Preface......Page 3
Contents......Page 6
1.1 Functions on the Sphere......Page 8
1.2 Spherical Harmonics......Page 11
1.3 Exponential and Legendre Functions......Page 20
1.4 Spherical Fourier Transform......Page 24
1.5 Some Useful Functions......Page 28
1.6 Rotation of Functions......Page 30
1.7 Spherical Convolution and Correlation......Page 36
References......Page 38
2.1 The Acoustic Wave Equation......Page 39
2.2 Spherical Bessel and Hankel Functions......Page 42
2.3 A Single Plane Wave......Page 46
2.4 Plane-Wave Composition......Page 52
2.5 Point Sources......Page 54
2.6 Sound Pressure Around a Rigid Sphere......Page 55
2.7 Translation of Fields......Page 61
References......Page 64
3.1 Sampling Order-Limited Functions......Page 65
3.2 Equal-Angle Sampling......Page 67
3.3 Gaussian Sampling......Page 72
3.4 Uniform and Nearly-Uniform Sampling......Page 73
3.5 Numerical Computation of Sampling Weights......Page 76
3.6 The Discrete Spherical Fourier Transform......Page 80
3.7 Spatial Aliasing......Page 81
References......Page 86
4 Spherical Array Configurations......Page 87
4.1 Single Open Sphere......Page 88
4.2 Rigid Sphere......Page 91
4.3 Open Sphere with Cardioid Microphones......Page 93
4.4 Dual-Radius Open Sphere......Page 96
4.5 Robustness to Errors and Numerical Array Design......Page 98
4.6 Design Examples with Robustness Analysis......Page 101
4.7 Spherical Shell Configuration......Page 104
4.8 Other Configurations......Page 107
References......Page 108
5.1 Beamforming Equations......Page 109
5.2 Axis-Symmetric Beamforming......Page 114
5.3 Directivity Index......Page 116
5.4 White Noise Gain......Page 118
5.5 Simple Axis-Symmetric Beamformers......Page 121
5.6 Beamforming Example......Page 123
5.7 Steering Non Axis-Symmetric Beam Patterns......Page 129
References......Page 131
6.1 Maximum Directivity Beamformer......Page 132
6.2 Maximum WNG Beamformer......Page 137
6.3 Example: Directivity Versus WNG......Page 140
6.4 Mixed-Objective Design......Page 142
6.5 Maximum Front-Back Ratio......Page 147
6.6 Dolph-Chebyshev Beam Pattern......Page 149
6.7 Multiple-Objective Design......Page 155
References......Page 160
7.1 Beamforming Equations Including Noise......Page 162
7.2 Minimum Variance Distortionless Response......Page 167
7.3 Example: MVDR with Sensor Noise and Disturbance......Page 171
7.4 Example: MVDR with Correlated Disturbance......Page 173
7.5 Linearly Constrained Minimum Variance......Page 177
7.6 Example: LCMV with Beam Pattern Amplitude Constraints......Page 179
7.7 LCMV with Derivative Constraints......Page 182
7.8 Example: Robust LCMV with Derivative Constraints......Page 186
References......Page 190
Glossary......Page 192
Index......Page 195