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382 IEEE SIGNAL PROCESSING LETTERS, VOL. 9, NO. 11, NOVEMBER 2002

Fast and Exact Synthesis for 1-D Fractional BrownianMotion and Fractional Gaussian Noises

Emmanuel Perrin, Rachid Harba, Rachid Jennane, and Ileana Iribarren

AbstractIn this letter, it is shown that fast and exact fractionalBrownian motion (fBm) and fractional Gaussian noise (fGn)signals can be synthesized by the circulant embedding method(CEM).CEMconsists inembeddingthe covariancematrixof the stationary fGn process in a larger circulantmatrix such that

. CEM is exact, since second-orderstatistics of the generated data are those of the Gaussian fGn.CEM is fast, since the optimal case

can be reached.Fast and exact fBm sequences can be easily recovered from fGnones.

Index TermsFast algorithms, fractional Brownian motion,fractional Gaussian noises, synthesis.

I. INTRODUCTION

FRACTIONAL Brownian motion (fBm) is a stochasticmodel for nonstationary fractal data [1]. This processhas stationary increments, namely fractional Gaussian noises

(fGn). FBM and FGN are often helpful for modeling numerous

real-world phenomena [2], [3].

The precise simulation of such signals is of great interest.

The most commonly used approaches can be split in two cate-

gories. The first one, related to theoritically exact methods, has

so far been composed only of a matrix factorization technique

based on the Cholesky decomposition of the fGn covariance

matrix [4]. Unfortunately, this technique has a complexity

of and requires high computational resources even

for moderate data length. The other category is composed of

nonexact techniques [2], [5]. All of the above methods have

their particular drawbacks and advantages. The choice between

them boils down to a tradeoff between speed and accuracy.

Dietrich and Newsam [6] have proposed a fast and exact

synthesis method for stationary Gaussian processes, called

the circulant embedding method (CEM). Since based on the

fast Fourier transform (FFT) algorithm, its complexity is only

. In this letter, we prove that CEM can be applied

to synthesize fast and exact fGn signals. Fast and exact fBm

data can be simply recovered from fGn ones. In Section II, fBm

and fGn are briefly presented.Manuscript receivedApril 3, 2002; revised July10, 2002. The associate editor

coordinating the review of this manuscript and approving it for publication wasDr. Xi Zhang.

E. Perrin was with the Laboratory of Electronics, Signal, Images, IPO-LESI,Universit dOrlans, BP 6745, 45067 Orleans Cedex, France. He is nowwith the Nuclear Magnetic Resonance Laboratory, UMR CNRS 5012,University Claude BernardLyon 1, Villeurbanne Cedex, France, (e-mail:Emmanuel.Perrin@univ-lyon1.fr.).

R. Harba and R. Jennane are with the Laboratory of Electronics, Signal, Im-ages, IPO-LESI, Universit dOrlans, BP 6745, 45067 Orleans Cedex, France.

I. Iribarren is with the Mathematical Department, Universidad Central deVenezuela, Caracas, Venezuela.

Digital Object Identifier 10.1109/LSP.2002.805311

II. FBM AND FGN

FBM of parameter in ]0,1[, denoted , is defined asan

extension of Brownian motion [1]. FBMis zero mean, Gaussian,

and second-order nonstationary. FGN, denoted , are defined

as

(1)

FGN is zero mean, Gaussian, and stationary, since its autocor-

relation can be written as

(2)

is the variance of . It should be noted that for ,

the function isalwaysnonnegative. Thissequence isalso

decreasing and convex, i.e., second differences are positive. Fi-

nally, for , one obtains for any integer

. Section III describes the CEM that will be used for gen-

erating fast and exact fBm and fGn signals.

III. CIRCULANTEMBEDDINGMETHOD

As fBm is nonstationary, it is more convenient to first

generate stationary fGn sequences and then to recover fBm

time series from these signals. For this reason, we shall take

as a canonical problem that of generating realizations of aone- dimensional (1-D) discrete stationary Gaussian process

of samples with zero mean and prescribed autocorrelation

function .

A. Description of CEM

Dietrich and Newsam [6] have proposed CEM to synthesize

stationary Gaussian processes over a regularly sampled domain.

This fast and exact method factorizes an extension of the

covariance matrix of the target process to produce random

vectors with exactly the required correlation structure via FFT.

The elements of are such that for

. CEM consists in embedding in a largermatrix such that . The optimal case

is called minimal embedding. The first row of ,

denoted , consists in the entries

(3)

If , the entries are

arbitrary, or conveniently chosen. is circulant, and thus any

matrix extracted along its diagonal is a copy of .

Being circulant, can be decomposed as where

1070-9908/02$17.00 2002 IEEE

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384 IEEE SIGNAL PROCESSING LETTERS, VOL. 9, NO. 11, NOVEMBER 2002

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