Differential geometry in array processing

Preface......Page 8
Contents......Page 10
1. Introduction......Page 14
1.1 Nomenclature......Page 16
1.2 Main Abbreviations......Page 17
1.3 Array of Sensors — Environment......Page 18
1.4.1 Spaces/Subspaces......Page 21
1.4.2 Projection Operator......Page 22
1.6 Modelling the Array Signal-Vector and Array Manifold......Page 24
1.7 Significance of Array Manifolds......Page 30
1.8 An Outline of the Book......Page 32
2.1 Manifold Curve Representation — Basic Concepts......Page 35
2.2.1 Number of Curvatures and Symmetricity in Linear Arrays......Page 39
2.2.2 “Moving Frame” and Frame Matrix......Page 41
2.2.3 Frame Matrix and Curvatures......Page 42
2.2.4 Narrow and Wide Sense Orthogonality......Page 44
2.3 “Hyperhelical” Manifold Curves......Page 46
2.3.1 Coordinate Vectors and Array Symmetricity......Page 50
2.3.2 Evaluating the Curvatures of Uniform Linear Array Manifolds......Page 51
2.4 The Manifold Length and Number of Windings (or Half Windings)......Page 54
2.5 The Concept of “Inclination” of the Manifold......Page 58
2.6 The Manifold-Radii Vector......Page 59
2.7.1 Proof of Eq. (2.24)......Page 66
2.7.2 Proof of Theorem 2.1......Page 67
3. Differential Geometry of Array Manifold Surfaces......Page 72
3.1 Manifold Metric......Page 74
3.2 The First Fundamental Form......Page 75
3.3 Christoffel Symbol Matrices......Page 77
3.4 Intrinsic Geometry of a Surface......Page 78
3.4.1 Gaussian Curvature......Page 79
3.4.2 Curves on a Manifold Surface: Geodesic Curvature......Page 81
3.4.3 Geodesic Curvature......Page 82
3.5 The Concept of “Development”......Page 85
3.7.1 Proof of Eq. (3.36) — Geodesic Curvature......Page 87
4. Non-Linear Arrays: (θ, φ)-Parametrization of Array Manifold Surfaces......Page 90
4.1 Manifold Metric and Christoffel Symbols......Page 91
4.2 3D-grid Arrays of Omnidirectional Sensors......Page 93
4.3 Planar Arrays of Omnidirectional Sensors......Page 94
4.4 Families of θ- and φ-curves on theManifold Surface......Page 96
4.5 “Development” of Non-linear Array Geometries......Page 100
4.7.1 Proof that the Gaussian Curvature of an Omni-directional Sensor Planar Array Manifold is Zero......Page 106
4.7.2 Proof of the Expression of det G for Planar Arrays in Table 4.2......Page 107
4.7.3 Proof of “Development” Theorem 4.6......Page 108
5.1 Mapping from the (θ, φ) Parameter Space to Cone-Angle Parameter Space......Page 110
5.2 Manifold Vector in Terms of a Cone-Angle......Page 113
5.3 Intrinsic Geometry of the Array Manifold Based on Cone-Angle Parametrization......Page 114
5.4 Defining the Families of  - and  -parameter Curves......Page 117
5.5.1 Geodecity......Page 118
5.5.3 Shape of α- and β-curves......Page 120
5.6 “Development” of α- and β-parameter Curves......Page 123
6. Array Ambiguities......Page 126
6.1 Classification of Ambiguities......Page 127
6.2 The Concept of an Ambiguous Generator Set......Page 131
6.3 Partitioning the Array Manifold Curve into Segments of Equal Length......Page 134
6.3.1 Calculation of Ambiguous Generator Sets of Linear (or ELA) Array Geometries......Page 143
6.4 Representative Examples......Page 145
6.5 Handling Ambiguities in Planar Arrays......Page 148
6.5.1 Ambiguities on φ-curves......Page 149
6.5.2 Ambiguities on α-curves/β-curves......Page 152
6.5.3 Some Comments on Planar Arrays......Page 159
6.5.4 Ambiguous Generator Lines......Page 162
6.6 Ambiguities and Manifold Length......Page 165
6.7.1 Proof of Theorem 6.1......Page 168
7.1 Symmetric Linear Arrays and det(AN(s))......Page 170
7.2 Characteristic Points on the Array Manifold......Page 172
7.3 Array Symmetricity and Non-Uniform Partitions of Hyperhelices......Page 176
7.4 Ambiguities of Rank-(N – 1) and Array Pattern......Page 180
7.5 Planar Arrays and ‘Non-Uniform’ Ambiguities......Page 183
7.6 Conclusions......Page 185
8.1 Circular Approximation of an Array Manifold......Page 187
8.2 Accuracy and the Cramer Rao Lower Bound......Page 192
8.2.1 Single Emitter CRB in Terms of Manifold’s Differential Geometry......Page 193
8.2.2 Two Emitter CRB in Terms of Principal Curvature......Page 197
8.2.2.1 Elevation Dependence of Two Emitters’ CRB......Page 199
8.2.2.2 Azimuth Dependence of Two Emitters’ CRB......Page 201
8.3 “Detection” and “Resolution” Thresholds......Page 203
8.3.1 Estimating the Detection Threshold......Page 204
8.3.2 Estimating the Resolution Threshold......Page 207
8.4 Modelling of the Uncertainty Sphere......Page 211
8.5 Thresholds in Terms of (SNR × L)......Page 212
8.6.1 Schmidt’s Definition of Resolution......Page 216
8.6.2 CRB at the Resolution Threshold......Page 217
8.6.3 Directional Arrays......Page 218
8.7 Array Capabilities Based on α- and β-curves......Page 219
8.8 Summary......Page 221
8.9.1 Radius of Circular Approximation......Page 222
8.9.3 Proof: CRB of Two Sources in Terms of κ1......Page 223
Bibliography......Page 228
Index......Page 230