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: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] 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 > 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. 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 [email protected] https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts _______________________________________________ time-nuts mailing list [email protected] https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts
