma^2_B and sigma^2_C like this

sigma^2_AB = sigma^2_A + sigma^2_B
sigma^2_AC = sigma^2_A + sigma^2_C
sigma^2_BC = sigma^2_B + sigma^2_C

these can be solves using normal linear algebra since there is three
relationships and three unknown.

All these being the AVAR of each source, and processed independently for
each tau.

The trouble with this is that signal is noisy before the square, so you
can get biases and even resolve into negative numbers as consequence.

There is essentially two ways to leverage that. One is to use the
Groslamberth processing or you use a cross-correlation technique. Both
of these works the problem by average multiple measurements before the
squaring. That way you build a stable measure before the processing and
avoid problematic situations.

For Groslamberth processing, see the work of Prof. Francois Vernotte in
recent years.

This approach has only recently been added to IEEE Std 1139.

Stable32 can even calculate the stability for each measure. You can also
use the TimeLab to do 3-cornered hat.

Cheers,
Magnus

Ah, welcome to the slippery slope of being a time nut. Some hints:

1. Don't expect any one oscillator to be "the best". In a pile of
   oscillators one may have best ADEV at tau 100 s, but poor ADEV at tau
   0.1 s. One may have lowest drift per day. One may have best phase noise
   at 100 Hz; one may have best phase noise at 100 kHz. So if you are
   looking for a reference oscillator be prepared to have several, each
   with its own region of proven excellent performance. Note also that some
   oscillators may have worse tempco than others, so one might be your best
   tau 1000 s reference on a cloudy day but not the one to trust on a sunny
   day.

2. A single input frequency counter is sufficient. Sure, 2 or 3 or N
   channels is nice but it's not necessary. Pick a tau, say 10 s. Then
   collect an hour of data measuring oscillator #1. Then do the same for
   #2. If their ADEV differs you have found the better one. If the ADEV is
   the same, then a) the oscillators are in fact the same, or b) they
   differ but are both better than the reference clock of the counter. It
   doesn't take much time to sort through a pile of oscillators in this
   way. At some point you will then use one of the better ones as the
   reference for your counter.

3. Let's assume that neither your reference nor your instrument is the
   limiting factor. Make ADEV plots of each oscillator and that tells you
   everything you need to know about which oscillator is best over some
   range of tau.

The 3-hat trick works with this method too. ADEV is a log-log plot and
noise tends to add like RMS, so the gaps between different ADEV lines
reveals their relative noise difference. You can take the pair-wise ADEV
readings and directly compute oscillator stability using 3-hat. In other
words, you don't need to perform 3-hat on the phase data itself, you can
just perform it on the ADEV table. It's as easy as the Pythagorean Theorem:

http://leapsecond.com/tools/3hat1.c

/tvb

On 7/19/2022 10:25 AM, Erik Kaashoek via time-nuts wrote: