[time-nuts] Re: Two questions about counters

[Magnus Danielson]

Hi Eric.

On 2023-09-05 17:21, Erik Kaashoek via time-nuts wrote:
> Pardon me for asking two highly specialized questions.
>
> In his may 2012 presentation  (see 
> http://athome.kaashoek.com/time-nuts/Counter%20Principles.pdf ) 
> Rubiola describes on slide 40 the enhanced resolution counter and 
> compares it to a Linear-regression counter on slide 45. Unfortunately 
> my match capabilities have disappeared in the 40 years since I left 
> university and I'm not able to understand the statement he makes ("The 
> linear regression estimator is asymptotically equivalent to the 
> enhanced resolution count") and the math is too complex for me to 
> understand if the enhanced resolution counter has any advantages over 
> the linear regression counter.
> I think I understand that the enhanced resolution counter has a well 
> defined decimation approach allowing a step wise decimation.
> The enhanced resolution counter however seems to require a lot more 
> memory related to the amount of decimation where a  linear regression 
> only needs a small set of running sums
> Is there someone in this community that can  explain the 
> (dis)advantages of these counter approaches in a simple way?

The enhancement of frequency estimation is there to provide a frequency 
estimation of higher resolution for the same update rate. Both the 
Delta-counter of avergage phase and Linear Regression counters achieve 
that. If getting a frequency estimation is what you search for, then 
using these methods is usually the right thing. If you attempt to 
measure *the* ADEV, it's not.

It is not the estimation itself which requires much processing. The 
Delta-counter enhancement requires one accumulator, the Omega-counter 
can be reduced to two integrators if doing the Least Square approach 
rather than the Linear Regression approach. What will increase memory 
need is the overlapping processing enabling higher read-out rate, which 
requites duplication by the read-out-rate multiplier, of say 10 times. 
So, the higher memory needs is not because of the base method, but how 
it has been implemented in particular products. The original 
Delta-counter did not do such overlapping processing and was hence very 
simple.

A particular issue is that counters producing outputs which is 
overlapping, will need to have post-processing respecting this, or 
further processing will fail.

Further, a delta-counter can has its output post-processed to change 
it's "software bandwidth" (as it was called in original MDEV article). 
However, you cannot do that for Omega-counters, and in the PDEV articles 
is says it cannot be decimated. Later decimation methods was provided, 
and to do that, you need to expose the two accumulating counters 
separately and not through the estimated value.

>
> In their 2016 paper "On temporal correlations in high-resolution 
> frequency counting" ( 
> http://athome.kaashoek.com/time-nuts/Measuring%20Counter%20Noise.pdf ) 
> the authors describe how using the undocumented RCON mode instead of 
> the CONT mode when using a Keysight 53230A can have certain 
> advantages, one seems to be the phase stability realized when using 
> the RCON mode
> The CONT mode is described as continuous resolution enhanced gap free 
> measurements and, although it is not formally known, the method used 
> could be a linear regression or enhanced resolution approach given the 
> slope of the ADEV when measuring the noise floor of the counter in 
> that mode.
> The RCON method without using internal averaging seems to have the 
> advantage for phase measurement that there is no cumulative error, 
> although the frequency resolution may be lower and it may be purely 
> based on the interpolated timestamps thus avoiding a cumulative phase 
> error.
> It is however possible to use the linear regression method to 
> calculate a phase relative to the counter reference so the output of 
> the counter (phase instead of frequency) is resolution enhanced but 
> the phase output does not have an accumulative error as would occur 
> when using the frequency output after resolution enhancement as it is 
> always referred to the counter reference.
You should be aware that this article was based on Ole observing these 
things, and then it was a good case for Tim to analyze it.
> For a counter that should be usable for ADEV and/or phase 
> measurements, would having a phase output mode without cumulative 
> error, either using a method similar to RCON or a linear regression 
> approach with phase hard linked to the reference, be an advantage?

You cannot use linear regression for ADEV, you need to use raw phase 
measures. RCON is doing the raw phase measurements rather than the 
delta-counting of CONT for the particular instrument, as given by the 
article.

I strongly suggest you read the IEEE Std 1139 as well as Allan Deviation 
Wikipedia article.

Cheers,
Magnus