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
On Sat, Oct 27, 2018 at 11:26 PM Magnus Danielson <
> 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
> > 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
> > Is this some elemental effect of integration (sqrt(n) or some such), or
> > I seeing the effects of bandwidth and/or bias-functions or other
> > 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:
> 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. :)
> > Thanks for any help!
> > Ole
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