# [time-nuts] AVAR <-> S_Y conversion

Wolfgang Wallner wolfgang-wallner at gmx.at
Thu Mar 12 06:05:03 EDT 2015

```Hello Tom, hello time-nuts,

> Have a look at the 20 plots in:
> http://leapsecond.com/pages/allan/Exploring_Allan_Deviation_v2.pdf

Thanks for sharing the data and the PDF! It's good to have reference values.

I also had a look at the stable32 user manual  to see how it
calculates the PSD graphs and compare it to my approach.

On page 195 there is a paragraph about "PSD Averaging", and it describes
basically the same averaging procedure that I use, so I guess this
approach is justified. The same page also has a paragraph about "PSD
Windowing", which I currently do not apply. I will have to read further
on that topic. What I could not find in the manual is how the green
lines in your PSD plots are calculated.

The manual also states that stable32 uses the noise generation as
described by Todd Walter, so it's the same approach that I apply.

> See if your results agree.

I have tried to analyze your data with my methods. To compare our
results I made screenshots of your graphs and used them as background
images in my plots. The PDF with my results is available at .
It is not explicitly stated in your PDF file, but from the graph
axis I assume a sampling frequency of 1Hz.

The images in the left column of my PDF show the estimated ADEV, and i
mostly agrees with your plots (at least below Tau = 10^2s).

If I use an averaging factor of 2 in the PSD estimation, the PSD plots
match yours very closely. These are the images in the center column in
my PDF. What was the averaging configuration for your plots?

The images on the right show the estimated PSD if I use an averaging
factor of 200. In these plots my estimates converge to single lines, but
they do not match the green lines of your graphs for any of the noise
types. It looks like they would be too large by a factor of 2 for any
noise type.

I have used both the frequency and phase data you provided for these
plots, and results agree (so at least my FFD/TD conversion scripts work :) )

I would interpret the comparison graphs as follows (please correct me if
you think otherwise):

*) The tool I use for ADEV calculation should be fine.
*) The script I use for PSD estimations returns usable results for an
averaging number of 2.

However, I'm still not sure how I this helps me to apply the domain
conversion formulas found in IEEE 1139 formulas to relate these plots
together.

Questions about the PSD plot generation:
*) How is the green line in your plots calculated? Or more generally:
>From a given set of noise data, how does one estimate h_alpha?
*) Is averaging the PSD until it converges to a line the wrong way to go?

General question about the IEEE1139 formulas:
*) What values do I need to use for f_h and Tau_0 when calculating the
terms D and E for WPM and FPM noise. If I have noise sampled at f_s =
1kHz, the smallest Tau value would be 1ms. Does that mean that Tau_0 =
1ms? And would f_h be f_s/2?

Thanks for your help so far,
Wolfgang

 http://www.wriley.com/Manual150.pdf
 http://leapsecond.com/tmp/graphs.pdf

```