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

djl djl at montana.com
Tue Nov 28 14:23:48 EST 2017

```True that the models depend on the noise statistics to be iid, that is
ergodic. That's the first assumption, and, while making the math
tractable, is the worst assumption.
Don

On 2017-11-28 01:52, Mattia Rizzi wrote:
> 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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--
Dr. Don Latham
PO Box 404, Frenchtown, MT, 59834
VOX: 406-626-4304

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