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

[dnessett]

I have an advanced question that I decided to post here in case there are some who contribute to this forum who have expertise in signal measurement. I am developing a test procedure for characterizing the stability of some hobbyiest 10 MHz oscillators. This is in preparation for a study of a couple of 10 MHz distribution amplifiers. (For the record, I intend to make the results of this study available under the creative commons attribution-share alike license and I get no financial gain from the project - I am doing it to satisfy my curiosity.) While researching background information for this project, a question arose that I have been unable to answer through Googling appropriate words and phrases.

An oscillator is mathematically characterized as:

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.

For any practical oscillator, the stochastic processes e(t) and phi(t) are cyclostationary, which means their moments (e.g., mean and variance) are normally not constant (which would be true for a stationary process), but periodic. That means over time they change in value, but are periodic over some timeframe.

My problem is how to properly sample cyclostationary processes such as e(t) and phi(t). My uneducated gut feeling is that if I am attempting to characterize short term oscillator stability, I would want the time during which multiple samples are drawn (one sample being, for example, the number of zero crossings during an interval) to be defined so the moments of the probability density functions of e(t) and phi(t) vary only slightly. Otherwise, the samples at the beginning of the sampling period (i.e., the total time during which samples are drawn) would be influenced by one set of moment values and the samples drawn later would be influenced by another significantly different set of moment values.

To characterize long-term stability, my gut feeling is to use a sampling interval during which the moments of e(t) and phi(t) "cycle" several times. This would allow the calculation of an "average" of these moments.

Of course, even if my gut feelings are correct, it isn't clear how to determine an appropriate sampling period. Without getting into a lot of experimental work attempting to characterize the cyclostationary process associated with each oscillator (which is probably beyond the reach of my equipment and skill), I was hoping some general rules of thumb have been developed for common oscillators.

In addition, I was wondering what parameters might control the cyclostationary processes. The ones I could come up with after googling a bit are (with no distinction between short-term and long-term): temperature, humidity, power supply ripple, crystal and electrical component aging, mechanical vibrations, variations in loading. Some of these can be controlled in the short-term (e.g., temperature, humidity, mechanical vibrations), while others are probably only long-term factors (e.g., crystal and electrical component aging).

Given this background, my questions are: is my gut feeling about short-term and long-term sampling intervals correct, and if so, are there any general guidelines that would help me to develop an experimental design in regards to this question? Are there other parameters that would contribute to variation in the cyclostationary processes?

[tomato]

NBS Monograph 140 (Time and Frequency: Theory and Fundamentals)

[tautech]

Might be interesting to see if a narrow Mask could detect drift and small amplitude variations on a DSO.
I might have a play with this and pop up some screenshots of results.

[JohnnyMalaria]

If I understand correctly, you want to measure deviations in amplitude and phase for a nominally ideal 10MHz sinusoid?

If you have a stable (enough) 10MHz reference, why not just multiply the two signals together which will give you the sum and difference of the two (acting as a demodulator). The difference component contains information about the amplitude and phase differences. This concept is used for phase-sensitive detection (such as for lock-in amplifiers).


I'm not sure it is necessary to delve into stochastic processes etc.

[ap]

The variability of the phase is called phase noise and related to this is what is called Allen variance. You will find a lot of information on technics how to measure phase noise in older HP documents.

[tomato]

The variability of the phase is called phase noise and related to this is what is called Allen variance. You will find a lot of information on technics how to measure phase noise in older HP documents.

Allan Variance

[dnessett]

If I understand correctly, you want to measure deviations in amplitude and phase for a nominally ideal 10MHz sinusoid?

If you have a stable (enough) 10MHz reference, why not just multiply the two signals together which will give you the sum and difference of the two (acting as a demodulator). The difference component contains information about the amplitude and phase differences. This concept is used for phase-sensitive detection (such as for lock-in amplifiers).


I'm not sure it is necessary to delve into stochastic processes etc.

Thanks for your comment. However, you are describing a particular technique to make a measurement. I have read quite a bit about that topic and have a good handle on how to do it. However, in order to obtain statistically valid results you need to make multiple measurements and then process them. My question is about the interval over which these multiple measurements are made.

The variability of the phase is called phase noise and related to this is what is called Allen variance. You will find a lot of information on technics how to measure phase noise in older HP documents.

Yes, I understand that. In fact there are more modern versions of the Allan variance (e.g., the modified Allan variance, the Hadamard variance) that provide better confidence intervals than the Allan variance. I have read quite a few NBS/NIST technical reports about processing the sample data of an oscillator once you have obtained it, but none of them (except perhaps the one referenced by Tomato above - I haven't completely read its relevant sections) discuss how to select the sample interval during which you make N measurements.

[tomato]

In fact there are more modern versions of the Allan variance (e.g., the modified Allan variance, the Hadamard variance) that provide better confidence intervals than the Allan variance.

You want to characterize "hobbyist 10 MHz oscillators".  The good 'ole vanilla (2-pt.) Allan variance will give you all the information you need. There's no need to mess with modified Allan Variance.

I have read quite a few NBS/NIST technical reports about processing the sample data of an oscillator once you have obtained it, but none of them (except perhaps the one referenced by Tomato above - I haven't completely read its relevant sections) discuss how to select the sample interval during which you make N measurements.

You choose your sampling interval based on the shortest time period you are interested in characterizing your oscillator.  You choose the number (N) of data points based on the longest time period you are interested in characterizing your oscillator.

[awallin]

Thanks for your comment. However, you are describing a particular technique to make a measurement. I have read quite a bit about that topic and have a good handle on how to do it. However, in order to obtain statistically valid results you need to make multiple measurements and then process them. My question is about the interval over which these multiple measurements are made.

depends on the quality of the clock I guess.
FWIW with active hydrogen masers that start out at 1e-13 @ 1s it usually makes sense to look at only one phase-measurement per day. A typical time-interval counter has white phase-noise of around 2e-11/tau(s), so by measuring more often you only see the noise of the counter, not the maser.
The masers usually have linear frequency drift and it takes at least 30 days to get a reliable number for the drift, usually something like 1e-15/day.
With Cs-clocks it could make sense to measure a bit more often, since the cs-clocks are noisier wrt. the counter.
AFAIK most timing labs take phase-measurements every 6/10/12 minutes or so - but as mentioned above for a good clock one point per day has the useful information.

On the other hand when characterizing poor oscillators or e.g. AC-mains feedthru at 50/60 Hz it is useful to have a phase-meter (like microsemi 3120A) that outputs 1000 phase-measurements per second so you easily catch e.g. 50Hz spurs as oscillations in the ADEV.

for poorer clocks you can guess at what causes instability - and match the measurement to that. For example AC-mains oscillations or poor voltage regulation or DC-DC switchers, GNSS constellation diurnals, temperature/air-conditioning effects, ambient pressure variation (Rb-clocks!), and so on.

[dnessett]

There seems to be some confusion about the questions I asked in my first post. Both Tomato and awallin understood it and I thank them for their help. However, for the benefit of others, I will briefly elaborate.

An ideal oscillator is mathematically characterized as:

v(t) = V0 * cos[w0*t], where V0 and w0 are defined as in the original post. One characteristic of an ideal oscillator is its signal has exactly the same value at t and t+(2*PI).

However, ideal oscillators exist only in the minds of mathematicians. The signal generated by a real physical oscillator obeys the equation given in the first post, i.e., v(t) = [V0 + e(t)] * cos[w0*t + phi(t)]. The quantities e(t) and phi(t) are not deterministic functions. Rather, for a given value of t, say t=10, they are random variables. That is, e(10) and phi(10) supply a value controlled by a probability distribution function (pdf). The symbols e(t) and phi(t) represent (in this case) an uncountably infinite number of random variables, indexed by the variable t representing time. We say e(t) and phi(t) are stochasitic processes.

If two random variables have the same pdf, they are said to be independent and identically distributed (often abreviated as "i.i.d."). If the random variables comprising a stochastic process are all i.i.d., then that process is said to be stationary. Stationary processes lend themselves to the experimental determination of the moments of their pdfs (e.g., their mean and variance). Since all random variables associated with the stochasitic process are i.i.d., sampling their values a different times and computing statistics corresponding to their mean and variance is mathematically justified. All samples are from the same pdf, so such computations are valid.

However, the stochastic processes e(t) and phi(t) are not stationary, so you can't blindly use samples from different times to compute the underlying moments of their pdfs. Fortunately, these processes normally have a weaker property known as cyclostationarity. This means that periodically the random variables associated with them are i.i.d., i.e., for some value p, e(t) and phi(t) are the same random variable as e(t+p) and phi(t+p). This periodicity is normally not represented by a sine wave, so p is not a constant value for all values t; the periodicity is controlled by quite a few physical variables and generally its shape is unknown.

Characterizing the frequency stability of an oscillator consists of using sample data generated by the oscillator signal to estimate the variance of phi(t). But, since phi(t) is cyclostationary, the expermenter must take into account the change of its random varaible pdfs. Normally, it is assumed: 1) over very short time periods the pdfs change only slightly, and 2) over very long time periods, the pdfs cycle. In the first case, samples taken over a short time period approximate samples from i.i.d. random variables. In the second case, a large number of samples taken over a long period approximate those of a random variable that is a complex combination of the underlying stochastic process random variables. Thus, computing a variance estimate using these samples is a kiind of averaging of the underlying pdf variances. (Don't ask me about the mathematics of this "averaging" - I imagine it is pretty complicated).

The questions in my first post basically sought guidance on how to determine the appropriate values for short and long sampling periods; and in addition wondered what factors control the cyclostationarity of e(t) and phi(t).

[tomato]

However, the stochastic processes e(t) and phi(t) are not stationary ... these processes normally have a weaker property known as cyclostationarity.

You're going to have to convince me of the validity of this as a general statement.  The only non-stationary noise I've ever observed in an oscillator involved failing components or an improper test setup, and I've never observed cyclo-stationary noise in an oscillator.

Can you give an example of a cycle-stationary noise process in an oscillator?

[dnessett]

You're going to have to convince me of the validity of this as a general statement.  The only non-stationary noise I've ever observed in an oscillator involved failing components or an improper test setup, and I've never observed cyclo-stationary noise in an oscillator.

Can you give an example of a cycle-stationary noise process in an oscillator?

I'm just quoting what I have read here (see slide 4), here, here (2nd paragraph in section 1), and here (1st partial paragraph under figure 3). Intuitively, it makes sense. Consider just one parameter - temperature - that controls oscillator noise and for argument sake, ignore all others. For a given temperature, the noise should be pretty much identical each time the environment cycles through that value.

[tomato]

I'm just quoting what I have read here (see slide 4), here, here (2nd paragraph in section 1), and here (1st partial paragraph under figure 3).

The second and fourth links are discussing noise that is cyclic at the period of the oscillator, which would be 100 ns for a 10 MHz oscillator. I assume you will be making measurements on time scales many orders of magnitude larger than this, so the cyclo-stationary aspect is not important.

Intuitively, it makes sense. Consider just one parameter - temperature - that controls oscillator noise and for argument sake, ignore all others. For a given temperature, the noise should be pretty much identical each time the environment cycles through that value.

I don't think temperature dependent noise would be considered cyclo-stationary. Regardless, I've never heard of phase noise being a strong function of temperature variations with any well designed oscillator.

[dnessett]

The second and fourth links are discussing noise that is cyclic at the period of the oscillator, which would be 100 ns for a 10 MHz oscillator. I assume you will be making measurements on time scales many orders of magnitude larger than this, so the cyclo-stationary aspect is not important.

OK. It makes it much easier to design a sampling experiment if I can assume phase noise is a stationary process. That is the type of advice I am looking for. Thanks.

Following up, let me ask your advice on averaging and sampling times for short-term stability. To define short-term, I was thinking that most hobbyists use a time standard (e.g., ocxo or rubidium oscillator) to synchronize test equipment like frequency counters, oscilloscopes, spectrum analyzers, ... when conducting measurement experiments. So, my first guess is they would be interested in short-term stability on the order of several minutes to several hours. So, following your advice in a previous post, I should make the averaging interval on the order of a minute and the sampling interval on the order of several hours. Have I understood you correctly?

[tomato]

Following up, let me ask your advice on averaging and sampling times for short-term stability. To define short-term, I was thinking that most hobbyists use a time standard (e.g., ocxo or rubidium oscillator) to synchronize test equipment like frequency counters, oscilloscopes, spectrum analyzers, ... when conducting measurement experiments. So, my first guess is they would be interested in short-term stability on the order of several minutes to several hours. So, following your advice in a previous post, I should make the averaging interval on the order of a minute and the sampling interval on the order of several hours. Have I understood you correctly?

Are you asking what time frames are of interest to users of a "hobbyist" oscillator? I don't know the answer to that question.

[rhb]

Get a copy of "Random Data" by Bendat and Piersol.  They treat the data analysis very thoroughly including how to deal with non-stationary series.  That's been my go to for weird questions that walked into my office for 30 years and 3 editions.  The 4th is the final one as Piersol passed away.

Fundamentally you take long samples, window them and average.  The details depend upon what you want to characterize.  In reflection seismology one is usually doing this to characterize attentuation so one is averaging amplitude spectra.

[tomato]

Get a copy of "Random Data" by Bendat and Piersol.  They treat the data analysis very thoroughly including how to deal with non-stationary series.  That's been my go to for weird questions that walked into my office for 30 years and 3 editions.  The 4th is the final one as Piersol passed away.

Fundamentally you take long samples, window them and average.  The details depend upon what you want to characterize.  In reflection seismology one is usually doing this to characterize attentuation so one is averaging amplitude spectra.

This is not seismology.  There are specific analysis techniques (i.e. Allan Variance) that have been developed for characterizing clocks and oscillators. NBS Monograph 140 is the correct reference for this problem.

[dnessett]

Get a copy of "Random Data" by Bendat and Piersol.  They treat the data analysis very thoroughly including how to deal with non-stationary series.  That's been my go to for weird questions that walked into my office for 30 years and 3 editions.  The 4th is the final one as Piersol passed away.

Fundamentally you take long samples, window them and average.  The details depend upon what you want to characterize.  In reflection seismology one is usually doing this to characterize attentuation so one is averaging amplitude spectra.

Thanks. I have ordered a copy. When I get some data (I am building the test setup right now), I can analyze it to see if it represents a stationary or non-stationary process.

[KE5FX]

The book you're looking for may be this one by Rubiola.

[dnessett]

The book you're looking for may be this one by Rubiola.

Looks like a good book, but its expensive ($62.75 in paperback). I looked through the table of contents and there seems not to be an in depth treatment of sampling oscillator signals to derive stability characteristics. In the index, measurement of random processes is given a single page right at the beginning of the book (13), which suggests only a superficial treatment. However, if I am wrong and sampling is given a thorough treatment, let me know and I will consider buying it.

[rhb]

Having spent my career in signal analysis and having a long standing hobby interest in all aspects of electronics, I'm quite intrigued by your question.  So I'll be contemplating the measurement problem in the context of the applicable system errors.

One approach that occurs to me is to interpolate the times of the zero crossings relative to a high precision clock.  Then look at the variance from the mean as a function of time (aka Allan variance).  If you do a sparse L1 pursuit to obtain the frequency and phase over a cycle or two of a sine wave  and then solve that for the central zero crossing you should avoid problems with quantization and mismatch between the sampling period and the frequency.

I'll need to study the oscillator model you posited to say more.

[GerryBags]

A few docs that may be of interest, if you haven't read them already:

http://tycho.usno.navy.mil/ptti/1985papers/Vol%2017_05.pdf "Characterization, Optimum Estimation, and Time Prediction
of Precision Clocks"

https://fenix.tecnico.ulisboa.pt/downloadFile/3779572188799/Tn296.pdf "Characterization of Clocks & Oscillators" Covers portions of Monograph 140 mentioned above by Tomato. Co-authored by Allan.

http://www.photonics.umbc.edu/Menyuk/Phase-Noise/rutman_ProcIEEE_910601.pdf "Characterization of Frequency Stability
In Precision Frequency Sources" By J. Rutman & F. A. Wall

[dnessett]

NBS Monograph 140 (Time and Frequency: Theory and Fundamentals)

I finished reading section 8 of the Technical Report, which is the material germane to this thread. It is a good clear explanation of the problem/approaches and I found it very useful.

However, there is something missing from it and parenthetically from all of the material I have read so far. Specially, how do you use the Allan variance (or its square root, the Allan deviation) once you have computed it. I imagine someone setting up a system might want to know if a particular oscillator is suitable for use in that system. For example, a hobbyiest may wish to measure signal characteristics of some radio transmission system he is building. He wants to use a 10 MHz reference clock passed through a distribution amplifier to synchronize the instruments he is using to test his system.

I understand from the reading that I have done that computing the traditional variance of fractional frequency data doesn't work because it doesn't converge as the sample size increases. That is one of the reasons Allan created his variance measure. But, with the traditional variance (actually its square root, the standard deviation), if you assume a gaussian distribution of the fractional frequency process, 99.7% of the values lie within a 3 sigma band around the mean. So, if the designer had a traditional variance to work with, he could look at the range of frequencies within the 3 sigma band and decide whether that sort of jitter was acceptible for testing his system.

So far, I have found nothing like this for the Allan variance. How do you use it in a practical situation?

[dnessett]

A few docs that may be of interest, if you haven't read them already:

Thanks for the references. Comments below.

http://tycho.usno.navy.mil/ptti/1985papers/Vol%2017_05.pdf "Characterization, Optimum Estimation, and Time Prediction
of Precision Clocks"

I have read several reports by Allan, although I don't think I have read this particular one. I briefly looked it over and I think it doesn't contain anything I haven't read before.

https://fenix.tecnico.ulisboa.pt/downloadFile/3779572188799/Tn296.pdf "Characterization of Clocks & Oscillators" Covers portions of Monograph 140 mentioned above by Tomato. Co-authored by Allan.

Looks interesting and it is more recent than Monograph 140. I will take a look at it.

http://www.photonics.umbc.edu/Menyuk/Phase-Noise/rutman_ProcIEEE_910601.pdf "Characterization of Frequency Stability
In Precision Frequency Sources" By J. Rutman & F. A. Wall

This one also looks interesting.

[rhb]

My "I read it on the internet" understanding of the Allan variance is it simply is treating the variance as a random variable. So one has then the mean and variance of the variance.  One can take the variance and compute the FFT to look for periodicities in the variance. I *think* that would be the cyclostationarity, but I've never encountered that term before.  So I'm just guessing.

I rather suspect that this is a lexical minefield where different specialties define slightly different meaning and scaling conventions to the same words.