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

[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