Skip to main content

Spatial interpolation methods for virtual rotating array beamforming with arbitrary microphone configurations

Abstract

Virtual rotating array (VRA) beamforming is a robust technique in the identification of rotating sound sources in frequency-domain. Under normal circumstances, the configuration of microphone array is established in ring geometry centred around the rotating axis. Two interpolation methods for arbitrary microphone configurations are proposed by Jekosch and Sarradj (Acoustics, 2020). One is to construct a mesh between all stationary microphones using Delaunay-triangulation, another one is a meshless technique based on radial basis function. However, whether other spatial interpolation methods are available in VRA beamforming with arbitrary microphone configurations is still unclear. This paper adds several new spatial interpolation methods in VRA beamforming and detailedly compares the performances of these interpolation methods in simulations. The simulating results demonstrated that all these interpolation methods are successfully applied in VRA beamforming with arbitrary microphone configurations. Inverse distance weighting interpolation method owns the best performance in rotating sound source localization. In addition, all these interpolation methods have poor spectrum construction capability and sound source strength precision.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Availability of data and materials

Not applicable.

Code availability

Not applicable.

References

  1. Michel, U.: History of acoustic beamforming. In: Proceedings of the 1st Berlin beamforming conference, 21 - 22 November 2006, Berlin, Germany, BeBeC-2006-01, pp. 1–17 (2006)

  2. Merino-Martínez R, Sijtsma P, Snellen M, Ahlefeldt T, Antoni J, Bahr CJ, Blacodon D, Ernst D, Finez A, Funke S, Geyer TF, Haxter S, Herold G, Huang X, Humphreys WM, Leclère Q, Malgoezar A, Michel U, Padois T, Pereira A, Picard C, Sarradj E, Siller H, Simons DG, Spehr C (2019) A review of acoustic imaging methods using phased microphone arrays. CEAS Aeronaut J 10(1):197–230. https://doi.org/10.1007/s13272-019-00383-4

    Article  Google Scholar 

  3. Brooks TF, Humphreys WM (2006) A deconvolution approach for the mapping of acoustic sources (damas) determined from phased microphone arrays. J Sound Vib 294:856–879. https://doi.org/10.1016/j.jsv.2005.12.046

    Article  Google Scholar 

  4. Sijtsma P (2007) Clean based on spatial source coherence. Int J Aeroacoustics 6(4):357–374. https://doi.org/10.1260/147547207783359459

    Article  Google Scholar 

  5. Ma W, Liu X (2018) Compression computational grid based on functional beamforming for acoustic source localization. Appl Acoustics 134:75–87. https://doi.org/10.1016/j.apacoust.2018.01.006

    Article  Google Scholar 

  6. Ma W, Liu X (2017) Damas with compression computational grid for acoustic source mapping. J Sound Vib 410:473–484. https://doi.org/10.1016/j.jsv.2017.03.027

    Article  Google Scholar 

  7. Ma W, Liu X (2017) Improving the efficiency of damas for sound source localization via wavelet compression computational grid. J Sound Vib 395:341–353. https://doi.org/10.1016/j.jsv.2017.02.005

    Article  Google Scholar 

  8. Sijtsma, P., Oerlemans, S., Holthusen, H.: Location of rotating sources by phased array measurements. In: 7th AIAA/CEAS aeroacoustics conference and exhibit, 28 - 30 May 2001, Maastricht, Netherlands, AIAA 2001-2167, pp. 1–11 (2001). https://doi.org/10.2514/6.2001-2167

  9. Ma W, Zhang C (2020) Doppler effect in the time-domain beamforming for rotating sound source identification. J Acoustical Soc Am 148(1):430. https://doi.org/10.1121/10.0001570

    Article  Google Scholar 

  10. Lowis, C.R., Joseph, P.: Inversion technique for determining the strength of rotating broadband sources in ducts. 11th AIAA/CEAS aeroacoustics conference (26th AIAA Aeroacoustics Conference), 23-25 May 2005, Monterey, California, AIAA 2005-3018 (2005). https://doi.org/10.2514/6.2005-3018

  11. Lowis CR, Joseph P (2006) Determining the strength of rotating broadband sources in ducts by inverse methods. J Sound Vib 295:614–632. https://doi.org/10.1016/j.jsv.2006.01.031

    Article  Google Scholar 

  12. Pannert W, Maier C (2014) Rotating beamforming - motion-compensation in the frequency domain and application of high-resolution beamforming algorithms. J Sound Vib 333(7):1899–1912. https://doi.org/10.1016/j.jsv.2013.11.031

    Article  Google Scholar 

  13. Herold G, Sarradj E (2015) Microphone array method for the characterization of rotating sound sources in axial fans. Noise Control Eng J 63(6):546–551. https://doi.org/10.3397/1/376348

    Article  Google Scholar 

  14. Ma, W., Bao, H., Zhang, C., Liu, X.: Beamforming of phased microphone array for rotating sound source localization. J Sound Vib 467 (2020). https://doi.org/10.1016/j.jsv.2019.115064

  15. Sarradj, E.: A generic approach to synthesize optimal array microphone arrangements. In: Proceedings of the 6th Berlin beamforming conference, 29 February - 1 March 2016, Berlin, Germany, BeBeC-2016-S4, p. 4 (2016)

  16. Jekosch, S., Sarradj, E.: An extension of the virtual rotating array method using arbitrary microphone configurations for the localization of rotating sound sources. Acoustics, 330–342 (2020). DOI: https://doi.org/10.3390/acoustics2020019

  17. Sarradj E (2012) Three-dimensional acoustic source mapping with different beamforming steering vector formulations. Adv Acoustics Vib 2012:1–12. https://doi.org/10.1155/2012/292695

    Article  Google Scholar 

  18. Renka, R.J., Renka, R., Cline, A.: A triangle-based c\(^1\) interpolation method. The Rocky Mountain journal of mathematics, 223–237 (1984). https://doi.org/10.1216/rmj-1984-14-1-223

  19. Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Volume 12 (2003). https://doi.org/10.1017/CBO9780511543241

  20. Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 23rd ACM National Conference, 27 - 29 August 1968, pp. 517–524 (1968). https://doi.org/10.1145/800186.810616

  21. Lu GY, Wong DW (2008) An adaptive inverse-distance weighting spatial interpolation technique. Comput Geosci 34(9):1044–1055. https://doi.org/10.1016/j.cageo.2007.07.010

    Article  Google Scholar 

  22. Cressie N (1990) The origins of kriging. Math Geol 22(3):239–252. https://doi.org/10.1007/bf00889887

    MathSciNet  Article  MATH  Google Scholar 

  23. Herold, G., Ocker, C., Sarradj, E., Pannert, W.: A comparison of microphone array methods for the characterization of rotating sound sources. In: Proceedings of the 7th Berlin Beamforming Conference, 5-6 March, 2018, Berlin, Germany, BeBeC-2018-D22, pp. 1–12 (2018). https://doi.org/10.3397/1/376348

Download references

Acknowledgements

The authors gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation. This work is supported by the National Science and Technology Major Project of China (2017-II-003-0015).

Funding

This study was funded by National Science and Technology Major Project of China (2017-II-003-0015).

Author information

Authors and Affiliations

Authors

Contributions

Conceptualization and design: [Wei Ma, Jiacheng Yang, Ce Zhang]; Methodology: [Jiacheng Yang]; Formal analysis and investigation: [Jiacheng Yang, Ce Zhang]; Material preparation, data collection and analysis: [Jiacheng Yang]; Original draft preparation: [Jiacheng Yang]; Writing, review and editing: [Wei Ma, Jiacheng Yang, Ce Zhang]; Funding acquisition: [Wei Ma]; Supervision: [Wei Ma]. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Jiacheng Yang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethics approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Yang, J., Zhang, C. & Ma, W. Spatial interpolation methods for virtual rotating array beamforming with arbitrary microphone configurations. AS 5, 149–158 (2022). https://doi.org/10.1007/s42401-021-00117-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42401-021-00117-7

Keywords

  • Beamforming
  • Rotating sound source localization
  • Virtual rotating array
  • Spatial interpolation method