High-order properties of M-band wavelet packet decompositions

Abstract

In many applications it is necessary to characterize the statistical properties of the wavelet/wavelet packet coefficients of a stationary random signal. This problem is typically encountered in denoising methods using wavelet packets. Then, in a stationary non-Gaussian noise scenario, it may be useful to determine the high-order statistics of the wavelet packet coefficients. In this work, we prove that this task may be performed through multidimensional filter banks. In particular we show how the cumulants of the M-band wavelet packet coefficients of a strictly stationary signal are derived from those of the signal and we provide recursive decomposition and reconstruction formulae to compute the cumulants of these coefficients. High-order wavelet packets, associated to multidimensional filter banks, are presented along with some of their properties. Finally, the asymptotic normality of the coefficients is proved.