Discrete Simulated Power Law Data in the Time Domain Using the Kasdin and Walter (1992) Method

This document will show graphically some discrete simulated power-law noise sequences as generated with the new program powlnsim.c on a Toshiba 1555 CDS laptop computer (380 MHz, 64 MB RAM, 4.3 GB HDD).

The Kasdin and Walter (1992) method works for any arbitrary exponent (Beta) and as an appendix they give core program c code. The Fourier transform and its inverse, uniform distribution random number generation (range 0-1), and generation of Gaussian random deviates were performed using c routines (Press et al., 1988). One programming change was made in that the user should put in a standard deviation rather than a variance. It reduces a square root computation, if nothing else! With comments, there are 634 lines of code in four programs.

The program was executed as follows to generate the simulated data:

powlnsim -n 2048 -slope < Beta > -scatter 0.250 -seed 1000

Each run of the program generated 2048 points, with the chosen Beta, with the indicated scatter, and with the same random number generator seed for each Beta. The Beta were selected to run from 0.0 (White PM) to -4.0 (Random Walk FM) at a step of 0.25, where "white" denotes non-frequency-dependent, PM is phase-modulated noise, and FM is frequency-modulated noise.

The results are shown below. Most interesting is to watch the noise grow up out of White PM dominate around White FM and then smooth out by Random Walk FM (leaving only the low-frequency/long period) structure. This indicates a scale change of the dominant noise from very high frequency to very low frequency. Also, there is an amplitude change going from the standard deviation value at White PM to the largest amplitudes for the Random Walk FM case. Continuous wavelet transform analysis should show these characteristic scale and amplitude changes very nicely, and show the fluctuations with time--this work has yet to be added.

Several of the standard noise exponent data were run through the program Stable32 and the characteristic slopes were indicated (TDEV) with some slight, but not unexpected, deviations.

Animated loop of all time series images. (154 KB)
Time to download
14.4 kbps: 1 m 54 s
28.8 kbps: 58 s;
56 kbps: 30 s;
and ISDN: 13 s.

Kasdin, N.J. and Walter, T. (1992), "Discrete Simulation of Power Law Noise," in Proceedings of the 1992 IEEE Frequency Control Symposium, pp. 274-283.
Press, W.H, et al. (1988). Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, Cambridge), p. 735.