Wave Field Analysis Using Virtual Circular Microphone Arrays

Transcription

1 **i Achim Kuntz таг] Ш 5 Wave Field Analysis Using Virtual Circular Microphone Arrays га [W] та

2 Contents Abstract Zusammenfassung v vii 1 Introduction l 2 Multidimensional Signals and Wave Fields One Dimensional Signals Fourier Transform Periodic Signals (3+l)D Signals and the MD Fourier Transform MD Signals Temporal Fourier Transform (3+l)D Signals in Cartesian Coordinates (3+l)D Signals in Cylindrical Coordinates Spatial Fourier Transform in Cylindrical Coordinates Fourier Series Representations and the Modified Hankel Transform (3+l)D Signals in Spherical Coordinates Spatial Fourier Transform in Spherical Coordinates Spherical Harmonics Representation and the Modified Spherical Hankel Transform Spatio-temporal MD Fourier Transform for (3+l)D Signals Relations Between Different Representations (2+l)D Signals and (2+l)D Representations (2+l)D Signals С (3+l)D Signals (2+l)D Signals in 2D Cartesian Coordinates Spatial Fourier Transform for (2+l)D Signals (2+l)D Signals in Polar Coordinates (2+l)D Signals in Spherical Coordinates 24

3 x Contents Relations Between (2+l)D Representations and (3+l)D Representations Fundamentals of Acoustic Wave Fields Homogeneous Wave Equation Derivation Fourier Transforms of the Wave Equation Inhomogeneous Wave Equation Point Sources Line Sources Plane Wave Sources Arbitrary Source Distributions Huygens' Principle and the Helmholtz Integral Equation Neumann and Dirichlet Green Functions D Wave Fields as (3+l)D Signals Generic Derivation of Wave Field Expansions Wave Fields in 3D Cartesian Coordinates Plane Wave Decomposition in 3D Cartesian Coordinates Spatio-Temporal Spectrum of Wave Fields in Cartesian Coordinates Wave Fields in Cylindrical Coordinates Cylindrical Harmonics Decomposition Plane Wave Decomposition in Cylindrical Coordinates Plane Wave Decomposition Derived From the Cylindrical Harmonics Decomposition Interpretation of the Plane Wave Decomposition and Its Inverse as Angular Convolutions Wave Fields with Limited Modal Bandwidth Relations of Wave Fields to MD Signals in Cylindrical Coordinates Wave Fields in Spherical Coordinates Spherical Harmonics Decomposition Plane Wave Decomposition in Spherical Coordinates Plane Wave Decomposition Derived From the Spherical Harmonics Decomposition Relations Between Representations of 3D Wave Fields in Different Coordinate Systems D Wave Field Representations and the Helmholtz Integral Equation Transitions from the Helmholtz Integral Representation Transitions to the Helmholtz Integral Representation... 67

4 Contents xi 2.6 2D Wave Fields Properties of 2D Wave Fields D Wave Fields in 2D Cartesian Coordinates D Wave Fields in Polar Coordinates D Wave Fields and the Helmholtz Integral Equation Relations of 2D Wave Field Representations 75 3 Wave Field Analysis Wave Field Analysis Concept and Measurement Effects Analysis of 2D Fields Analysis Based on Circular Apertures Principle of 2D Wave Field Analysis Based on the Circular Harmonics Decomposition Circular Harmonics Decomposition Derived Using Microphones with First Order Directivity Circular Harmonics Decomposition Derived Using a Scatterer Inside the Array Separation of Converging and Diverging Waves Using Pressure and Velocity Signals Circular Harmonics Analysis Using Measurements on Two Radii Wave Field Analysis Based on Different 2D Measurement Geometries Helmholtz Integral Based Approach Wave Field Analysis Based on 2D Fourier Transforms D HOA Based on Spherical Harmonics Modal Beamforming Analysis of 3D Fields Analysis Using Spherical Arrays Wave Field Analysis Based on Spherical Harmonics Spherical Harmonics Decomposition for Ambisonics Systemsl Plane Wave Decomposition Based on Spherical Measurements Beamforming Based on Spherical Harmonics Decomposition Analysis Using Arbitrary Arrays Existing Approaches to 3D Field Analysis Using Circular Measurements Extraction of 3D Plane Waves Using Cylindrical Harmonics Decomposition with Effective Radius Extraction of 3D Plane Waves Using the Radon Transform 114

5 xii Contents Least Squares Spherical Harmonics Decomposition Summary Wave Field Analysis Using Virtual Circular Arrays Sequential Circular Measurement Setups Circular Measurements Virtual Arrays Analysis of 2D Wave Fields Consequences of Measuring Effects on the Circular Harmonics Decomposition Modal Aliasing Sensor Noise and Array Aperture Positioning Errors Wave Field Extrapolation From Circular Harmonics Decomposition Extrapolation Inside the Effective Aperture Extrapolation Outside the Effective Aperture Mode Selection Simple Mode Selection Strategies Modal Optimal Filtering Combining Strategies for Measurements on Multiple Radii Combining of Measurements on Two Radii Using Hankel Functions Selection Combining for Measurements on Multiple Radii Maximum Ratio Combining for Measurements on Multiple Radii Wave Field Analysis Using the Plane Wave Decomposition Discrete Plane Wave Decomposition Extrapolation from the Discrete Plane Wave Decompositionl Modally Band Limited Plane Wave Decomposition Errors in 2D Analysis Introduced by 3D Field Components Amplitude Errors Elevated Plane Waves Errors in Speed of Propagation Circular Harmonics Decomposition and Extrapolation Spherical Harmonics Decomposition Derived from Circular Measurements Order Aliasing Parameterized Spherical Harmonics Decomposition Derivation of Elevation Angles 171

6 Contents xiii D Wave Field Extrapolation Using Parameterized Spherical Harmonics Decomposition Conclusion Summary Realization and Applications of Wave Field Analysis Using Virtual Circular Arrays Realization Measurement System Virtual Circular Array Room Impulse Response Measurements Failsafe Operation Cardioid Pattern Optimization Application to Room Acoustic Analysis Measurement and Direct Interpretation Of RIRs D Wave Field Decomposition and Extrapolation Circular Harmonics Decomposition and Extrapolation Plane Wave Decomposition Extended Analysis using Complementary 3D Simulation Simulation Method Complementary Analysis D Wave Field Decomposition and Extrapolation Application to Auralization Auralization by Wave Field Synthesis Extrapolation From the Plane Wave Decomposition WFS Results Application to Binaural Auralization Techniques Direct Calculation of Ear Signals Inclusion of Head Related Transfer Functions Conclusions Summary 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

7 XIV Contents В The Inverse Modified Spherical Hankel Transform 237 B.l Derivation 237 B.2 The (Inverse) Modified Spherical Hankel Transform as a Transform Pair С Spatial Fourier Transform of the 3D Free Space Green Function 239 D Notations 243 Bibliography 249