Design of Circular Differential Microphone Arrays (eBook)

Abstract 6
Contents 7
1 Introduction 10
1.1 Introduction 10
1.2 General Principles of Microphone Arrays 11
1.3 From Linear Differential Arrays to Circular Differential Arrays 19
1.4 Organization of the Book 20
References 21
2 Problem Formulation 23
2.1 Signal Model 23
2.2 Beampattern 27
2.3 Gain in Signal-to-Noise Ratio (SNR) 34
References 38
3 Design of First-Order Circular Differential Arrays 40
3.1 Principle 40
3.2 Design Examples 43
3.2.1 First-Order Dipole 43
3.2.2 First-Order Cardioid 46
3.2.3 General First-Order Circular Differential Array 49
References 59
4 Design of Second-Order Circular Differential Arrays 60
4.1 Principle and Design Examples with Four Microphones 60
4.1.1 Second-Order Dipole 61
4.1.2 Second-Order Cardioid 64
4.1.3 Second-Order Hypercardioid 67
4.1.4 Second-Order Supercardioid 70
4.2 Principle and Design Examples with Five Microphones 73
4.2.1 Second-Order Dipole 74
4.2.2 Second-Order Cardioid 78
4.2.3 Second-Order Hypercardioid 81
4.2.4 Second-Order Supercardioid 84
References 87
5 Design of Third-Order Circular Differential Arrays 88
5.1 Principle and Design Example with Six Microphones 88
5.2 Principle and Design Example with Seven Microphones 92
5.3 Generalization 96
References 97
6 Superdirective Beamforming with Circular Arrays 98
6.1 Superdirective Beamforming with ULAs 98
6.2 Superdirective Beamforming with UCAs 100
6.2.1 Without the Symmetry Constraint 101
6.2.2 With the Symmetry Constraint 103
6.2.3 With the Maximum of Nulls 105
6.3 Design Examples 105
6.3.1 Superdirective Beamforming with a ULA 105
6.3.2 Superdirective Beamforming with a UCA –without the Symmetry Constraint 106
6.3.3 Superdirective Beamforming with a UCA – with the Symmetry Constraint 112
6.3.4 Superdirective Beamforming with a UCA – with the Maximum of Nulls 114
References 118
7 Minimum-Norm Solution for Robust Circular Differential Arrays 119
7.1 Adaptive Beamforming versus Circular Differential Arrays 119
7.2 Minimum-Norm Filter 122
7.3 Design Examples 123
7.3.1 First-Order Cardioid 124
7.3.2 Second-Order Cardioid 130
7.3.3 Third-Order Cardioid 136
7.3.4 Third-Order Pattern with Three Distinct Nulls 142
References 147
8 Design of Circular Differential Arrays with the Jacobi-Anger Expansion 149
8.1 Beampattern Definitions Revisited 149
8.2 Design Principle 152
8.3 First-Order Differential Arrays 153
8.4 Second-Order Differential Arrays 155
8.5 Third-Order Differential Arrays 157
8.6 Adaptive Beamforming versus Differential Arrays 165
8.7 Design of the Ideal Beampattern 169
References 170
Index 171