Modal Analysis of Rigid Microphone Arrays using Boundary Elements Fabio Kaiser.
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Transcript of Modal Analysis of Rigid Microphone Arrays using Boundary Elements Fabio Kaiser.
Modal Analysis of Rigid Microphone Arrays using Boundary Elements
Fabio Kaiser
Compact Microphone Arrays
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 2
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
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 Elements 3
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
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 Elements 4
N=3 N=8
Introduction BEM Modal Analysis Spatial Resolution Conclusions
This work
• Alternative array shapes
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 5
• DSP for alternative...
• Array modes? Frequency independence? Real-valued?
• Spatial resolution? Discrimination of incidence directions?
Introduction BEM 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 6
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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
Boundary Element Method
• Discretization of boundary and sound pressure (collocation)
• HIE becomes Matrix Equation
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 8
• where
• using standard collocation (p and pn constant on element)
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Rigid Scattering with BEM
• Solution for the scattering on a rigid body
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 9
• Implementation:- OpenBEM, http://www.openbem.dk/- By Peter Juhl (Phd thesis, 1993) and Vicente Cutanda
Henriquez
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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 Elements 10
• Computations for one m only
• Solutions assembled afterwards (truncation)
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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 Elements 11
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Acoustic Radiation Modes (2/2)
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 12
• Radiation Operator
• Singular value decomposition (SVD)
Introduction BEM Modal Analysis Spatial Resolution Conclusions
• uj and vj are „ARMs“ and σj are „radiation efficiencies“
Modal Analysis of Free-Field Scatterers
• The scattering problem
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 13
• Neumann boundary condition (rigid case)
Introduction BEM Modal Analysis Spatial Resolution Conclusions
The Scattering Operator
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 14
• where P: Ω -> S
• Using operator notation
Introduction BEM Modal Analysis Spatial Resolution Conclusions
• SVD of operator P- Array modes, modal strength
Spherical Source Distribution
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 15
• Continuous plane wave distribution - Ambisonics
• where
• Ω is a sphere and
• is a single spherical basis function
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Scattering Matrix
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 16
• Using the BEM with pn=0
• In matrix form
• and
• with the scattering matrix
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Scattering Matrix
• ...is the scattering response to spherical basis functions
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 17
Introduction BEM Modal Analysis Spatial Resolution Conclusions
SVD of the Scattering Matrix
• Singular value decomposition
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 18
• 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!
Joint SVD
• Joint SVD via Joint eigendecomposition of Pz
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 19
• 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
Summary
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 20
Using a surrounding spherical source distribution
Regular and high density mesh
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Simulation Results
• Sphere and Cylinder
• k=0.1-10
• Axisymmetric bodies
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 21
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Sphere (R=1), Σ
• Singular values over k, Black dashed
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 22
0
1-2
3-5
6-9
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Sphere (R=1), U
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 23
• 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
Sphere (R=1), V
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 24
• V for several kR
• Below kr≈1, V is identity
• Above, modes start to mix
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Cylinder (R=1,L=0.5), Σ
• Singular values over k, Black dashed
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 25
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Cylinder (R=1,L=0.5), U
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 26
• 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
Cylinder (R=1,L=0.5), V
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 27
• V for several kR
• Below kr≈1, V is identity
• Above, modes start to mix
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Cylinder (R=1,L=1), Σ
• Singular values over k, Black dashed
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 28
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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
Cylinder (R=1,L=1), V
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 30
• V for several kR
• Below kr≈1, V is identity
• Above, modes start to mix
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Cylinder (R=1,L=2), Σ
• Singular values over k, Black dashed
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 31
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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
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
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 34
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Analysis of Spatial Resolution (1/3)
• Sound pressure distribution due to incoming plane waves
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 35
Fig.: Vertical and horizontal resolution angle with regard to a reference zenith angle ϑ0
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Analysis of Spatial Resolution (2/3)
• Decomposition into two plane waves
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 36
• 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
Analysis of Spatial Resolution (3/3)
• Ragarding only the matrix
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 37
• Use determinant
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Example: Rigid sphere
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 38
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Simulation Results
• Compared arrays
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 39
• 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
Ring Array
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 40
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Ribbon Array
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 41
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Full Array
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 42
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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 Elements 43
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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 44
Introduction BEM Modal Analysis Spatial Resolution Conclusions
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 Elements 45
Introduction BEM Modal Analysis Spatial Resolution Conclusions
Thank you!
Question!?
Fabio Kaiser - Modal Analysis of Rigid Microphone Arrays using Boundary Elements 46