[time-nuts] Re: Comparing 3 oscillators using a 2 channel frequency counter?

[Magnus Danielson]

Hi Erik,

On 7/19/22 19:25, Erik Kaashoek via time-nuts wrote:
> As a good engineer I want to check the short and long term stability 
> of all my oscillators but to do that a need a better oscillator...
> But I've read that it should be possible to measure 3 independent 
> oscillators together and use some kind of statistical tools or voting 
> to get some better insight in the performance of each of the 3 
> oscillators.
> Now the problem is: where to find a 3 input frequency counter? I don't 
> have one, but I do have a 2 input frequency counter that can use an 
> external reference. So I connected 2 oscillators to the two inputs and 
> the 3rd to the counter reference and for each measurement of the two 
> frequencies I also calculated also the ratio of the two measured 
> frequencies scaled back to the oscillator frequencies (all 10MHz) and 
> imported in Timelab. [1]

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.

> Looking at frequency difference chart two oscillators (DOCXO-RB) seem 
> to be long term more stable with respect to each other compared to the 
> two other combinations (OCXO-Rb and DOCXO-OCXO), these two act as if 
> they are each others opposite. Of course this is just looking at the 
> plot so my questions are:
> - Is this the correct way to use a 2 channel frequency counter to get 
> info on 3 oscillators?
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.
>
> - What (mathematical) tools can be used to get insight in the 
> performance of the 3 oscillators individually?

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.

>
> I've read about 3-cornered hat and using the three frequency 
> measurements I know how to calculate the taus of each pair and import 
> into stable 32 to do the 3-cornered hat calculation but what would 
> that tell?

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

Cheers,
Magnus