Fast and Exact Synthesis for 1D fractional Brownian Motion and Fractional Gaussian Noises.pdf
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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:[email protected].).
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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[6] C. R. Dietrich and G. N. Newsam, Fast and exact simulation of sta-tionary Gaussian processes through circulant embedding of the covari-ance matrix,SIAM J. Sci. Comput., vol. 18, pp. 10881107, 1997.
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