[time-nuts] Is there a good web page introducing ADEV?

[djl]

attaboy for Attila. Reality always strikes.

On 2020-10-12 03:07, Attila Kinali wrote:
> On Thu, 24 Sep 2020 13:11:26 -0700
> "Richard (Rick) Karlquist" <richard at karlquist.com> wrote:
> 
>> 1.  The statistics of clocks are (take your pick)
>> 
>> a.  Not gaussian, central limit theorem doesn't apply
>> b.  Not stochastic
>> c.  Not stationary
>> d.  Not ergodic
>> e.  Contain flicker of frequency processes that do not
>> average to zero; AKA 1/f noise.
> 
> There is a small thing I like to add here: We all fool ourselves when
> we look at ADEV.
> Ok, that's slightly bigger than small, but let me explain.
> 
> Looking at noise processes in different frequency sources, one can 
> identify
> two regions:
> 1) a close in region where the dominant noise's origin is intrinsic to
> the system and
> normal/Gauss distributed
> 2) a far out region, where the dominant "noise" is mediated through 
> changes in
> the environment or the aparatus itself, which is decidedly not 
> Gaussian.
> 
> For the close in region, our statistical tools (*DEV) do work and
> deliver the answers
> that we were looking for. For the far out region, the assumptions of
> our tools fail
> and we are basically tricking ourselves that we understand what's going 
> on.
> 
> Let me first go in into the far out noise as this has a more intuitive
> explanation:
> The main contributors to this noise are temperature, air pressure, air 
> humidity,
> vibration (a quiet office building has 0.1g to 1g of acceleration
> above 100Hz, constantly)
> for the environmental noises and chemical absorption/desorption,
> material creep/deformation
> (including stress relaxation), and general aging of components, both 
> electronic
> and mechanical for the aparatus changes.
> 
> It is easy to tell that (almost) none of the effects above can have a
> Gaussian distribution
> (would either need something inherently Gaussian or averaging over
> many non-Gaussian events).
> E.g., temperature has a distinct periodicity at different frequencies
> (daily, seasonal, etc).
> Even the amount of vibration in a building has a diurnal and seasonal
> varation, due to
> how many people are active in and around the building. For some of
> these, approximating
> them by a Gaussian source is ok (e.g. steady state
> absorption/desorption in equilibrium),
> as they are close to being Gaussian already, for others only after the
> main trend has been
> removed (e.g. temperature after daily/seasonal variation removed). It
> still does not make
> the math correct, it just makes it a good enough approximation. A word
> of caution here:
> while removal of trends can make effects behave like Gaussian noise,
> this has to be
> checked. Especially for long running measurements, where the removal
> might not be as
> good as it might seem.
> 
> It is not hard to see, why our statistical tools fail for these types 
> of noises,
> when the processes we are looking at are not Gaussian. And this is why 
> we fool
> ourselves when looking at ADEV, as ADEV assumes Gaussian distribution 
> in its
> machinery, which is not the case for these types of noises.
> 
> 
> Now to the hard part: the intrinsic noises.
> The source of these noises are usually either from the "thing"
> measured or by the electronics
> used to measure. E.g. a quartz crystal has thermal noise that feeds
> its white and 1/f noise
> processes, it also has phonon scattering due to crystal defects that
> again lead to white
> and 1/f noise. An active hydrogen maser detects the low power emission
> of the hydrogen atoms
> in the cavity (a few to a few 100s of pW of power, IIRC), so the noise
> of the detector
> circuit is quite substantial.
> 
> On a high level view, these noises seem to fall into two categories:
> white noise and 1/f noise.
> Both are Gaussian, meaning, if you would take many atomic clocks,
> start them at the same
> time with zero phase offset, let them run for some time, measure the
> phase differn and
> check the sample distribution, you would get a Gaussian bell shape.
> The difference
> between the two is their correlation in time: While white noise has no
> corrlation
> in time (often abreviated with i.i.d. = identically independent 
> distributed),
> 1/f noise has a 1/sqrt(t) decaying correlation in time. It is this 
> correlation
> in time, that makes things like mean and variance fail for 1/f noise, 
> because it
> breaks two other assumption we often make: stationarity and
> ergodicity. Ergodicity
> breaks because we have a non-stationary noise process. And 1/f noise
> is non-stationary
> because the expected value of the process is not independent of time
> (very short:
> the expected value for any future point of an 1/f process, is the last
> sampled value).
> 
> You might have noticed that I have written only of two types of noise,
> white and 1/f
> and left out all other noise processes 1/f^a with an exponent a > 1.
> The reason for
> this is, because I think they are "not real". I have no proof for
> this, but my conjecture
> from looking at many publications and too much data is, that only
> white and 1/f noise
> are actually physical processes and 1/f^a processes come into
> existence because we are
> integrating in some way or other over a white or 1/f noise process.
> Integration in time,
> if you remember your Fourier transform tables, adds an 1/f term to the
> Fourier transform
> of a function. As we are dealing with the power spectrum (square of
> the function/signal),
> this becomes a factor of 1/f^2 per integration. I.e. if we integrate,
> white noise
> becomes 1/f^2 and 1/f noise becomes 1/f^3. A nice example of this is
> the Leeson effect
> in harmonic oscillators. The resonator acts as an storage element and
> thus as an integrator
> for the noise. But similar things can be said for atomic clocks as
> well. E.g. all passive
> atomic clocks (Rb vapor cells, Cs beam/fountain standards, passive
> hydrogen masers) measure
> the frequency of the atoms in question. Thus the noise (detection
> noise and noise in the
> electronics) acts upon the frequency. And frequency is nothing but the
> time integral of phase.
> 
> So, why does ADEV and friends work when the noise in question does
> defy the tools we have.
> Because one property of 1/f noise is that the increments (difference
> between one sample
> and the next) are Gauss distributed and uncorrelated in time. I.e. if
> you look at the
> increments, you can apply your usual statistical tools and things will
> work out. It is
> even better, using the increments, mean and variance converge almost
> up to 1/f^3 noise
> (at 1/f^3 things break apart and we are back to square one). The ADEV
> now looks at
> the increments between two consecutive frequency samples. And because
> frequency is the
> time integral of phase, all noise up to 1/f^5 will be transformed to
> convergent mean
> and variances. (The above is a result from a branch of math called
> fractional Brownian
> motion. I am not sure whether David Allan was aware of this or not)
> 
> Comming back to Rick's list and trying to summarize the above:
> Depending on what time scale you are looking at and what type of
> frequency source,
> all of the points will be true. For short term measurements, 1/f^a
> noise will lead
> to non-stationary and thus non-ergodic noise whos variance will not 
> average out.
> For long term measurements central limit theorem might not apply and 
> thus the
> noise will not be Gaussian and is likely to have some considerable
> correlation in time.
> Be aware what you are measuring and what kind of numbers you are
> looking for. For
> some questions, ADEV & Co might be the right tool even though their
> base assumptions
> might be violated. For others, you just get random data... literally.
> 
> 
> 				Attila Kinali

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