[time-nuts] Re: Two questions about counters

[Bruce Griffiths]

How exactly does a least squares fit differ from the usual linear regression fit?
Unless the linear regression is done without using least squares perhaps?

Bruce
> On 08/09/2023 11:01 NZST Magnus Danielson via time-nuts <time-nuts at lists.febo.com> wrote:
> 
>  
> 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
> _______________________________________________
> time-nuts mailing list -- time-nuts at lists.febo.com
> To unsubscribe send an email to time-nuts-leave at lists.febo.com