nth-order fractional Brownian motion and fractional Gaussian noises

Emmanuel Perrin (1) , Rachid Harba (1) , Corinne Berzin-Joseph (2) , Ileana Iribarren (3) , Aline Bonami (4)

1 LESI - Laboratoire d'électronique, signaux, images
2 LABSAD - Laboratoire de Statistiques et Analyse des Données
3 UCV - Universidad Central de Venezuela
4 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

A generalization of fractional Brownian motion (fBm) of parameter H in ]0, 1[ is proposed. More precisely, this work leads to nth-order fBm (n-fBm) of H parameter in ]n-1, n[, where n is any strictly positive integer. They include fBm for the special case n=1. Properties of these new processes are investigated. Their covariance function are given, and it is shown that they are self similar. In addition, their spectral shape is assessed as $1/f^{\alpha}$ with $\alpha$ belonging to ]1; +∞[, providing a larger framework than classical fBm. Special interest is given to their nth-order stationary increments, which extend fractional Gaussian noises. Covariance function and power spectral densities are calculated. Properties and signal processing tasks such as a Cholesky-type synthesis technique and a maximum likelihood estimation method of the H parameter are presented. Results show that the estimator is efficient (unbiased and reaches the Cramér-Rao lower bound) for a large majority of tested values

Mots clés: self-similarity; 1/f processes; signal synthesis.; fractional Gaussian noises; fractional Brownian motion; signal synthesis; parameter estimation

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Emmanuel Perrin, Rachid Harba, Corinne Berzin-Joseph, Ileana Iribarren, Aline Bonami. nth-order fractional Brownian motion and fractional Gaussian noises. IEEE Transactions on Signal Processing, 2001, 49 (5), pp.1049-1059. ⟨10.1109/78.917808⟩. ⟨hal-00319136⟩