# 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
## 5 Discussion

The proposed L1-parSBL was used in this study to address the adverse effect of the basis mismatch in PW discretization and improve the approximation accuracy of acoustic modes compared with conventional methods. L1-parSBL suffered from smaller reconstruction errors at extrapolated field points outside the region in a local sampling region because it estimated the corresponding PWs more precisely. Note that no sparsity was assumed in Equation (1), and the finite sum of Equation (1) was an approximation required for computation [12]. The sparsity depends on the consistency between the enclosure geometry and the physical propagation model employed. For example, a cylindrical cavity will barely have a sparse representation with PWs (except in the \(z\)-axis) because PWs do not sparsely represent Bessel functions. The performance of CS-based methods deteriorates when the PWs of an acoustic mode are not very sparse. In this case, the selection threshold \(\beta\) should be set with further insight [34; 35]. The randomly distributed samplings must be arranged such that sufficient information of the sound field is captured. In addition, Stage 2 depended on the results of Stage 1. If the estimation was wrong in Stage 1 (for example, due to low SNR), it might be difficult to obtain good results in Stage 2. The IRLS was used here to calculate the coarse PWAs, \(\hat{\mathbf{w}}_{\text{L1}}\), in the first stage. Other algorithms need to be compared to evaluate an optimal paradigm, especially in terms of robustness. ## 6 Conclusions

We proposed a two-stage method, L1-parSBL, that combines \(\ell_{1}\)-norm relaxation with parametric SBL for the PW approximation of acoustic modes to reduce the adverse effect of basis mismatch. The first stage was used to select the sparse dominant PW directions from pre-discretized directions. The second stage was performed under the Bayesian framework to reformulate a parameterized dictionary of low dimensionality to re-estimate PWAs while optimizing the PW directions to approximate the actual directions. Numerical simulations were conducted using two models of an enclosure. The proposed L1-parSBL achieved a considerably higher approximation accuracy of acoustic modes than LS, L1, and L1-L1 for the rectangular enclosure with rigid walls. Based on an analysis of its (1, 3, 1) mode, it was illustrated that L1-parSBL could efficiently optimize the PW directions to reduce the effect of basis mismatch, albeit at the cost of a slightly increased computation time. A local measurement region was examined considering an enclosed aircraft cabin with a damping boundary. The results showed that L1-parSBL outperformed LS and L1, especially for extrapolated field points outside the region. L1-parSBL also exhibited much better robustness in low-SNR cases. The proposed L1-parSBL can further assist in enhancing the performance of reconstruction for sound fields, and related experimental work will be performed as part of future research. Figure 10: (Color online) Estimated PWs under different SNRs for the mode at approximately 106.5 Hz when using LS, L1, and L1-parSBL (\(R=400\)). (**a–c**) SNR = 30 dB. (**d–f**) SNR = 20 dB. (**g–i**) SNR = 10 dB. **Author Contributions:** Conceptualization, J.X. ; methodology, J.X. ; software, J.X. and L.W. ; validation, J.X. and L.W. ; investigation, J.X. ; writing--original draft preparation, J.X. ; writing--review and editing, J.X., L.W. and J.Z. ; visualization, J.X. ; supervision, K.C. ; funding acquisition, K.C. and J.Z. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Natural Science Foundation of China (Grant No. 11974287), the Open Fund of State Key Laboratory of Power Grid Environmental Protection (No. GYW51201001554) and the Research Funds of the Central Finance (MJ-2018-F-09). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.