# [time-nuts] ergodicity vs 1/f (was: Allan variance by sine-wave fitting)

Attila Kinali attila at kinali.ch
Thu Nov 30 09:40:03 EST 2017

```On Thu, 30 Nov 2017 12:44:13 +0100
Mattia Rizzi <mattia.rizzi at gmail.com> wrote:

> Let me emphasize your sentence:  "you will have a statistically significant
> number of samples of *one* realization of the random variable.".
> This sentence is the meaning of ergodic process [
> https://en.wikipedia.org/wiki/Ergodic_process]
> If it's ergodic, you can characterize the stochastic process using only one
> realization.
> If it's not, your measurement is worthless, because there's no guarantee
> that it contains all the statistical information.

You are mixing up ergodicity and reproducability.

Also, you are moving the goalpost.
We usually want to characterize a single clock or oscillator.
Not a production lot. As such the we only care about the statistical
properties of that single instance. If you want to verify that your
production lot has consistent performance metrics, then this is a
completely different goal and requires a different methodology. But
in the end it will boil down to measuring each clock/oscillator
individualy to make sure it fullfils the specs.

> >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
>
> Without ergodicity you cannot claim it. You have to suppose ergodicity.

If you demand ergodicity, you cannot have 1/f.
You can have only one or the other. Not both.
And if you choose ergodicity, you will not faithfully model a clock.

> If it's not stationary, it can change over time, therefore you are not
> authorized to use a SA. It's like measuring the transfer function of a
> time-varying filter (e.g. LTV system), the estimate doesn't converge.

Please take one of the SA's you have at CERN, measure an oscillator
for a long time and note down the center frequency with each measurement.
I promise you, you will be astonished.

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
```