[time-nuts] Allan Variance vs. Least squares

[Magnus Danielson]

Hi Olaf,

On 05/23/2011 12:23 PM, Olaf Bossen wrote:
> Hi there,
>
> I am looking for some advice on stability metrics for a slow oscillator.
> The oscillator is used to do a measurement of another quantity which is
> connected to the frequency of the Oscillator. I have taken generously
> oversampled data of the oscillator voltage and now I have two
> contradicting measures:
>
> 1) When I record the zero crossings and use them as phase data for the
> Allan variance, the minimum is 10^-4 and it initially decays with a
> slope of t^-0.5

Please notice that while we say that we measure Allan variance we plot 
the Allan deviation. The initial slope of t^-1 is expected from the 
measurement limitation of your measurement system.

> 2) On the other hand if I do least squares fits of the same data with
> consecutively longer runs the reported frequency uncertainty goes down
> to sigma_f/f = 10^-7. It also drops much faster with t^-1.5 wich seems
> to be due to the "Cramèr-Rao lower bound" (not that I really understand
> what it means), and it doesn't really go up again

Don't do that. You will perform a form of filtering and you no longer 
measure something attempting to be Allan deviation. Use overlapping or 
similar estimator to get good

> It seems to be common lore that the Allan variance minimum is the best
> obtainable frequency accuracy for an oscillator, but the least squares
> fits seem to be much better. I have problems understanding this.

Allan Variance/Deviation performs a particular filtering prior to 
squaring and averaging.

Modified Allan Variance/Deviation performs another particular filtering 
prior to squaring and averaging.

Hadamard Variance/Deviation performs yeat another particular filtering 
prior to squaring and averaging.

You just did a different filtering mechanism. Some of these filterings 
have known bias functions and biases can be significant. Standard 
variance/deviation can for some noise-types have a bias reaching for 
infinity.

So trying to be "smart" like this doesn't provide you with true Allan 
variance. You may however spend quite some time analysing these bias 
functions for the various noise-types and by using dominant noise 
identification algorithms selecting among bias functions for that 
particular tau.

As you notice, this is tedious and might not be as rewarding as one may 
initially think. I would warmly recommend another approach.

> I have a mental picture that might explain this, maybe you can tell me
> if it seems correct to you: Oscillators are running at a very precise
> frequency much better than what our measurement devices are capable of
> resolving. So the drop in the Allan variance, that is initially gained
> when the measurement is longer, is actually just the reduction of the
> measurement error for this precise frequency. Only when the Allan
> variance goes up again, we are on a timescale over which the true
> frequency of the oscillator varies.

Depending on tau your counter or your reference will provide a 
measurement floor. There are methods to measure below this. But prior to 
making bold statements relating to this, one should use methods known to 
be repeatable, bias free and providing low noise. You should be able to 
get essentially the same results regardless of setup, as long as it is 
not significantly worse than the clock you try to measure.

Ponder over this article:
http://en.wikipedia.org/wiki/Allan_variance

I don't know if this is the answer you seek, but I do hope it is 
interesting at least.

Cheers,
Magnus