Two questions about counters
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?
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.
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?
Erik.
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
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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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@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
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To unsubscribe send an email to time-nuts-leave@lists.febo.com