The Wavelet-Based Synthesis for Fractional Brownian Motion Proposed by F. Sellan and Y. Meyer: Remarks and Fast Implementation

Abstract

Abstract-The aim of this communication is to propose some complementary remarks and interpretation on the waveletbased synthesis technique for fractional Brownian motion proposed by Sellan in 1995. These comments will lead us to propose a fast and efficient pyramidal filter bank-based Mallattype algorithm, which permits an easy and efficient implementation of this synthesis technique. c 1996 Academic Press, Inc. MOTIVATIONFractional Brownian motion. Fractional Brownian motion (hereafter fBm) is a continuous-time random process proposed by Mandelbrot and Van Ness [11]. Basically, it consists in a fractional integration of a white Gaussian process and is therefore a generalization of Brownian motion (as defined by P. Lévy), which consists simply in a standard integration of a white Gaussian process. Because it presents deep connections with the concepts of self-similarity, fractal, long-range dependence or 1/f-processes, fBm quickly became a major tool for the various fields where such concepts are relevant. Many efforts have therefore been devoted to the possibility of performing numerical simulation for such a process (for a review, see [14]). None of these methods, however, was able to produce a process that possesses all the properties of fBm. Very recently, Sellan proposed [14, 15] a powerful wavelet-based analysis of fBm which also provides us with a general scheme to synthesize it.Scope of the communication. Our aim here is to propose a fast and efficient implementation of this synthesis technique that relies on the use of a fast filter bank-based Mallat-type pyramidal algorithm as well as to propose further remarks that give complementary viewpoints on this technique. While the next section restates the main ideas of the construction and theorems presented in [14], Section 3 clearly details how to derive the coefficients of the filter bank involved in the fast pyramidal algorithm. Section 4 addresses both practical issues and interpretation questions. Matlab routines can be obtained at the ACHA software ftp site, and are available upon request. WAVELET-BASED SYNTHESIS FOR fBmDefinition. Let us first recall the commonly used definition for fBm [11],where where σ 2 = Γ(1 − 2H) cos πH πH .377

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