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 <

***@rubidium.dyndns.org> 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 -- time-***@lists.febo.com

> > To unsubscribe, go to

> http://lists.febo.com/mailman/listinfo/time-nuts_lists.febo.com

> > and follow the instructions there.

> >

>

> _______________________________________________

> time-nuts mailing list -- time-***@lists.febo.com

> To unsubscribe, go to

> http://lists.febo.com/mailman/listinfo/time-nuts_lists.febo.com

> and follow the instructions there.

>

_______________________________________________

time-nuts mailing list -- time-***@lists.febo.com

To unsubscribe, go to http://lists.febo.com/mailman/listinfo/time-nuts_lists.febo.com

and follow the instructions there.