[time-nuts] Method for comparing oscillators

[Magnus Danielson]

Magnus Danielson wrote:
> Steve Rooke wrote:
>> 2009/8/6 Magnus Danielson <magnus at rubidium.dyndns.org>:
>>> Ulrich Bangert wrote:
>> ...
>>>> Well, stability over time is what exacly is displayed in a
>>>> tau-sigma-diagram
>>>> of an oscillator. Since only a few words before he is saying that he is
>>>> NOT
>>>> intersted into Allan Deviation plots, then he is perhaps interested 
>>>> into
>>>> something else?
>>> Yes. Sigma-Tau plots of the Allan Deviation fame (with friends) 
>>> addresses
>>> the instability of the noise part of things. For crystal oscillators and
>>> other non-atomic oscillators "linear" factors in frequency drift is 
>>> not best
>>> specified, described or measured using that method, which was invented
>>> purely to be able to handle the phase noise side of things, not the slow
>>> frequency drift.
>>
>> For these sorts of measurements on drifting oscillators would it not
>> be prudent to use the Hadamard Deviation?
> 
> Hadamard Deviation does not fully cancel the non-stable drift.
> 
> Just do the math... d(t) = AB/(B*t+1) and derive and you get what 
> infects the Hadamard Deviation, i.e.
> 
>             AB^2
> d'(t) = - ---------
>           (B*t+1)^2
> 
> So, regardless of which of the Allan Dev friends we have, identifying 
> drift mechanism, cancel that out of the data before Allan Dev friend 
> processing is the propper way to do it. Hadamard gets you closer in a 
> one-step process. I have also played with tricks to calculate the 
> constant drift in parallel with building the quadrature for Allan Dev 
> and it works out fine too. Gets some of the job done, but drift 
> post-processing remains an issue that needs to be handled to get propper 
> data out of the measurements. Just cancelling the average drift as 
> modeled as a constant drift gets part of the job done, regardless if 
> done separate or through Hadamard Deviation.

The Hadamard processing (Dev, ModDev or Tot) gives another B term, which 
considering that B is fairly small gives a drift-gain. As the series 
progresses, the drift derivate dies away faster (1/t^2 rather than 1/t) 
than for Allan processing so the longer time sequence used, the better 
suppression of the drift mechanism (which is true for Allan processing too).

So, in this context Hadamard is better... but it still does not nail it.
It may be sufficient however. Estimating A and B and remove the trend 
from the data isn't too hard.

It is an interesting exercise to estimate A and B, produce the drift 
only time-series and see what Allan and Hadamard time-series for that 
looks like and then compare them to the Allan and Hadamard time-series 
of the raw data. Cranking out a compensated time-series and produce the 
Allan and Hadamard time-series for that isn't that hard either.

Do plot the frequency or phase plot of the estimated curve along with 
the actual data, along with the difference to make sure you have a good 
match. This can give you a good hint that your drift estimate is either 
bad or has the wrong model.

That way you can really estimate if you where drift limited or not.

Cheers,
Magnus