Measuring periodic phase noise by phase structure and difference functions

[JohnnyMalaria]

[Please move if appropriate]

Hi,

I've been following a very interesting thread elsewhere in this forum concerning the measurement of oscillator stability and it has struck me how very similar the concepts and goals are to the mainstay of my work that seems at first to be a wholly unrelated discipline. However, the more I read about characterizing oscillator stability, the more I realize just how close it is to what I do.

I'd like to know if there is an unrecognized interdisciplinary overlap that may be of use.


Some background:

I am a physical chemist, specifically a colloid chemist, which means I study the behavior of nanoparticles dispersed in a material continuum. My particular focus is the measurement of the motion of charged nanoparticles in a liquid due to the application of an external electric field (electrophoresis). If a laser light is incident on the particles then the phase of the light scattered by them is determined by their position and the temporal change in the phase manifests itself as a Doppler frequency shift of the light.

The scattered light is mixed at a detector with unscattered light ('reference') that is frequency-shifted relative to the incident light by an amount w_o (typ. 250Hz - 20kHz). For a perfect stationary scatterer, this leads to a sinusoidal detector signal with the same frequency as the difference between the reference and scattered light. This can be considered the ideal oscillator.

In a version of this experiment that I developed for my PhD (phase analysis light scattering (PALS)), three separable components to the phase are considered:

1. Random brownian motion due to particle diffusion (gaussian)
2. Linearly changing with time due to a constant velocity (e.g., settling)
3. Periodic oscillation due to particle motion in an alternating field

Prior to the advent of PALS, attempts to separate these components were unsuccessful. Autocorrelation was the standard method. Unfortunately, because the photodetector signal is proportional to the intensity of the light, the phase cannot be obtained directly. With PALS, it can.

The detector signal, E(t), is just like for an oscillator:

E(t) = A(t)exp(i.phi(t))

with the optical phase being:

phi(t) = w_o(t) + phi_linear(t) + phi_gaussian(t) + B(t)cos(w_p(t))

Demodulating the photodetector signal about wo leads an IQ pair from which are readily transformed to A(t) and phi(t).

In this representation of phi(t), B(t)cos(w_p(t)) is akin to period phase noise with a frequency w_p and amplitude B(t). phi_gaussian represents the stochastic process from which the diffusion coefficient and, hence, particle size are estimated.

Determination of the three components uses statistical methods that are very similar in concept to those used for estimating Allan variance etc. For PALS, there are two functions that are constructed from the experimental amplitude and phase data: phase structure function and phase difference function. Computationally these are easy to construct esp. compared to autocorrelation/FFT.

The structure function allows for estimation of the gaussian term, and the amplitude and frequency of the periodic phase variation. No a priori knowledge of the frequency is needed. As with autocorrelation, the amplitude of the periodic phase variation can be dominated by the random noise. Unlike the autocorrelation function, the structure function does not decay and so coexisting fast and slow periodic phase variations can be revealed.

If the frequency is known then the phase difference function allows the amplitude and frequency of the periodic phase variation to be estimated in the absence of the random noise term.

In the realm of my experiments, I easily measure periodic phase variations of amplitudes of a few mrad in the presence of gaussian noise with sigma of the order of a few Hz. The periodic phase variation frequencies range from 1 to 1000Hz.

I hope you can see the parallels even though the scale of the numbers is doubtless quite different.

I'm very curious to see if this method can be used with IQ data obtained from oscillator stability experiments.

Here is a link to the pertinent and long chapter from my PhD thesis (file size too big to attach). The hard-core equations begin about half-way through but the first half should help show the similarities I have alluded to.

https://1drv.ms/b/s!AhmR5if7W0HCjccI0YLXWbEq5VTccA

In the light scattering world (which is very important in the pharmaceutical/food/cosmetic/paint/water treatment industries), PALS has replaced autocorrelation/FFT as the de facto data analysis method.

[rhb]

I think you should change the thread title to:

"Abandon All Hope, Ye who Enter Here"

On a serious note, it looks very interesting.  I just did a light skim through the chapter.  I'd forgotten how ugly typewriter math is.  Praise the lord for Donald Knuth.

My MS was in igneous petrology, so I was well trained in optical phenomena. And PhD studies in reflection seismology taught me classical Wiener analysis (not sure what else to call it).

The analysis looks applicable to the characterization of oscillators, particularly over short time periods.  The problem of acquiring suitable data is more vexing.

How would you collect the data without having a golden reference oscillator with which to compare?  The problem posited was deciding which oscillator was most golden.  And better phase measurement seems to be the big obstacle in horology.

[MasterT]

The scattered light is mixed at a detector with unscattered light ('reference') that is frequency-shifted relative to the incident light by an amount w_o (typ. 250Hz - 20kHz).

Never hear that light could be frequency-shifted. How? What kind of non-linear optics is that, I know they use some crystal to get blue and green laser beam, but its simply 2-nd and 3-rd harmonics

[JohnnyMalaria]

Thanks for being brave and taking the time to dip your toe into ye olde typewritten document

How would you collect the data without having a golden reference oscillator with which to compare?  The problem posited was deciding which oscillator was most golden.  And better phase measurement seems to be the big obstacle in horology.

Not sure I quite understand but maybe this will address it.


In the experiment, the frequency difference between the reference and scattered light can be generated in a couple of ways. The first is to use a vibrating piezo mirror to shift the frequency of the reference. This is a shit method used in all commercial instruments. The way I do it is to use acousto-optic modulation. The reference is modulated at, say, 40MHz, and the other at 40MHz + the desired shift frequency. Hence, the overall difference between the reference and the scattered light is the desired shift. Originally, I achieved this via SSBM but today I use a 4-channel DDS. Two provide the RF signals and two provide the I and Q components of the shift frequency. All the signals derive from the same master clock. I use analog quadrature detection of the detector signal to obtain my amplitude and phase information. Anyway, the relevance of this is that one of the IQ reference signals acts as the golden oscillator. I typically operate with the shift frequency 2 orders of magnitude higher that the frequencies I'm interested in measuring. My setup is basically a lock-in amplifier. The phase structure and difference functions can be constructed without knowing the oscillator frequency. A quadratic term or linear term, respectively, directly provide a frequency correction so that the demodulation frequency can be adjusted to match the center of the oscillator frequency spectrum. Because the stability of my DDS is much greater than that of the oscillator created by the particle motion, I can assume it to be golden though perhaps in need of polishing. So, in principle, the method can provide the following without a prior knowledge:

Oscillator center frequency
Periodic phase variation amplitude and frequency
Width of spectral line broadening due to gaussian noise


Of course, I do all of this <50kHz except for the RF signals for the acouso-optic modulation. How all this would translate into the RF world is one of the things I'm interested in. I'm not at all familiar with the typical orders of magnitude for things like signal sampling rate, random noise, period phase amplitude and frequency etc.

One interesting thing with the phase difference function is that it does require knowledge of the frequency of the phase oscillation which can be obtained from the structure function if it is dominant over the random noise. If the source of the phase oscillation is known and is "available", it can be used which overcomes the need to dominate the random noise. Again, I'm pretty much ignorant of this. Are typical period phase variations due to things like 60Hz interference from nearby power or are they of much higher frequencies? Also, how temporal coherent is that variation? i.e., how many periods would it remain within, say, 5%?


Bit of a long thought as is my habit.

[JohnnyMalaria]

The scattered light is mixed at a detector with unscattered light ('reference') that is frequency-shifted relative to the incident light by an amount w_o (typ. 250Hz - 20kHz).

Never hear that light could be frequency-shifted. How? What kind of non-linear optics is that, I know they use some crystal to get blue and green laser beam, but its simply 2-nd and 3-rd harmonics

In my case the shift is very small - a few 100Hz - 10kHz. The technique uses acousto-optic modulation that is used in laser printers, supermarket checkouts, wacky laser light shows and other applications. Here's a good explanation:

https://en.wikipedia.org/wiki/Acousto-optic_modulator

[KE5FX]

How would you collect the data without having a golden reference oscillator with which to compare?  The problem posited was deciding which oscillator was most golden.  And better phase measurement seems to be the big obstacle in horology.

From my reading, it's a homodyne application where the reference is an independent interferometer arm and the analysis is done at baseband. 

I don't immediately see how the frequency of the light can be shifted a few dozen kHz, though -- I'd think that the effect of the AOM would be to create PM sidebands around a carrier that remains at its original amplitude.  This stuff is very far from anything I've had experience with, but it does sound intriguing.  Beating the carrier under test with a frequency-shifted version of itself isn't something I've seen done before, just because at RF there is no straightforward way to shift the frequency of a carrier without mixing it with a local oscillator of some sort.

[rhb]

Because the stability of my DDS is much greater than that of the oscillator created by the particle motion, I can assume it to be golden though perhaps in need of polishing

This is the critical difference.  For the oscillator evaluation we don't have a golden reference.  We're trying to find one.

If one has 8 oscillators and compares phase among all 32 pairwise combinations we have an even determined system in 4 variables.  However we also have 2**32-1 possible states for which we need to collect statistics on their occurrence over time.   I *think* this can be solved, but it's pushing my boundaries quite hard.  I'm spending an hour or two each day contemplating this and scribbling on pieces of paper.  No assurance of progress though.

But it *is* interesting.  And I like the company :-)

[JohnnyMalaria]

How would you collect the data without having a golden reference oscillator with which to compare?  The problem posited was deciding which oscillator was most golden.  And better phase measurement seems to be the big obstacle in horology.

From my reading, it's a homodyne application where the reference is an independent interferometer arm and the analysis is done at baseband. 

I don't immediately see how the frequency of the light can be shifted a few dozen kHz, though -- I'd think that the effect of the AOM would be to create PM sidebands around a carrier that remains at its original amplitude.  This stuff is very far from anything I've had experience with, but it does sound intriguing.  Beating the carrier under test with a frequency-shifted version of itself isn't something I've seen done before, just because at RF there is no straightforward way to shift the frequency of a carrier without mixing it with a local oscillator of some sort.

Acoustooptic modulation exploits the Bragg effect. Yes, sidebands are created at spacings equal to the excitation frequency. The important thing is that in order not to violate the rule that energy cannot be created nor destroyed, the exit beam from the AOM is diffracted by an angle proportional to the excitation frequency. Hence, the output consists of multiple beams with each one being one of the sidebands. The relative intensities of each beam depends upon the angle of incidence of the incoming beam.

You say "independent interferometer arm". Just to clarify, both the reference light and the light from the sample come from the same laser source. One AOM operates at 40.000000MHz (say) and the other at 40.001000MHz to give a frequency difference between the beams of 1kHz. It isn't one beam that is shifted by this small amount. Laser-Doppler velocimetry is a similar method but because it is intended for much faster particle velocities it can use just one AOM.

[JohnnyMalaria]

Because the stability of my DDS is much greater than that of the oscillator created by the particle motion, I can assume it to be golden though perhaps in need of polishing

This is the critical difference.  For the oscillator evaluation we don't have a golden reference.  We're trying to find one.

If one has 8 oscillators and compares phase among all 32 pairwise combinations we have an even determined system in 4 variables.  However we also have 2**32-1 possible states for which we need to collect statistics on their occurrence over time.   I *think* this can be solved, but it's pushing my boundaries quite hard.  I'm spending an hour or two each day contemplating this and scribbling on pieces of paper.  No assurance of progress though.

But it *is* interesting.  And I like the company :-)

Part of my interest is to pragmatically compare two or more oscillators, e.g., Perhaps you need to select a matched pair or make sure that there isn't considerable periodic phase noise at a specific frequency or monitor an oscillator through its life.

[rhb]

I'm afraid I'm confused.

What is the application?    Clocks are different from frequency references which are different still from VFOs.

For my purposes, the GPSDO from Leo Bodnar should serve me well.  I'm interested in the metrology of oscillators.  But I'm fascinated by metrology in general, so no big deal there.

[tomato]

Clocks are different from frequency references ...

That's a puzzling sentence.

[JohnnyMalaria]

I'm afraid I'm confused.

What is the application?    Clocks are different from frequency references which are different still from VFOs.

For my purposes, the GPSDO from Leo Bodnar should serve me well.  I'm interested in the metrology of oscillators.  But I'm fascinated by metrology in general, so no big deal there.

Me, too!

Is this meant for the other thread on this subject?

[rhb]

The comment was meant for this thread, but they are closely related.  I assumed you want to split off the serious mathematics to this thread which seems to me appropriate.

In radar applications, phase noise is the critical metric.  In horology, long term frequency stability is the critical metric.  For a laboratory frequency reference I'd expect that the critical metrics are some weighted combination of those and other criteria.  Not having considered the problem before, what those criteria are is unclear.  It probably depends in part upon the characteristics of the instruments which use the reference.  As we are interested in metrology, we cannot assume a golden reference.  So finding a basis for measurement must be independent of the errors in physical instances.

I'd like to suggest getting a copy of "A Mathematical Introduction to Compressive Sensing" by Foucart and Rauhut.  Donoho and Candes work in the 2004-2010 time frame is in my view the biggest advance in data analysis since Wiener and Shannon.  Foucart and Rauhut do a good job of summarizing a lot of literature.  Compressive sensing is but one of many applications of sparse L1 pursuits.  The following quote  from https://statweb.stanford.edu/~donoho/Reports/2004/l1l0EquivCorrected.pdf should make clear why I hold the work in such high regard.

Many situations in science and technology call for solutions to underdetermined systems of equations, i.e. systems of linear equations with fewer equations than unknowns. Examples in array signal processing, inverse problems, and genomic data analysis all come to mind. However, any sensible person working in such fields would have been taught to agree with the statement:  “you have a system of linear equations with fewer equations than unknowns. There are infinitely many solutions.” And indeed, they would have been taught well. However, the intuition imbued by that teaching would be misleading.

On closer inspection, many of the applications ask for sparse solutions of such systems, i.e. solutions with few nonzero elements; the interpretation being that we are sure that ‘relatively few’ of the candidate sources, pixels, or genes are turned ‘on’, we just don’t know a priori which ones those are. Finding sparse solutions to such systems would better match the real underlying situation. It would also in many cases have important practical benefits, i.e. allowing us to install fewer antenna elements, make fewer measurements, store less data, or investigate fewer genes.

The search for sparse solutions can transform the problem completely, in many cases making unique solution possible (Lemma 2.1 below, see also [7, 8, 16, 14, 26, 27]). Unfortunately, this only seems to change the problem from an impossible one to an intractable one! Finding the sparsest solution to an general underdetermined system of equations is NP-hard [21]; many classic combinatorial optimization problems can be cast in that form.

In this paper we will see that for ‘most’ underdetermined systems of equations, when a sufficiently sparse solution exists, it can be found by convex optimization. More precisely, for a given ratio m/n of unknowns to equations, there is a threshold ρ so that most large n by m matrices generate systems of equations with two properties:

(a) If we run convex optimization to find the L1-minimal solution, and happen to find a solution with fewer than ρn nonzeros, then this is the unique sparsest solution to the equations; and

(b) If the result does not happen to have ρn nonzeros, there is no solution with < ρn nonzeros.

 In such cases, if a sparse solution would be very desirable – needing far fewer than n coefficients – it may be found by convex optimization. If it is of relatively small value – needing close to n coefficients – finding the optimal solution requires combinatorial optimization.

So far as I know, this is the only instance of solving an NP-hard L0 problem in L1 time.  At some point I need to investigate what the complexity crowd have done with it.  I only learned of it 9 years after the fact which stuns me as I should have expected to have learned of it within a year of publication.  Solving NP-hard problems is of major importance.

The mathematical niceties laid out in Foucart and Rauhut are extremely complex.  Donoho published a single proof that took 15 pages! However a similarly rigorous treatment of the Fourier transform would be equally long winded and difficult.  In practice one simply sets up Ax=y and tries to solve it.  You might not be able to solve it, but if you can, you have the correct answer.

I'm not suggesting that a sparse L1 pursuit is the answer, but there are some important concepts for which I know of no other treatment.

[nfmax]

A very similar technique is used in fibre-optic distributed vibration (or acoustic) sensing. In that case, however, there are no 'particle' scatterers: instead, the scatterers are the random, frozen-in thermal fluctuations of the glass refractive index within the fibre. This is Rayleigh scattering.

Vibration alternately stretches and compresses the length of fibre between two scatterers, phase modulating what is in effect a random carrier wave. This can be demodulated using a frequency-shifted version of the transmitted laser and the vibration signal recovered from the I & Q IF components (all complicated by optical polarisation effects). A pulsed laser source is used so that backscatter from different parts of the fibre arrives at different time delays. This gives the effect of an array of sampled vibration sensors along the fibre. Typically, you can get 5 to 10 m spacing for a few tens of km. The maximum sample rate is set by the fibre length.

While it doesn't have the dynamic range of a geophone, it is good enough to be used in seismic applications. The ability to install a fibre in a well allows for VSP applications, using either wireline or permanently installed fibre. It has also seen applications in microseismic monitoring, flow- and leak-detection, and perimeter security.

My colleague Arthur Hartog covers the technique & its applications in his recent book:

A. H. Hartog, An introduction to distributed optical fibre sensors. Boca Raton: CRC Press, Taylor & Francis Group, 2017. ISBN 978-1-4822-5957-5

[rhb]

If we take this equation as the model of the general case:

v(t) = [V0 + e(t)] * cos[w0*t + phi(t)]

Then I find that solving it without requiring a perfect oscillator as reference requires 8 oscillators giving us a total of 32 unknowns.  This in turn requires 32 independent linear equations.  The only way I know of to obtain those equations is by making pairwise comparisons among all 8 oscillators and solving the resulting set of simultaneous linear equations.

By superposition, we can make phi(t) arbitrarily complex to account for both PM and FM noise processes.  But I can see no way to proceed until the above has been solved.

I'll add the derivation later.  For now I'd like an independent confirmation of my result.

If anyone knows of a paper which discusses an analysis which requires 8 oscillators please supply a link or reference.  The "three corner hat" makes assumptions about some of the terms in the model above and only addresses FM noise.  So far as I can see, this is the simplest realistic model possible for arriving at phi(t) so the analysis in the dissertation chapter can be applied.

[JohnnyMalaria]

What is the typical form of phi(t)? If you write it as a linear combination of different noise terms, are the general forms of each term unique. What I mean is that in the PALS case phi(t) is of the general form a + b.t + c.t² + d.cos(wt) where w is the frequency of the periodic phase noise and d is its amplitude. It is easy to separate them when fitting the phase structure function. I haven't tried it but it should be possible to separate multiple periodic phase frequencies. This is similar to autocorrelation except that the structure function doesn't decay with the random noise and so much slower periodic phase variation should be detectable.

Can you explain the difference between a clock and an oscillator, and the significance in terms of applying different measurement approaches?

[rhb]

The principal figure of merit for a clock is that at various times in the future it is correct.  This requires in turn  a high degree of frequency stability.  So long as the phase noise is zero mean over the shortest period you wish to measure with the clock and this holds true over the full range from the shortest to the longest, the phase noise is irrelevant to time keeping.  The Allan variance was designed with this in mind as are MVAR and TVAR.  This can be seen in the plots of the frequency domain transfer functions.

An electronic oscillator is a means of realizing many things, clocks, radios, synchronous data transmission, etc.  The list is long.  Clocks  with mechanical oscillators existed for many years before electronic oscillators were developed.

One has in an electronic circuit many types of noise, flicker (i/f), shot, thermal, current plus possibly some more.  It gets hard to tell because of the tendency for the same phenomenum to be encountered in different contexts and given different names.  All of these noise sources have characteristic traits which generally exhibit themselves in the frequency spectra.  These are trivially separable from the amplitude spectrum using a sparse L1 pursuit.   I know of no other method that will make that separation.

I am not well versed in the details of electronic noise sources.  I have only a rudimentary grasp of the subject as I have not spent a lot of time on it.  So take the following with a pound or two of salt.

Flicker noise and shot noise are not correlated with current flow.  Thermal and current noise are.  So in an electronic oscillator the current and thermal noise are cyclostationary.   I commented on this previously in one of the two threads.  Measuring these using regular sampling per Shannon and Nyquist poses the problem that the noise is highly correlated with whatever clock is being used for the sampling.  Hence my repeated emphasis on the need for random sampling derived from a provably random source such as radioactive decay.

That requirement in turn poses a complication in that the correlation will creep back in via the sample time clock.  One can mitigate that to some extent by simply counting events over a time window, but the time window is correlated with the noise.  I've not calculated the degree of suppression.

In my "applications of sparse L1 pursuits" thread I presented a frequency plot of the effect of aliasing of  thermal noise in an integrating ADC such as is used in the HP 3458A.

I've had lunch and am getting sleepy.  So I'll stop for now.  Hopefully this has pointed in the right direction.  It is *very* closely related to you dissertation.  But the physics are different and neither of us is as good at the physics as needed.  Yet.

But to quote Harry Callahan, "A man's got to know his limitations."  If you do, you can remove them.  But woe if you don't.

[tomato]

The principal figure of merit for a clock is that at various times in the future it is correct.  This requires in turn  a high degree of frequency stability.  So long as the phase noise is zero mean over the shortest period you wish to measure with the clock and this holds true over the full range from the shortest to the longest, the phase noise is irrelevant to time keeping.

100% incorrect.

The Allan variance was designed with this in mind as are MVAR and TVAR.  This can be seen in the plots of the frequency domain transfer functions.

No, the Allan Variance was developed to link the time domain with the frequency domain picture. Your understanding and pronouncements about Allan Variance in these posts indicates a superficial understanding.

One has in an electronic circuit many types of noise, flicker (i/f), shot, thermal, current plus possibly some more.  It gets hard to tell because of the tendency for the same phenomenum to be encountered in different contexts and given different names.  All of these noise sources have characteristic traits which generally exhibit themselves in the frequency spectra.  These are trivially separable from the amplitude spectrum using a sparse L1 pursuit.   I know of no other method that will make that separation.

Except, of course, the Allan Variance.

I am not well versed in the details of electronic noise sources.  I have only a rudimentary grasp of the subject as I have not spent a lot of time on it.  So take the following with a pound or two of salt.

This is the most accurate statement you've made.

Flicker noise and shot noise are not correlated with current flow.

Shot noise scales with current.

Thermal and current noise are.

Thermal noise does not depend on current.

So in an electronic oscillator the current and thermal noise are cyclostationary.   I commented on this previously in one of the two threads.  Measuring these using regular sampling per Shannon and Nyquist poses the problem that the noise is highly correlated with whatever clock is being used for the sampling.  Hence my repeated emphasis on the need for random sampling derived from a provably random source such as radioactive decay.

Random sampling timed from radioactive decay?  Are you serious???

But to quote Harry Callahan, "A man's got to know his limitations."

Ironic statement from a guy who previously posted:

I'm not a time nut. I have never had justification for digging into the details of Allan variance.  I have less than a metrology lab technician knowledge of the subject.

[rhb]

Thermal and current noise are.

Thermal noise does not depend on current.

Heating is proportional to current and resistance.  Or at least in the universe the rest of us live in.  Therefore thermal noise depends upon current.  Apparently that is somewhat beyond your educational level.  So be a good lad and toddle off now.

Thirty years as a metrology lab tech does not make you an expert on anything.  If you actually understood what you claim, you could explain the logic behind your assertions.  But since you don't and there *isn't* any logic, all you do is make pompous claims of authority.  Sorry, but I don't bow to authority.

The Harry Callahan quote was directed at you.  You need to learn your limitations.

[tomato]

Heating is proportional to current and resistance.  Or at least in the universe the rest of us live in.  Therefore thermal noise depends upon current.  Apparently that is somewhat beyond your educational level.  So be a good lad and toddle off now.

Thermal noise depends on temperature. Claiming that it therefore depends on current because current can lead to temperature change is silly.

Thirty years as a metrology lab tech does not make you an expert on anything.  If you actually understood what you claim, you could explain the logic behind your assertions.  But since you don't and there *isn't* any logic, all you do is make pompous claims of authority.  Sorry, but I don't bow to authority.

The Harry Callahan quote was directed at you.  You need to learn your limitations.

I have made no claims of authority.  I have only pointed out several errors in your post.  For the record, however, I am not a lab tech and I am not someone new to the field of frequency standards.

[JohnnyMalaria]

The principal figure of merit for a clock is that at various times in the future it is correct.  This requires in turn  a high degree of frequency stability.  So long as the phase noise is zero mean over the shortest period you wish to measure with the clock and this holds true over the full range from the shortest to the longest, the phase noise is irrelevant to time keeping.

That makes sense. So, in my mind, a clock is a specific type of oscillator and so has specific measurement requirements enshrined in international standards whereas other types of oscillator have their own measurement requirements dependent on the applications.

That requirement in turn poses a complication in that the correlation will creep back in via the sample time clock.  One can mitigate that to some extent by simply counting events over a time window, but the time window is correlated with the noise.  I've not calculated the degree of suppression.

Could you not measure a given clock with multiple sampling devices such as multiple ADCs sampling the same signal? I would expect that the noise from the clock under test could be better estimated since the noise from each ADC's oscillator/clock(?) would be different.

I've had lunch and am getting sleepy.  So I'll stop for now.  Hopefully this has pointed in the right direction.  It is *very* closely related to you dissertation.  But the physics are different and neither of us is as good at the physics as needed.  Yet.

Thanks for the explanations. It's very interesting to see how similar concepts exist in multiple disciplines but are applied for very different purposes. It's a shame that it takes a crazy Australian dude's blog to see the parallels

[RoGeorge]

The principal figure of merit for a clock is that at various times in the future it is correct.  This requires in turn  a high degree of frequency stability.  So long as the phase noise is zero mean over the shortest period you wish to measure with the clock and this holds true over the full range from the shortest to the longest, the phase noise is irrelevant to time keeping.

That makes sense.

The claim in bold is false.

In a clock, the oscillator's phase noise will accumulate over time as an imprecision in time keeping, even if the average value of the phase noise is zero.

[rhb]

That requirement in turn poses a complication in that the correlation will creep back in via the sample time clock.  One can mitigate that to some extent by simply counting events over a time window, but the time window is correlated with the noise.  I've not calculated the degree of suppression.

Could you not measure a given clock with multiple sampling devices such as multiple ADCs sampling the same signal? I would expect that the noise from the clock under test could be better estimated since the noise from each ADC's oscillator/clock(?) would be different.

Under Gaussian assumptions you'd get a 1/sqrt(n) reduction, but I'm not convinced that sprinkling Gauss water on the problem is a good idea.  I have done that on occasion, but always with a great deal of nervousness.   The alternative would be to measure the clock with 7 ADCs which were all independently clocked. The larger problem is that a 10 MHz reference is expensive to sample at more than 3-4 degree intervals.  That's why I like tabulating the phase relationships.

This really gets deeply into your dissertation work.

Here's a more detailed outline of the hardware setup I'm proposing:

One has 8 input channels.  Each channel has a set of 7 low noise  op amp voltage followers feeding a fast comparator like the ADCMP581.  The voltage followers are there to isolate the comparators from each other.  The output of all the comparators is presented as a 32 bit word to a moderately fast processor such as  BeagleBone X15.

An additional comparator tracks the zero mean output of a physical noise source which has been low pass filtered so that the maximum frequency is less than the time required to read the word, fetch the count at that address, increment it and store it.  At every clock tick on the GPIO bus, you read the random noise comparator.  If it is high you read the phase word and increment the appropriate counter.

So for each sample window you are collecting the expectations for 4 billion states.  At the end of a sample window, you increment the base address of the array of counters and initiate a DMA transfer from memory to disk.  The ADCMP581 is not cheap, $1500/100 from Digikey.  But they are blisteringly fast.

From there it becomes an analysis problem. Which, naturally, is where we should start.

What we would have is the ratio of  A<B and A>B for all pairings over alarge number of time windows. If we can work out how to set up the mathematics we should be able to create a virtual perfect oscillator for which the uncertainties would be the physically irreducible noise.

It seems to me that there is high likelihood someone has already done this.  I've reinvented the wheel many times. But perhaps not.

The random sampling guarantees that if we find a periodicity, it is not a consequence of the sampling clock.  Which is why an analog random noise source with constant properties is so desirable.  Radioactive decay is trivially characterized and completely unaffected by temperature.  It is also a provably Gaussian event stream.

[rhb]

The principal figure of merit for a clock is that at various times in the future it is correct.  This requires in turn  a high degree of frequency stability.  So long as the phase noise is zero mean over the shortest period you wish to measure with the clock and this holds true over the full range from the shortest to the longest, the phase noise is irrelevant to time keeping.

That makes sense.

The claim in bold is false.

In a clock, the oscillator's phase noise will accumulate over time as an imprecision in time keeping, even if the average value of the phase noise is zero.

I did not make the statement in bold.   My statement was a sentence with an important constraint which you wish to ellide and then claim I am wrong.   I included that constraint precisely because the statement in bold is false.

Would you please show mathematically why a clock meeting the constraint I imposed would lead to an accumulation of errors?  I shall be very interested in your argument (aka mathematical proof).

[JohnnyMalaria]

This is really very interesting.

An additional comparator tracks the zero mean output of a physical noise source which has been low pass filtered so that the maximum frequency is less than the time required to read the word, fetch the count at that address, increment it and store it.  At every clock tick on the GPIO bus, you read the random noise comparator.  If it is high you read the phase word and increment the appropriate counter.

So, in effect, you end up sampling on average at half the GPIO clock rate with a lot of what I'd call jitter or spread, right?

So for each sample window you are collecting the expectations for 4 billion states.

Now I understand the source of the 2³²-1.

It seems to me that there is high likelihood someone has already done this.  I've reinvented the wheel many times. But perhaps not.

I've developed a lot of instruments (not just light scattering) over the years often preceded by people telling me "it won't work" or "it's been tried before, the results are crap." Most of those instruments turned out to out-perform the tried and tested (and expensive) ways of measuring a given physical process. Hence my signature