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Astron. Astrophys. 328, 670-681 (1997)

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2. Wavelets

An efficient multi-scale analysis is essential to studying solar activity records. Fourier analysis fails when the time-dependence of scale properties is to be considered. Wavelet analysis is more appropriate to a local scale analysis with variable resolution. Such a method was developed in the last decade and has already been described in reviews, see e.g. Farge, 1992, and books, see e.g. Meyer, 1992, Daubechies, 1992, Torresani (1995).

Wavelets, unlike sinusoidals, are localized near time [FORMULA] and decay if [FORMULA] exceeds the characteristic scale a. The wavelet representation can be compared to a kind of mathematical microscope with variable position and magnification. The wavelet transform represents one-dimensional signals as a function of both time and time-scale, and is similar to a local, filtered Fourier transform obtained by expanding and translating the wavelet, and then convolving it with the signal. The windowed Fourier transform contains three parameters (position, scale, frequency), while the wavelet contains two only.

As in the case of the Fourier transform, there are two basic kinds of wavelet transforms: continuous and discrete, that provide orthonormal basis. In this paper we are dealing with the continuous transform, although many of the properties will be valid for both the continuous and the discrete transforms.

2.1. What is a wavelet ?

A wavelet's family is generated from a mother wavelet function [FORMULA] by translations and dilatations,

[EQUATION]

where a and b correspond to the dilatation and the translation respectively.

There are certain requirements in order for the function [FORMULA] to be a wavelet. First, the function must have a zero mean (the admissibility condition)

[EQUATION]

Second, the function should be localized in both physical and Fourier space (time and frequency), i.e. its time spread [FORMULA] and its frequency spread [FORMULA] must satisfy the condition [FORMULA]. In addition, at least one reconstruction formula is needed for reconstructing the signal from its wavelet coefficients and for deducing the energy (or other invariants).

2.2. Continuous wavelet transform

The wavelet transform [FORMULA] of signal [FORMULA] is defined as

[EQUATION]

where [FORMULA] is a real or complex valued analyzing wavelet, satisfying conditions of subsection 2.1. and [FORMULA] stands for complex conjugate.

[EQUATION]

and [FORMULA] is the Fourier transform of [FORMULA]

[EQUATION]

If [FORMULA] the wavelet transform can be inverted (Grossmann and Morlet, 1984) as

[EQUATION]

One important property of the continuous wavelet transform is a generalization of the Parseval's theorem (Grossmann and Morlet, 1984)

[EQUATION]

as a consequence of which the equality is established between energy in physical and wavelet spaces.

The wavelet transform can be also related to the Fourier transform [FORMULA] of a signal [FORMULA] as

[EQUATION]

[EQUATION]

We also define the global wavelet spectrum, i.e. the energy contained in all wavelet coefficients of the same scale a, as a function of a:

[EQUATION]

which is related with the Fourier spectrum [FORMULA] as

[EQUATION]

and is actually a smoothed version of Fourier spectrum. Due to the normalization defined in (3), if the Fourier spectrum follows a power law [FORMULA] the global wavelet spectrum displays the same power law ([FORMULA]), as long as the wavelet presents enough cancellation (Perrier et al. 1995).

2.3. Some examples

It is evident that the wavelet representation depends upon the choice of the wavelet [FORMULA]. Two kinds of wavelets will be used in this paper: the real-valued wavelet, called the "Mexican hat" (Fig. 1a)

[EQUATION]

and the most widely used complex valued Morlet wavelet (Fig. 1b)

[EQUATION]

with [FORMULA] (chosen to approximately satisfy the condition (1)). The wavelet technique can be illustrated by canonical examples (see Appendix).

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© European Southern Observatory (ESO) 1997

Online publication: March 26, 1998

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