Hi Hal,

ADEV turns out to be pretty hard to understand. I took me years for it
to sink in. There are web pages, articles, even books. Part of the
problem is matching the explanation with the audience.

It's hard to take a clock phase / frequency / measurement example and
use it to teach people who want to use ADEV for stock market or climate
analysis. It's hard to take quartz and atomic clock examples and use it
to teach a mechanical clock & watch audience. Another example is
applying ADEV to sensor noise [1]. Even though it is technically correct
it's not likely to help explain the guts of a GPSDO or NTP.

I can point you to a list of references but in my experience they are
too shallow or way too deep. What you want is something specific to your
NTP case. How about ignoring ADEV completely. Maybe just try to explain
why there is polling, why there is a poll interval, how the poll
interval could be less or more, why that may or may not affect
performance, how it might improve performance, or why it might actually
make it worse. That gets the concept of the "cross-over point" across
without all the rest of ADEV baggage.

Another challenge is that an audience needs to unlearn what they were
taught in school. We were told: the more data the better; the more data
the more precision you get; averaging data always gives better results;
get rid of outliers; stuff like that. Those lessons are valid for
measurements of something that doesn't change. But they are not valid if
what you're measuring is changing while you are making all those
measurements. Hence the ADEV "V" you mentioned. So maybe try to explain
NTP using words and analogies rather than using heavy math and
statistics; try not mentioning ADEV, slopes, and power law noise at all.

/tvb

[1] https://www.phidgets.com/docs/Allan_Deviation_Primer

On 9/23/2020 5:26 PM, Hal Murray wrote:

Another great post, Tom.

Amplifying Tom's last paragraph:

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.

IOW, most of what you think you know about statistics doesn't
apply.  For example, "variance" is undefined because "mean" is
undefined.  It is important to get inside Dave Allan's head in
terms of why he invented this in the first place.

2. Many poorly informed practitioners are in denial about the
   above and resort the measures known as "jitter" or "wander".
   These further muddy the water.  They are IMHO, even more difficult
   to understand than ADEV.  They have their place (a very narrow one)
   but should be disregarded if you want to understand ADEV.

3. Unlike phase noise, ADEV is a fairly non-intuitive concept.
   While there are methods to convert phase noise to ADEV, you
   can't go in the other direction.

You sometimes hear the phrase "this is not rocket science".
Well in the case of ADEV, it IS rocket science (or at least
rocket statistics).

Rick N6RK

On 9/24/2020 12:35 PM, Tom Van Baak wrote:

Good evening Hal-

I found a pretty good one from Rakon.  I'll attach it here.  It was one of
the better descriptions I've seen.  They also make some of the best current
manufactured new what I would call Oscilloquartz/Brandywine BVA
replacements the HSO14.  Lots more good docs on their website under:

http://www.rakon.com/products/technical-resources/tech-docs

Bill

On Thu, Sep 24, 2020, 2:02 PM Richard (Rick) Karlquist <
richard@karlquist.com> wrote:

I found http://www.leapsecond.com/pages/adev/ being pretty helpful to
get the concept.
Best, Ulli

Am 24.09.2020 um 21:35 schrieb Tom Van Baak:

On 2020-09-24 22:11, Richard (Rick) Karlquist wrote:

Actually flicker noise is a border-case of average to zero or not. Seems
to, but cannot be shown, but it has great deviations from this.

f. Integrated white and flicker noise. These will for sure not average
to zero.

This is all true, except for the last part, since you really need to
consider the duo of Jim Barnes and David Allan. If you look at the early
work, their work and contribution overlap. Some of the important math
was actually discovered by Jim Barnes. They had to work hard to figure
out a way to overcome the lack of a meaningful mean value, and they did
that by reducing the mean to a minimum, over 2 frequency values and then
average over that.

Well, there is a context where these terms have a very clear and
practical meaning, and the narrow place they have is wide scale telecom
and broadcast use, and anything else falling into that category, which
is more and more. Using these terms where they are not meant to is
however just asking for trouble. However, in their telecom context they
have a very useful meaning with very practical and important implications.

ADEV is poorly explained many times, I did try at one point in time to
make the Wikipedia article fairly clear, and I think it may need to be
revisited because of many edits later things may have been lost. ADEV is
really the very specific estimation of two-point frequency estimation
stability as measured over two points of frequency measures. If this is
the type of measurement you want to do, it's meaningful. You can
understand some properties in those terms. At the time, measuring this
way was the only meaningful way they felt they had, they had counters to
handle the "long term stability", so they had to make sense of the
statistics that came out of those counters, that's how we ended up with
the ADEV, that was what was on David Allans mind at the time.

Cheers,
Magnus

On 9/30/2020 12:47 PM, Magnus Danielson wrote:

Wow, I never realized that.  My bad.  It does possibly explain the
fact that Dave never used the term "Allan variance" but rather
always called it the "two-sample variance."  Although I suspect
that he would have done that anyway even if he were the undisputed
inventor of ADEV.  He also used "sigma sub-y of (2,tau)" rather
than the usual "sigma sub-y of tau" nomenclature.

Rick N6RK

On Thu, 24 Sep 2020 13:11:26 -0700
"Richard (Rick) Karlquist" richard@karlquist.com wrote:

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

attaboy for Attila. Reality always strikes.

On 2020-10-12 03:07, Attila Kinali wrote:

--
Dr. Don Latham  AJ7LL
PO Box 404, Frenchtown, MT, 59834
VOX: 406-626-4304