[time-nuts] ergodicity vs 1/f

[Magnus Danielson]

On 11/30/2017 03:40 PM, Attila Kinali wrote:
> 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.

After tons of measurements and attempts on theory a model was formed 
that was sufficiently consistent with measurements.

The model that fits observation makes much of the traditional 
statistical measures and definitions "tricky" to apply.

Flicker, that is PSD of 1/f, still is tricky to hunt down the real root 
and model it, so we just use approximation in it's place because we need 
to have something to work with. Even without flicker, the white 
frequency noise messes with us.

This thread seems to lost contact with these aspects.

Cheers,
Magnus