[time-nuts] Omega counters and Parabolic Variance (PVAR)

Don Latham djl at montana.com
Tue Jul 14 15:37:21 EDT 2015


I look forward to the article, Magnus!
Don

Magnus Danielson
> Fellow time-nuts,
>
> Since I haven't seen any reports on this, I though I would write down a
> few lines.
>
> While normal counters use a pair of phase-samples to estimate the
> frequency, now called Pi counters (big pi, which has the shape of the
> weighing function of frequency samples), counter vendors have been
> figuring out how to improve the precision of the frequency estimation
> for the given observation time. One approach is to overlay multiple
> measurements in blocks, which for the frequency estimation looks like a
> triangle-shape weighing, so this type of counter is referred to as Delta
> counters (again to resemble the shape).
>
> Classical counters of the Pi shape is HP5370A, SR620 etc.
> Classical counter of the Delta shape is the HP53132A.
>
> However, counters using the Linear Regression methodology does not fit
> into either of those categories. Enrico Rubiola derived the parabolic
> shape of the weighing function (which I then independently verified
> after we spoke during EFTF 2014), and he then passed on the results to
> Francois Vernotte and other colleagues to continue the analysis.
>
> The new weighing function is a parabolic, looking like an Omega sign, so
> that is the name for this type of counter.
>
> Counters using the Omega shape is HP5371A, HP5372A, Pendelum CNT-90,
> CNT-91 etc.
>
> These weighing shapes acts like filters, and the block variant of the Pi
> weighing has no real filtering properties, where as both the Delta and
> Omega shapes has strong low-pass properties, which is beneficial in that
> they will suppress white phase noise strongly, and that is the typical
> measurement limitation of counters. The counter resolution limit also
> acts like white phase noise even if it is a systematic noise, which can
> interact in interesting ways as we have seen when signal frequencies has
> interesting relationships to the reference frequency. However, for cases
> when this is not true, the weighing helps to reduce that noise too from
> the measurements.
>
> For frequency estimation this is good improvements. This technique was
> actually introduced in optical measurements, as illustrated by J.J.
> Snyder in his 1980 and 1981 articles. This inspired further development
> of the Allan Variance to include the filtering technique of Snyder, and
> that resulted in the Modified Allan Variance (MVAR). Today we refer to
> the Snyder technique as the Delta counter.
>
> What Rubiola, Vernotte et. al discovered was that using a Linear
> Regression (LR) type of frequency estimated for variance estimation
> forms a new measure which they ended up calling Parabolic Variance
> (PVAR). They have done a complete analysis of PVAR properties (noise
> response and EDF) and it has benefits over MVAR.
>
> Variance made by a Delta counter thus becomes MVAR, but only as a
> special case.
> Variance made by a Omega counter becomes PVAR, but only as a special case.
>
> This is my main critique of their work, if you have access to the full
> stream of phase samples, you can form MVAR and PVAR using the two
> shaping techniques. However, if you use counters that perform these
> frequency estimations, then you can only correctly estimate variance of
> the two methods for the tau0 of the measurement result rate (and
> assuming that you know if they are back to back or interlaced, which is
> a mistake that was done at one time). If you have an Omega counter that
> produce frequency estimates and then process it further, the parabolic
> filtering shape does not change with m as it should for propper PVAR.
> This is exactly the same as using a Delta counter for frequency
> estimates and then perform variance estimation. For both cases, the
> counter will provide a fixed filtering bandwidth, but as you increase
> the m*tau0 for your analysis, the frequencies of your sample series will
> move into the pass-band of the low-pass filter and eventually the
> filtering effect is completely lost. The result is the hockey-puck
> response where the low-tau part of the ADEV/MDEV/PDEV curve first
> increases and then bends down to the white phase noise of the input as
> if it was not filtered.
>
> While Vernotte et al does not provide guidance for how to extend the
> PVAR from shorter measurements, I have proposed such a solution to them.
> Unfortunatly none of the existing counters will support that today.
>
> Why then, should one use PVAR? Well, PVAR does give good suppression of
> white flicker noise, and just as MVAR does has a 1/tau^3 curve rather
> than 1/tau^2 curve. This means that the measurement noise can be
> suppressed more effectively and the source noise can be reached for a
> lower tau. PVAR will have a 3/4 of MVAR for the white phase nosie, so
> there is a 1.25 dB improvement there.
>
> So, while it may read it from their papers that you get the PVAR from
> Omega counters, it's not the same in their analysis where the filtering
> function changes with m as you have with a typical counter which runs at
> fixed m. This is not to say that the PVAR technique is not useful.
>
> Getting proper results with these types of techniques takes care in the
> detail, but if you do you can harvest their benefits.
>
> For further reading, please check these articles:
>
> E. Rubiola, On the measurement of frequency and of its sample variance
> with high-resolution counters (PDF, 130 kB), Rev. Sci. Instrum. vol.76
> no.5 article no.054703, May 2005. ©AIP. Open preprint
> arXiv:physics/0411227 [physics.ins-det], December 2004 (14 pages, PDF
> 220 kB).
> http://rubiola.org/pdf-articles/journal/2005rsi-hi-res-freq-counters.pdf
>
> The Omega Counter, a Frequency Counter Based on the Linear Regression
> http://www.researchgate.net/publication/278419387_The_Omega_Counter_a_Frequency_Counter_Based_on_the_Linear_Regression
>
> Least-Square Fit, Ω Counters, and Quadratic Variance
> http://www.researchgate.net/publication/274732320_Least-Square_Fit__Counters_and_Quadratic_Variance
>
> The Parabolic variance (PVAR), a wavelet variance based on least-square fit
> http://www.researchgate.net/publication/277665360_The_Parabolic_variance_%28PVAR%29_a_wavelet_variance_based_on_least-square_fit
>
> I should probably shape this up into a proper article.
>
> Cheers,
> Magnus
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>


-- 
"If you don't know what it is,
don't poke it."
Ghost in the Shell
-------------------------------
"Noli sinere nothos te opprimere"

Dr. Don Latham, AJ7LL
Six Mile Systems LLC, 17850 Six Mile Road
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