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

[Attila Kinali]

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

-- 
Science is made up of so many things that appear obvious 
after they are explained. -- Pardot Kynes