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

Mattia Rizzi mattia.rizzi at gmail.com
Sun Dec 17 09:09:03 EST 2017


Hi,

Finally I have time to answer it properly.
Let's do a quick recap. Topic is flicker noise, statistics theory vs
experimental hypothesis. I am aware that flicker noise, in stochastic
theory, it's not ergodic nor stationary.

Mattia#1: (If you striclty apply the stochastic theory) you are not allowed
to take a realization, make several fft and claim that that's the PSD of
the process. But that's what the spectrum analyzer does, because it's not a
multiverse instrument.
Every experimentalist suppose ergodicity on this kind of noise (i.e.
flicker noise), otherwise you get nowhere.

Attila#1: Err.. no. Even if you assume that the spectrum tops off at some
very low frequency and does not increase anymore, ie that there is a finite
limit to noise power, even then ergodicity is not given.
Ergodicity breaks because the noise process is not stationary. And assuming
so for any kind of 1/f noise would be wrong.  the reason why this is wrong
is because assuming noise is ergodic means it is stationary. But the reason
why we have to
treat 1/f noise specially is exactly because it is not stationary.


Mattia:It's not so simple. If you don't assume ergodicity, your spectrum
analyzer does not work, because:
1) [...]
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.


Attila#2: I do not see how ergocidity has anything to do with a spectrum
analyzer.
You are measuring one single instance. Not multiple.
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.

Mattia: 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. 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.

End of recap.

Let's start again with Attila#2 in the recap. You say that a flat signal
cannot be a realization of flicker process. Well, you're using one
assumption "At least not for a statisticially significant number of
time-samples". This property is true only for an ergodic process.
Definition of ergodic process (from wikipedia): "a stochastic process is
said to be ergodic if its statistical properties can be deduced from a
single, sufficiently long, random sample of the process".
You applied ergodicy to dismiss my mental experiment. If you striclty apply
the stochastic theory, from an experimental point of view, you cannot proof
or dismiss hypothesis, which is the core of scientific research.


>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.


Nope. I was always talking about *a single* realization, coming from a
single DUT.
Due to the complex nature of flicker noise, you have just a single
realization in this Universe (thats why I am talking about multiverse in
Mattia#1).

> 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.

I am talking about the issues of flicker noise processes for an
experimentalist. I know that the (current) theory is incompatible with
ergodicy, but for an experimentalist ergodicity is an assumption that you
have to do. You did as well, in Attila#2.

>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.

Let's keep the focus on flicker noise, for instance, flicker noise of an
amplifier. Noise in oscillators is more fuzzy.



cheers,
Mattia




2017-11-30 15:40 GMT+01:00 Attila Kinali <attila at kinali.ch>:

> 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
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