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

Mattia Rizzi mattia.rizzi at gmail.com
Tue Nov 28 03:52:37 EST 2017

```Hi

>This is true. But then the Fourier transformation integrates time from
minus infinity to plus infinity. Which isn't exactly realistic either.

That's the theory. I am not arguing that it's realistic.

>Ergodicity breaks because the noise process is not stationary.

I know but see the following.

>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.
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.

2017-11-27 23:50 GMT+01:00 Attila Kinali <attila at kinali.ch>:

> Hoi Mattia,
>
> On Mon, 27 Nov 2017 23:04:56 +0100
> Mattia Rizzi <mattia.rizzi at gmail.com> wrote:
>
> > >To make the point a bit more clear. The above means that noise with
> > > a PSD of the form 1/f^a for a>=1 (ie flicker phase, white frequency
> > > and flicker frequency noise), the noise (aka random variable) is:
> > > 1) Not independently distributed
> > > 2) Not stationary
> > > 3) Not ergodic
> >
> > I think you got too much in theory. If you follow striclty the statistics
> > theory, you get nowhere.
> > You can't even talk about 1/f PSD, because Fourier doesn't converge over
> > infinite power signals.
>
> This is true. But then the Fourier transformation integrates time from
> minus infinity to plus infinity. Which isn't exactly realistic either.
> The power in 1/f noise is actually limited by the age of the universe.
> And quite strictly so. The power you have in 1/f is the same for every
> decade in frequency (or time) you go. The age of the universe is about
> 1e10 years, that's roughly 3e17 seconds, ie 17 decades of possible noise.
> If we assume something like a 1k carbon resistor you get something around
> of 1e-17W/decade of noise power (guestimate, not an exact calculation).
> That means that resistor, had it been around ever since the universe was
> created, then it would have converted 17*1e-17 = 2e-16W of heat into
> electrical energy, on average, over the whole liftime of the universe.
> That's not much :-)
>
> > In fact, 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.
>
> Well, any measurement is an estimate.
>
> > Every experimentalist suppose ergodicity on this kind of noise, otherwise
> > you get nowhere.
>
> 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.
>
>
>                         Attila Kinali
> --
> <JaberWorky>    The bad part of Zurich is where the degenerates
>                 throw DARK chocolate at you.
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```