# [time-nuts] A philosophy of science view on the tight pll discussion

Magnus Danielson magnus at rubidium.dyndns.org
Sat Jun 5 07:07:44 EDT 2010

```On 06/03/2010 02:15 PM, Ulrich Bangert wrote:
> Gentlemen,
>
> the discussion between Bruce and Warren concerning Warren's implementation
> of NIST's "Tight PLL Method" has caused quite a stir in our group.
>
> My scientifical knowledge about the discussed topic is so much inferior
> compared to Bruce's one that I don't have the heart to enter a contribution
> to the discussion itself. It may however be helpful to have a look at the
> discussion from a "philosophy of science" point of view.
>
> The most basic form of logic is the propositional logic.

I think even attempting propositional logic has a basic flaw, namely can
the tentative goal be reached at all?

In this case, can we get a "True Allan variance" measure?

The answer is simply no. We can't get it. We can get close to it thought.

First of all, the definition for Allan variance comes with a set of
assumptions. It assumes that dead-time is zero. If it is very near zero
(i.e. just a fraction of tau0), you will get values very near the true
Allan variance, and it may be handled using either the B2 or B3 bias
function. The bias functions was invented to translate a non-zero
dead-time measurement into a zero-dead-time measurement. To do this, the
dominant noise-form for the intended tau needs to be identified, this is
where reading NIST SP1065 becomes useful and actually very simple to
implement.

Second, the bandwidth of the measurement system needs to known and
documented with the measurement, as the WPM and FPM noise forms will
have Allan variance measures depending on the system bandwidth.

Third, the bandwidth limit itself is assumed to be far away from the
taus of interest, or else the traditional formulas for various
noiseforms is not valid.

Fourth, the slope of the system bandwidth is assumed to be brick-wall.
Again, for WPM and FPM noises, this will have a noticeable effect, but
the other noise forms will also be affected if they are too close the
limit. The theoretical formulas often replicated for the noise types
does not include include the slope tail, but is simply integrated over f
from 0 to f_H and then ignores the slope.

Fifth, the definition assumes an infinit average from minus infinity to
plus infinity. We can't wait that long and we just wasn't there to setup
the measurement to start with, we have to revert to statistical
estimators. Statistical estimators can then be biased (scale or offset
values) and have different efficiency in using the available data to
come arbitrarilly close to the true value, without reaching it.

Sixth, the definition assumes a system of no systematic drift,
environmental effects and such which will limit the measurement as it is
intended to be used for noise only.

Seventh, all measurements includes imperfections such as trigger jitter,
stability of reference(s), stability of circuit, non-linearity of
circuit, cross-talk, dependence on temperature, resolution, etc. etc.

... and as you probably got by now, I can keep going on.

So, the basic assumption of being able to get the "True" value is false,
so we have to revert to second best... close enought approximation. If
you look into the roots of Allan variance you will discover that it
forms a tentative base-case for a number of measurements, with many
strings attached to it. Additional details have been worked out over the
years. The field is complex and diversed.

I think one has to be humble when relating to "True Allan variance" in
that there will always be flaws in the data one has collected and the
methods one is using. One needs to be open-minded to see that regardless
of how I collect it, I need to be able to re-evaluate it, compare it and
essentially acknowledge "that it is to the best of my current
understanding". In this hunt for the unobtainable, trying to remove
error sources becomes a matter of art.

Cross-correlation gains is among the tricks in the hat we pull out to
get below some limits.

Cheers,
Magnus

```

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