Wavefield Decomposition Using Microphone Arrays and Its Application to Acoustic Scene Analysis

279 Pages·2006·12.2 MB·English

by  Heinz Teutsch

Wave(cid:12)eld Decomposition Using Microphone
Arrays and Its Application to Acoustic
Scene Analysis
Der Technischen Fakulta(cid:127)t der
Friedrich-Alexander-Universita(cid:127)t Erlangen-Nu(cid:127)rnberg
zur Erlangung des Grades
Doktor-Ingenieur
vorgelegt von
Heinz Teutsch
Erlangen, 2005
Als Dissertation genehmigt von
der Technischen Fakulta(cid:127)t der
Friedrich-Alexander-Universita(cid:127)t
Erlangen-Nu(cid:127)rnberg
Tag der Einreichung: 04.10.2005
Tag der Promotion: 25.01.2006
Dekan: Prof. Dr.-Ing. Alfred Leipertz
Berichterstatter: Prof. Dr.-Ing. Walter Kellermann
Dr. James L. Flanagan, Rutgers University, NJ, USA
Acknowledgments
I would like to thank my supervisor, Prof. Walter Kellermann of the Friedrich-Alexander
University in Erlangen, Germany, who, with his inspirations and great patience has sup-
ported me throughout my career in his research group. He showed faith in my work even
during times where research funding was critical. Thanks go to Dr. Rudolf Rabenstein
for funding my (cid:12)rst two and a half years throughout the CARROUSO project.
I am very indebted to Dr. James L. Flanagan of Rutgers University, New Jersey, USA,
who was willing to dedicate a big chunk of his busy schedule to the review of this thesis.
Thanks also go to Prof. Wolfgang Koch, Prof. em. Adolf W. Lohmann, Prof. Reinhard
Lerch, and Dr. Manfred Kaltenbacher, all from the University of Erlangen, Germany, for
showing interest in my work and for (cid:12)nding the time to participate in the defense of this
thesis.
I am grateful to Dr. Gary Elko for many fruitful discussions and for making my
memorable stay at Bell Laboratories, New Jersey, USA, possible.
I would also like to thank all my former colleagues at the Telecommunications Lab-
oratory for creating a very enjoyable and inspiring atmosphere. I am thankful for the
unresting dedication of the supportive sta(cid:11). In particular, I would like to thank Mrs.
Ursula Arnold for making the impossible possible, as well as Ru(cid:127)diger Na(cid:127)gel and Manfred
Lindner for helping me with the design of numerous hardware components.
I am very grateful to my friends and family for their unconditional support and for
sharing unforgettable moments with me.
And (cid:12)nally, my deepest thanks go to Debbie for giving me her heart and for accepting
the countless days that we had to spend so far apart.

v
Contents
Abbreviations and Acronyms ix
Notation x
1 Introduction 1
2 Acoustic Wave(cid:12)elds 5
2.1 Mathematical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Euler’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 The acoustic wave equation . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Point sources and plane-waves . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Point sources in three dimensions . . . . . . . . . . . . . . . . . . . 9
2.2.2 Plane-waves in three dimensions . . . . . . . . . . . . . . . . . . . . 12
2.2.3 Point sources and plane-waves in two dimensions . . . . . . . . . . 13
2.3 Acoustic wave equation in cylindrical coordinates . . . . . . . . . . . . . . 15
2.3.1 General solution of the acoustic wave equation in cylindrical coor-
dinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 The cylindrical radiator . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.2.1 The cylindrical radiator of in(cid:12)nite length . . . . . . . . . . 19
2.3.2.2 The cylindrical radiator of (cid:12)nite length . . . . . . . . . . . 22
2.3.3 The cylindrical scatterer . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.3.1 The in(cid:12)nite-length cylinder as a rigid scatterer . . . . . . 25
2.3.3.2 The (cid:12)nite-length cylinder as a rigid scatterer . . . . . . . 29
2.3.3.3 In(cid:12)nite-length cylindrical scatterer with pressure-release
boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Acoustic wave equation in spherical coordinates . . . . . . . . . . . . . . . 33
2.4.1 Generalsolutionoftheacousticwaveequationinsphericalcoordinates 33
2.4.2 The spherical radiator . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.3 The spherical scatterer . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.3.1 The rigid spherical scatterer . . . . . . . . . . . . . . . . . 38
2.4.3.2 The pressure-release spherical scatterer . . . . . . . . . . . 40
vi Contents
3 Wave(cid:12)eld Analysis 41
3.1 WFA using circular apertures and arrays . . . . . . . . . . . . . . . . . . . 42
3.1.1 Continuous circular apertures . . . . . . . . . . . . . . . . . . . . . 42
3.1.1.1 The unba(cid:15)ed circular aperture . . . . . . . . . . . . . . . 43
3.1.1.2 The ba(cid:15)ed circular aperture . . . . . . . . . . . . . . . . 48
3.1.2 Wave(cid:12)eld decomposition using directional circular apertures . . . . 53
3.1.2.1 Circular dipole aperture . . . . . . . . . . . . . . . . . . . 54
3.1.2.2 Circular cardioid aperture . . . . . . . . . . . . . . . . . . 56
3.1.3 Circularly symmetric microphone arrays . . . . . . . . . . . . . . . 57
3.1.4 Representation of a 2D wave(cid:12)eld using a (cid:12)nite number of harmonics 62
3.1.5 Circular apertures and near(cid:12)eld sources . . . . . . . . . . . . . . . . 65
3.2 WFA using spherical apertures and arrays . . . . . . . . . . . . . . . . . . 68
3.2.1 Continuous spherical apertures . . . . . . . . . . . . . . . . . . . . 68
3.2.1.1 The unba(cid:15)ed spherical aperture . . . . . . . . . . . . . . 69
3.2.1.2 The ba(cid:15)ed spherical aperture . . . . . . . . . . . . . . . . 72
3.2.1.3 Additional properties of spherical apertures . . . . . . . . 74
3.2.2 Wave(cid:12)eld decomposition using directional spherical apertures . . . 74
3.2.2.1 Spherical dipole aperture . . . . . . . . . . . . . . . . . . 76
3.2.2.2 Spherical cardioid aperture . . . . . . . . . . . . . . . . . 78
3.2.3 Spherical microphone arrays . . . . . . . . . . . . . . . . . . . . . . 78
3.2.3.1 The (cid:30) -Gaussian method for spherical sampling . . . . . 79
equ
3.2.3.2 The t-design method for spherical sampling . . . . . . . . 82
3.2.4 Representation of a 3D wave(cid:12)eld using a (cid:12)nite number of harmonics 86
3.2.5 Spherical apertures and near(cid:12)eld sources . . . . . . . . . . . . . . . 87
3.3 WFA using other types of apertures . . . . . . . . . . . . . . . . . . . . . . 88
3.3.1 Linear apertures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3.2 Cylindrical apertures . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4 Acoustic Scene Analysis Using Classical Array Signal Processing 95
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 Signal models and assumptions . . . . . . . . . . . . . . . . . . . . . . . . 97
4.2.1 Sensor-related assumptions . . . . . . . . . . . . . . . . . . . . . . . 97
4.2.2 Time-domain signal model . . . . . . . . . . . . . . . . . . . . . . . 98
4.2.3 Frequency-domain signal model . . . . . . . . . . . . . . . . . . . . 100
4.3 Waveform Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3.1 Space-time (cid:12)ltering and beamforming . . . . . . . . . . . . . . . . . 102
4.3.2 Performance measures . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3.3 Data-independent waveform estimation . . . . . . . . . . . . . . . . 112
4.3.3.1 The ULA beamformer . . . . . . . . . . . . . . . . . . . . 112
Contents vii
4.3.3.2 The constant-directivity beamformer . . . . . . . . . . . . 113
4.3.3.3 Di(cid:11)erential and superdirective beamformer . . . . . . . . . 114
4.3.4 Data-dependent waveform estimation . . . . . . . . . . . . . . . . . 119
4.4 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.4.1 Performance measure { the Cram(cid:19)er-Rao lower bound . . . . . . . . 122
4.4.1.1 Limitations of the CRLB . . . . . . . . . . . . . . . . . . 124
4.4.2 TDOA-based algorithms . . . . . . . . . . . . . . . . . . . . . . . . 124
4.4.2.1 Generalized cross-correlation-based algorithms . . . . . . . 124
4.4.2.2 The adaptive eigenvalue decomposition algorithm . . . . . 126
4.4.2.3 Performance of TDOA-based algorithms . . . . . . . . . . 128
4.4.3 Subspace-based DOA estimation algorithms . . . . . . . . . . . . . 131
4.4.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.4.3.2 Source localization using MUSIC-based algorithms . . . . 134
4.4.3.3 Source localization using ESPRIT-based algorithms . . . . 137
4.4.3.4 Performance of subspace-based DOA estimation . . . . . . 143
4.4.4 Detection of the number of active sources . . . . . . . . . . . . . . . 147
5 Acoustic Scene Analysis Using Modal Array Signal Processing 151
5.1 Waveform estimation using eigenbeam processing . . . . . . . . . . . . . . . 151
5.1.1 The modal beamformer { pattern synthesis . . . . . . . . . . . . . . 152
5.1.1.1 Pattern synthesis using circular harmonics . . . . . . . . . 152
5.1.1.2 Pattern synthesis using spherical harmonics . . . . . . . . 152
5.1.2 Optimum beampattern design using eigenbeams . . . . . . . . . . . 154
5.1.3 The adaptive modal beamformer . . . . . . . . . . . . . . . . . . . 158
5.2 Parameter estimation using eigenbeam processing . . . . . . . . . . . . . . . 159
5.2.1 Eigenbeam array manifold vectors . . . . . . . . . . . . . . . . . . . 159
5.2.2 Eigenbeam signal model and modal signal subspaces . . . . . . . . 160
5.2.3 Eigenbeam processing and the CRLB . . . . . . . . . . . . . . . . . 161
5.2.4 Eigenbeam-based DOA estimation using circular apertures . . . . . 162
5.2.4.1 ESPRIT-based algorithm for single-source localization tasks163
5.2.4.2 ESPRIT-based algorithm for multiple-source localization
tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.2.5 Eigenbeam processing using spherical apertures . . . . . . . . . . . 171
5.2.6 Resolution capacity and DOA estimation of more than two wide-
band sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.2.7 Detection of the number of active sources . . . . . . . . . . . . . . . 176
5.2.7.1 Detection algorithm for circular apertures . . . . . . . . . 176
5.2.7.2 Detection algorithm for spherical apertures . . . . . . . . 183
viii Contents
6 A Practical Acoustic Scene Analysis System 187
6.1 System details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.1.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.1.2 Algorithm implementation . . . . . . . . . . . . . . . . . . . . . . . 189
6.1.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.2 Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.2.1 Evaluation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.2.2 Waveform Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.2.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 196
7 Summary and Conclusions 201
A Signal Transforms 205
A.1 One- and multi-dimensional Fourier transforms . . . . . . . . . . . . . . . 205
A.2 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
A.3 Spherical harmonics transform . . . . . . . . . . . . . . . . . . . . . . . . . 206
B Special Functions 207
B.1 Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
B.2 Spherical Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
B.3 Legendre polynomials, associated Legendre functions, and spherical har-
monics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
C Microphone Arrays and Near(cid:12)eld/Far(cid:12)eld Sources 217
D Eigenbeam-CRLB for a Single Source 221
E Frequency-independence of EB-ESPRIT for a Single Source 223
F A Practical Acoustic Scene Analysis System { Further Results 227
G Titel, Inhaltsverzeichnis, Einleitung und Zusammenfassung 239
G.1 Titel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
G.2 Inhaltsverzeichnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
G.3 Einleitung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
G.4 Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Bibliography 251
Abbreviations and Acronyms ix
Abbreviations and Acronyms
ACC All curtains closed
ACO All curtains open
AED Adaptive eigenvalue decomposition
AIC Akaike information criteria
ASA Acoustic scene analysis
BM Blocking matrix
CC Cross-correlation
CDB Constant directivity beamformer
CRLB Cramer-Rao lower bound
DMA Di(cid:11)erential microphone array
DOA Direction-of-arrival
DSB Delay-and-sum beamformer
EBF Eigen-beamformer
EB-GSC Eigenbeam generalized sidelobe canceler
ESPRIT Estimation of signal parameters via rotational invariance techniques
FFT Fast Fourier transform
FIR Finite impulse response
FBF Fixed beamformer
FBSS Forward-backward spatial smoothing
FSB Filter-and-sum beamformer
GCC Generalized cross-correlation
GSC Generalized side-lobe canceler
IC Interference canceler
LCMP Linearly constrained minimum power
LCMV Linearly constrained minimum variance
LMS Least-mean square
LS Least-squares
MDL Minimum description length
MIMO Multiple-input-multiple-output
MLS Maximum-length sequence
MUSIC Multiple signal classi(cid:12)cation
NOS Number of sources
PE Parameter estimation
PHAT Phase transform
SCOT Smoothed coherence transform
SDB Superdirective beamformer
SNR Signal-to-noise ratio
x Notation
SVD Singular value decomposition
TDOA Time-di(cid:11)erence-of-arrival
TLS Total least-squares
ULA Uniform linear array
WE Waveform estimation
WFA Wave(cid:12)eld analysis
WFS Wave(cid:12)eld synthesis
WNG White noise gain
WSS Wide-sense stationary
Notations and Conventions
Conventions
The following conventions are used throughout this thesis.
Time-domain scalar quantities are denoted by lowercase characters, e.g. a(t).
(cid:15)
Frequency-domain scalar quantities are denoted by uppercase characters, e.g. A(!).
(cid:15)
Time-domain vector quantities are denoted by boldface lowercase characters, e.g.
(cid:15)
a(t).
Frequency-domain vector quantities are denoted by boldface uppercase characters,
(cid:15)
e.g. A(!).
Time-domain matrix quantities are denoted by underlined, boldface lowercase char-
(cid:15)
acters, e.g. a(t).
Frequency-domain matrix quantities are denoted by underlined, boldface uppercase
(cid:15)
characters, e.g. A(!).
All vectors are assumed to be column vectors.
(cid:15)
The denotation 0(1)M means 0;1;:::;M, where M is an integer.
(cid:15)
Mathematical operations
( )T Vector or matrix transposition
(cid:1)
( ) Conjugate complex of a vector or matrix
(cid:3)
(cid:1)
( )H Hermitian operation, i.e. conj. compl. transposed, of a vector or matrix
(cid:1)
( )( 1) Matrix inverse
(cid:0)
(cid:1)