## 2. WaveletsAn 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. Wavelets, unlike sinusoidals, are localized near time
and decay if exceeds the
characteristic scale 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 by translations and dilatations, where There are certain requirements in order for the function to be a wavelet. First, the function must have a zero mean (the admissibility condition) Second, the function should be localized in both physical and Fourier space (time and frequency), i.e. its time spread and its frequency spread must satisfy the condition . 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 transformThe wavelet transform of signal is defined as where is a real or complex valued analyzing
wavelet, satisfying conditions of subsection and is the Fourier transform of If the wavelet transform can be inverted (Grossmann and Morlet, 1984) as One important property of the continuous wavelet transform is a generalization of the Parseval's theorem (Grossmann and Morlet, 1984) 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 of a signal as We also define the global wavelet spectrum, i.e. the energy
contained in all wavelet coefficients of the same scale which is related with the Fourier spectrum as and is actually a smoothed version of Fourier spectrum. Due to the normalization defined in (3), if the Fourier spectrum follows a power law the global wavelet spectrum displays the same power law (), as long as the wavelet presents enough cancellation (Perrier et al. 1995). ## 2.3. Some examplesIt is evident that the wavelet representation depends upon the choice of the wavelet . Two kinds of wavelets will be used in this paper: the real-valued wavelet, called the "Mexican hat" (Fig. 1a) and the most widely used complex valued Morlet wavelet (Fig. 1b) with (chosen to approximately satisfy the condition (1)). The wavelet technique can be illustrated by canonical examples (see Appendix). © European Southern Observatory (ESO) 1997 Online publication: March 26, 1998 |