Preface -- Contents -- 1. Introduction -- 1.1 Nomenclature -- 1.2 Main Abbreviations -- 1.3 Array of Sensors - Environment -- 1.4 Pictorial Notation -- 1.4.1 Spaces/Subspaces -- 1.4.2 Projection Operator -- 1.5 Principal Symbols -- 1.6 Modelling the Array Signal-Vector and Array Manifold -- 1.7 Significance of Array Manifolds -- 1.8 An Outline of the Book -- 2. Differential Geometry of Array Manifold Curves -- 2.1 Manifold Curve Representation - Basic Concepts -- 2.2 Curvatures and Coordinate Vectors in CN -- 2.2.1 Number of Curvatures and Symmetricity in Linear Arrays -- 2.2.2 "Moving Frame" and Frame Matrix -- 2.2.3 Frame Matrix and Curvatures -- 2.2.4 Narrow and Wide Sense Orthogonality -- 2.3 "Hyperhelical" Manifold Curves -- 2.3.1 Coordinate Vectors and Array Symmetricity -- 2.3.2 Evaluating the Curvatures of Uniform Linear Array Manifolds -- 2.4 The Manifold Length and Number of Windings (or Half Windings) -- 2.5 The Concept of "Inclination" of the Manifold -- 2.6 The Manifold-Radii Vector -- 2.7 Appendices -- 2.7.1 Proof of Eq. (2.24) -- 2.7.2 Proof of Theorem 2.1 -- 3. Differential Geometry of Array Manifold Surfaces -- 3.1 Manifold Metric -- 3.2 The First Fundamental Form -- 3.3 Christoffel Symbol Matrices -- 3.4 Intrinsic Geometry of a Surface -- 3.4.1 Gaussian Curvature -- 3.4.2 Curves on a Manifold Surface: Geodesic Curvature -- 3.4.2.1 Arc Length -- 3.4.2.2 The Concept of Geodicity -- 3.4.3 Geodesic Curvature -- 3.5 The Concept of "Development" -- 3.6 Summary -- 3.7 Appendices -- 3.7.1 Proof of Eq. (3.36) - Geodesic Curvature -- 4. Non-Linear Arrays: (θ, φ)-Parametrization of Array Manifold Surfaces -- 4.1 Manifold Metric and Christoffel Symbols -- 4.2 3D-grid Arrays of Omnidirectional Sensors -- 4.3 Planar Arrays of Omnidirectional Sensors -- 4.4 Families of θ- and φ-curves on theManifold Surface.

4.5 "Development" of Non-linear Array Geometries -- 4.6 Summary -- 4.7 Appendices -- 4.7.1 Proof that the Gaussian Curvature of an Omni-directional Sensor Planar Array Manifold is Zero -- 4.7.2 Proof of the Expression of det G for Planar Arrays in Table 4.2 -- 4.7.3 Proof of "Development" Theorem 4.6 -- 5. Non-Linear Arrays: (α, β)-Parametrization -- 5.1 Mapping from the (θ, φ) Parameter Space to Cone-Angle Parameter Space -- 5.2 Manifold Vector in Terms of a Cone-Angle -- 5.3 Intrinsic Geometry of the Array Manifold Based on Cone-Angle Parametrization -- 5.4 Defining the Families of - and -parameter Curves -- 5.5 Properties of α- and β-parameter Curves -- 5.5.1 Geodecity -- 5.5.2 Length of Parameter Curves -- 5.5.3 Shape of α- and β-curves -- 5.6 "Development" of α- and β-parameter Curves -- 6. Array Ambiguities -- 6.1 Classification of Ambiguities -- 6.2 The Concept of an Ambiguous Generator Set -- 6.3 Partitioning the Array Manifold Curve into Segments of Equal Length -- 6.3.1 Calculation of Ambiguous Generator Sets of Linear (or ELA) Array Geometries -- 6.4 Representative Examples -- 6.5 Handling Ambiguities in Planar Arrays -- 6.5.1 Ambiguities on φ-curves -- 6.5.2 Ambiguities on α-curves/β-curves -- 6.5.3 Some Comments on Planar Arrays -- 6.5.4 Ambiguous Generator Lines -- 6.6 Ambiguities and Manifold Length -- 6.7 Appendices -- 6.7.1 Proof of Theorem 6.1 -- 7. More on Ambiguities: Symmetrical Arrays -- 7.1 Symmetric Linear Arrays and det(AN(s)) -- 7.2 Characteristic Points on the Array Manifold -- 7.3 Array Symmetricity and Non-Uniform Partitions of Hyperhelices -- 7.4 Ambiguities of Rank-(N - 1) and Array Pattern -- 7.5 Planar Arrays and 'Non-Uniform' Ambiguities -- 7.6 Conclusions -- 8. Array Bounds -- 8.1 Circular Approximation of an Array Manifold -- 8.2 Accuracy and the Cramer Rao Lower Bound.

8.2.1 Single Emitter CRB in Terms of Manifold's Differential Geometry -- 8.2.2 Two Emitter CRB in Terms of Principal Curvature -- 8.2.2.1 Elevation Dependence of Two Emitters' CRB -- 8.2.2.2 Azimuth Dependence of Two Emitters' CRB -- 8.3 "Detection" and "Resolution" Thresholds -- 8.3.1 Estimating the Detection Threshold -- 8.3.2 Estimating the Resolution Threshold -- 8.4 Modelling of the Uncertainty Sphere -- 8.5 Thresholds in Terms of (SNR × L) -- 8.6 Comments -- 8.6.1 Schmidt's Definition of Resolution -- 8.6.2 CRB at the Resolution Threshold -- 8.6.3 Directional Arrays -- 8.7 Array Capabilities Based on α- and β-curves -- 8.8 Summary -- 8.9 Appendices -- 8.9.1 Radius of Circular Approximation -- 8.9.2 "Circular" and "Y" Arrays - Sensor Locations -- 8.9.3 Proof: CRB of Two Sources in Terms of κ1 -- Bibliography -- Index.