Magnus Danielson magnus at rubidium.dyndns.org
Sun Oct 28 06:51:59 EDT 2018

```Hej Ole!

On 10/28/18 9:19 AM, Ole Petter Ronningen wrote:
> 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

:-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?)

The trouble is that the mathematical difference as you see in the table
is very very hard to make use of, as you have two functions with almost
the same shape, and the difference between them is small, so small that
the confidence interval for the noisy type of data make it hard to
extract the difference with any form of useful trust in the numbers. Add
that you have other forms of disturbances which isn't pure noise. For
most practical purposes they are indistinguishable using ADEV and this
annoyed David Allan for 15 years until he finally could present the
modified Allan, which is "the one" for him, as it is more complete in
the noise separation aspect, which was the driver for his work in the
first place.

So, the table is correct, but not very useful in this regard.

Remember that it is noisy data, and for any finite series of noisy data,
there is practical limits to how much we can derive out of them. We keep
inventing better tools to gain precision, reduce processing, reduce
length of measurement etc. Thus, theoretical differences may turn out
not be very useful in the practical world, so we need to do things in a
way that is practical.

When using MDEV you has an algorithmic bandwidth that change, which so
for higher tau you have a more narrow-band filter, which makee white
phase noise change amplitude much faster than flicker phase noise, and
hence the distinction can be made.

As you move over to parabolic deviation, it has even steeper filter and
thus suppress noise even more. This helps to explain the improved
performance of regression based frequency estimation.

So, ADEV is far from the right tool for everything. In fact, it is
greatly misused.

> 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]?

If you say that your samples is at tau_0=1s, then the sample-rate
becomes f_S = 1/tau_0 = 1 Hz and the f_H becomes f_H = f_S * 1/2 = 1/2 Hz.

Cheers,
Magnus

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