Spectral representation of fractional Brownian motion in n dimensions and its properties

  • Authors:

  • I. S. Reed;P. C. Lee;T. K. Truong

  • Affiliations:

  • Dept. of Electr. Eng. Syst., Univ. of Southern California, Los Angeles, CA;-;-

  • Venue:
  • IEEE Transactions on Information Theory
  • Year:
  • 2006

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Abstract

Fractional Brownian motion (fBm) provides a useful model for processes with strong long-term dependence, such as 1/fβ spectral behavior. However, fBm's are nonstationary processes so that the interpretation of such a spectrum is still a matter of speculation. To facilitate the study of this problem, another model is provided for the construction of fBm from a white-noise-like process by means of a stochastic or Ito integral in frequency of a stationary uncorrelated random process. Also a generalized power spectrum of the nonstationary fBm process is defined. This new approach to fBm can be used to compute all of the correlations, power spectra, and other properties of fBm. In this paper, a number of these fBm properties are developed from this model such as the TH law of scaling, the power law of fractional order, the correlation of two arbitrary fBm's, and the evaluation of the fractal dimension under various transformations. This new treatment of fBm using a spectral representation is extended also, for the first time, to two or more topological dimensions in order to analyze the features of isotropic n-dimensional fBm