Abstract
Titre : Analyse multirésolution des signaux aléatoires
Autheurs : A.Cohen, J.Froment, J.Istas
This paper deals with the multiscale approximation
of a stationnary random signal. The results associate the regularity
in quadratic mean of this signal and the number of vanishing moments
for the analysing wavelet.
We consider a stationary random signal of the second order
and we examine its multiresolution approximation in the sense given by
Yves Meyer in [1], i.e. its projections on a ladder of embedded spaces
V_j representing its components at the scale 2^j.
We are in particular interested in the mean square error between the
signal and its approximation at a certain dyadic scale 2^j, and how this
error evoluates asymptotically when the scale gets finer. This problem can
be studied by using the wavelet basis associated to the multiresolution
analysis since these elements characterize at each scale the
missing information between a level of approximation and the next finer
level. By studying the expectation of the square modulus for each wavelet
coefficient, we obtain a formula for the mean square error which involves
both the wavelet which is used and the decay at infinity of the power
spectum for this random signal.
This result can be viewed as the statistical version of the Hölder
regularity characterization by the wavelet coefficients for deterministic
function which is presented in [1] and [2]. It is indeed well known that
the wavelet coefficient of a smooth function decay at a geometric rate
which is proportional to the degree of smoothness, provided that the wavelet
has enough vanishing moments. Here the Hölder regularity is replaced by
the ``statistical regularity'' expressed by the decay of the power spectrum.
[1] Yves Meyer, ``Ondelettes et Opérateurs'', ed Hermann, Paris, 1990.
[2] Stéphane Jaffard, ``Exposants de Hölder en des points donnés et
coefficients d'ondelettes'', C.R.A.S. Paris 308 série I, pp.79-81, 1989.