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

Attila Kinali attila at kinali.ch
Mon Nov 27 17:50:22 EST 2017

```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.
```