[time-nuts] AVAR <-> S_Y conversion

Wolfgang Wallner wolfgang-wallner at gmx.at
Thu Mar 5 08:35:51 EST 2015

Hello time-nuts community,

I hope this is the right place for the following question :)

I'm dealing with the simulation of powerlaw noise, and I stumbled upon
something I cannot explain when I tried out some formulas of IEEE 1139 [1]:

According to Table B.2 in [1] the one-sided power spectral density of
fractional frequency data and the Allan variance of this data can be
related as follows:

PSD of FFD:   S_y(f) = h_alpha * f1^alpha

AVAR:    Sigma_y(Tau) = K * h_alpha * f ^ x

where K and x depend on the type of noise (alpha).

For your convenience I made a screenshot of the formulas I'm referring
to: http://postimg.org/image/6qcx3ggu9/

I have generated data sets with simulated powerlaw noise for different
values of alpha, and did the following for each of these noise vectors:

*) Calculated and plotted the Allan Variance
*) Used to formulas of [1] to calculate h_alpha
*) Calculated and plotted the FFD-PSD (the PSD plot is averaged the get
better visual results)
*) Added colored lines to both plots according to the calculated h_alpha

I would have expected that the colored lines would match each plot.
However, this is only the case for White PM, Flicker PM, White FM and
Flicker FM noise. To my surprise the calculated line for random walk
noise does not match the PSD plot.

For the random walk noise the expected line is off by a factor of
exactly 2 from the calculated plot, and I don't know how to explain this

I supposed that maybe the factor A is twice as large as it should be,
and thus I searched in other powerlaw noise publications for different
formulas. However, as far as I can see they agree with the definition
given in [1]. I could only find one paper with another definition: [2]
In that other paper A is defined as 2 pi^2/6 instead of 2 pi^2/3
(equation 24). Using this definition would result in a plot that matches
what I would have expected.

These are the graphs I'm referring to:

  White PM:   http://postimg.org/image/fk059s243/full/
  Flicker PM: http://postimg.org/image/6q71l03cp/full/
  White FM:   http://postimg.org/image/mxhxeszqx/full/
  Flicker FM: http://postimg.org/image/3vzpj7jmf/full/

  Random Walk: http://postimg.org/image/hxad6okwv/full/ <-- the bad guy

Does anyone know the reason for the behavior I see?

best regards, Wolfgang Wallner

PS: I tried to keep this mail short. If I have left out any information
that would be useful feel free to ask, please :)

[1] 1139-2008  -  IEEE Standard Definitions of Physical Quantities for
Fundamental Frequency and Time Metrology
[2] Gaderer, et al - Achieving a Realistic Notion of Time in Discrete
Event Simulation

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