7 research outputs found
Central Limit Theorems for Wavelet Packet Decompositions of Stationary Random Processes
This paper provides central limit theorems for the wavelet packet
decomposition of stationary band-limited 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 -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.Comment: Submitted to the IEEE Transactions on Signal Processing, October 200
On the statistical decorrelation of the wavelet packet coefficients of a band-limited wide-sense stationary random process
Abstract This paper is a contribution to the analysis of the statistical correlation of the wavelet packet coefficients resulting from the decomposition of a random process, stationary in the wide-sense, whose power spectral density (PSD) is bounded with support in ½Àp; p. Consider two quadrature mirror filters (QMF) that depend on a parameter r, such that these filters tend almost everywhere to the Shannon 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-Lemarie´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 packet associated with a large enough resolution level, provided that the QMF order is large enough and above a value that depends on this wavelet packet. Another consequence of our derivation is that, when the coefficients associated with a given wavelet packet are approximately decorrelated, the value of the autocorrelation function of these coefficients at lag 0 is close to the value taken by the PSD of the decomposed process at a specific point. This specific point depends on the path followed in the wavelet packet tree to attain the wavelet packet under consideration. Some simulations highlight the good quality of the ''whitening'' effect that can be obtained in practical cases.
Noise Covariance Properties in Dual-Tree Wavelet Decompositions
Dual-tree wavelet decompositions have recently gained much popularity, mainly
due to their ability to provide an accurate directional analysis of images
combined with a reduced redundancy. When the decomposition of a random process
is performed -- which occurs in particular when an additive noise is corrupting
the signal to be analyzed -- it is useful to characterize the statistical
properties of the dual-tree wavelet coefficients of this process. As dual-tree
decompositions constitute overcomplete frame expansions, correlation structures
are introduced among the coefficients, even when a white noise is analyzed. In
this paper, we show that it is possible to provide an accurate description of
the covariance properties of the dual-tree coefficients of a wide-sense
stationary process. The expressions of the (cross-)covariance sequences of the
coefficients are derived in the one and two-dimensional cases. Asymptotic
results are also provided, allowing to predict the behaviour of the
second-order moments for large lag values or at coarse resolution. In addition,
the cross-correlations between the primal and dual wavelets, which play a
primary role in our theoretical analysis, are calculated for a number of
classical wavelet families. Simulation results are finally provided to validate
these results
Some Results on the Wavelet Packet Decomposition of Nonstationary Processes
Wavelet/wavelet packet decomposition has become a very useful tool in describing nonstationary processes. Important examples of nonstationary processes encountered in practice are cyclostationary processes or almost-cyclostationary processes. In this paper, we study the statistical properties of the wavelet packet decomposition of a large class of nonstationary processes, including in particular cyclostationary and almost-cyclostationary processes. We first investigate in a general framework, the existence and some properties of the cumulants of wavelet packet coefficients. We then study more precisely the almost-cyclostationary case, and determine the asymptotic distributions of wavelet packet coefficients. Finally, we particularize some of our results in the cyclostationary case before providing some illustrative simulations.</p