----- Original Message ----- From: "Tom Van Baak" To: "Time Nuts" Sent: Thursday, November 30, 2006 7:37 AM Subject: Re: [time-nuts] Predicting clock stability from the various characterization methods 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 32. The actual P2 was 35 . 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.

----- Original Message ----- From: "Tom Van Baak" To: "Time Nuts" Sent: Thursday, November 30, 2006 11:26 AM Subject: Re: [time-nuts] Predicting clock stability from the various characterization methods ---- 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 - 3*P2 + 3*P1 - 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