IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL. ?, NO. ?, ???? 1 Investigating Hyperhelical Array Manifold Curves Using the Complex Cartan Matrix Athanassios Manikas, Senior Member, IEEE, Harry Commin, Student Member, IEEE, and Adham Sleiman Abstract—The differential geometry of array manifold curves where the (N 3) real matrix [rx, ry, rz] denotes the array has been investigated extensively in the literature, leading to sensor locations: numerous applications. However, the existing differential geo- T N 3 metric framework restricts the Cartan matrix to be purely real [rx, ry, rz] = [r1, r2, . , rN ] (2) and so the vectors of the moving frame U(s) are found to be 2 R orthogonal only in the wide sense (i.e. only the real part of their For the purposes of this paper, it is convenient to define the inner product is equal to zero). Imaginary components are then phase reference at the centroid of the array. In Equation 1, accounted for separately using the concept of the inclination angle k(, ) is the wavenumber vector: of the manifold. The purpose of this paper is therefore to present an alternative u(,φ) theoretical framework which allows the manifold curve in CN to 2 k(, ) [cos() cos(), sin() cos(), sin()] (3) be characterised in a more convenient and direct manner. A , continuously differentiable strictly orthonormal basis is estab- z }| { lished and forms a platform for deriving a generalised complex where is the carrier signal wavelength and u(, ) is the Cartan matrix with similar properties to those established under (3 1) real unit vector pointing from (, ) towards the origin. the previous framework. Concepts such as the radius of circular As is common practice, the notation used for a(, ) in approximation, the manifold curve radii vector and the frame matrix are also revisited and rederived under this new frame- Equation 1 ignores dependence upon the constant, known work. quantities rx, ry, rz and . However, it is important to note that the manifold vector is by no means restricted to Index Terms—Array manifold, differential geometry, array processing. a (, )-parameterisation. Firstly, this is because the (, )- parameterisation is not unique; other valid directional parame- NOTATION terisations exist (with cone-angle parameterisation particularly a, A Scalar useful for the analysis of planar arrays [1]). Secondly, many a,A Column vector other variable parameters of interest (besides direction) can be A Matrix incorporated into the response vector modelling. When these ( )T , ( )H Transpose, conjugate transpose additional parameters (such as delay, Doppler, polarisation, · · Absolute value subcarrier and spreading/scrambling codes) are included, the jj Euclidean norm of vector resulting response vector is referred to as an extended manifold ak·kb Element-by-element power vector, the properties of which have been investigated in [2]. exp(a) Element-by-element exponential Therefore, for the sake of generality, the response vector will diag a Diagonal matrix whose diagonal entries are a simply be denoted as a(p), where the vector p may comprise f g any or all of the aforementioned variable parameters (and/or IN (N N) identity matrix others). The central purpose of this paper then lies in exploring 0N (N 1) vector of zeros Tr Matrix trace operator the geometrical nature of the mathematical object which is fg Hadamard (element-by-element) product traced out by a(p) when p is evaluated across the range of all R The set of real numbers feasible parameter values (denoted by the parameter space, ). C The set of complex numbers This resulting object is called the array manifold. It completely characterises the array system and is formally defined as: I. INTRODUCTION a(p), p (4) ONSIDER an array of N sensors (residing in three- A , 8 2 C dimensional real space) receiving one or more narrow- In the single-parameter case (i.e. p = p), it can be seen band plane waves. The response of the array to a signal inci- that a(p) traces out a curve in CN . For two parameters, the dent from azimuth [0, 360) and elevation ( 90, 90) manifold is a surface and, similarly, for larger numbers of is commonly modelled2 using the (N 1) complex2 array parameters, the manifold is some higher dimensional object. response vector, or manifold vector: A branch of mathematics dedicated to the investigation of these kinds of (differentiable) manifolds is differential geome- a(, ) , exp j rx, ry, rz k (, ) (1) try [3,4]. The tools of differential geometry have already been The authors are with the Department of Electrical and Electronic Engi- applied extensively in the array signal processing literature [5]. neering, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK e-mail: ([email protected]). The main body of this existing research relates to the study Manuscript received XX XX, 20XX; revised XX XX, 20XX. of manifold curves and surfaces. Although the geometrical IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL. ?, NO. ?, ???? 2 properties of array manifold surfaces have been investigated Under the traditional approach, the manifold vector a(p), directly [6], this is a more complicated task than for curves. residing in CN , is instead considered to comprise 2N real To simplify matters, it has been shown that array manifold components. For this reason, the moving frame, Uw(s), con- surfaces can alternatively be considered to consist of families sists of up to 2N complex vectors: of constant-parameter curves [7]. Therefore, the theoretical framework introduced in this paper will be done so with a Uw(s) , uw,1(s), uw,2(s), . , uw,d(s) (8) focus on array manifold curves. A particularly important class of array manifold curves where N 1 d 2N (depending on the level of is the hyperhelix. Hyperhelices are especially convenient to symmetricity exhibited≤ ≤ by the sensor array). Together with analyse since all their curvatures are constant (do not vary from Uw(s), up to 2N 1 non-zero real curvatures can be defined. point to point) and may be calculated recursively. Although all These allow the motion of the moving frame to be expressed linear arrays of isotropic sensors have hyperhelical manifold as: curves, the useful properties of the hyperhelix have also been Uw0 (s) = Uw(s)Cr(s) (9) exploited for the analysis of planar arrays using cone-angle parameterisation and the concept of the “equivalent linear where Cr(s) is the purely real Cartan matrix, which contains array” [5]. Furthermore, hyperhelical manifold curves have the curvatures, i, according to the following skew-symmetric been identified in the analysis of extended array manifolds structure: [2]. Therefore, in this paper, particularly close attention will be paid to the analysis of hyperhelical manifold curves. 0 1 0 0 In order to highlight the motivation of this paper, it is ···. .. useful at this point to briefly review the existing differential 2 1 0 2 . 3 . geometric framework that has been applied to the analysis of Cr(s) = 6 0 .. .. 0 7 (10) 6 2 7 array manifold curves. A short summary of some of its key 6 . .. .. .. 7 6 . d 1 7 applications in the literature will then be given. 6 7 6 0 0 d 1 0 7 6 ··· 7 A. Geometry of Array Manifold Curves: Traditional Approach 4 5 As a consequence of constraining the entries of r(s) to be N C Given a curve a(p) C as a function of a single real purely real, it is found that the moving frame, (s), can only 2 Uw parameter p R (such as azimuth, , or elevation, ), the arc be orthonormal in the “wide sense” (i.e. applying only to the 2 length along the manifold curve, s(p), and its rate of change, real part): s_(p), are defined, respectively, as: Re H (s) (s) = (11) p Uw Uw Id s(p) , s_(p0)dp0 (5) As a result, it has been proven [5, Eq. 2.23] that this leads to Z0 ds (s) taking the form: s_(p) = a_(p) (6) Cr , dp k k H A useful feature of parameterising a manifold curve in terms Cr(s) = Re Uw (s)Uw0 (s) (12) of arc length, s, is that it is an invariant parameter. This means that the tangent vector to the curve (expressed in terms of s): This means that Cr(s) does not, in general, uniquely describe the shape and size of the manifold curve (since any imaginary d dp d a_(p) components are ignored). Only in the special case of arrays a0(s) , a(s) = a(p) = (7) ds ds dp a_(p) with sensors located symmetrically about the origin is the k k is always unit length. (Note that differentiation with respect to manifold seen to admit a representation entirely in RN . In s has been denoted by “prime” and differentiation with respect the general case, therefore, imaginary components must be to p by “dot”. This convention will be followed throughout this accounted for separately, using the concept of the inclination paper). angle of the manifold (defined as the angle between the The tangent vector a0(s0) provides useful local information manifold vector and a certain subset of the even-numbered about the manifold curve in the neighbourhood of s0 (it is frame vectors [5, p.45]). a geometric approximation of the first order). However, in The purpose of the new theoretical framework presented order to build up a full (local) characterisation of the manifold in this paper is to allow the manifold curve in CN to be curve, it is necessary to attach some additional vectors to the characterised in a more convenient and direct manner. The running point, s0.