# Sparse Plane Wave Approximation of Acoustic Modes to Address Basis Mismatch
## 3 Proposed Combined Two-Stage Method
### Stage 2: Estimation of PWAs and Directions Using Parametric SBL
#### 3.2.3 Hidden Random Variable Inference
Accordingly, the single-step update of \(\mathbf{x}\) is derived as  
\[\mathbf{x}^{(\text{new})}=\mathbf{x}^{(\text{old})}-[\
abla^{2}\mathcal{ L}(\mathbf{x})]^{-1}\
abla\mathcal{L}(\mathbf{x})|_{\mathbf{x}=\mathbf{x}^{(\text{old})}},\] \[\
abla\mathcal{L}(\mathbf{x})=2\Re[\text{diag}(\mathbf{\mu}^{(i)})(\mathbf{H _{y}})^{\text{T}}(\mathbf{H}(\mathbf{x})\mathbf{\mu}^{(i)}-\mathbf{\psi})^{*}+\text{diag}(\bm {\Sigma}^{(i)})\circ\text{diag}((\mathbf{H_{y}})^{\text{H}}\mathbf{H}(\mathbf{x}))],\] \[\ln\hat{Q}_{j}(\mathcal{H}_{j})=\mathbb{E}_{Q(\mathcal{H}\setminus\mathcal{H}_ {j})}[\ln P(\mathcal{H},\mathbf{\psi})], \tag{13}\]0  
where \(\{\mathbf{x},\mathbf{y}\}\) represents \(\{\mathbf{\vartheta},\mathbf{\vartheta}\}\) or \(\{\mathbf{\varphi},\mathbf{\phi}\}\); \(\
abla\mathcal{L}(\mathbf{x})\) and \(\
abla^{2}\mathcal{L}(\mathbf{x})\) denote the gradient and Hessian matrix of function \(\mathcal{L}\) at \(\mathbf{x}\), respectively; \(\Re[\cdot]\```
Initialize \(\mathbf{\Sigma}^{(0)}\leftarrow\mathbf{O}_{R^{\prime}}\), \(\mathbf{\phi}\leftarrow\mathbf{0}_{R^{\prime}}\), \(\mathbf{\varphi}\leftarrow\mathbf{0}_{R^{\prime}}\), \(\Delta_{w}\leftarrow\infty\), \(\mathbf{w}^{(0)}=\mathbf{\mu}^{(0)}\leftarrow\mathbf{0}_{R^{\prime}}\); set the parameters \(\{a,b,c,d\}\gets 10^{-6}\); set the maximum iteration \(i_{\text{max}}\) and stop tolerance \(\delta\); set the iteration index \(i=0\). while\(\Delta_{w}\geq\delta\) and \(i<i_{\text{max}}\)do \(i=i+1\);  Step 1 - Hidden random variable update:  Update \(\mathbf{\alpha}^{(i)}\) based on Equation (14);  Update \(\tau^{(i)}\) based on Equation (15);  Update \(\mathbf{w}^{(i)}\) based on Equation (17);  Step 2 - Unknown parameters update:  Update \(\mathbf{\theta}\) based on Equation (19) with \(\mathbf{x}=\mathbf{\vartheta}\);  Update \(\mathbf{\varphi}\) based on Equation (19) with \(\mathbf{x}=\mathbf{\varphi}\); \(\Delta_{\mathbf{w}}=||\mathbf{w}^{(i)}-\mathbf{w}^{(i-1)}||_{\infty}/||\mathbf{w}^{(i-1)}||_ {\infty}\)  end while  Perform Step 1 again based on the current \(\mathbf{H}(\mathbf{\theta},\mathbf{\varphi})\). The optimized PW directions are \(\{\mathbf{s}(\theta_{r}+\theta_{r},\phi_{r}+\varphi_{r})\}_{r=1}^{R^{\prime}}\), denoted as the set \(\mathcal{S}^{\prime\prime}\). If \(\theta_{r}+\theta_{r}<0^{\circ}\) or \(>180^{\circ}\), the inclination is adjusted to \(-(\theta_{r}+\theta_{r})\) or \(360^{\circ}-(\theta_{r}+\theta_{r})\). ```

**Algorithm 1** Stage 2: Estimation of PWs using parametric SBL (parSBL). ## 4 Simulations

### Validation Case: A Rigid-Walled Rectangular Enclosure

In our numerical simulation, we considered a rigid-walled rectangular enclosure of \(3.7~{}\text{m}\times 1.8~{}\text{m}\times 1.2~{}\text{m}\). Each mode can be evaluated as a sum of eight PWs whose directions can be expressed analytically [32]. The results obtained by the proposed L1

\[\ln\hat{Q}_{j}(\mathcal{H}_{j})=\mathbb{E}_{Q(\mathcal{H}\setminus\mathcal{H}_ {j})}[\ln P(\mathcal{H},\mathbf{\psi})], \tag{13}\]1 Flowchart of the combined two-stage method, L1-parSBL. parSBL were compared with conventional methods to examine the effectiveness and performance superiority of our method. We employed 100 sets of \(M=40\) randomly distributed samplings to measure the modes to eliminate the influence of random sampling positions. As it is difficult to calculate the estimation error of the PWAs directly, we used them further to reconstruct the mode and indirectly evaluate the relative mean error (RME) between the reconstructed mode values (\(\mathbf{\psi}_{\text{rec}}\)) and those desired (\(\mathbf{\psi}_{\text{des}}\)) at predefined field points in the volume as follows [14]:

\[\ln\hat{Q}_{j}(\mathcal{H}_{j})=\mathbb{E}_{Q(\mathcal{H}\setminus\mathcal{H}_ {j})}[\ln P(\mathcal{H},\mathbf{\psi})], \tag{13}\]2

In this case, the field points were set as \(30\times 15\times 10\) equally-spaced grid points in the enclosure. The selection threshold \(\beta\) was set to 0.1 because we clearly know that there are definite PWs objectively in this case.