An advanced question - sampling an oscillator's signal for analysis

[DimitriP]

To asnwer your question:  Best stability and phase noise for a given working voltage and power consumption in a size that fits and doesn't cost an arm a a leg. Depending on what you are building and who is paying for it.

Best stability measured by what metric?

When going window shopping I look at published specs.
This one is 0.1ppb and it's just under $1800 (so just an arm, you get to keep the leg ), and if you need more that three, you gotta wait 17 weeks
https://www.digikey.com/product-detail/en/abracon-llc/AOCJY6-10.000MHZ-1/535-11919-ND/3641391

[rhb]

I've got my 8648C and 33622A within 0.01Hz which as close as I can adjust the 33622A.  So it takes a minute or two to go through a complete cycle.  I can see the Lissajous figure "breathe" as it rotates slowly.  So phase noise close in is clearly visible.  The "breathing" is at 0.1-1 Hz.  However, I have no way of knowing if it's real or an artifact of the DSO timebase.

I'm going to look into setting up my GPSDO.  The first task is to decide where to mount the antenna and how to bring the coax into the house.  The 8648C has the high stability timebase option, but the 33622A does not.

If the GPS antenna installation proves troublesome, I'll reference the 33622A to the 8648C for a few tests.

[dnessett]

When going window shopping I look at published specs.
This one is 0.1ppb and it's just under $1800 (so just an arm, you get to keep the leg ), and if you need more that three, you gotta wait 17 weeks
https://www.digikey.com/product-detail/en/abracon-llc/AOCJY6-10.000MHZ-1/535-11919-ND/3641391

I looked at the detailed spec and a frequency stability of +/- 0.1ppb seems to be an aging factor; specifically, +/- 0.1ppb per day. (There is no frequency stability figure given in the detailed spec.) Also, phase noise values are given for 1, 10, 100, 1K, 10K and 100K Hz. You didn't mention them. So for the application you had in mind, are you saying the phase noise data isn't a factor and frequency stability means for you the upper limit on frequency drift per day?

What application did you have in mind when responding?

[dnessett]

I've got my 8648C and 33622A within 0.01Hz which as close as I can adjust the 33622A.  So it takes a minute or two to go through a complete cycle.  I can see the Lissajous figure "breathe" as it rotates slowly.  So phase noise close in is clearly visible.  The "breathing" is at 0.1-1 Hz.  However, I have no way of knowing if it's real or an artifact of the DSO timebase.

I looked at the specs for the 8648C and 33622A. The former gives a frequency stability of 3*10^-6 * carrier frequency and an aging factor of +/- 2 ppm/year and +/- .0005/day. The latter specifies a combined stability/aging value of +/- (1ppm + 15 pHz) * carrier frequency over a year period.

These are very precise units with a concomitant price tag. On eBay, a used 8648C is advertised for $2,555 and a 33622A for $3,373. If people want to invest in this class of equipment, that's great. Either of these units could be used to generate the reference signal against which DUT oscillators are measured.

But, I am more interested in getting as much information as possible from more affordable gear. So, I will use a $89 GPS disciplined 10 MHz oscillator as the reference signal and the other oscillators will be measured against it. Both the reference and DUT oscillators will be fixed frequency units (with the exception of the Rigol 1022, which will be one of the DUT oscillators); there is no way to adjust their frequency to get a Lissajous figure.

My objective is to develop or use already developed techniques to measure phase noise and fluctuations with fixed frequency oscillators. I'm not saying your approach is wrong; it may be better. But, it is not a path available to me.

[rhb]

I got my 8648C for under $1k, but I paid Keysight's eBay store about what you quoted for the 33622A.

I'm *not* interested in measurement methods that require that level of equipment. I'm interested in what can be done on the cheap.  I only used them because *I* don't have two OXCOs and a Rb reference to use as signal sources. ;-)

Besides which, I'm measuring them with a cheap Chinese DSO.  I happen to have my $1700 MSO-2204EA hooked up at present, but I was using my $244 GDS-2072E yesterday.

I did a little preliminary work today for mounting the GPSDO antenna and hopefully I'll be able to finish in the morning before it gets hot.

As I know that there are issues in setting up the GPSDO, I'll probably reference the 33622A to the 8648C before I reference the two of them to the GPSDO.

A side note on the cost of gear.  I just turned 65.  I'm "retired" only because no one in the oil industry will employ anyone my age.  All my life I have squeezed every penny with a pair of vice grips.  Despite a tremendous desire for a good electronics lab, I went without because of the cost.   In recent years I've had two friends die of cancer and two of a heart attack.  Another friend is currently undergoing chemo.  But the biggest impact has been watching my brother in-law deteriorate from Parkinson's over the last 9 months.  That has made me an old man in a hurry.  And loosened my purse strings.  I still have the one new vehicle I ever bought for myself in 1993, a base model Toyota pickup for which I paid $7800 with A/C.  So I'm allowing myself the cost of a new midrange car for my lab.  I've never bought a boat, RV, motorcycle, taken a $3000 vacation or any of the other things most people do.

My primary interest is low cost test gear.  My major project is writing FOSS FW for Zynq based scopes.  Unfortunately for that project, I am easily side tracked by good problems such as the one you presented.  For which I am grateful.  Not quite as much fun as a similar class problem in my specialty, but still a great pleasure.

[dnessett]

Given the fiasco with figure 4 in a recent post of mine, I thought I should provide a more useful image of the phase noise between the FEI FE-5650 and Rigol DG1022. I finally got my scope to synch both signals by increasing the sample size to its maximum for 3 active channels (3Mpoints). The resulting image shows the FEI in yellow, the Rigol in blue and the phase difference signal in purple.



There is an obvious conclusion from this trace (concentrate on the purple plot). There is significant linear drift between the two oscillators (not something that will astonish anyone), specifically 74.63 Hz. Surrounding this is a bunch of fuzz representing the phase noise. So, the trick will be to get the numerical data, analyze it for linear drift, subtract the linear drift, and then subject the resulting signal to phase noise analysis. Again, this is somewhat obvious, but since the object of the current exercise is to develop a technique for analyzing the phase difference data, stating the obvious once shouldn't be too annoying.

[rhb]

Looks pretty good.  But where's the Lissajous?

I don't think stating the obvious in a project like this is bad.  I think it needs to be repeated regularly.  Otherwise there is a great tendency to run off into the ditch.

[dnessett]

Looks pretty good.  But where's the Lissajous?

I don't think stating the obvious in a project like this is bad.  I think it needs to be repeated regularly.  Otherwise there is a great tendency to run off into the ditch.

I finally got the Lissajous to work, but on my scope it was basically useless. I could get it to almost stop, but the lines defining it were so thick that you couldn't really see any "breathing".

[rhb]

Looks pretty good.  But where's the Lissajous?

I don't think stating the obvious in a project like this is bad.  I think it needs to be repeated regularly.  Otherwise there is a great tendency to run off into the ditch.

I finally got the Lissajous to work, but on my scope it was basically useless. I could get it to almost stop, but the lines defining it were so thick that you couldn't really see any "breathing".

The width of the Lissajous figure should be the phase noise from cycle to cycle.

I saw that on the MSOX3104T.  I *think* it's caused by a waveform update rate which exceeds the response time of the LCD.  That was a factor in my decision to return it.  I've been considering getting a fast analog scope.  I found a nice refurbished 485, but to my horror they are socketed semiconductors throughout.  While that's fine in a dry climate like southern California, it's a nightmare in a humid climate such as Houston or central Arkansas.

I'd like to return to the equation from your original post:

v(t) = V0 + A(t)*(cos( wt + phi(t))

That seems to me an adequate formulation.  I can't think of any real oscillator behavior that can't be shoved into either A(t) or phi(t).  The linear drift can be accommodated by writing phi(t) = a + b*t + c*rand(t).  Once we have phi(t) we can pick it apart as needed.  So the first order problem is to separate the variables.

As a perfect reference does not exist, the only way to do that is to set up a system of equations with as many equations as there are unknowns.  We can do that with 8 oscillators by comparing phase pair wise for all pairs.  However that leads to 32 equations.

The A(t) term is the harmonic content. So there is the potential to simplify the problem by low pass filtering the oscillator outputs.  That would reduce the problem to 18 equations in 3 variables.  But 120 dB filters are not easy and require very careful shielding to be effective.  To get 120 dB down by the 2nd harmonic requires a 20 pole filter.  I rather fear trying to build even one of those.

So on balance, from a physical implementation perspective I think the easiest way to proceed is with 8 oscillators and 32 comparators.   However, this is 4 bytes per comparison.  With a 10 MHz oscillator frequency, a single comparison per cycle would produce 40 MB/S.  So even this is not easy.

However, the quantities of interest are the expectations of random processes.  So if we can guarantee that the samples are not correlated with any oscillator a few samples per cycle should suffice.

The preceding is a national lab level measurement.  We need to determine if it has been tried or studied and found to have a fatal flaw other than the problem of output data volume.  I don't think it sensible to move on implementing it without having completed a thorough analysis.

While not adequate for absolute measurements, mixing the oscillator ouput with the  10 MHz output of a GPSDO and using that to produce a baseband IQ stream seems an appropriate first step.  The STM32F4 series have three  2.4 MSa/S 12 bit ADCs.  They can be interleaved to get 7.2 MSa/S which allows for a fairly simple anti-alias filter.  Another option would be an LPC4370 which offers an 80 MSa/S single channel ADC and is available on a $20 eval board from Digikey.

A NOTE TO OTHER READERS:  I shall ignore all comments which do not include carefully stated, mathematically sound proof of any assertions made.  So if you can't do the math, don't bother to comment.  And I should like to caution that any use of the word "clock" is not in the sense of time keeping.  This is not a discussion of time keeping.  It is a discussion of the physics and metrology of electrical oscillators.

[tomato]

As a perfect reference does not exist, the only way to do that is to set up a system of equations with as many equations as there are unknowns.  We can do that with 8 oscillators by comparing phase pair wise for all pairs.  However that leads to 32 equations.

Shouldn't that give you 28 equations? n(n-1)/2?

The preceding is a national lab level measurement.  We need to determine if it has been tried or studied and found to have a fatal flaw other than the problem of output data volume.  I don't think it sensible to move on implementing it without having completed a thorough analysis.

You're in luck -- this has not been done before.  You will pioneer the field.

[rhb]

Thank you for calling attention to my blunder.

The correct equation is (N-1)*(N/2) = 4*N.  So one needs 9 oscillators to produce 36 equations with 36 unknowns.

I am rather surprised by your assertion that this has not been investigated.  While the problem was for a long time computationally difficult, modern computers can handle it with ease.

Do you know the name of anyone at NIST who works on such things?

[tomato]

I am rather surprised by your assertion that this has not been investigated.  While the problem was for a long time computationally difficult, modern computers can handle it with ease.

It has nothing to do with the computational difficulty.

Do you know the name of anyone at NIST who works on such things?

Yes, but I'm not dragging them into this, because they are my customers.  Check the NIST personnel directory.

[awallin]

The correct equation is (N-1)*(N/2) = 4*N.  So one needs 9 oscillators to produce 36 equations with 36 unknowns.

The conventional N-cornered hat and N-clock ensemble time-scale papers make N-1 independent measurements between the oscillators. This leaves the 'absolute' frequency of all oscillators unknown (N unknowns, N-1 measurements) but you can estimate and predict the frequency/phase differences between oscillators.
I think I lost track of what you're trying to achieve, and your oscillator model with 4? unknowns per oscillator....

[rhb]

From the first post in the thread:

v(t) = [V0 + e(t)] * cos[w0*t + phi(t)], where V0 is the base oscillator amplitide, w0 is the base oscillator frequency (in radians/sec), and both e(t) and phi(t) are stochasitic processes that respecitively add amplitude noise and phase noise to the oscillator's output.

It's not my project, I just got interested in the problem.  Making the separation above is the first step.  Then the real fun begins dissecting  phi(t) into the sum of multiple noise sources.  The latter is what interests me as it seems an excellent application of a sparse L1 pursuit.  In particular, a basis pursuit with a large dictionary of possible models.

Curiously for someone with a BA in English lit and an MS in geology, as I have gotten older I have become increasingly interested in mathematics and the analysis of random data in particular.  I think it's because I now see so many parallels to familiar problems from 30 years of reflection seismic signal processing in the oil industry.

I ran what I jokingly call "the orphan home for lost problems".  Lots of strange, arcane questions would come to my office door. It was lots of fun solving them.

[rhb]

R&S sent me a link to a primer on spectrum analyzers.  It's actually quite well written and provides references relevant to this topic.

http://www.rohde-schwarz-usa.com/rs/324-UVH-477/images/SpecAnFundamentalsPrimer.pdf

[dnessett]

RoGeorge provided a very nice explanation why the traditional variance of time-keeping devices diverges as sample sizes increase. However, it is in another thread (here). I responded in that thread and asked several questions. One specifically relates to Figure 3 in Rutman and Wall's paper Characterization of Frequency Stability In Precision Frequency Sources that shows a knee in a plot of log(sqrt(<y²(t)>)) against log(tau). RoGeorge couldn't comment on the question, since he was unfamiliar with the paper and had other things he needed to focus on rather than reading it.

I don't want to continue the discussion of this question within that other thread, because that would be hijacking it for a purpose other than that which it was created. So, I am raising the question in this post and suggesting why I think that knee may be important.

After following the discussion in the other thread, it seems now clear (at least to me) that time-keeping has very different requirements for oscillators than other applications, such as doppler radar and spread spectrum communications. This can be discussed subsequently, but I would like to focus on time-keeping and suggest that Allan Variance/Deviation may be important for long-term time-keeping applications, such as national clock references, but may not be critical for short- to medium-term time-keeping applications.

To illustrate, suppose there is an application that needs to time the flight of frisbees in some frisbee sport competition. Frissbee flight times are relatively short, so the clocks used likely need not be designed/selected with their Allan Variance/Deviation in mind. Similarly, suppose there is a need to time the growth of fungus on moist leather. Again, the Allan Variance/Deviation of these clocks is probably not a selection parameter when considering which to use.

So, when does it become necessary to consider the Allan Variance/Deviation of a clock for a particular application? I hypothesize in order to start discussion that it has something to do with the knee shown in Figure 3 of the paper referenced above. As long as the local variance and Allan variance are for all practical purposes equal, then traditional oscillator stability measures (e.g., parts-per-whatever/(minute, hour, day) frequency error rates, phase noise values/plots in rad²/Hz) are sufficient. Applications needing clocks to operate for sufficiently long periods of time (that period being related to the knee in Figure 3), on the other hand, probably should consider the Allan Variance/Deviation of the clocks they select for use.

[tomato]

So, when does it become necessary to consider the Allan Variance/Deviation of a clock for a particular application? I hypothesize in order to start discussion that it has something to do with the knee shown in Figure 3 of the paper referenced above. As long as the local variance and Allan variance are for all practical purposes equal, then traditional oscillator stability measures (e.g., parts-per-whatever/(minute, hour, day) frequency error rates, phase noise values/plots in rad²/Hz) are sufficient. Applications needing clocks to operate for sufficiently long periods of time (that period being related to the knee in Figure 3), on the other hand, probably should consider the Allan Variance/Deviation of the clocks they select for use.

Question: How do you know where the knee is?

Answer:  Allan Variance

General: 

1. There is nothing about the Allan Variance that restricts it's use to the "long term." 
2. There is no information that can be derived from the standard deviation that cannot be derived in a more unambiguous way from the Allan Variance. 
3. The Allan Variance tells you when the standard deviation is a useful parameter.

[rhb]

So, when does it become necessary to consider the Allan Variance/Deviation of a clock for a particular application? I hypothesize in order to start discussion that it has something to do with the knee shown in Figure 3 of the paper referenced above. As long as the local variance and Allan variance are for all practical purposes equal, then traditional oscillator stability measures (e.g., parts-per-whatever/(minute, hour, day) frequency error rates, phase noise values/plots in rad²/Hz) are sufficient. Applications needing clocks to operate for sufficiently long periods of time (that period being related to the knee in Figure 3), on the other hand, probably should consider the Allan Variance/Deviation of the clocks they select for use.

Question: How do you know where the knee is?

Answer:  Allan Variance

General: 

1. There is nothing about the Allan Variance that restricts it's use to the "long term." 
2. There is no information that can be derived from the standard deviation that cannot be derived in a more unambiguous way from the Allan Variance. 
3. The Allan Variance tells you when the standard deviation is a useful parameter.

Perhaps you would be so good as to produce a concrete mathematical example.  So far all you have done is thump your chest claiming great authority and expertise.  Anyone who actually understands anything can explain it to a 12 year old.  So how about explaining it to PhDs.  That should be even easier.

Someone providing janitorial services to NIST has NIST as a customer.  But they're still just a janitor.

[tomato]

Perhaps you would be so good as to produce a concrete mathematical example.

Regurgitating mathematics to correct a general misstatement isn't a very good use of anyone's time.

So far all you have done is thump your chest claiming great authority and expertise. 

I didn't realize correcting mistakes or misinterpretations fell under the category of "thumping one's chest."  And I certainly have not claimed great authority in any of my posts. In fact, I have been careful not to list my credentials.

Anyone who actually understands anything can explain it to a 12 year old.  So how about explaining it to PhDs.  That should be even easier.

Explaining things to 12 year olds is often easier.

Someone providing janitorial services to NIST has NIST as a customer.  But they're still just a janitor.

I do not provide janitorial services to NIST.

[dnessett]

Recently, here in a different thread, thermistor-guy responded to a comment I made about Allan Variance in that thread. Specifically, I stated that I had not seen mentioned anywhere how someone is supposed to use Allan Variance in practice (other than as an abstract measure of "goodness"). He pointed me to the Interpretation of value section of the Wikipedia article on Allan Variance. I am responding in this thread to that post so that I do not hijack the other thread in order to discuss this issue, which is really not related to the other thread's topic. I invite thermistor-guy to reply here, rather than in the other thread.

I will provide the quote from the Wikipedia article to which I believe thermistor-guy is referring to save people the trouble of finding it. It is short, so I think this is appropriate.

"An Allan deviation of 1.3×10⁻⁹ at observation time 1 s (i.e. tau = 1 s) should be interpreted as there being an instability in frequency between two observations 1 second apart with a relative root mean square (RMS) value of 1.3×10⁻⁹. For a 10 MHz clock, this would be equivalent to 13 mHz RMS movement. If the phase stability of an oscillator is needed, then the time deviation variants should be consulted and used."

It is not entirely clear, at least to me, what this means. Variance is a measure related to a probability distribution function (pdf), specifically that distribution's second moment. The statement in the above quote seems to suggest a deterministic "movement" in the signal's frequency. This is more apparent in the example, where it is suggested that for a 10 MHz clock, the movement would be equivalent to "13 mHz RMS movement". I think the problem is the idea of "movement" is left undefined. Given that Allan Variance is a variance, I would have expected an interpretation that referenced a probability bounds on the signal's frequency at the end of the 1 s period.

I would like to make one other point. When the article states, "If the phase stability of an oscillator is needed, then the time deviation variants should be consulted and used.", does this mean that the traditional variance of frequency fluctuations is the appropriate measure to use when computing phase stability? If someone knows the answer to this question, would they respond?

In order to forestall any misinterpretation of my response, I am grateful to thermistor-guy for pointing me to this information, so this post is not intended as a snotty rebuke. Rather, I just don't understand the explanation given in the Wikipedia article.

[awallin]

"An Allan deviation of 1.3×10⁻⁹ at observation time 1 s (i.e. tau = 1 s) should be interpreted as there being an instability in frequency between two observations 1 second apart with a relative root mean square (RMS) value of 1.3×10⁻⁹. For a 10 MHz clock, this would be equivalent to 13 mHz RMS movement. If the phase stability of an oscillator is needed, then the time deviation variants should be consulted and used."

That's a reasonable explanation, although note that conventional RMS is 'global' i.e. you compare each data-point to the mean of all data and compute the root-mean-square, while ADEV is 'local' in the sense that you only take consecutive frequency-points/pairs from the time series in order to build the sum.

It is not entirely clear, at least to me, what this means. Variance is a measure related to a probability distribution function (pdf), specifically that distribution's second moment. The statement in the above quote seems to suggest a deterministic "movement" in the signal's frequency. This is more apparent in the example, where it is suggested that for a 10 MHz clock, the movement would be equivalent to "13 mHz RMS movement". I think the problem is the idea of "movement" is left undefined. Given that Allan Variance is a variance, I would have expected an interpretation that referenced a probability bounds on the signal's frequency at the end of the 1 s period.

maybe the wikipedia use of 'movement' is not the best here - I don't think any deterministic movement should be understood.
In the example if you take a time-series of frequency-points, each averaged for 1s, and histogram the difference between consecutive points, you should get some (not necessarily known..) distribution with a width of 1.3e-9 in relative units (13 mHz if the time-series in in Hz).

I would like to make one other point. When the article states, "If the phase stability of an oscillator is needed, then the time deviation variants should be consulted and used.", does this mean that the traditional variance of frequency fluctuations is the appropriate measure to use when computing phase stability? If someone knows the answer to this question, would they respond?

TVAR is just MVAR scaled with the averaging-time (usually 'tau'), and thus TDEV has units of time (seconds). It predicts how much variance in phase (in units of time) to expect (in an RMS-sense) from one phase point to the next (where the spacing between points is tau).
In practice there are technical problems with measuring a (gap-free!) frequency time-series and then predicting (integrating) phase from that - not recommended. For timekeeping measure phase with a time-interval counter.

[rhb]

I just came across this by way of a mailing list.  A little surprised it had not been mentioned before.

http://www.ke5fx.com/gpib/pn.htm

[dnessett]

That's a reasonable explanation, although note that conventional RMS is 'global' i.e. you compare each data-point to the mean of all data and compute the root-mean-square, while ADEV is 'local' in the sense that you only take consecutive frequency-points/pairs from the time series in order to build the sum.

....

maybe the wikipedia use of 'movement' is not the best here - I don't think any deterministic movement should be understood.
In the example if you take a time-series of frequency-points, each averaged for 1s, and histogram the difference between consecutive points, you should get some (not necessarily known..) distribution with a width of 1.3e-9 in relative units (13 mHz if the time-series in in Hz).

....

TVAR is just MVAR scaled with the averaging-time (usually 'tau'), and thus TDEV has units of time (seconds). It predicts how much variance in phase (in units of time) to expect (in an RMS-sense) from one phase point to the next (where the spacing between points is tau).
In practice there are technical problems with measuring a (gap-free!) frequency time-series and then predicting (integrating) phase from that - not recommended. For timekeeping measure phase with a time-interval counter.

Thanks for the replies, but my question remains unanswered. I think I need some time to try to work it out myself, which seems silly because I am sure someone has already done so. Oh well, I suppose the attempt will have the benefit of clarifying the concepts in my mind even further.

[rhb]

I don't know why I didn't realize it earlier, but hands down the best way to make the measurements is to get an SDRplay RSP2 and a GPSDO that provides the  reference frequency required by the RSP2.  Leo Bodnar sells one which is why I bought it.  That combination will provide 12 bit data limited only by the available disk space.

Record the IQ stream to a stereo WAV file and import into Octave.

I'll try to get my GPSDO setup and collect some data from several sources later today.

Edit:  Probably won't happen today.  I got the GPSDO installed and producing 10 & 24 MHz, but the SDRplay RSP2 is not cooperating.  Windows says it is there, but SDRUno and SDR Console say it isn't :-(

[dnessett]

For me clarity about Allan Variance was greatly increased when RoGeorge posted his explanations (here and here) of the conceptual reasoning behind the increase of traditional variance as tau increases. I have thought about this and think I can supply a more mathematically inclined argument for this property. The concepts behind the argument are only roughly accurate, as should become apparent during their presentation, but it does have the advantage of a bit more rigour than RoGeorge's metaphorical description. It also supports a result that I have not yet seen mentioned.

Before presenting the mathematics, it is prudent to make sure everyone is on the same page in regards to terminology. I have read a lot of papers about Allan Variance and have noted the terminology is not quite consistent.

Figure 1 shows a sine wave generated by an oscillator. Periods of this sine wave are measured from one rising edge to the next. For the purposes of this presentation, at each rising edge the preceding period is determined to be either longer or shorter than the nominal period of an ideal oscillator. So, period 1 may be shorter than the nominal period (and hence averaged over the period, the oscillator has a greater frequency than the nominal ideal frequency). Or period 1 may be longer than the nominal period (and therefore when averaged over the period, the oscillator has a smaller frequency than the ideal oscillator). While it would be more accurate to include a third possibility, i.e., period 1 has the same frequency as the nominal oscillator, that would overly complicate the model and would not provide any significant clarity to the narrative.

Figure 1 -

Figure 2 shows mulitple periods of the oscillator. An averaging interval defined by the value tau measures whether the oscillator is of higher or lower frequency than a nominal ideal oscillator. For the purposes of the following mathematics, tau is always a multiple of the nominal ideal oscillator period. The averaging measurement (conceptually) occurs by noting the "polarity" of each oscillator period. Here "polarity" means whether for that period the oscillator was of greater frequency (G) or lower frequency (L) than the nominal ideal oscillator. There are m oscillator periods in each averaging interval.

Figure 2 -

Figure 3 shows how each averaging interval is utilized to create a total measurement. In particular n intervals of length tau are statistically analyzed to produce the measurement. The sample time (ST) is the value tau times n.

Figure 3 -

Consider the situation in Figure 1. Each period produces a result - either G or L. These results are analyzed over the averaging interval. If the probability of obtaining G is p, then the probability of obtaining an L is 1-p. For simplicity it is assumed that p=1-p=.5.

For measuring oscillator stability the statistic of interest is not how many Gs or Ls appear in an averaging interval, but the difference between these values. The process represented by an averaging interval is well-known and is called a bernoulli trial. The expected value of the difference between the number of Gs and Ls is presented here, specifically: 2mp - m = m(2p-1) = 0. [Note: the referenced web page uses n as the number of trials, whereas here that value is m. The value n is used here to represent the number of averaging intervals. Also, the problem solved there is stated in terms of successes and failures. The logic is exactly the same. Simply substitute L for success and G for failure.]

The variance of the difference between the two random variables in a Bernoulli trial (see above reference) is: 4mp² = m. Notice (!) that the variance depends on m. So, as the value of tau increases, so does the variance.

This has an interesting side-effect. The sample time equals tau * n. So, the traditional variance does not diverge as sample time increases. It diverges as the averaging time increases. Given the capabilities of computers in the 1960s and 1970s, when Allan Variance was developed, it was necessary to increase tau in order to obtain long-term measures of clock stability. Today, computers are much more powerful. So, it would be interesting to determine the sample_time/tau ratio above which an analyst would be forced to increase tau in order to obtain practical clock evaluation results. This would, of course, depend on the computer available. However, I would guess most desktop systems these days could analyze a very long data set in a practical amount of time.