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Achim Kuntz

Wave Field Analysis Using Virtual Circular Microphone Arrays

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Contents

Abstract v

Zusammenfassung vii

1 Introduction l

2 Multidimensional Signals and Wave Fields 9 2.1 One Dimensional Signals 9

2.1.1 Fourier Transform 10 2.1.2 Periodic Signals 11

2.2 (3+l)D Signals and the MD Fourier Transform 11 2.2.1 MD Signals 12 2.2.2 Temporal Fourier Transform 12 2.2.3 (3+l)D Signals in Cartesian Coordinates 12 2.2.4 (3+l)D Signals in Cylindrical Coordinates 13

2.2.4.1 Spatial Fourier Transform in Cylindrical Coordinates . . . 13 2.2.4.2 Fourier Series Representations and the Modified Hankel

Transform 15 2.2.5 (3+l)D Signals in Spherical Coordinates 16

2.2.5.1 Spatial Fourier Transform in Spherical Coordinates . . . . 17 2.2.5.2 Spherical Harmonics Representation and the Modified Spher­

ical Hankel Transform 17 2.2.6 Spatio-temporal MD Fourier Transform for (3+l)D Signals 19 2.2.7 Relations Between Different Representations 19

2.3 (2+l)D Signals and (2+l)D Representations 20 2.3.1 (2+l)D Signals С (3+l)D Signals 21 2.3.2 (2+l)D Signals in 2D Cartesian Coordinates 22

2.3.2.1 Spatial Fourier Transform for (2+l)D Signals 22 2.3.3 (2+l)D Signals in Polar Coordinates 23 2.3.4 (2+l)D Signals in Spherical Coordinates 24

x Contents

2.3.5 Relations Between (2+l)D Representations and (3+l)D Represen­tations 25

2.4 Fundamentals of Acoustic Wave Fields 26 2.4.1 Homogeneous Wave Equation 26

2.4.1.1 Derivation 26 2.4.1.2 Fourier Transforms of the Wave Equation 28

2.4.2 Inhomogeneous Wave Equation 29 2.4.2.1 Point Sources 29 2.4.2.2 Line Sources 30 2.4.2.3 Plane Wave Sources 31 2.4.2.4 Arbitrary Source Distributions 33 2.4.2.5 Huygens' Principle and the Helmholtz Integral Equation . 35 2.4.2.6 Neumann and Dirichlet Green Functions 37

2.5 3D Wave Fields as (3+l)D Signals 40 2.5.1 Generic Derivation of Wave Field Expansions 40 2.5.2 Wave Fields in 3D Cartesian Coordinates 41

2.5.2.1 Plane Wave Decomposition in 3D Cartesian Coordinates . 42 2.5.2.2 Spatio-Temporal Spectrum of Wave Fields in Cartesian

Coordinates 45 2.5.3 Wave Fields in Cylindrical Coordinates 46

2.5.3.1 Cylindrical Harmonics Decomposition 47 2.5.3.2 Plane Wave Decomposition in Cylindrical Coordinates . . 49 2.5.3.3 Plane Wave Decomposition Derived From the Cylindrical

Harmonics Decomposition 51 2.5.3.4 Interpretation of the Plane Wave Decomposition and Its

Inverse as Angular Convolutions 52 2.5.3.5 Wave Fields with Limited Modal Bandwidth 54 2.5.3.6 Relations of Wave Fields to MD Signals in Cylindrical

Coordinates 56 2.5.4 Wave Fields in Spherical Coordinates 57

2.5.4.1 Spherical Harmonics Decomposition 57 2.5.4.2 Plane Wave Decomposition in Spherical Coordinates . . . 59 2.5.4.3 Plane Wave Decomposition Derived From the Spherical

Harmonics Decomposition 60 2.5.5 Relations Between Representations of 3D Wave Fields in Different

Coordinate Systems 62 2.5.6 3D Wave Field Representations and the Helmholtz Integral Equation 64

2.5.6.1 Transitions from the Helmholtz Integral Representation . . 64 2.5.6.2 Transitions to the Helmholtz Integral Representation . . . 67

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2.6 2D Wave Fields 68 2.6.1 Properties of 2D Wave Fields 68 2.6.2 2D Wave Fields in 2D Cartesian Coordinates 69 2.6.3 2D Wave Fields in Polar Coordinates 71 2.6.4 2D Wave Fields and the Helmholtz Integral Equation 74 2.6.5 Relations of 2D Wave Field Representations 75

3 Wave Field Analysis 79 3.1 Wave Field Analysis Concept and Measurement Effects 80 3.2 Analysis of 2D Fields 84

3.2.1 Analysis Based on Circular Apertures 85 3.2.1.1 Principle of 2D Wave Field Analysis Based on the Circular

Harmonics Decomposition 86 3.2.1.2 Circular Harmonics Decomposition Derived Using Micro­

phones with First Order Directivity 88 3.2.1.3 Circular Harmonics Decomposition Derived Using a Scat-

terer Inside the Array 93 3.2.1.4 Separation of Converging and Diverging Waves Using Pres­

sure and Velocity Signals 95 3.2.1.5 Circular Harmonics Analysis Using Measurements on Two

Radii 97 3.2.2 Wave Field Analysis Based on Different 2D Measurement Geometries 98

3.2.2.1 Helmholtz Integral Based Approach 98 3.2.2.2 Wave Field Analysis Based on 2D Fourier Transforms . . . 99 3.2.2.3 2D HOA Based on Spherical Harmonics 101

3.2.3 Modal Beamforming 104 3.3 Analysis of 3D Fields 105

3.3.1 Analysis Using Spherical Arrays 105 3.3.1.1 Wave Field Analysis Based on Spherical Harmonics . . . . 106 3.3.1.2 Spherical Harmonics Decomposition for Ambisonics Systemsl07 3.3.1.3 Plane Wave Decomposition Based on Spherical Measure­

ments 107 3.3.1.4 Beamforming Based on Spherical Harmonics Decomposition 109

3.3.2 Analysis Using Arbitrary Arrays 109 3.3.3 Existing Approaches to 3D Field Analysis Using Circular Measure­

ments 112 3.3.3.1 Extraction of 3D Plane Waves Using Cylindrical Harmon­

ics Decomposition with Effective Radius 112 3.3.3.2 Extraction of 3D Plane Waves Using the Radon Transform 114

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3.3.3.3 Least Squares Spherical Harmonics Decomposition . . . . 1 1 5

3.4 Summary 115

4 Wave Field Analysis Using Virtual Circular Arrays 117 4.1 Sequential Circular Measurement Setups 117

4.1.1 Circular Measurements 118

4.1.2 Virtual Arrays 119

4.2 Analysis of 2D Wave Fields 120

4.2.1 Consequences of Measuring Effects on the Circular Harmonics De­

composition 120

4.2.1.1 Modal Aliasing 121

4.2.1.2 Sensor Noise and Array Aperture 123

4.2.1.3 Positioning Errors 126

4.2.2 Wave Field Extrapolation From Circular Harmonics Decomposition 128

4.2.2.1 Extrapolation Inside the Effective Aperture 129

4.2.2.2 Extrapolation Outside the Effective Aperture 130

4.2.3 Mode Selection 133

4.2.3.1 Simple Mode Selection Strategies 135

4.2.3.2 Modal Optimal Filtering 139

4.2.4 Combining Strategies for Measurements on Multiple Radii 143

4.2.4.1 Combining of Measurements on Two Radii Using Hankel

Functions 143

4.2.4.2 Selection Combining for Measurements on Multiple Radii . 148

4.2.4.3 Maximum Ratio Combining for Measurements on Multiple

Radii 150

4.2.5 Wave Field Analysis Using the Plane Wave Decomposition 153

4.2.5.1 Discrete Plane Wave Decomposition 154

4.2.5.2 Extrapolation from the Discrete Plane Wave Decompositionl57

4.2.5.3 Modally Band Limited Plane Wave Decomposition . . . . 159

4.3 Errors in 2D Analysis Introduced by 3D Field Components 161

4.3.1 Amplitude Errors 162

4.3.2 Elevated Plane Waves 164

4.3.2.1 Errors in Speed of Propagation 164

4.3.2.2 Circular Harmonics Decomposition and Extrapolation . . 165

4.4 Spherical Harmonics Decomposition Derived from Circular Measurements . 168

4.4.1 Order Aliasing 168

4.4.2 Parameterized Spherical Harmonics Decomposition 170

4.4.2.1 Derivation of Elevation Angles 171

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4.4.2.2 3D Wave Field Extrapolation Using Parameterized Spher­ical Harmonics Decomposition 173

4.4.3 Conclusion 175 4.5 Summary 177

5 Realization and Applications of Wave Field Analysis Using Virtual Cir­cular Arrays 179 5.1 Realization 179

5.1.1 Measurement System 180 5.1.2 Virtual Circular Array 180 5.1.3 Room Impulse Response Measurements 183

5.1.3.1 Failsafe Operation 184 5.1.4 Cardioid Pattern Optimization 185

5.2 Application to Room Acoustic Analysis 187 5.2.1 Measurement and Direct Interpretation Of RIRs 188 5.2.2 2D Wave Field Decomposition and Extrapolation 191

5.2.2.1 Circular Harmonics Decomposition and Extrapolation . . 191 5.2.2.2 Plane Wave Decomposition 194

5.2.3 Extended Analysis using Complementary 3D Simulation 195 5.2.3.1 Simulation Method 195 5.2.3.2 Complementary Analysis 196

5.2.4 3D Wave Field Decomposition and Extrapolation 203 5.3 Application to Auralization 205

5.3.1 Auralization by Wave Field Synthesis 207 5.3.1.1 Extrapolation From the Plane Wave Decomposition . . . . 209 5.3.1.2 WFS Results 210

5.3.2 Application to Binaural Auralization Techniques 213 5.3.2.1 Direct Calculation of Ear Signals 214 5.3.2.2 Inclusion of Head Related Transfer Functions 215

5.3.3 Conclusions 216 5.4 Summary 218

6 Conclusion 221

A Special Functions 225 A.l Spherical Harmonics 225 A.2 Bessel Functions 227 A.3 Spherical Bessel Functions 230 A.4 Delta Distributions in Two and Three Dimensions 235

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В The Inverse Modified Spherical Hankel Transform 237 B.l Derivation 237 B.2 The (Inverse) Modified Spherical Hankel Transform as a Transform Pair . . 238

С Spatial Fourier Transform of the 3D Free Space Green Function 239

D Notations 243

Bibliography 249