On the asymptotic decorrelation of the wavelet packet coefficients of a wide-sense stationary random process
Résumé
This paper is a contribution to the analysis of the statistic al correla- tion of the wavelet packet coefficients resulting from the dec omposition of a random process, stationary in the wide-sense, whose pow er spectral density is bounded with support in [ − π, π ]. Consider two quadrature mirror filters (QMF) that depend on a pa- rameter r , such that these filters tend almost everywhere to the Shan- non QMF when r increases. The parameter r is called the order of the QMF under consideration. The order of the Daubechies filters (resp. the Battle-Lemari ́e filters) is the number of vanishing moments of the wavelet function (resp. the spline order of the scaling function). Given any decomposition path in the wavelet packet tree, the wavelet packet coefficients are proved to decorrelate for every packe t associated with a large enough resolution level, provided that the QMF o rder is large enough and above a value that depends on this wavelet packet. Another consequence of our derivation is that, when the coeffi cients associated with a given wavelet packet are approximately de correlated, the value of the autocorrelation function of these coefficients a t lag 0 is close to the value taken by the power spectral density of the decompos ed process at a specific point. This specific point depends on the path fol lowed in the wavelet packet tree to attain the wavelet packet under co nsideration. Some simulations highlight the good quality of the "whiteni ng" effect that can be obtained in practical cases.
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