# Sparse Plane Wave Approximation of Acoustic Modes to Address Basis Mismatch

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. PW Approximation of Acoustic Modes

## 3. Proposed Combined Two-Stage Method

#### 3.1. Stage 1: Selection of Dominant PW Directions Using ${\ell}_{1}$-Norm Relaxation

#### 3.2. Stage 2: Estimation of PWAs and Directions Using Parametric SBL

#### 3.2.1. Parameterized Dictionary

#### 3.2.2. Probabilistic Model for SBL

#### 3.2.3. Hidden Random Variable Inference

Algorithm 1: Stage 2: Estimation of PWs using parametric SBL (parSBL). |

## 4. Simulations

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

#### 4.2. Case: An Aircraft Cabin with Damping Floor

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Murillo Gómez, D.M.; Astley, J.; Fazi, F.M. Low frequency interactive auralization based on a plane wave expansion. Appl. Sci.
**2017**, 7, 558. [Google Scholar] [CrossRef] - Mazur, R.; Katzberg, F.; Mertins, A. Robust room equalization using sparse sound-field reconstruction. In Proceedings of the ICASSP 2019–2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, UK, 12–17 May 2019. [Google Scholar]
- Bai, M.; Hsu, H.; Wen, J. Spatial sound field synthesis and upmixing based on the equivalent source method. J. Acoust. Soc. Am.
**2014**, 135, 269–282. [Google Scholar] [CrossRef] - Jin, W.; Kleijn, W. Theory and design of multizone soundfield reproduction using sparse methods. IEEE/ACM Trans. Audio Speech Lang. Process.
**2015**, 23, 2343–2355. [Google Scholar] - Caviedes-Nozal, D.; Heuchel, F.; Brunskog, J.; Riis, N.; Fernandez-Grande, E. A Bayesian spherical harmonics source radiation model for sound field control. J. Acoust. Soc. Am.
**2019**, 146, 3425–3435. [Google Scholar] [CrossRef] - Ajdler, T.; Sbaiz, L.; Vetterli, M. The plenacoustic function and its sampling. IEEE Trans. Signal Process.
**2006**, 54, 3790–3804. [Google Scholar] [CrossRef] - Candès, E.J.; Wakin, M.B. An introduction to compressive sampling. IEEE Signal Process. Mag.
**2008**, 25, 21–30. [Google Scholar] [CrossRef] - Gerstoft, P.; Mecklenbräuker, C.F.; Seong, W.; Bianco, M.J. Introduction to compressive sensing in acoustics. J. Acoust. Soc. Am.
**2018**, 143, 3731–3736. [Google Scholar] [CrossRef] [Green Version] - Koyama, S. Sparsity-based sound field reconstruction. Acoust. Sci. Technol.
**2020**, 41, 269–275. [Google Scholar] [CrossRef] - Wang, Y.; Chen, K. Compressive sensing based spherical harmonics decomposition of a low frequency sound field within a cylindrical cavity. J. Acoust. Soc. Am.
**2017**, 141, 1812–1823. [Google Scholar] [CrossRef] - Wang, Y.; Chen, K. Sound field reconstruction within an entire cavity by plane wave expansions using a spherical microphone array. J. Acoust. Soc. Am.
**2017**, 142, 1858–1870. [Google Scholar] [CrossRef] [PubMed] - Mignot, R.; Chardon, G.; Daudet, L. Low frequency interpolation of room impulse responses using compressed sensing. IEEE/ACM Trans. Audio Speech Lang. Process.
**2014**, 22, 205–216. [Google Scholar] [CrossRef] [Green Version] - Antonello, N.; De Sena, E.; Moonen, M.; Naylor, P.A.; Van Waterschoot, T. Room impulse response interpolation using a sparse spatio-temporal representation of the sound field. IEEE/ACM Trans. Audio Speech Lang. Process.
**2017**, 25, 1929–1941. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.; Chen, K. Sparse plane wave decomposition of a low frequency sound field within a cylindrical cavity using spherical microphone arrays. J. Sound Vib.
**2018**, 431, 150–162. [Google Scholar] [CrossRef] - Verburg, S.A.; Fernandez-Grande, E. Reconstruction of the sound field in a room using compressive sensing. J. Acoust. Soc. Am.
**2018**, 143, 3770–3779. [Google Scholar] [CrossRef] [PubMed] - Fernandez-Grande, E. Sound Field Reconstruction in a Room from Spatially Distributed Measurements. In Proceedings of the 23rd International Congress on Acoustics, Aachen, Germany, 9–13 September 2019; pp. 4983–4990. [Google Scholar]
- Pham Vu, T.; Hervé, L. Low frequency sound field reconstruction in a non-rectangular room using a small number of microphones. Acta Acust.
**2020**, 4, 5. [Google Scholar] [CrossRef] - Vekua, I. New Methods for Solving Elliptic Equations; North-Holland Publishing Co.: Amsterdam, The Netherlands, 1967. [Google Scholar]
- Moiola, A.; Hiptmair, R.; Perugia, I. Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys.
**2011**, 62, 809–837. [Google Scholar] [CrossRef] [Green Version] - Chi, Y.; Scharf, L.L.; Pezeshki, A.; Calderbank, A.R. Sensitivity to basis mismatch in compressed sensing. IEEE Trans. Signal Process.
**2011**, 59, 2182–2195. [Google Scholar] [CrossRef] - Murata, N.; Koyama, S.; Takamune, N.; Saruwatari, H. Sparse sound field decomposition with parametric dictionary learning for super-resolution recording and reproduction. In Proceedings of the 2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), Cancun, Mexico, 13–16 December 2015; pp. 69–72. [Google Scholar]
- Wang, L.; Liu, Y.; Zhao, L.; Wang, Q.; Zeng, X.; Chen, K. Acoustic source localization in strong reverberant environment by parametric Bayesian dictionary learning. Signal Process.
**2018**, 143, 232–240. [Google Scholar] [CrossRef] - You, K.; Guo, W.; Peng, T.; Liu, Y.; Zuo, P.; Wang, W. Parametric sparse Bayesian dictionary learning for multiple sources localization with propagation parameters uncertainty and nonuniform noise. IEEE Trans. Signal Process.
**2020**, 68, 4194–4209. [Google Scholar] [CrossRef] - Yang, Y.; Chu, Z.; Yang, Y.; Yin, S. Two-dimensional Newtonized orthogonal matching pursuit compressive beamforming. J. Acoust. Soc. Am.
**2020**, 148, 1337–1348. [Google Scholar] [CrossRef] - Beal, M. Variational Algorithms for Approximate Bayesian Inference; University of London: London, UK, 2004. [Google Scholar]
- Buchgraber, T. Variational Sparse Bayesian Learning: Centralized and Distributed Processing; Graz University of Technology: Graz, Austria, 2013. [Google Scholar]
- Tibshirani, R. Regression selection and shrinkage via the lasso. J. R. Stat. Soc. Ser. B
**1994**, 58, 267–288. [Google Scholar] [CrossRef] - Rish, I.; Grabarnik, G.Y. Sparse Modeling: Theory, Algorithms, and Applications; CRC Press: Boca Raton, FL, USA, 2014. [Google Scholar]
- Hald, J. A comparison of iterative sparse equivalent source methods for near-field acoustical holography. J. Acoust. Soc. Am.
**2018**, 143, 3758–3769. [Google Scholar] [CrossRef] [Green Version] - Yang, Z.; Xie, L.; Zhang, C. Off-grid direction of arrival estimation using sparse bayesian inference. IEEE Trans. Signal Process.
**2013**, 61, 38–43. [Google Scholar] [CrossRef] [Green Version] - Fletcher, R. Practical Methods of Optimization, Volume 1: Unconstrained Optimization; John Wiley & Sons Ltd.: Hoboken, NJ, USA, 1980. [Google Scholar]
- Jacobsen, F.; Juhl, P.M. Fundamentals of General Linear Acoustics; John Wiley & Sons Ltd.: Hoboken, NJ, USA, 2013. [Google Scholar]
- Semechko, A. S2-Sampling-Toolbox. Available online: https://github.com/AntonSemechko/S2-Sampling-Toolbox (accessed on 16 December 2021).
- Piironen, J.; Vehtari, A. On the hyperprior choice for the global shrinkage parameter in the horseshoe prior. Artif. Intell. Stat.
**2017**, 54, 905–913. [Google Scholar] - Bush, D.; Xiang, N. A model-based Bayesian framework for sound source enumeration and direction of arrival estimation using a coprime microphone array. J. Acoust. Soc. Am.
**2018**, 143, 3934–3945. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Probabilistic model of parametric SBL represented as a Bayesian network (for conventional SBL, the parameters a, b, c, and d, are often set to be very small values, such as ${10}^{-6}$, in this study).

**Figure 3.**(Color online) Relative mean error (RME) of different methods in different modes (—: $R=400$; - -: $R=1000$ ).

**Figure 4.**(Color online) Results of the PW approximation for the (1, 3, 1) mode with $R=400$. (

**a**) Desired mode shape. (

**b**–

**d**) Spatial distribution of relative errors of the reconstructed mode when using (

**b**) LS, (

**c**) L1, and (

**d**) L1-parSBL.

**Figure 5.**(Color online) Comparison between the theoretical and estimated PWs obtained using (

**a**) LS ($R=400$), (

**b**) L1 ($R=400$), (

**c**) L1-L1 (${R}^{\prime}=23$), and (

**d**) L1-parSBL (${R}^{\prime}=23$) for the (1, 3, 1) mode. The arrows indicate the PW directions, and their lengths are equal to the absolute PWAs.

**Figure 6.**(Color online) (

**a**) Geometry of the enclosure, with the dashed box denoting the virtual parallelepiped and the black dots representing the randomly distributed samplings. (

**b**) Desired mode shapes of the 20 modes around 107.5 Hz.

**Figure 7.**(Color online) RME of L1-parSBL versus the selection threshold $\beta $ in different modes, (

**a**) inside the virtual region (i.e., for interpolated field points) and (

**b**) outside the virtual region (i.e., for extrapolated field points).

**Figure 8.**(Color online) RME of LS, L1, and L1-parSBL in different modes (—: $R=400$; - -: $R=1000$), (

**a**) inside the virtual region and (

**b**) outside the virtual region.

**Figure 9.**(Color online) RME of LS, L1, and L1-parSBL (—: Inside; - -: Outside ) (

**a**) versus M (the number of samplings) with SNR = 25 dB, and (

**b**) versus SNR with $M=40$ for the mode at approximately 106.5 Hz with $R=400$.

**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.

R | L1 | L1-L1 | L1-parSBL |
---|---|---|---|

400 | 0.95 s | 0.96 s | 1.17 s |

1000 | 6.23 s | 6.24 s | 6.43 s |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xu, J.; Chen, K.; Wang, L.; Zhang, J.
Sparse Plane Wave Approximation of Acoustic Modes to Address Basis Mismatch. *Appl. Sci.* **2022**, *12*, 837.
https://doi.org/10.3390/app12020837

**AMA Style**

Xu J, Chen K, Wang L, Zhang J.
Sparse Plane Wave Approximation of Acoustic Modes to Address Basis Mismatch. *Applied Sciences*. 2022; 12(2):837.
https://doi.org/10.3390/app12020837

**Chicago/Turabian Style**

Xu, Jian, Kean Chen, Lei Wang, and Jiangong Zhang.
2022. "Sparse Plane Wave Approximation of Acoustic Modes to Address Basis Mismatch" *Applied Sciences* 12, no. 2: 837.
https://doi.org/10.3390/app12020837