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Modal Analysis of Rigid Microphone Arrays using Boundary Elements

Modal Analysis of Rigid Microphone Arrays using Boundary Elements. Fabio Kaiser. Introduction BEM Modal Analysis Spatial Resolution Conclusions. Compact Microphone Arrays. Sound field analysis. Tasks: Source localization Beamforming

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Modal Analysis of Rigid Microphone Arrays using Boundary Elements

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  1. Modal Analysis of Rigid Microphone Arrays usingBoundary Elements Fabio Kaiser

  2. IntroductionBEM Modal Analysis Spatial Resolution Conclusions Compact Microphone Arrays • Sound fieldanalysis • Tasks: • Source localization • Beamforming • 3D soundrecording • Applications: • Acousticsurveillance • Speech recognition • Telecommunication Fig.: Model of an acoustic scene Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

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

  4. IntroductionBEM Modal Analysis Spatial Resolution Conclusions Spatialresolution • Practicalmicrophonearrays • Continuouspressure sensitive surfacewouldbenice but... • Finite numberofsamplingpoints (microphones) • Finite numberofarraymodes • Finite spatialresolution N=8 N=3 Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  5. IntroductionBEM Modal Analysis Spatial Resolution Conclusions This work • Alternative arrayshapes • DSP for alternative... • Array modes? Frequencyindependence? Real-valued? • Spatialresolution? Discriminationofincidencedirections? Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  6. IntroductionBEM Modal Analysis Spatial Resolution Conclusions 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 Elements

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

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

  9. IntroductionBEM Modal Analysis Spatial Resolution Conclusions Rigid Scatteringwith BEM • Solution forthescattering on a rigid body • Implementation: • OpenBEM, http://www.openbem.dk/ • By Peter Juhl (Phdthesis, 1993) and Vicente CutandaHenriquez Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  10. IntroductionBEM Modal Analysis Spatial Resolution Conclusions Axisymmetric BEM • Formulationforrotationallysymmetricbodies • Axis ofsymmetryisthez-axis • Representacoustic variables by • Computationsforone m only • Solutions assembledafterwards (truncation) Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  11. IntroductionBEMModal AnalysisSpatial Resolution Conclusions Acoustic Radiation Modes (ARMs) (1/2) • Modal analysisoffree-fieldradiators (Borgiotti,1990, Cunefare, 2004) • Goal is a representationofsurfacevibrationpatterns • ARMs loudandlow • ARMs of a continuoussphere • Low ordersphericalharmonicsare: loud! • Applications • Activenoisecontrol (Nelson, 1994) • Loudspeakerdirectivitycontrol (Pasqual, 2010) Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  12. IntroductionBEMModal AnalysisSpatial Resolution Conclusions Acoustic Radiation Modes (2/2) • Radiation Operator • Singular valuedecomposition (SVD) • uj and vj are „ARMs“ and σj are „radiation efficiencies“ Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  13. IntroductionBEMModal AnalysisSpatial Resolution Conclusions Modal Analysis of Free-Field Scatterers • The scatteringproblem • Neumann boundarycondition (rigid case) Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  14. IntroductionBEMModal AnalysisSpatial Resolution Conclusions The Scattering Operator • Usingoperatornotation • where P: Ω -> S • SVD ofoperator P • Array modes, modal strength Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  15. IntroductionBEMModal AnalysisSpatial Resolution Conclusions Spherical Source Distribution • Continuous plane wavedistribution - Ambisonics • where • Ω is a sphere and • is a singlesphericalbasisfunction Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  16. IntroductionBEMModal AnalysisSpatial Resolution Conclusions Scattering Matrix • Usingthe BEM withpn=0 • and • In matrix form • withthescatteringmatrix Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  17. IntroductionBEMModal AnalysisSpatial Resolution Conclusions Scattering Matrix • ...isthescatteringresponsetosphericalbasisfunctions Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  18. IntroductionBEMModal AnalysisSpatial Resolution Conclusions SVD oftheScattering Matrix • Singular valuedecomposition • Oreigenvectors (arraymodes) • oreigenvectors (fieldmodere-combinations) • withthesingularvalues Analysis foronefrequencyonly! Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  19. IntroductionBEMModal AnalysisSpatial Resolution Conclusions Joint SVD • Joint SVD via Joint eigendecompositionofPz • and same for PzHP • Approximation necessary • Minimization of off-diagonal terms of Σz • Algorithms used from (Cardoso,1996) • http://perso.telecom-paristech.fr/~cardoso/jointdiag.html Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  20. IntroductionBEMModal AnalysisSpatial Resolution Conclusions Summary Using a surrounding spherical source distribution Regular and high density mesh Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  21. IntroductionBEMModal AnalysisSpatial Resolution Conclusions Simulation Results • SphereandCylinder • k=0.1-10 • Axisymmetricbodies Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  22. IntroductionBEMModal AnalysisSpatial Resolution Conclusions Sphere (R=1), Σ • Singular valuesoverk, Black dashed 0 1-2 3-5 6-9 Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

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

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

  25. IntroductionBEMModal AnalysisSpatial Resolution Conclusions Cylinder (R=1,L=0.5), Σ • Singular valuesoverk, Black dashed Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

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

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

  28. IntroductionBEMModal AnalysisSpatial Resolution Conclusions Cylinder (R=1,L=1), Σ • Singular valuesoverk, Black dashed Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

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

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

  31. IntroductionBEMModal AnalysisSpatial Resolution Conclusions Cylinder (R=1,L=2), Σ • Singular valuesoverk, Black dashed Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

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

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

  34. IntroductionBEMModal AnalysisSpatial Resolution Conclusions 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 Elements

  35. IntroductionBEM Modal AnalysisSpatial ResolutionConclusions Analysis ofSpatial Resolution (1/3) • Sound pressuredistribution due toincoming plane waves Fig.: Vertical and horizontal resolution angle with regard to a reference zenith angle ϑ0 Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  36. IntroductionBEM Modal AnalysisSpatial ResolutionConclusions Analysis ofSpatial Resolution (2/3) • Decompositionintotwo plane waves • whereis a measuredarrayresponse • Solve in a least-squares sense • yields • Weshalltake a lookclosed on PHP Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  37. IntroductionBEM Modal AnalysisSpatial ResolutionConclusions Analysis ofSpatial Resolution (3/3) • Ragardingonlythematrix • Usedeterminant Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  38. IntroductionBEM Modal AnalysisSpatial ResolutionConclusions Example: Rigid sphere Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  39. IntroductionBEM Modal AnalysisSpatial ResolutionConclusions Simulation Results • Comparedarrays Fig.: Different array shapes, (a) ring arrays, (b) ribbon arrays, (c) full arrays. • Short cylinder , longcylinder • Sound pressure on arrayusing BEM • Rth= 0.5 • High densitymesh, nospatialaliasing • Ribbonarrayheight +- 0.5R Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  40. IntroductionBEM Modal AnalysisSpatial ResolutionConclusions Ring Array Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  41. IntroductionBEM Modal AnalysisSpatial ResolutionConclusions Ribbon Array Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  42. IntroductionBEM Modal AnalysisSpatial ResolutionConclusions Full Array Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  43. IntroductionBEM Modal AnalysisSpatial ResolutionConclusions Conclusions • Rigid Microphone Arrays • Methods also valid for open arrays • Investigations on alternative arrayshapes • Cylinderas an example • Boundary Element Methodforscattering • Axisymmetricformulationadvantageconcerningsampling Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  44. IntroductionBEM Modal AnalysisSpatial ResolutionConclusions 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 Elements

  45. IntroductionBEM Modal AnalysisSpatial ResolutionConclusions Conclusion – Spatial Resolution • Measureforlocal horizontal andverticalresolution • Based on correlationofarrayresponses • Scatteringbyemploying BEM • In combinationwidelyapplicable CylindricalMicrophone Arrays: • Heigthofarrayinfluencesverticalresolution • Cylinderbehavessimilartosphere • -> Cylindricalequivalentof a sphericalmicrophonearray • Adcantage: Easiertobuild Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

  46. Thankyou! Question!? Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements

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