1. Modal Analysis of Rigid Microphone Arrays using Boundary Elements Fabio Kaiser

2. Compact Microphone Arrays Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements2 Fig.: Model of an acoustic scene Sound field analysis Tasks: Source localization Beamforming 3D sound recording Applications: Acoustic surveillance Speech recognition Telecommunication IntroductionBEM Modal Analysis Spatial Resolution Conclusions

3. Sound field analysis Model of sound propagation – Acoustic model Obtain model parameters by measuring or computing boundary values Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements3 Fig.: Sketch of modal processing for a spherical array Modal beamformer Orthogonal basis functions – modal functions – array modes Frequency independent beampatterns Operational frequency range IntroductionBEM Modal Analysis Spatial Resolution Conclusions

4. Spatial resolution Practical microphone arrays -Continuous pressure sensitive surface would be nice but... -Finite number of sampling points (microphones) -Finite number of array modes Finite spatial resolution Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements4 N=3 N=8 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

5. This work Alternative array shapes Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements5 DSP for alternative... Array modes? Frequency independence? Real-valued? Spatial resolution? Discrimination of incidence directions? IntroductionBEM Modal Analysis Spatial Resolution Conclusions

6. Outlook Boundary Element Method Modal Analysis of Free-Field Scatterers Spatial Resolution of Rigid Microphone Arrays Conclusions Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements6 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

7. Helmholtz Integral Equation Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements7 Fig.: Region of definition for HIE Solid angles: Sound pressure and its normal derivative Green‘s function and its normal derivative IntroductionBEM Modal Analysis Spatial Resolution Conclusions

8. Boundary Element Method Discretization of boundary and sound pressure (collocation) HIE becomes Matrix Equation Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements8 where using standard collocation (p and pn constant on element) IntroductionBEM Modal Analysis Spatial Resolution Conclusions

9. Rigid Scattering with BEM Solution for the scattering on a rigid body Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements9 Implementation: -OpenBEM, http://www.openbem.dk/http://www.openbem.dk/ -By Peter Juhl (Phd thesis, 1993) and Vicente Cutanda Henriquez IntroductionBEM Modal Analysis Spatial Resolution Conclusions

10. Axisymmetric BEM Formulation for rotationally symmetric bodies -Axis of symmetry is the z-axis Represent acoustic variables by Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements10 Computations for one m only Solutions assembled afterwards (truncation) IntroductionBEM Modal Analysis Spatial Resolution Conclusions

11. Acoustic Radiation Modes (ARMs) (1/2) Modal analysis of free-field radiators (Borgiotti,1990, Cunefare, 2004) Goal is a representation of surface vibration patterns ARMs loud and low ARMs of a continuous sphere -Low order spherical harmonics are: loud! Applications -Active noise control (Nelson, 1994) -Loudspeaker directivity control (Pasqual, 2010) Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements11 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

12. Acoustic Radiation Modes (2/2) Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements12 Radiation Operator Singular value decomposition (SVD) IntroductionBEM Modal Analysis Spatial Resolution Conclusions u j and v j are „ARMs“ and σ j are „radiation efficiencies“

13. Modal Analysis of Free-Field Scatterers The scattering problem Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements13 Neumann boundary condition (rigid case) IntroductionBEM Modal Analysis Spatial Resolution Conclusions

14. The Scattering Operator Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements14 where P: Ω -> S Using operator notation IntroductionBEM Modal Analysis Spatial Resolution Conclusions SVD of operator P -Array modes, modal strength

15. Spherical Source Distribution Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements15 Continuous plane wave distribution - Ambisonics where Ω is a sphere and is a single spherical basis function IntroductionBEM Modal Analysis Spatial Resolution Conclusions

16. Scattering Matrix Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements16 Using the BEM with p n =0 In matrix form and with the scattering matrix IntroductionBEM Modal Analysis Spatial Resolution Conclusions

17. Scattering Matrix...is the scattering response to spherical basis functions Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements17 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

18. SVD of the Scattering Matrix Singular value decomposition Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements18 Or eigenvectors (array modes) or eigenvectors (field mode re-combinations) with the singular values IntroductionBEM Modal Analysis Spatial Resolution Conclusions Analysis for one frequency only!

19. Joint SVD Joint SVD via Joint eigendecomposition of P z Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements19 and same for P z H P Approximation necessary Minimization of off-diagonal terms of Σ z Algorithms used from (Cardoso,1996) -http://perso.telecom-paristech.fr/~cardoso/jointdiag.html IntroductionBEM Modal Analysis Spatial Resolution Conclusions

20. Summary Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements20 Define scattering operator Compute scattering matrix via BEM SVD Analysis for single k Joint SVD Analysis for wider range of k Using a surrounding spherical source distribution Regular and high density mesh IntroductionBEM Modal Analysis Spatial Resolution Conclusions

21. Simulation Results Sphere and Cylinder k=0.1-10 Axisymmetric bodies Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements21 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

22. Sphere (R=1), Σ Singular values over k, Black dashed Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements22 0 1-2 3-5 6-9 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

23. Sphere (R=1), U Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements23 Six „strongest“ singular vectors Colors...U for kR=(0.1,0.5,1) ---- ass. Legendre function Plotted over the whole circumferential (polar plot) IntroductionBEM Modal Analysis Spatial Resolution Conclusions

24. Sphere (R=1), V Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements24 V for several kR Below kr≈1, V is identity Above, modes start to mix IntroductionBEM Modal Analysis Spatial Resolution Conclusions

25. Cylinder (R=1,L=0.5), Σ Singular values over k, Black dashed Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements25 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

26. Cylinder (R=1,L=0.5), U Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements26 Six „strongest“ singular vectors Colors...U for kR=(0.1,0.5,1) ---- ass. Legendre function Plotted over the whole circumferential (polar plot) IntroductionBEM Modal Analysis Spatial Resolution Conclusions

27. Cylinder (R=1,L=0.5), V Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements27 V for several kR Below kr≈1, V is identity Above, modes start to mix IntroductionBEM Modal Analysis Spatial Resolution Conclusions

28. Cylinder (R=1,L=1), Σ Singular values over k, Black dashed Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements28 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

29. Cylinder (R=1,L=1), U Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements29 Six „strongest“ singular vectors Colors...U for kR=(0.1,0.5,1) ---- ass. Legendre function Plotted over the whole circumferential (polar plot) IntroductionBEM Modal Analysis Spatial Resolution Conclusions

30. Cylinder (R=1,L=1), V Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements30 V for several kR Below kr≈1, V is identity Above, modes start to mix IntroductionBEM Modal Analysis Spatial Resolution Conclusions

31. Cylinder (R=1,L=2), Σ Singular values over k, Black dashed Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements31 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

32. Cylinder (R=1,L=2), U Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements32 Six „strongest“ singular vectors Colors...U for kR=(0.1,0.5,1) ---- ass. Legendre function Plotted over the whole circumferential (polar plot) IntroductionBEM Modal Analysis Spatial Resolution Conclusions

33. Cylinder (R=1,L=2), V Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements33 V for several kR Below kr≈1, V is identity Above, modes start to mix IntroductionBEM Modal Analysis Spatial Resolution Conclusions

34. Discussion – Modal Analysis Rotationally symmetric geometries (axisymmtric) Sphere vs. Cylinders Frequency dependent modes except for below kr≈1 Modes are real-valued (at least of constant-phase) Joint SVD was applied -Diagonalzation using a range of k=0.1-10 -Smaller range better Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements34 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

35. Analysis of Spatial Resolution (1/3) Sound pressure distribution due to incoming plane waves Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements35 Fig.: Vertical and horizontal resolution angle with regard to a reference zenith angle ϑ 0 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

36. Analysis of Spatial Resolution (2/3) Decomposition into two plane waves Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements36 where is a measured array response Solve in a least-squares sense yields We shall take a look closed on P H P IntroductionBEM Modal Analysis Spatial Resolution Conclusions

37. Analysis of Spatial Resolution (3/3) Ragarding only the matrix Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements37 Use determinant IntroductionBEM Modal Analysis Spatial Resolution Conclusions

38. Example: Rigid sphere Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements38 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

39. Simulation Results Compared arrays Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements39 Short cylinder, long cylinder Sound pressure on array using BEM R th = 0.5 High density mesh, no spatial aliasing Ribbon array height +- 0.5R Fig.: Different array shapes, (a) ring arrays, (b) ribbon arrays, (c) full arrays. IntroductionBEM Modal Analysis Spatial Resolution Conclusions

40. Ring Array Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements40 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

41. Ribbon Array Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements41 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

42. Full Array Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements42 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

43. Conclusions Rigid Microphone Arrays -Methods also valid for open arrays Investigations on alternative array shapes -Cylinder as an example Boundary Element Method for scattering -Axisymmetric formulation advantage concerning sampling Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements43 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

44. Conclusions – Modal Analysis Method for modal analysis of microphone arrays -Scattering operator and/or matrix -Axisymmetric BEM -SVD, Joint SVD Frequency independent modes -Just for frequencies below kr≈1 (e.g. r=0.1m -> k≈550Hz) --> Open arrays could have been used Simplification of DSP possible Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements44 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

45. Conclusion – Spatial Resolution Measure for local horizontal and vertical resolution Based on correlation of array responses Scattering by employing BEM In combination widely applicable Cylindrical Microphone Arrays: Heigth of array influences vertical resolution Cylinder behaves similar to sphere -> Cylindrical equivalent of a spherical microphone array -Adcantage: Easier to build Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements45 IntroductionBEM Modal Analysis Spatial Resolution Conclusions

46. Thank you! Question!? Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements46