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 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: 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 _______________________________________________ time-nuts mailing list -- time-nuts@febo.com To unsubscribe, go to https://www.febo.com/cgi-bin/mailman/listinfo/time-nuts and follow the instructions there.
