Magnus Danielson magnus at rubidium.dyndns.org
Sat Oct 27 17:25:30 EDT 2018

```Hi Ole,

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.

> 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
>
>
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```