Central Limit Theorems for Wavelet Packet Decompositions of Stationary Random Processes (original) (raw)

This paper provides central limit theorems for the wavelet packet decomposition of stationary bandlimited random processes. The asymptotic analysis is performed for the sequences of the wavelet packet coefficients returned at the nodes of any given path of the M-band wavelet packet decomposition tree. It is shown that if the input process is centred and strictly stationary, these sequences converge in distribution to white Gaussian processes when the resolution level increases, provided that the decomposition filters satisfy a suitable property of regularity. For any given path, the variance of the limit white Gaussian process directly relates to the value of the input process power spectral density at a specific frequency. Index Terms Wavelet transforms, Band-limited stochastic processes, Spectral analysis. I. INTRODUCTION This paper addresses the statistical properties of the M-Band Discrete Wavelet Packet Transform, hereafter abbreviated as M-DWPT. In [1], asymptotic analysis is given for the correlation structure and the distribution of the M-Band wavelet packet coefficients of stationary random processes. The limit autocorrelation functions and distributions are shown to be the same for every M-DWPT path. This seems to be a paradox because the M-DWPT paths are characterised by several sequences of wavelet filters. Two arbitrary sequences are different, and thus, do not have the same properties. In addition, the