## 2 Preamble
### Wavelet Transform and Logarithmic Observation Scales  
Due to the fact that the conventional Fast Fourier Transform (FFT)-based spectral analysis method provides poor representation of signals well localized in time, time-scale signal processing tools have to be used to provide a good description of signals. [PERSON] [30] proposed applying the wavelet approach to analyze the vibration of systems. The wavelet analysis transforms a signal into wavelets that are well localized in both frequency and time. The wavelet transform \(Wf\) can be expressed as  
\[Wf(s,x_{0})=\int_{-\infty}^{\infty}f(x)\sqrt{s}\phi(s(x-x_{0}))dx \tag{11}\]  
where \(\phi\) defines the mother wavelet. \(s\) corresponds to the expansion factor: it allows the function \(\phi\) to observe the studied signal \(f\) at a particular position \(x_{0}\). From the mother wavelet, it is possible to define a function \(\phi_{s}=\sqrt{s}\phi(sx)\), where the expansion factor \(s\) allows her to move the frequency of the bandpass filter of the base function and then to modify the width of the frequency band. By denoting \(<.\,,.>\) the usual scalar product of \(\mathbf{L}^{2}(\mathbb{R})\), it is then possible to rewrite the wavelet transform \(Wf\) as a convolution product between the function \(f\) and the wavelet \(\phi_{s}\), such as  
\[Wf(s,x_{0})=<f(x),\phi_{s}(x-x_{0})> \tag{12}\]  
Any function \(f\in\mathbf{L}^{2}(\mathbb{R})\) can thus be decomposed on the basis \(\left(\varphi_{j,k}\right)_{(j,k)\in\mathbb{Z}^{2}}\), such as  
\[f(x)=\sum_{(jk)\in\mathbb{Z}^{2}}<f(y),\varphi_{j,k}(y)>\varphi_{j,k}(x) \tag{13}\]  
with \(\varphi_{j,k}(x)=\sqrt{2^{-j}}\varphi_{2^{j}}(x-2^{-j}k)\) with \((j,k)\in\mathbb{Z}^{2}\). \(\left(\varphi_{j,k}\right)_{(jk)\in\mathbb{Z}^{2}}\) defines a wavelet family that corresponds to an orthonormal basis of \(\mathbf{L}^{2}(\mathbb{R})\). For a given observation scale \(2^{j}\), the set of functions \(\left(\varphi_{j,k}\right)_{k\in\mathbb{Z}}\) defines the vector subspace \(V_{2^{j}}\). Each \(V_{2^{j}}\) corresponds to a multiresolution vector space sequence. It can be noted that \(\forall j\in\mathbb{Z},\ V_{2^{j}}\subset V_{2^{j+1}}\), which reflects the fact that the successive sets of function approximations, allows for a more precise description of any function by a factor of 2. We can define the operator \(A_{2^{j}}\) which approximates some function \(f(x)\) at a resolution \(2^{j}\), such as  
\[\forall j\in\mathbb{Z},\ A_{2^{j}}f(x)=\sum_{k\in\mathbb{Z}}<f(y),\varphi_{j, k}(y)>\varphi_{j,k}(x) \tag{14}\]  
It should be noted that \(A_{2^{j}}\) is a linear operator. \(A_{2^{j}}f(x)\) is not modified if we approximate it again at the resolution \(2^{j}\) (i.e., \(A_{2^{j}}\) is a projection operator on a particular vector space \(V_{2^{j}}\subset\mathbf{L}^{2}(\mathbb{R})\)). Then, it is possible to extract the difference of information between the approximation of a function \(f(x)\) at the two resolutions \(2^{j}\) and \(2^{j+1}\). The discrete detail signal \(D_{2^{j}}f(x)\) is given by  
\[ \begin{split}\forall j\in\mathbb{Z},\ D_{2^{j}}f(x)=\sum_{k\in \mathbb{Z}}<f(y),\psi_{j,k}(y)>\psi_{j,k}(x)\\ \text{with }D_{2^{j}}f=\big{(}<f(y),\psi_{j,k}(y)>\big{)}_{k\in \mathbb{Z}}\end{split}  \tag{15}\]  
which represents the difference of information between the resolutions \(2^{j}\) and \(2^{j+1}\). It should be noted that the resolutions \(2^{j}\) and \(2^{j+1}\) of the function \(f(x)\) are, respectively, equal to its orthogonal projection on \(V_{2^{j}}\) and \(V_{2^{j+1}}\). So the discrete detail signal \(D_{2^{j}}f(x)\) is given by the orthogonal projection of the original signal on the orthogonal complement \(O_{2^{j}}\) of \(V_{2^{j}}\) in \(V_{2^{j+1}}\), such as \(O_{2^{j}}\circ V_{2^{j}}=V_{2^{j+1}}\). [PERSON] [28; 29] proposes an iterative algorithm allowing us to calculate the signal at the lower resolution (i.e., \(A_{2^{j}}\)) as well as the information between the resolutions \(2^{j}\) and \(2^{j+1}\) (i.e., \(D_{2^{j}}\)) from a signal \(A_{2^{j+1}}\).