modal analysis of rigid microphone arrays using boundary elements
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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Modal Analysis of Rigid Microphone Arrays using Boundary Elements
Fabio Kaiser
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Compact Microphone Arrays
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Fig.: Model of an acoustic scene
• Sound field analysis
Tasks:•Source localization•Beamforming•3D sound recording
Applications:•Acoustic surveillance•Speech recognition•Telecommunication
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Sound field analysis
• Model of sound propagation – Acoustic model
• Obtain model parameters by measuring or computing boundary values
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Fig.: Sketch of modal processing for a spherical array
Modal beamformer
• Orthogonal basis functions – modal functions – array modes• Frequency independent beampatterns• Operational frequency range
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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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
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N=3 N=8
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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This work
• Alternative array shapes
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• DSP for alternative...
• Array modes? Frequency independence? Real-valued?
• Spatial resolution? Discrimination of incidence directions?
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Outlook
• Boundary Element Method
• Modal Analysis of Free-Field Scatterers
• Spatial Resolution of Rigid Microphone Arrays
• Conclusions
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Helmholtz Integral Equation
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 7
Fig.: Region of definition for HIE
Solid angles:
Sound pressure and its normal derivative
Green‘s function and its normal derivative
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Boundary Element Method
• Discretization of boundary and sound pressure (collocation)
• HIE becomes Matrix Equation
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• where
• using standard collocation (p and pn constant on element)
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Rigid Scattering with BEM
• Solution for the scattering on a rigid body
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• Implementation:- OpenBEM, http://www.openbem.dk/- By Peter Juhl (Phd thesis, 1993) and Vicente Cutanda
Henriquez
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Axisymmetric BEM
• Formulation for rotationally symmetric bodies- Axis of symmetry is the z-axis
• Represent acoustic variables by
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• Computations for one m only
• Solutions assembled afterwards (truncation)
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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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)
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Acoustic Radiation Modes (2/2)
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• Radiation Operator
• Singular value decomposition (SVD)
Introduction BEM Modal Analysis Spatial Resolution Conclusions
• uj and vj are „ARMs“ and σj are „radiation efficiencies“
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Modal Analysis of Free-Field Scatterers
• The scattering problem
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• Neumann boundary condition (rigid case)
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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The Scattering Operator
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• where P: Ω -> S
• Using operator notation
Introduction BEM Modal Analysis Spatial Resolution Conclusions
• SVD of operator P- Array modes, modal strength
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Spherical Source Distribution
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• Continuous plane wave distribution - Ambisonics
• where
• Ω is a sphere and
• is a single spherical basis function
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Scattering Matrix
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• Using the BEM with pn=0
• In matrix form
• and
• with the scattering matrix
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Scattering Matrix
• ...is the scattering response to spherical basis functions
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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SVD of the Scattering Matrix
• Singular value decomposition
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• Or eigenvectors (array modes)
• or eigenvectors (field mode re-combinations)
• with the singular values
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Analysis for one frequency only!
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Joint SVD
• Joint SVD via Joint eigendecomposition of Pz
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• and same for PzH
P
• Approximation necessary
• Minimization of off-diagonal terms of Σz
• Algorithms used from (Cardoso,1996)- http://perso.telecom-paristech.fr/~cardoso/jointdiag.html
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Summary
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Using a surrounding spherical source distribution
Regular and high density mesh
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Simulation Results
• Sphere and Cylinder
• k=0.1-10
• Axisymmetric bodies
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Sphere (R=1), Σ
• Singular values over k, Black dashed
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0
1-2
3-5
6-9
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Sphere (R=1), U
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• Six „strongest“ singular vectors
• Colors...U for kR=(0.1,0.5,1)
• ---- ass. Legendre function
• Plotted over the whole circumferential (polar plot)
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Sphere (R=1), V
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• V for several kR
• Below kr≈1, V is identity
• Above, modes start to mix
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Cylinder (R=1,L=0.5), Σ
• Singular values over k, Black dashed
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Cylinder (R=1,L=0.5), U
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• Six „strongest“ singular vectors
• Colors...U for kR=(0.1,0.5,1)
• ---- ass. Legendre function
• Plotted over the whole circumferential (polar plot)
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Cylinder (R=1,L=0.5), V
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• V for several kR
• Below kr≈1, V is identity
• Above, modes start to mix
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Cylinder (R=1,L=1), Σ
• Singular values over k, Black dashed
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Cylinder (R=1,L=1), U
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 29
• Six „strongest“ singular vectors
• Colors...U for kR=(0.1,0.5,1)
• ---- ass. Legendre function
• Plotted over the whole circumferential (polar plot)
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Cylinder (R=1,L=1), V
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• V for several kR
• Below kr≈1, V is identity
• Above, modes start to mix
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Cylinder (R=1,L=2), Σ
• Singular values over k, Black dashed
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Cylinder (R=1,L=2), U
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 32
• Six „strongest“ singular vectors
• Colors...U for kR=(0.1,0.5,1)
• ---- ass. Legendre function
• Plotted over the whole circumferential (polar plot)
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Cylinder (R=1,L=2), V
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 33
• V for several kR
• Below kr≈1, V is identity
• Above, modes start to mix
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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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
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Analysis of Spatial Resolution (1/3)
• Sound pressure distribution due to incoming plane waves
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Fig.: Vertical and horizontal resolution angle with regard to a reference zenith angle ϑ0
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Analysis of Spatial Resolution (2/3)
• Decomposition into two plane waves
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• where is a measured array response
• Solve in a least-squares sense
• yields
• We shall take a look closed on PHP
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Analysis of Spatial Resolution (3/3)
• Ragarding only the matrix
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• Use determinant
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Example: Rigid sphere
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Simulation Results
• Compared arrays
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• Short cylinder , long cylinder
• Sound pressure on array using BEM
• Rth = 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.
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Ring Array
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Ribbon Array
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Full Array
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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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
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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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
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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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
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Introduction BEM Modal Analysis Spatial Resolution Conclusions
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Thank you!
Question!?
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 46