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

Mattia Rizzi mattia.rizzi at gmail.com
Thu Nov 30 06:44:13 EST 2017

```Hi,

@All
>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.

hypothesis vs theory.
I said that if you follow strictly the theory, you cannot claim anything on
any stuff you're measuring that may have a flicker process.
Therefore, any experimentalist suppose ergodicity in his measurements.

@Attila
>I do not see how ergocidity has anything to do with a spectrum analyzer.
You are measuring one single instance. Not multiple [...] 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.

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.

>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.

>And no, you do not need stationarity either. The spectrum analyzer has
a lower cut of frequency, which is given by its update rate and the
inner workings of the SA.

You need stationarity. Your SA takes several snapshots of the realization,
with an assumption: the characteristics of the stochastic process are not
changing over time. If the stochastic process is stationary, the
autocorrelation function doesn't depend over time. So you are authorized to
take several snapshots, compensate for the obseveration time (low cut-off
frequency) (*), and be sure that the estimated PSD will converge to
something meaningful.
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.

cheers,
Mattia

(*) You can compensate the measured PSD to mimic the stochastic process
PSD, because the SA is a LTI system.

2017-11-28 20:23 GMT+01:00 djl <djl at montana.com>:

> 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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