Hi Erik,
On 2022-01-30 12:46, Erik Kaashoek wrote:
In a timestamping counter I'm trying to calculate phase and frequency
using statistical techniques.
The counter has two counters, one for the input events and one for an
internal clock.
The capturing of these counters happens synchronized with an event.
The counter takes the timestamps at more or less regular, but not
identical, intervals
An example of 20 captures is:
Events Clock
280707207 1693452332
280708207 1693473665
280709207 1693494999
280710207 1693516332
280711207 1693537665
280712207 1693558999
280713207 1693580332
280714206 1693601644
280715207 1693622999
280716206 1693644310
280717206 1693665644
280718206 1693686977
280719206 1693708311
280720206 1693729644
280721206 1693750977
280722206 1693772311
280723206 1693793644
280724206 1693814977
280725206 1693836310
280726206 1693857644
By subtracting the counts at the first capture one gets:
Events Clock
0 0
1000 21333
2000 42667
3000 64000
4000 85333
5000 106667
6000 128000
6999 149312
8000 170667
8999 191978
9999 213312
10999 234645
11999 255979
12999 277312
13999 298645
14999 319979
15999 341312
16999 362645
17999 383978
18999 405312
It is visible the clock count interval is not completely constant and
the mathematical method used should be able to deal with these
interval variations.
To calculate the ratio between the frequency of the events and the
frequency of the clock a linear regression is calculated where the
event capture is the X and the clock capture is the Y.
The slope of the linear regression is the ratio between the frequency
of the events and the frequency of the clock.
To calculate the phase between the clock and the events it is assumed
that the regression also provides the Y intersect for X is zero as the
fractional correction of the clock count at event count zero as the
capture happened synchronized with the event (and NOT with the clock)
I think you got off to a great start there.
This leads me to my questions:
1: Is using linear regression as described above a good method to
calculate the phase relation between events and clock? If not, what
method to use?
Modern counters use linear regression. However, the usual formulas can
run into numerical precission issues. An alternative approach is to do
least square analysis, that gives the same benefit as linear regression,
but the calculation can be made to avoid or control numerical precission
issues.
To get started, classic linear regression is doing just fine.
You should be aware that the effective bandwidth you have will depend on
the number of samples, so as you later perform ADEV it will be affected
by the filtering of the linear regression/least square, the same
filtering helping you do good measures, as it supress white noise.
It is often incorrectly claimed that linear regression will give optimum
frequency estimation, this is not true.
2: For highest accuracy of the calculation output, is it best the
captures are at (almost) regular intervals (as above) or is some form
of dithering of the interval better? And what form of dithering is best?
That will not help you in a good way. Keep it as close to the same
interval as you can, and then let the linear regression work for you.
The supression slope will be the optimum 1/(N^1.5) until you hit the
noise floor of other noises.
There is a systematic noise you want to fight, and the best way to fight
it is with dither-signal onto the quantization, and the sample points on
a large scale is not the best method. Typically a "white noise" is
better. However, the other filtering method does most of this anyway so
it is rarely done.
3: Assuming it is possible to have a large amount (1e+5) of captures
per measurement interval, are there other or additional methods to
further improve the accuracy?
Linear regression / least square at a high rate can let you decimate
data for good precision and high read-out rate.
Maybe this [1] article of mine can be of inspiration.
Cheers,
Magnus
[1]
https://www.academia.edu/63957662/Memory_efficient_high_speed_algorithm_for_multi_%CF%84_PDEV_analysis
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