[time-nuts] AVAR <-> S_Y conversion
Tom Van Baak
tvb at LeapSecond.com
Thu Mar 5 22:29:10 EST 2015
List -- I apologize for letting Wolfgang's posting earlier today slip though the filter. Normally we pre-screen all URL's to prevent exposure to questionable "free image hosting" sites.
Wolfgang -- I have taken your 6 images and attached them to this posting. This way members can view your equations and plots without being exposed to postimg.org and its third party affiliates. The original text of your excellent posting is included below, with attachments instead of spam URL's.
Thanks,
/tvb
----- Original Message -----
From: "Wolfgang Wallner" <wolfgang-wallner at gmx.at>
To: "Time-Nuts" <time-nuts at febo.com>
Sent: Thursday, March 05, 2015 5:35 AM
Subject: [time-nuts] AVAR <-> S_Y conversion
>
> 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: [IEEE1139_Table_B_2.png]
>
> 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
> values
>
> 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
> behavior.
>
> 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: [White_PM.png]
> Flicker PM: [Flicker_PM.png]
> White FM: [White_FM.png]
> Flicker FM: [Flicker_FM.png]
>
> Random Walk: [Random_Walk.png] <-- 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
> _______________________________________________
> time-nuts mailing list -- time-nuts at febo.com
> To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts
> and follow the instructions there.
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