# Sparse Plane Wave Approximation of Acoustic Modes to Address Basis Mismatch
## 2 PW Approximation of Acoustic Modes  
An acoustic mode function \(\psi_{\mathbf{n}}\) of order \(\mathbf{n}\) can be approximated by a finite sum of PWs sharing the same wavenumber \(k_{\mathbf{n}}\). The \(R\)-order approximation of \(\psi_{\mathbf{n}}\) at position \(\mathbf{r}=[x,y,z]^{\mathrm{T}}\) is described as follows [12]:  
\[\psi_{\mathbf{n}}(\mathbf{r})\approx\sum_{r=1}^{R}w_{\mathbf{n},r}\exp(-\mathrm{i}k_{\mathbf{n }}s_{r}^{\mathrm{T}}\mathbf{r}), \tag{1}\]  
where the unit vector \(\mathbf{s}_{r}=\mathbf{s}(\theta_{r},\phi_{r})=[\sin\theta_{r}\cos\phi_{r},\sin\theta_ {r}\sin\phi_{r},\cos\theta_{r}]^{\mathrm{T}}\) represents the direction of the \(r\)th PW with inclination \(\theta_{r}\in[0^{\circ},180^{\circ}]\) and azimuth \(\phi_{r}\in[0^{\circ},360^{\circ})\); \((\cdot)^{\mathrm{T}}\) denotes the transpose operation; \(k_{\mathbf{n}}\mathbf{s}_{r}\) is the \(r\)th wave vector of the \(n\)th mode; and \(w_{\mathbf{n},r}\) is the corresponding PWA. For a complex \(k_{\mathbf{n}}\) (damping wall), we assume that Equation (1) remains valid at locations far from the wall [12]. The problem is to estimate the values of \(\{w_{\mathbf{n},r}\}_{r=1}^{R}\) from limited measurements. Assuming that \(M\) samples are employed, Equation (1) can be converted to a matrix form as follows:  
\[\mathbf{\psi}_{\mathbf{n}}=\tilde{\mathbf{H}}_{\mathbf{n}}\tilde{\mathbf{w}}_{\mathbf{n}}, \tag{2}\]  
where \(\tilde{\mathbf{H}}_{\mathbf{n}}\) is the \(M\times R\) matrix, with its \((m,r)\)th entry being \(h_{\mathbf{n},m}(\mathbf{s}_{r})=\exp(-\mathrm{i}k_{\mathbf{n}}s_{r}^{\mathrm{T}}\mathbf{r}_{ m})\), and \(\tilde{\mathbf{w}}_{\mathbf{n}}\) is the \(R\times 1\) vector of the PWAs. The subscript \((\cdot)_{\mathbf{n}}\) is omitted below for simplicity of notation.  
The discretized PW directions, \(\mathcal{S}=\{\mathbf{s}_{r}\}_{r=1}^{R}\), are uniformly sampled on the surface of a unit sphere. Generally, \(R>M\), and \(\tilde{\mathbf{w}}\) can be solved as  
\[\tilde{\mathbf{w}}_{\mathrm{LS}}=\tilde{\mathbf{H}}^{\mathrm{H}}(\tilde{\mathbf{H}}\tilde {\mathbf{H}}^{\mathrm{H}})^{-1}\mathbf{\psi}, \tag{3}\]  
where \((\cdot)^{\mathrm{H}}\) denotes the conjugate transpose operation, which is the analytic solution of the underdetermined least-squares (LS) evaluated using the Tikhonov regularization. However, this solution tends to generate spatial aliasing artifacts. An alternative solution based on the CS framework exploits the prior knowledge that \(\tilde{\mathbf{w}}\) is sparse. In this case, \(\ell_{1}\)-norm relaxation can be used to formulate an \(\ell_{1}\)-norm-regularized LS, also known as Lasso [27]. Its solution,  
\[\tilde{\mathbf{w}}_{\mathrm{L1}}=\arg\min_{\tilde{\mathbf{w}}}\{\frac{1}{2}||\mathbf{\psi }-\tilde{\mathbf{H}}\tilde{\mathbf{w}}||_{2}^{2}+\lambda||\tilde{\mathbf{w}}||_{1}\}, \tag{4}\]  
where \(||\cdot||_{2}\) and \(||\cdot||_{1}\) represent the \(\ell_{2}\)-norm and \(\ell_{1}\)-norm, respectively, has few non-zero values and can be solved using iterative algorithms [28]. In our study, we applied the _iterative reweighted least squares_ (IRLS) algorithm [29].  
\(R\) is usually set to be considerably large to reduce the basis mismatch and obtain sufficiently dense PW directions. Thus, a high-dimensional dictionary was constructed. However, implementing this method is computationally expensive and even leads to failure of approximation.