Central Limit Theorems for Wavelet Packet Decompositions of Stationary Random Processes (original) (raw)
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HAL (Le Centre pour la Communication Scientifique Directe), 2006
This paper is a contribution to the analysis of the statistical correlation of the wavelet packet coefficients issued from the decomposition of a random process, stationary in the wide-sense, whose power spectral density is bounded with support in [−π, π]. 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-Lemarié 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 to 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 to 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 power spectral density 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.
Some Results on the Wavelet Packet Decomposition of Nonstationary Processes
Eurasip Journal on Advances in Signal Processing, 2002
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.
ó Consider the wavelet packet coefcients issued from the decomposition of a random process stationary in the wide- sense. We address the asymptotic behaviour of the autocorre- lation of these wavelet packet coefcients. In a rst step, we explain why this analysis is more intricate than that already achieved by several authors in the case of the standard discrete orthonormal wavelet decomposition. In a second step, it is shown that the autocorrelation of the wavelet packet coefcients can be rendered arbitrarily small provided that both the decomposition level and the regularity of the quadrature mirror lters are large enough.
IEEE Transactions on Information Theory, 2000
The paper addresses the analysis and interpretation of second order random processes by using the wavelet packet transform. It is shown that statistical properties of the wavelet packet coefficients are specific to the filtering sequences characterizing wavelet packet paths. These statistical properties also depend on the wavelet order and the form of the cumulants of the input random process. The analysis performed points out the wavelet packet paths for which stationarization, decorrelation and higher order dependency reduction are effective among the coefficients associated with these paths. This analysis also highlights the presence of singular wavelet packet paths: the paths such that stationarization does not occur and those for which dependency reduction is not expected through successive decompositions. The focus of the paper is on understanding the role played by the parameters that govern stationarization and dependency reduction in the wavelet packet domain. This is addressed with respect to semi-analytical cumulant expansions for modeling different types of nonstatonarity and correlation structures. The characterization obtained eases the interpretation of random signals and time series with respect to the statistical properties of their coefficients on the different wavelet packet paths.
Digital Signal Processing, 2014
We present a second order statistical analysis of the 2D Discrete Wavelet Transform (2D DWT) coefficients. The input images are considered as wide sense bivariate random processes. We derive closed form expressions for the wavelet coefficients' correlation functions in all possible scenarios: inter-scale and inter-band, inter-scale and intra-band, intra-scale and inter-band and intra-scale and intraband. The particularization of the input process to the White Gaussian Noise (WGN) case is considered as well. A special attention is paid to the asymptotical analysis obtained by considering an infinite number of decomposition levels. Simulation results are also reported, confirming the theoretical results obtained. The equations derived, and especially the inter-scale and intra-band dependency of the 2D DWT coefficients, are useful for the design of different signal processing systems as for example image denoising algorithms. We show how to apply our theoretical results for designing state of the art denoising systems which exploit the 2D DWT.
Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significance testing
Physical Review E, 2007
We propose an equivalence class of nonstationary Gaussian stochastic processes defined in the wavelet domain. These processes are characterized by means of wavelet multipliers and exhibit well-defined timedependent spectral properties. They allow one to generate realizations of any wavelet spectrum. Based on this framework, we study the estimation of continuous wavelet spectra, i.e., we calculate variance and bias of arbitrary estimated continuous wavelet spectra. Finally, we develop an areawise significance test for continuous wavelet spectra to overcome the difficulties of multiple testing; it uses basic properties of continuous wavelet transform to decide whether a pointwise significant result is a real feature of the process or indistinguishable from typical stochastic fluctuations. This test is compared to the conventional one in terms of sensitivity and specificity. A software package for continuous wavelet spectral analysis and synthesis is presented.
Decorrelation of Wavelet Coefficients for Long-Range Dependent Processes
IEEE Transactions on Information Theory, 2007
We consider a discrete-time stationary long-range dependent process (X) such that its spectral density equals '(jj) , where ' is a smooth function such that '(0) = ' (0) = 0 and '() c for 2 [0; ]. Then for any wavelet with N vanishing moments, the lag k within-level covariance of wavelet coefficients decays as O(k) when k ! 1. The result applies to fractionally integrated autoregressive moving average (ARMA) processes as well as to fractional Gaussian noise.
In this paper, we give a strong motivation, based on new statistical problems mostly concerned with high frequency data, for the construction of second generation wavelets. These new wavelets basically differ from the classical ones in the fact that, instead of being constructed on the Fourier basis, they are associated with different orthonormal bases such as bases of polynomials. We give in the introduction three statistical problems where these new wavelets are clearly helpful. These examples are revisited in the core of the paper, where the use of the wavelets are enlightened. The construction of these new wavelets is given as well as their important concentration properties in spectral and space domains. Spaces of regularity associated with these new wavelets are studied, as well as minimax rates of convergence for nonparametric estimation over these spaces.