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

[Mattia Rizzi]

I'm talking about flicker noise processes

2017-11-27 23:04 GMT+01:00 Mattia Rizzi <mattia.rizzi at gmail.com>:

> Hi,
>
> >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.
> 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.
> Every experimentalist suppose ergodicity on this kind of noise, otherwise
> you get nowhere.
>
> cheers,
> Mattia
>
> 2017-11-27 22:50 GMT+01:00 Attila Kinali <attila at kinali.ch>:
>
>> On Mon, 27 Nov 2017 19:37:11 +0100
>> Attila Kinali <attila at kinali.ch> wrote:
>>
>> > X(t): Random variable, Gauss distributed, zero mean, i.i.d (ie PSD =
>> const)
>> > Y(t): Random variable, Gauss distributed, zero mean, PSD ~ 1/f
>> > Two time points: t_0 and t, where t > t_0
>> >
>> > Then:
>> >
>> > E[X(t) | X(t_0)] = 0
>> > E[Y(t) | Y(t_0)] = Y(t_0)
>> >
>> > Ie. the expectation of X will be zero, no matter whether you know any
>> sample
>> > of the random variable. But for Y, the expectation is biased to the last
>> > sample you have seen, ie it is NOT zero for anything where t>0.
>> > A consequence of this is, that if you take a number of samples, the
>> average
>> > will not approach zero for the limit of the number of samples going to
>> infinity.
>> > (For details see the theory of fractional Brownian motion, especially
>> > the papers by Mandelbrot and his colleagues)
>>
>> 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
>>
>>
>> Where 1) means there is a correlation between samples, ie if you know a
>> sample, you can predict what the next one will be. 2) means that the
>> properties of the random variable change over time. Note this is a
>> stronger non-stationary than the cyclostationarity that people in
>> signal theory and communication systems often assume, when they go
>> for non-stationary system characteristics. And 3) means that
>> if you take lots of samples from one random process, you will get a
>> different distribution than when you take lots of random processes
>> and take one sample each. Ergodicity is often implicitly assumed
>> in a lot of analysis, without people being aware of it. It is one
>> of the things that a lot of random processes in nature adhere to
>> and thus is ingrained in our understanding of the world. But noise
>> process in electronics, atomic clocks, fluid dynamics etc are not
>> ergodic in general.
>>
>> As sidenote:
>>
>> 1) holds true for a > 0 (ie anything but white noise).
>> I am not yet sure when stationarity or ergodicity break, but my guess
>> would
>> be, that both break with a=1 (ie flicker noise). But that's only an
>> assumption
>> I have come to. I cannot prove or disprove this.
>>
>> For 1 <= a < 3 (between flicker phase and flicker frequency, including
>> flicker
>> phase, not including flicker frequency), the increments (ie the difference
>> between X(t) and X(t+1)) are stationary.
>>
>>                                 Attila Kinali
>>
>>
>> --
>> May the bluebird of happiness twiddle your bits.
>>
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>
>