On Thu, 24 Sep 2020 13:11:26 -0700
"Richard (Rick) Karlquist" <[email protected]> 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
