[time-nuts] Re: Two questions about counters

Erik Kaashoek erik at kaashoek.com
Fri Sep 8 13:42:44 UTC 2023 

Hi Magnus,
Thanks again for the input. I do understand the LR counter does filter 
the phase noise close to the smallest tau and thus can not be used as 
input to calculate a correct ADEV.
Now I am trying to build the calculations you specify in your paper on 
"Least square estimation of ...." but I'm stuck
What I understood (I think at least)
An array of phase samples is used: x0 .. xN-1 with t0 time between the 
samples. (formula 11) and (formula 12)
You calculate the
C = sum n=0..N-1 (xn)                (formula 15)
D = sum n=0..N-1 (n*xn)            (formula 16)

the x (phase) estimator is now given in (formula 19) and can be 
calculated using N,C and D
the y (frequency) estimator is now given in (formula 20) and can be 
calculated using N,C and D and t0

I've create an excel spreadsheet (attached) with simulated x input and 
this calculates C,D, x est and y est.
However I do not understand the outcome.
If I use LINEST from excel I get the expected phase and frequency 
numbers and I've implemented LINEST (or LR) in my counter, which 
understandably does not create the right input for ADEV processing.

Guess I have to study a lot more before being able to understand all this.
Erik.


On 8-9-2023 1:01, Magnus Danielson via time-nuts 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
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