Modal Analysis of Rigid Microphone Arrays using Boundary Elements

Fabio Kaiser

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

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

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

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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

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

Helmholtz Integral Equation

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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

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• 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

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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

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

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

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“

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

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

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

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

Scattering Matrix

• ...is the scattering response to spherical basis functions

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Introduction BEM Modal Analysis Spatial Resolution Conclusions

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!

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

Summary

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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

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Introduction BEM Modal Analysis Spatial Resolution Conclusions

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

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

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

Cylinder (R=1,L=0.5), Σ

• Singular values over k, Black dashed

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Introduction BEM Modal Analysis Spatial Resolution Conclusions

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

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

Cylinder (R=1,L=1), Σ

• Singular values over k, Black dashed

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Introduction BEM Modal Analysis Spatial Resolution Conclusions

Cylinder (R=1,L=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

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

Cylinder (R=1,L=2), Σ

• Singular values over k, Black dashed

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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

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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

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

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

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

Analysis of Spatial Resolution (3/3)

• Ragarding only the matrix

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• Use determinant

Introduction BEM Modal Analysis Spatial Resolution Conclusions

Example: Rigid sphere

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Introduction BEM Modal Analysis Spatial Resolution Conclusions

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

Ring Array

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Introduction BEM Modal Analysis Spatial Resolution Conclusions

Ribbon Array

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Introduction BEM Modal Analysis Spatial Resolution Conclusions

Full Array

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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

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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

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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

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Introduction BEM Modal Analysis Spatial Resolution Conclusions

Thank you!

Question!?

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