## 2 Preamble
### Application for Squeal Noise and Convergence Results  
#### 3.4.1. Preamble: On the Use of the Multiresolution Signal Decompostion for Squeal Noise and Definition of Convergence Criteria  
The acoustic results for the prediction of squeal noise can be defined by matrices of coefficients representing the acoustic intensity on a discretized two-dimensional space (see for example the three images in 2D-dimension from Figure 3). These images are bounded patterns with a given resolution. There is therefore a maximum resolution beyond which it is no longer possible to extract detail from the signal. This resolution is linked to the mesh of the surface. Therefore, it defines the starting resolution of the decomposition algorithm. Since the acoustic observation surface is bounded, there is a minimum observation scale, beyond which the decomposition algorithm reaches a resolution which is that of the size of the surface itself. Assuming that the image of acoustic radiation is given by a matrix of size \(N\times M\) with \(N\geqslant M\), this minimum resolution is equal to \(l_{opt}=\lceil log_{2}(N)\rceil\) where \(\lceil\.\rceil\) sets rounding to the  
Figure 6: Illustration of the 2D wavelet decomposition of a pattern of size \(N\) for three iterations, according to [PERSON]’s algorithm. next higher unit. In this case, the two-dimensional image is completely represented by the \(3J_{opt}+1\) discrete images with the decomposition \(\mathcal{N}(f)_{I_{opt}}\) given in Equation (24). The matrices given in Equations (20)-(23) that correspond to the first iterations of \(j\) are those which analyze details at a fine resolution. So they can be associated with a small wavelength which characterizes the fine details of the pattern. On the contrary, the matrices associated with a larger \(j\) are those that reflect the fluctuations at high wavelength, characterizing the overall appearance of the two-dimensional studied image. So, it is possible to quantify the differences in the representation of the acoustic fields for different reduced models by analyzing and comparing details of a pattern for different C&B and DMS reductions, and this is for different observation scales ranging from the pixel to the size of the complete acoustic pattern. In order to quantify the quality of the convergence based on C&B or DMS reductions for the two-dimensional acoustic radiation, a first criterion \(\epsilon_{j}^{ac}\) based on the relative error between a signal \(f\) and the reference signal \(f_{ref}\) is proposed, such as  
\[\epsilon_{j}^{ac}(f)=\frac{\delta_{j}^{ac}(f_{ref}-f)}{\delta_{j}^{ac}(f_{ref} )},1\leqslant j\leqslant J_{opt} \tag{25}\]  
with  
\[\left\{ \begin{aligned} \delta_{J_{opt}}^{ac}(f)=\|\mathbf{A}_{ \mathbf{2}^{-I_{opt}}}f\|_{F}+\frac{1}{3}\sum_{i=1}^{3}\|\mathbf{D}_{2^{-I_{ opt}}}^{\mathbf{i}}f\|_{F}\\ \delta_{j}^{ac}(f)=\frac{1}{3}\sum_{i=1}^{3}\frac{\|\mathbf{D}_{2 ^{i}-j}^{i}f\|_{F}}{2^{2(I_{opt}-j)}},\quad\text{ for }\quad 1\leqslant j<J_{opt} \end{aligned} \right. \tag{26}\]  
where \(\|\,.\,\|_{F}\) defines the Frobenius norm. It should be noted that one of the major advantages of this criterion is that it makes it possible to have an appreciation of the quality of the convergence for different scales of observations (i.e., for each \(j\in\llbracket 1;J_{opt}\rrbracket\)) on the complete 2D image of the estimated acoustic field.