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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