[time-nuts] Allan deviation and modified Allan deviation

Magnus Danielson magnus at rubidium.dyndns.org
Mon Feb 18 23:56:52 EST 2013


On 18/02/13 20:59, cdelect at juno.com wrote:
> Ok I've been plotting my oscillators for years using Allan Deviation.
> That way all my records can be compared easily.
> Is there any advantage to using Modified Allan Deviation?
> It seems to give a better stability plot but that just seems to be
> cheating!

It's not.

> If I plot both ways what do the differences mean?

When the Modified Allan Deviation was introduced, two things happen:

1) It fixes the "bug" in the Allan Deviation which can't make useful 
differentiation between white phase modulation and flicker phase 
modulation noises. This is achieved by using a tau-long averager prior 
to the Allan Deviation calculation. If you are in a region where these 
noises exist, use MDEV to separate them. Dave Allan himself is eager to 
push MDEV over ADEV.

2) The traditional (non-overlapping) Allan Deviation estimator has poor 
usage of the samples taken. The MDEV estimator given uses the 
overlapping strategy to increase (but not maximize) the use of samples, 
especially for longer taus. This gives improved confidence intervals.

These steps improved the state of art processing significantly in the 
early 80thies. Since then have the total and Theo strategies for even 
further improved use of samples to achieve good confidence intervals.

Using the traditional non-overlapping Allan deviation estimator for the 
Allan Deviation measure is throwing a lot of useful data in the trash 
and suffer by bad precision or long measurement-times.

Allan Deviation is frequency drift limited in its upper end, where 
linear (and higher order) drift will obscure the output and hide the 
noise processes that Allan Deviation is meant to analyse. Drift is a 
systematic component that needs to be taken out of the data in order for 
the noise processes to be visible.

The Allan Deviation has a cousin in the Hadamard Deviation, both is 
normalized to the same scale (i.e. same as RMS white frequency 
modulation noise). The Hadamard beats Allan for real-time processing, as 
it removes the first degree frequency drift. However, for 
post-processing a better drift model can be used, so drift removal on 
data prior to Allan Deviation is preferred. Then, both Allan and 
Hadamard Deviations can exist in their normal and modified variation, 
with the tau-long averager. For degrees of freedom treatment, you can do 
non-overlapping, overlapping, TOTAL and Theo processing, where TOTAL and 
Theo is the best.

There is also FFT based methods to process data, but I need to refresh 
myself on that.

So, it sounds like you can do a lot more with your data without cheating.

Magnus - from LA

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