Hi,

Sorry about that - only have the question came through earlier. Strange..

For a while now I've got a growing interest in clocks, stability and
specifically GPSDOs (mostly quartz). My education started out with the NIST
Tech Note 1337. After studying the largest part of the document I felt that
this was probably not a bad starting point. This document explained the
various issues involved with characterizing clocks (Alan deviation and phase
noise) and I bet that most of you are familiar with it.

and frequency domain. Also, that the Alan deviation and phase noise plots
are the recommended performance measures to compare clocks in the respective
domains. Another cool thing is that there is an empirical link between the
two and that one can switch between the two with reasonable accuracy.

If one took two Allan deviation (or phase noise) plots one can compare two
clocks directly. Both the phase noise and Allan deviation measures are
statistical in nature and are respectively frequency and time dependent. I
also know that one can get a grip on the phase noise plot by integrating
under the curve for a specific frequency range. This answer can then be
scaled to seconds to give the integrated jitter over the frequency range
involved. What I don't understand is how does one use the Alan deviation to
predict how much a clock will drift on a certain time scale. For example,
500ps over a period of 1us. Or 1s for each day.

Initially I was convinced that the Alan deviation is very nifty because one
can easily identify the different noise types. And, of course to directly
compare clocks in the time domain. However, there seems to be an easy to
read off by how much a clock will drift after a certain period in time? It
would be much appreciated if someone could elaborate a bit on this topic. Or
point me to a previous thread that already did.

Regards,

Stephan Sandenbergh

one

Or

Stephan,

Sounds like you've done some good research already.

Ignore Allan deviation for a moment and work through the
process with me for a minute. Imagine checking the time
error of a nice quartz clock each minute.

Let's say your first phase reading, P0, is 10 us, and the
second reading a minute later, P1, is 21 us. What do
you know so far?

Well, you know the time error (also called phase error)
between your clock and your reference is a couple tens
of microseconds. That tells you how "on time" the clock
is. You now know your clock isn't perfectly on time.
Close, but not perfect. What else do you know?

With just two points, you know that your clock has drifted
in time by 11 us in a minute. Congratulations, you have
now determined the frequency error of the clock. It is
11 us / 1 minute = 11e-6 s / 60 s = 0.18 ppm = 1.83e-7.
A drift in time is the same thing as frequency offset, also
called frequency error. F1 = (P1 - P0) / 60 s. So your
clock is not only not perfectly on time, it is also not
keeping perfect time. Close, but not perfect.

Now, based on just those two readings, what would you
expect; what would you guess; what would you bet that
reading P2 will be?

I think you would agree that since your clock appears to
be drifting in time by 11 us per reading that P2 should be
about 32 us, right? The expected gain is P1-P0, or 11 us.
The last reading was P1=21 us, so your guess is simply
P1 + (P1 - P0) = 2 P1 - P0 = 32 us. Right?

OK, you wait a minute and P2 is 35 us. Your guess was
close. That's good. If the clock were perfectly stable, it
should have read 32 us, but it was off by a bit. Not only
is your clock off a bit in time, and off a bit in frequency,
it is also off a bit in predictability, in stability. Close, but
not perfect.

What do you know now? Well, based on points P1 and P2
the frequency error for this reading, F2 = 35-21 = 14us/min =
0.23 ppm = 2.33e-7. So you now have two frequency
readings. You can no longer boldly claim the frequency
error of your clock is exactly 1.83e-7; you are more inclined
to say it is 2e-7 because you realize both readings differ,
and are imprecise, but both close to 2e-7. You sense an
average would be a better measure.

You also know that your prediction was off by 3 us. Why?
Your prediction P2' was 35. The actual P2 was 32. The
error in your guess, E2 =P2 - P2' is
E2 = P2 - [ P1 + (P1 - P0) ] = P2 - 2 P1 + P0

Are you with me so far? Imagine keeping this up for a while
and making many predictions and collecting many actual
phase readings. Each new phase reading gives you a new
frequency measure; you hope they continue to average to
a nice value that you can write on your oscillator. Each
new phase reading gives you another chance to see how
well your prediction matches. You hope the errors of your
prediction stay pretty small. This time it was 3 us. Next
perhaps 2, or 4, or -3, or -1, or 5, etc. These are the small
errors in your ability to predict the phase error of the next
reading.

After a batch of N phase readings you have collected
N-1 Fi and so your average frequency error is the sum
of all Fi divided by N-1. You are also curious how confident
you are in your frequency average. You could compute
the standard deviation.

You are also curious how small your errors of prediction
are. You have collected N-2 Ei and it would be good to
compute the standard deviation of this too.

When it comes to an oscillator like this, the initial phase
error is usually no problem (you can correct for this). And
even a frequency error is not a problem (you can correct
for this in hardware or software).

What really gets you is the uncertainty in the frequency;
the jitter; the instability; the limitations of the clock in
meeting your predictions. This, you cannot correct for
and so it is a measure of how intrinsically good your
clock is.

Do you remember the square root sum of squares formula
for stdev? Take a look now at the formula for Allan Variance
or Allan Deviation. Can you see that it is just the standard
deviation of all those P2 - 2 P1 + P0 phase prediction error
terms? So Allan Deviation is not some magic formula; it's
just a regular old standard deviation formula used in a
special case.

And this is why the Allan Deviation can be used as a
predictor of time drift; by definition, it is a measure of
the expected deviation of time drift.

/tvb

See also these ADEV links, in order:

An non-technical ADEV summary from USNO:
Clock Performance and Performance Measures
http://tycho.usno.navy.mil/mclocks2.html

A scholarly paper on ADEV is found here:
The Basics of Frequency Stability Analysis
http://www.wriley.com/paper2ht.htm

This is an all-time classic:
The Science of Timekeeping. Application Note 1289
http://www.allanstime.com/Publications/DWA/Science_Timekeeping/
TheScienceOfTimekeeping.pdf

This a nice write-up from NIST:
Properties of Oscillator Signals and Measurement Methods
http://tf.nist.gov/phase/Properties/main.htm

Some free ADEV source code:
http://www.leapsecond.com/tools/adev1.htm

Many of my plots are made with Bill Riley's Stable32:
http://www.wriley.com
But even if you don't need to buy his software you can
enjoy all his papers.

Tom -

Excellent description of the process. Glad you took the time to explain this
so clearly. While I do understand the process, I do not believe I could have
stated it so well. Not to nit pick, but you did make a small typo in that
you interchanged the predicted and measured value of P2 in your example. For
most of us that will be obvious, and non relevant, but, to some it may be
confusing. Regards - Mike

Mike B. Feher, N4FS
89 Arnold Blvd.
Howell, NJ, 07731
732-886-5960

-----Original Message-----
From: time-nuts-bounces@febo.com [mailto:time-nuts-bounces@febo.com] On
Behalf Of Tom Van Baak
Sent: Thursday, November 30, 2006 9:38 AM
To: Discussion of precise time and frequency measurement
Subject: Re: [time-nuts] Predicting clock stability from
thevariouscharacterization methods

one

Or

Stephan,

Sounds like you've done some good research already.

Ignore Allan deviation for a moment and work through the
process with me for a minute. Imagine checking the time
error of a nice quartz clock each minute.

Let's say your first phase reading, P0, is 10 us, and the
second reading a minute later, P1, is 21 us. What do
you know so far?

Well, you know the time error (also called phase error)
between your clock and your reference is a couple tens
of microseconds. That tells you how "on time" the clock
is. You now know your clock isn't perfectly on time.
Close, but not perfect. What else do you know?

With just two points, you know that your clock has drifted
in time by 11 us in a minute. Congratulations, you have
now determined the frequency error of the clock. It is
11 us / 1 minute = 11e-6 s / 60 s = 0.18 ppm = 1.83e-7.
A drift in time is the same thing as frequency offset, also
called frequency error. F1 = (P1 - P0) / 60 s. So your
clock is not only not perfectly on time, it is also not
keeping perfect time. Close, but not perfect.

Now, based on just those two readings, what would you
expect; what would you guess; what would you bet that
reading P2 will be?

I think you would agree that since your clock appears to
be drifting in time by 11 us per reading that P2 should be
about 32 us, right? The expected gain is P1-P0, or 11 us.
The last reading was P1=21 us, so your guess is simply
P1 + (P1 - P0) = 2 P1 - P0 = 32 us. Right?

OK, you wait a minute and P2 is 35 us. Your guess was
close. That's good. If the clock were perfectly stable, it
should have read 32 us, but it was off by a bit. Not only
is your clock off a bit in time, and off a bit in frequency,
it is also off a bit in predictability, in stability. Close, but
not perfect.

What do you know now? Well, based on points P1 and P2
the frequency error for this reading, F2 = 35-21 = 14us/min =
0.23 ppm = 2.33e-7. So you now have two frequency
readings. You can no longer boldly claim the frequency
error of your clock is exactly 1.83e-7; you are more inclined
to say it is 2e-7 because you realize both readings differ,
and are imprecise, but both close to 2e-7. You sense an
average would be a better measure.

You also know that your prediction was off by 3 us. Why?
Your prediction P2' was 35. The actual P2 was 32. The
error in your guess, E2 =P2 - P2' is
E2 = P2 - [ P1 + (P1 - P0) ] = P2 - 2 P1 + P0

Are you with me so far? Imagine keeping this up for a while
and making many predictions and collecting many actual
phase readings. Each new phase reading gives you a new
frequency measure; you hope they continue to average to
a nice value that you can write on your oscillator. Each
new phase reading gives you another chance to see how
well your prediction matches. You hope the errors of your
prediction stay pretty small. This time it was 3 us. Next
perhaps 2, or 4, or -3, or -1, or 5, etc. These are the small
errors in your ability to predict the phase error of the next
reading.

After a batch of N phase readings you have collected
N-1 Fi and so your average frequency error is the sum
of all Fi divided by N-1. You are also curious how confident
you are in your frequency average. You could compute
the standard deviation.

You are also curious how small your errors of prediction
are. You have collected N-2 Ei and it would be good to
compute the standard deviation of this too.

When it comes to an oscillator like this, the initial phase
error is usually no problem (you can correct for this). And
even a frequency error is not a problem (you can correct
for this in hardware or software).

What really gets you is the uncertainty in the frequency;
the jitter; the instability; the limitations of the clock in
meeting your predictions. This, you cannot correct for
and so it is a measure of how intrinsically good your
clock is.

Do you remember the square root sum of squares formula
for stdev? Take a look now at the formula for Allan Variance
or Allan Deviation. Can you see that it is just the standard
deviation of all those P2 - 2 P1 + P0 phase prediction error
terms? So Allan Deviation is not some magic formula; it's
just a regular old standard deviation formula used in a
special case.

And this is why the Allan Deviation can be used as a
predictor of time drift; by definition, it is a measure of
the expected deviation of time drift.

/tvb

See also these ADEV links, in order:

An non-technical ADEV summary from USNO:
Clock Performance and Performance Measures
http://tycho.usno.navy.mil/mclocks2.html

A scholarly paper on ADEV is found here:
The Basics of Frequency Stability Analysis
http://www.wriley.com/paper2ht.htm

This is an all-time classic:
The Science of Timekeeping. Application Note 1289
http://www.allanstime.com/Publications/DWA/Science_Timekeeping/
TheScienceOfTimekeeping.pdf

This a nice write-up from NIST:
Properties of Oscillator Signals and Measurement Methods
http://tf.nist.gov/phase/Properties/main.htm

Some free ADEV source code:
http://www.leapsecond.com/tools/adev1.htm

Many of my plots are made with Bill Riley's Stable32:
http://www.wriley.com
But even if you don't need to buy his software you can
enjoy all his papers.

time-nuts mailing list
time-nuts@febo.com
https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts

this

have

For

Ah, right. In the example, the prediction, P2', should
be 32 and the actual, P2, is 35; a prediction error of
3 us. Thanks.

By the way, here's extra credit for some of you:

(1) With one point you get phase, or time error.

(2) With two points you get change in phase over time,
or frequency.

(3) With three points you get change in frequency over
time, or drift. The standard deviation of the frequency
prediction errors is called the Allan Deviation.

This is a measure of frequency stability; the better the
predicted frequency matches the actual frequency the
lower the errors. A little bit of noise or any drift causes
the errors to increase; the ADEV to increase. In the
summation you'll see terms like P2 - 2*P1 + P0. You
can see why constant phase offset or frequency offset
doesn't affect the sum.

(4) With four points you get change in drift over time.
The standard deviation of the drift prediction errors is
called the Hadamard Deviation.

This is a measure of stability where even drift, as long
as it's constant, is not a bad thing. In the summation
you'll see P3 - 3P2 + 3P1 - P0. You can see why
constant phase, frequency, or even drift doesn't affect
the sum.

So imagine a situation where you're making a GPSDO
and very long-term holdover performance is a key design
feature. What OCXO spec is important?

In this application phase error is easy to fix - you just
reset the epoch.

Frequency error is easy to fix. After some minutes or
perhaps hours you get a good idea of the frequency
offset. You then just set the EFC DAC to a calculated
value and maintain it during hold-over. In this case the
OCXO with the lowest drift rate (best Allan Deviation)
is the one to choose.

But with a little programming even drift is also easy to
fix. After some days or perhaps weeks you get a pretty
good idea of frequency drift over time and so you ramp
the EFC DAC over time to compensate.

The only limitation to extended hold-over performance
in such a GPDO is irregularity in drift rate.

In this example, the Hadamard Deviation would be a
good statistic to use to qualify the OCXO you need.
Drift, as long as it's constant (e.g., fixed, linear, even
log, or other prediction model) is not the limitation.

/tvb

Hello Tom,
many thanks as well from my side, in lieu of a lot of interested people,
for your short but splendid teaching excursion!
I had realized this crossing of values, a minor point, but now it is fully
clarified.
It will be hard to beat the explanation.

greetings,

Arnold

On Thu, 30 Nov 2006 10:26:20 -0800, Tom Van Baak wrote:

Hi Tom:

Is there a way to use the Allan plot to predict the variation in a reading?
For example if you use the plot comparing the 1 PPS from a GPS receiver
to a good Cesium frequency standard, then:
(1) what size of variation would you expect if the Cesium standard was
divided down to 1 kHz and that was compared to the GPS 1 PPS, or
(2) what size of variation would you expect if the Cesium standard was
divided down to 1 Pulse/1,000 seconds?

Have Fun,

Brooke Clarke

w/Java http://www.PRC68.com
w/o Java http://www.pacificsites.com/~brooke/PRC68COM.shtml
http://www.precisionclock.com

Tom Van Baak wrote:

From: Brooke Clarke brooke@pacific.net
Subject: Re: [time-nuts] Predicting clock stability from thevariouscharacterization methods
Date: Thu, 30 Nov 2006 13:31:53 -0800
Message-ID: 456F4DC9.9080402@pacific.net

First on the phase variations as they are divided:

1. The phase deviation frequency is unchanged, unless wrapped around the
   clock (Nyqvist wrapping).
2. The phase deviation amplitude is divided by the carrier divide ratio, i.e.
   dividing a 10 MHz down to 1 MHz will make the phase deviation go from say
   -130 dBc/Hz to -150 dBc/Hz (on the same offset from the carrier, say 100 Hz)
   assuming the divider chain does not add phase noise.

Allan deviation experience about the same of properties, with the natural
conversion of the above to tau language rather than frequency speak.

Then, on your GPS/Caesium issue, no. The reason being that when you measure
between two clocks, you don't know how much noise come from one or the other.
You need more sources of sufficiently low phase noise to match case to make a
three-cornered hat resolution of the individual sources phase noise. Once you
have done that there is no magic on predicting the divided down variants.

You can however make educated guesses, but that is another issue.

Cheers,
Magnus

Brooke Clarke wrote:

Brooke

To answer this you will need to know the Allan deviation as a function
of tau for both the caesium frequency standard and the GPS receiver output.
The Allan variance of the measurement will be equal to the sum of the
Allan variances of the GPS receiver and the Caesium standard for the
each tau.
The resultant Allan deviation will be equal to the square root of the
Allan variance.
This assumes that the 2 sources are uncorrelated. Eventually for a long
enough tau some correlation will be found as variations in room
temperature produces correlated phase shifts/delays in the caesium
standard (and any associated dividers) and the GPS receiver.
For most GPS timing receivers the Allan deviation of the receiver output
will dominate until tau is around 1 day or so for a high quality GPS
timing receiver.
The value of tau at which the Allan deviations of both the caesium
standard and the GPS timing receiver  are comparable will  be much
longer for most older generation GPS timing receivers.

For example a Caesium frequency standard with a standard tube will have
a worst case timing errors as listed below:
Tau      timing error
1        120ps
10        85ps
100      270ps
1000  850ps
1 day  17ns

Whereas a high quality GPS timing receiver will have an rms timing error
under favourable conditions of a little better than 10ns for tau < 1 day.

A carrier phase tracking GPS receiver will do much better than a typical
high quality GPS timing receiver for small values of tau (< 1000 sec or
so), however it will eventually be limited by the stability of the
transmitted GPS carrier. A high quality quartz oscillator disciplined by
a GPS carrier phase tracking receiver will outperform a Caesium
frequency standard with a standard tube, for tau < 1 day, eventually it
is limited by the GPS Allan deviation floor of around 5E-14 at tau = 1 day.

Bruce

Hi folks,

you know that i like to advertise from time to time for it:

My PLOTTER utility does compute the normal as well as the overlapping
Hadamard deviation. It may be downloaded from

http://www.ulrich-bangert.de/plotter.zip

Cheers
Ulrich, DF6JB