# [time-nuts] Allan variance by sine-wave fitting

Attila Kinali attila at kinali.ch
Tue Nov 28 08:12:54 EST 2017

```On Tue, 28 Nov 2017 09:52:37 +0100
Mattia Rizzi <mattia.rizzi at gmail.com> wrote:

> >Well, any measurement is an estimate.
>
> It's not so simple. If you don't assume ergodicity, your spectrum analyzer
> does not work, because:
> 1) The spectrum analyzer takes several snapshots of your realization to
> estimate the PSD. If it's not stationary, the estimate does not converge.

I do not see how ergocidity has anything to do with a spectrum analyzer.
You are measuring one single instance. Not multiple.
And no, you do not need stationarity either. The spectrum analyzer has
a lower cut of frequency, which is given by its update rate and the
inner workings of the SA.

> 2) It's just a single realization, therefore also a flat signal can be a
> realization of 1/f flicker noise. Your measurement has *zero* statistical
> significance.

A flat signal cannot be the realization of a random variable with
a PSD ~ 1/f. At least not for a statisticially significant number
of time-samples. If it would be, then the random variable would not
have a PSD of 1/f. If you go back to the definition of the PSD of
a random variable X(ω,t), you will see it is independent of ω.

And about statistical significance: yes, you will have zero statistical
significance about the behaviour of the population of random variables,
but you will have a statistically significant number of samples of *one*
realization of the random variable. And that's what you work with.

Attila Kinali
--
It is upon moral qualities that a society is ultimately founded. All
the prosperity and technological sophistication in the world is of no
use without that foundation.
-- Miss Matheson, The Diamond Age, Neil Stephenson
```