Hi, Magnus! Thank you for this! I am of course nowhere near really comprehending it from reading it a couple of times, but I will still show my ignorance by asking a couple of questions with respect to the power law noise table. Counting on your day being still non-grumpy.. :D
1. I notice the formulas for ADEV is different for W PM and F PM - ADEV does not distinguish between the two, as is pointed out in the article. Does this not imply a fixed relationship between the power coefficients h1 and h2, such that the results of those to formulas are the same? Or am I misunderstanding the point of the table? (Also, what is the parameter y/gamma in the FPM formulas?) 2. I am not sure I understand the concept of f_H correctly, particularly as it applies to synthetic data. What is the corner frequency of a random sequence [0-1]? Ole On Sat, Oct 27, 2018 at 11:26 PM Magnus Danielson < [email protected]> wrote: > Hi Ole, > > I saw this post and thread, but waited until I had the time to address > it sufficiently, as it is an important topic. As such, I really enjoy > you asking the question as I am sure it will be a relevant question for > many more on this list. > > On 10/26/18 11:34 AM, Ole Petter Ronningen wrote: > > Hi, all > > > > I'm simulating some noise to try to improve my somewhat sketchy > > understanding of what goes on with the various noise types as shown on an > > ADEV plot. Nothing fancy, ~3600 points of gaussian random numbers > between 0 > > and 1 in excel, imported into Timelab as phase data, scaled to ns. > > I can recommend you and everyone else to use Stable32. You can download > it for free from IEEE UFFC. It not only do analysis, it also do noise > simulations for you. > > There is some work to be done on the source code. Uhm, that time. > > > I mostly get what I expect; "pure" random noise, gives the expected slope > > for W/F PM, -1. Integrating the same random data gives the expected slope > > for W FM -1/2. Integrating the same random data yet again gives a slope > > of +1/2, again as expected for RW FM. > > As expected from ADEV yes. > > > However, looking at the data, I am somewhat baffled by a difference in > the > > starting point of the slopes. Given that this is exactly the same random > > sequence, I would expect the curves to have the same startingpoint at > > tau0.. Clearly not (see attached), but I do not understand why. Any > clues? > > > > Is this some elemental effect of integration (sqrt(n) or some such), or > am > > I seeing the effects of bandwidth and/or bias-functions or other > esoterica? > > > > In case the screenshot does not make it though; > > W PM starts at 1.69e-9 > > W FM starts at 9.74e-10 > > RW FM starts at 6.92e-10 > > It depends on how the phase-noise slope as multiplied by the Allan > kernel and integrated over all frequencies behave. Each noise type > integrates up to different values for the same type due to the slope. > > I prepared a handy table for you when I completely rewrote the poor > excuse of a Wikipedia article that I found for Allan Deviation: > > https://en.wikipedia.org/wiki/Allan_variance#Power-law_noise > > As you simulate, you need to be careful to ensure that your simulated > noise matches that of the phase-noise slope so you do not get a bias there. > > Take a good look at the right-most column. Assume that h_2 to h_-2 all > have the same amplitude, that is the same energy at 1 Hz and we analyze > at the same tau=1s, the numbers will still be different and those comes > from how the integration of those slope works. > > The integration is very important aspect, as a number of assumptions > becomes embedded into it, such as the f_H frequency which is the Nyquist > frequency for counters, so sampling interval is also a relevant > parameter for expected level. > > I spent quite a bit of time trying to replicate these formulas, and it > taught me quite a bit. If I where a grumpy university professor holding > class on time and frequency, my students would be tortured with them up > and down to really understand them. > > For the not so grumpy and non-uni-professor me, I would easily spend a 2 > hour lecture on them. > > In short, they are not expected to start of at the same level, as the > homework was done we learned that they are not at all expected to start > at the same point. Do use the table as your reference for expected, and > adjust things to learn how to make numbers match up. > > The formulas that pops out from all the different variants of Allan > deviation and friends is different for the same slope, tau and f_H > parameters. As we then use say MDEV instead of ADEV, MDEV would fit the > MDEV expected values, but that would have an algorithmic bias to that of > ADEV, which can be estimated quite accurately separately if needed. > > The grumpy professor would say, and I would agree, that there is > fundamental differences and they are probably best understood by > studying the many different forms of representations there is for these > measures. Do study the cause of biases, as a sea of mistakes can be > avoided by understanding them. > > With that being said, good you caught me on a non-grumpy day. :) > > Cheers, > Magnus > > > Thanks for any help! > > Ole > > > > > > _______________________________________________ > > time-nuts mailing list -- [email protected] > > To unsubscribe, go to > http://lists.febo.com/mailman/listinfo/time-nuts_lists.febo.com > > and follow the instructions there. > > > > _______________________________________________ > time-nuts mailing list -- [email protected] > To unsubscribe, go to > http://lists.febo.com/mailman/listinfo/time-nuts_lists.febo.com > and follow the instructions there. > _______________________________________________ time-nuts mailing list -- [email protected] To unsubscribe, go to http://lists.febo.com/mailman/listinfo/time-nuts_lists.febo.com and follow the instructions there.
