## 3 Plane Wave Expansion from Finite Element Data
### Reference Case
#### 3.1.1 Regularization in the Formulation of the Inverse Problem  
A well-established technique to improve the stability of the solutions of inverse methods is the use of regularization in the inversion of the propagation matrix \(\mathbf{H}(\omega)\)[29; 30]. Tikhonov regularization is used for this purpose. It is based on the concept of changing the cost function \(J(\omega)\) by the inclusion of an additional term [26], that is:  
\[J(\omega)=\mathbf{e}^{H}(\omega)\mathbf{e}(\omega)+\beta(\omega)\mathbf{q}^{H }(\omega)\mathbf{q}(\omega), \tag{17}\]  
where \(\beta(\omega)\) is the regularization parameter. The minimization of the cost function \(J(\omega)\) of Equation (17) is given by [26]:  
\[\mathbf{q}(\omega)=\left[\mathbf{H}^{H}(\omega)\mathbf{H}(\omega)+\beta( \omega)\mathbf{I}\right]^{-1}\mathbf{H}^{H}(\omega)\mathbf{p}(\omega). \tag{18}\]  
Equation (17) indicates that the minimization of the cost function takes into account the sum of the squared errors between the reconstructed and target acoustic pressure and, in addition, the sum of the squared norm of plane wave amplitude vector. Figures 8-11 show the results obtained for the reference problem when Tikhonov regularization is applied. The value of \(\beta\), which is given in each case, was calculated as:  
\[\beta=\left\|\mathbf{H}(\omega)\right\|^{2}\Gamma, \tag{19}\]  
in which \(\left\|\cdot\right\|\) is the spectral norm (the largest singular value) of the propagation matrix \(\mathbf{H}(\omega)\) and \(\Gamma\) is an arbitrary constant whose value is selected between \(1\times 10^{-3}\) and \(1\times 10^{-6}\). Figure 8: Target (**a**) and reconstructed (**b**) field of the reference room, regularized \(\beta=0.33\). Figure 7: Phase error of the reference room at 63 Hz (**a**) and 250 Hz (**b**). Figures 8-11 indicate that the implementation of regularization reduces the area of accurate reconstruction at 63 Hz compared to the non-regularized case. In contrast, regularization does not have a significant effect in the sound field reconstruction at 250 Hz. This can be explained because the condition number at this frequency is low (45.9), and regularization therefore has little effect on the inversion of the matrix \(\mathbf{H}(\omega)\). Nevertheless, according to Figures 6 and 10, the amplitude of the acoustic field tends to be quieter for the regularized case at 63 Hz. This particular result is convenient for an interactive auralization system because the translation operator can lead to a zone with high acoustic pressure if no regularization is applied, which affects the stability and robustness of the implementation. An additional analysis of the energy distribution of the plane wave density \(q_{l}\) was conducted to evaluate the effects of regularization in the formulation of the inverse problem. Figures 12 and 13 show an interpolated distribution of the complex amplitude of the plane waves. This is plotted in two dimensions by unwrapping the unit sphere onto a 2D plane whose axes represent the elevation and azimuth angle in degrees. The total energy of the plane wave expansion is calculated from the expression:  
\[\mathbf{q}_{\text{total}}(\omega)=\sum_{l=1}^{L}\left|q_{l}(\omega)\right|^{2} \tag{20}\]  
Figure 11: Phase error of the reference room at 63 Hz (**a**) and 250 Hz (**b**), regularized \(\beta=0.33\). Figure 10: Amplitude error of the reference room at 63 Hz (**a**) and 250 Hz (**b**), regularized \(\beta=0.33\). Figure 9: Target (**a**) and reconstructed (**b**) field of the reference room, regularized \(\beta=0.33\). in which \(q_{l}\) is the complex amplitude of the \(l\)-th plane wave. This value is noted in the figure captions. These figures indicate that regularization has an important effect on the spatial distribution of the energy of the plane wave density at 63 Hz. For this frequency, a more concentrated directional representation was found when regularization is implemented. Figure 14 shows the energy of each plane wave component for the regularized and non-regularized solution.