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

AbstractSThe aim of this communication is to proposesome complementary remarks and interpretation on the wavelet-based synthesis technique for fractional Brownian motionproposed by Sellan in 1995. These comments will lead us topropose a fast and ecient pyramidal lter bank-based Mallat-type algorithm, which permits an easy and ecient imple-mentation of this synthesis technique. 1996 Academic Press, Inc.c 1. MOTIVATION Fractional Brownian motion. Fractional Brownian mo-tion (hereafter fBm) is a continuous-time random processproposed by Mandelbrot and Van Ness [11]. Basically, itconsists in a fractional integration of a white Gaussian pro-cess and is therefore a generalization of Brownian motion(as de ned by P. L·evy), which consists simply in a standardintegration of a white Gaussian process. Because it presentsdeep connections with the concepts of self-similarity, frac-tal, long-range dependence or 1=f-processes, fBm quicklybecame a major tool for the various elds where such con-cepts are relevant. Many e orts have therefore been devotedto the possibility of performing numerical simulation forsuch a process (for a review, see [14]). None of these meth-ods, however, was able to produce a process that possessesall the properties of fBm.