Piecewise fractional Brownian motion

Starting from fractional Brownian motion (fBm) of unique parameter H, a piecewise fractional Brownian motion (pfBm) of parameters H/sub o/, H/sub i/, and /spl gamma/ is defined. This new process has two spectral regimes: It behaves like an fBm of parameter H/sub o/ for low frequencies |/spl omega/|</spl gamma/ and like an fBm of parameter H/sub i/ for high frequencies |/spl omega/|/spl ges//spl gamma/. When H/sub o/=H/sub i/, or for limit cases /spl gamma//spl rarr/0 and /spl gamma//spl rarr//spl infin/, pfBm becomes classical fBm. It is shown that pfBm is a continuous, Gaussian, and nonstationary process having continuous, Gaussian, and stationary increments, namely, piecewise fractional Gaussian noises. The asymptotic self-similarity of pfBm is shown according to the considered regime: At large scale, the process is self-similar with parameter H/sub o/ and with parameter H/sub i/ at low scale.

[1]  Rachid Harba,et al.  Fast and exact synthesis for 1-D fractional Brownian motion and fractional Gaussian noises , 2002, IEEE Signal Processing Letters.

[2]  R. Harba,et al.  Fractal estimation in a given frequency range. Application to smectite images , 1998, 9th European Signal Processing Conference (EUSIPCO 1998).

[3]  A. Yaglom.  Correlation Theory of Stationary and Related Random Functions I: Basic Results , 1987 .

[4]  Rachid Harba,et al.  DETERMINATION OF FRACTAL SCALES ON TRABECULAR BONE X-RAY IMAGES , 1994 .

[5]  G. T. Zunich.  Modified MUSIC in the presence of random phase errors , 1993, 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[6]  Trieu-Kien Truong,et al.  Spectral representation of fractional Brownian motion in n dimensions and its properties , 1995, IEEE Trans. Inf. Theory.

[7]  C.-C. Jay Kuo,et al.  Extending self-similarity for fractional Brownian motion , 1994, IEEE Trans. Signal Process..

[8]  Mohamed A. Deriche,et al.  Signal modeling with filtered discrete fractional noise processes , 1993, IEEE Trans. Signal Process..

[9]  Dale W. Schaefer,et al.  Fractal geometry of colloidal aggregates , 1984 .

[10]  Petre Stoica,et al.  MUSIC, maximum likelihood, and Cramer-Rao bound , 1989, IEEE Transactions on Acoustics, Speech, and Signal Processing.

[11]  Rachid Harba,et al.  nth-order fractional Brownian motion and fractional Gaussian noises , 2001, IEEE Trans. Signal Process..

[12]  J. L. Véhel,et al.  The Generalized Multifractional Brownian Motion , 2000 .

[13]  C.-C. Jay Kuo,et al.  Texture Roughness Analysis and Synthesis via Extended Self-Similar (ESS) Model , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[14]  B. Mandelbrot,et al.  Fractals: Form, Chance and Dimension , 1978 .

[15]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[16]  A. Lee Swindlehurst,et al.  A Performance Analysis of Subspace-Based Methods in the Presence of Model Errors: Part &-Multidimensional Algorithms , 1993 .

[17]  Rangasami L. Kashyap,et al.  Synthesis and Estimation of Random Fields Using Long-Correlation Models , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[18]  B. Hofmann-Wellenhof,et al.  Introduction to spectral analysis , 1986 .

[19]  K. Buckley,et al.  Robust high-resolution eigenspace DOA estimation , 1992, [1992] Conference Record of the Twenty-Sixth Asilomar Conference on Signals, Systems & Computers.