1 Application of Differential Geometry in Spherical Harmonics Domain Saurabh Kataria and Arun Parthasarathy with Lalan Kumar and Prof. Rajesh M. Hegde CONTENTS I INTRODUCTION 2 II Differential geometry of manifold curves 3 II-A Data Model . 3 II-B Array Manifold . 4 II-C Dimensionality Reduction . 5 II-D Orthonormal Coordinate System . 5 II-E Curvatures, Cartan matrix and orthogonality . 5 II-F Significance of rate of change of arc length . 7 II-G Manifold Length, Uodd and Ueven ................................ 8 II-H Inclination angle . 8 II-I Manifold Radii Vector . 9 II-J MUSIC algorithm using differential geometry . 9 III Differential Geometry of Array Manifold Surfaces 10 III-A Manifold Metric . 10 III-B Christoffel Symbol Matrices . 11 III-C Intrinsic Geometry of a Surface . 11 III-C1 Gaussian Curvature . 11 III-C2 Geodesic Curvature . 12 IV Non-Linear Arrays: Manifold Parameters 12 IV-A 3D-grid Arrays of Omnidirectional Sensors . 12 IV-B Development of manifold surfaces . 13 V Array Bounds 14 V-A Circular approximation of manifold curves . 14 V-B Cramer Rao Lower Bound (CRLB) . 15 V-C Single emitter CRB . 16 V-D Two emitter CRB . 17 V-E Detection and resolution thresholds . 17 V-F Estimating detection threshold . 18 V-G Estimating resolution threshold . 19 V-H Uncertainty sphere model . 20 VI Spherical Harmonics Domain (SHD) 21 VI-A Signal Processing in Spherical Harmonics Domain . 21 VI-B Differential Geometry of SHD Manifold . 24 VI-C Detection and Resolution Thresholds . 25 VI-D Source Localization in Spherical Harmonics Domain . 26 Saurabh Kataria and Arun Parthasarathy are currently junior undergraduate students of Department of Electrical Engineering, Indian Institute of Technology, Kanpur. Lalan Kumar, who is the project guide, is currently a Ph.D. scholar under Prof. Rajesh M. Hegde, who is the project supervisor. SUMMER PROJECT REPORT 2 VII Simulation Results 26 VIII CONCLUSION 26 IX FUTURE WORK/ ADVANCEMENTS 27 Appendix A: Proof of Eq.117 28 Appendix B: φ-curve Differential Geometry Parameters 28 SUMMER PROJECT REPORT 3 NOTATIONS A; a Scalar A,a Column vector A Matrix AT Transpose AH Hermitian transpose A Norm of a matrix k k a ; A Magnitude of a number j j j j Hadamard product exp(A) Element wise exponential of a matrix expm(A) Matrix exponential Tr(A) Trace of a matrix det(A) Determinant of a matrix IN Identity matrix of dimension N N × R Set of real numbers C Set of complex numbers [A] Subspace spanned by columns of A L ABBREVIATIONS CRB Cramer Rao Bound DF Direction Finding DOA Direction Of Arrival DSHD Differential Spherical Harmonics Domain DSH-MUSIC Differential Spherical Harmonics-MUSIC FOV Field Of View MVDR Minimum Variance Distortion-less Response NLDR Non-Linear Dimensionality Reduction SHD Spherical Harmonics Domain SNR Signal-to-Noise Ratio ULA Uniform Linear Array UCA Uniform Circular Array I. INTRODUCTION IFFERENTIAL geometry is a well-known branch of mathematics that uses the techniques of differential D calculus, integral calculus and linear algebra in geometry problems. The use of differential geometry in array signal processing gives us tools to investigate manifold in terms of parameters like curvature and rate of change of arc-length (in case of manifold curve) to calculate detection and resolution thresholds and even implement DF algorithms like MUSIC for a given sensor geometry. Array of various geometries are available now-a-days of which spherical array enjoys a full 3D FOV and symmetry, due to which it is one of the most widely used array. In this report, tools of differential geometry are employed to investigate spherical array which are conventionally studied in spherical harmonics domain. Some useful insights have been formed by performing CRB-analysis and MUSIC on a 32-sensor spherical array. II. DIFFERENTIAL GEOMETRY OF MANIFOLD CURVES A. Data Model Consider the following data model with N omnidirectional sensors and M sources: x(t) = Am(t) + n(t) (1) where x(t) is the (N 1) vector containing output from all sensors, m(t) is the (M 1) vector containing amplitude × × of signals, n(t) is the (N 1) vector consisting of additive baseband white noise from each sensor and A (N M) × × SUMMER PROJECT REPORT 4 is called the steering matrix. T A = exp( jr k) (2) − Position vector matrix (3 N), as the name suggests, comprises position vectors of each sensor while assuming the origin of the coordinate× system to be at the centroid of the array. Throughout the report, this assumption is strictly followed. T r = [rx; ry; rz] (3) Wavenumber matrix is represented by the following expression: k = [k1; k2; :::; kM ] (4) th Each column km of the above matrix is a wavenumber vector pointing towards the DOA of the m signal. Wavenumber vector is formally defined as: 2π T 1 k = [sin(θ ) cos(φ ); sin(θ ) sin(φ ); cos(θ )] m− (5) m λ m m m m m where λ is the wavelength of the carrier wave of mth signal. However, we can get the following expression by changing its unit: T 1 km = π[sin(θm) cos(φm); sin(θm) sin(φm); cos(θm)] (λ/2 m)− (6) To maintain dimensional consistency in equation (2), position vector of the sensors must be taken in a different unit which is (λ/2) m. Coordinate system used throughout the report is shown here in Fig. 1. Fig. 1. Elevation: θ 2 [0; π] and azimuth: φ 2 [0; 2π] B. Array Manifold Steering matrix defined in previous section has M columns. Each column am of that matrix is called steering vector corresponding to the mth DOA. Since the number of DOAs are generally finite, there will be finitely many columns in the steering matrix. In general, steering vector can be found out for any direction (θ; φ). The locus of steering vector while varying (θ; φ) across every point in the FOV is a geometrical object called array manifold. Since the dimension of steering vector is N 1, the manifold will be embedded in an N-dimensional complex space. In other words, manifold consists of steering× vectors corresponding to every possible DOA and therefore, every time some points (vectors) in the manifold will always correspond to the actual DOA. In DF algorithms, the main aim is to identify those vectors. The shape of the manifold, whether it is a curve or a surface, depends on the number of directional parameters involved in the expression for steering vector. SUMMER PROJECT REPORT 5 Steering vector (N 1) can be mathematically expressed as: July 6, 2004× 9:29 WSPC/Book Trim Size for 9in x 6in chap02 a (θ ; φ ) = exp j(r sin(θ ) cos(φ ) + r sin(θ ) sin(φ ) + r cos(θ )) (7) m m m {− x m m y m m z m g For any linear array placed on x-axis, FOV is φ [0; π] which is independent of θ because of front-back ambiguity 2 (a drawback of linear arrays). For a planar array placed on x-y plane, FOV is φ [0; 2π]; θ [0; π=2] (again there 2 2 is a limitation in FOV due to its planarDifferential structure). Geometry And for of Array a three-dimensional Manifold Curves array, FOV is23φ [0; 2π]; θ [0; π]. 2 2 (That is, any value of (θ; φ)) is allowed as a DOA.) From the above it is clear that in the case of linear array, there is only one parameter involved i.e. φ whereas in the case of planar or three-dimensional array, there are two parameters involved, φ and θ. This means that for a linear array, there will be only one parameter in the steering vector and its locus will generate a curve which is called the manifold curve. Similar arguments can be made to conclude that for a two- or three-dimensional array, manifold will be a surface (known as manifold surface). Manifold curve is formally defined as: a(φ) CN ; φ : φ Ω (8) A , f 2 8 2 φg where Ωφ is the parameter space or the FOV. There is a need for re-parametrisation of the manifold curve in terms of a new parameter s, which denotes the arc-length of the curve (see Fig. 2) An appropriate reference point is chosen which is s = 0; φ = 0. The expression for arc-length is as follows: s s(φ) _a(φ) dφ (9) , k k Z0 Fig. 2.1 Manifold curve embedded in CN . A Fig. 2. Manifold parameterization in termsFig. 2.2 of arc Manifold length s parameterization in terms of arc length s. The benefit of re-parametrisationThe array manifold of a( isφ) conventionallyin terms of s parametrizedis that s is an in invariant terms of parameter a generic and the norm of the first derivative of a w.r.t. to s is a constant s as shown below: bearing parameter p. Forinstanceforalineararray,8 p may represent the azimuth angle θ. However,da(s) parametrizationda(φ)=dp in terms_a(φ of) the arcs_(φ length) s (see a0(s) = = = k k = = 1 (10) Fig. 2.2),k whichk is theds most basic featureds=dp of a curves_( andφ) a naturals_(φ) parameter representing the actual physical length of a segment of the manifold curve in Parametrisation can be physically understood through following statements. The first derivative of a(φ) can be thought of as velocity. Then, its norm should signify speed. Also, if φ is considered to be analogous with time, it can be said that speed is being integrated w.r.t to time to give distance covered, which is nothing here but the arc-length.