# [time-nuts] ergodicity vs 1/f

Bob kb8tq kb8tq at n1k.org
Thu Nov 30 12:04:30 EST 2017

```Hi

> On Nov 30, 2017, at 11:10 AM, Magnus Danielson <magnus at rubidium.dyndns.org> wrote:
>
>
>
> 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.

I believe that was roughly the third thing the prof said when he introduced 1/F noise back when I was in school. It *might* have been
the fourth thing …. that was a long time ago ….

Bob

> Even without flicker, the white frequency noise messes with us.
>
> This thread seems to lost contact with these aspects.
>
> Cheers,
> Magnus
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