Singularity spectrum of fractal signals from wavelet analysis: Exact results - Centre de mathématiques appliquées (CMAP) Accéder directement au contenu
Article Dans Une Revue Journal of Statistical Physics Année : 1993

Singularity spectrum of fractal signals from wavelet analysis: Exact results

Résumé

The multifractal formalism for singular measures is revisited using the wavelet transform. For Bernoulli invariant measures of some expanding Markov maps, the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima. The generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures. It is demonstrated that the Hausdorff dimension $D(h)$ of the set of singularities of Hölder exponenth can be directly determined from the wavelet transform modulus maxima. The singularity spectrum so obtained is shown to be not disturbed by the presence, in the signal, of a superimposed polynomial behavior of ordern, provided one uses an analyzing wavelet that possesses at least $N>n$ vanishing moments. However, it is shown that a $C^∞$ behavior generally induces a phase transition in the $D(h)$ singularity spectrum that somewhat masks the weakest singularities. This phase transition actually depends on the number $N$ of vanishing moments of the analyzing wavelet; its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal. These theoretical results are illustrated with numerical examples. They are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional Brownian motions and turbulent velocity signals.
Fichier principal
Vignette du fichier
BMA.pdf (1.84 Mo) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01557137 , version 1 (05-07-2017)

Licence

Paternité

Identifiants

Citer

Emmanuel Bacry, Jean-François Muzy, Alain Arneodo. Singularity spectrum of fractal signals from wavelet analysis: Exact results. Journal of Statistical Physics, 1993, 70 (3-4), pp.635-674. ⟨10.1007/bf01053588⟩. ⟨hal-01557137⟩
240 Consultations
849 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More