Some Results on the Wavelet Packet Decomposition of Nonstationary Processes
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Some Results on the Wavelet Packet Decomposition of Nonstationary Processes Sami Touati Equipe Signal pour les Communication, Institut d’´electronique et d’informatique Gaspard-Monge and URA 820, Universit´e de Marne-la-Vall´ee, 5 boulevard Descartes, Champs sur Marne, 77 454 Marne la Vall´ee C´edex 2, France Email: [email protected]
Jean-Christophe Pesquet Equipe Signal pour les Communication, Institut d’´electronique et d’informatique Gaspard-Monge and URA 820, Universit´e de Marne-la-Vall´ee, 5 boulevard Descartes, Champs sur Marne, 77 454 Marne la Vall´ee C´edex 2, France Email: [email protected] Received 20 July 2001 and in revised form 15 July 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. Keywords and phrases: wavelets, wavelet packets, cyclostationary processes, nonstationarity, higher-order statistics, central limit, asymptotic statistics.
1.
INTRODUCTION
Nonstationary processes have recently received an increasing attention in the signal processing community, due to their wide applicability in modeling natural phenomena generated by physical systems with time-varying parameters. Let x(t), t ∈ R, be such a zero-mean random process. Its translated cumulants (if existing) cumn+1 x,u (t) = cum(x(u), x(t1 + u), . . . , x(tn + u)) depend in general on the lag u. When these functions in u are uniformly almost periodic (UAP) [1], we say that the process is nth-order almost-cyclostationary. This means that the cumulants have generalized Fourier series n (t)e2πηs u , with Ω countable [2, 3] and A s∈Ω s
1 D→+∞ D
Ans t = lim
D 0
−2πηs u cumn+1 du. x,u t e
(1)
If the dependence on u of the translated cumulants is periodic, the process is said to be nth-order cyclostationary. Cyclostationary and almost-cyclostationary processes are very useful for modeling many real signals which appear in communication, telemetry, radar, sonar, and economics [4]. In practice, the nth-order translated cumulants may also be assumed to have (n − 1)th-order finite energy, that is, to
belong to L2 (Rn−1 ). Moreover, they often correspond to regular functions. This structure within the data suggests the use of a multiscale analysis, for example, a wavelet analysis, for the extraction of these higher-order statistical informations, combine
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