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 ∈ 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(θ, φ) is the wavenumber vector: of the manifold. The purpose of this paper is therefore to present an alternative u(θ,φ) theoretical framework which allows the manifold curve in CN to 2π k(θ, φ) [cos(θ) cos(φ), sin(θ) cos(φ), sin(φ)] (3) be characterised in a more convenient and direct manner. A , λ continuously differentiable strictly orthonormal basis is estab- z }| { lished and forms a platform for deriving a generalised complex where λ is the carrier signal wavelength and u(θ, φ) is the Cartan matrix with similar properties to those established under (3 1) real unit vector pointing from (θ, φ) towards the origin. the previous framework. Concepts such as the radius of circular ×As is common practice, the notation used for a(θ, φ) 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 λ. 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 (θ, φ)-parameterisation. Firstly, this is because the (θ, φ)- 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 |·| 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 { } 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 {·} 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, Ω). 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 Ω (4) ONSIDER an array of N sensors (residing in three- A , ∀ ∈ 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 θ [0, 360◦) and elevation φ ( 90, 90◦) manifold is a surface and, similarly, for larger numbers of is commonly modelled∈ using the (N 1) complex∈ − 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(θ, φ) , exp j rx, ry, rz k (θ, φ) (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, κi, 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 κ1 0 0 In order to highlight the motivation of this paper, it is − ···. . .. . useful at this point to briefly review the existing differential  κ1 0 κ2 .  −. . geometric framework that has been applied to the analysis of Cr(s) =  0 κ .. .. 0  (10)  2  array manifold curves. A short summary of some of its key  ......   . . . . κd 1  applications in the literature will then be given.  − −   0 0 κd 1 0   ··· −  A. Geometry of Array Manifold Curves: Traditional Approach   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 ∈ Uw parameter p R (such as azimuth, θ, or elevation, φ), the arc be orthonormal in the “wide sense” (i.e. applying only to the ∈ 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. Specifically, an orthonormal moving frame key to doing this lies in reformulating Equations 8 - 12. In of (local) coordinate vectors can be constructed. Then, the particular, a continuously differentiable strictly orthonormal N N manner in which this frame twists and turns as the running basis, U(s) C × , is established and forms a basis for de- ∈ point progresses along the curve will be shown to provide a riving a generalised complex Cartan matrix, C(s), with similar profoundly meaningful description of the curve. However, the properties to those established under the previous framework. way in which these frame vectors are chosen is of central Concepts such as the radius of circular approximation, the importance to this paper. A subtly different approach will be manifold curve radii vector and the frame matrix are also seen to provide significant new insight. revisited and rederived under this new framework. IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL. ?, NO. ?, ???? 3

B. Exemplary Results and Motivation and then all subsequent coordinate vectors (for k 2) are ≥ The usefulness and importance of applying the tools of obtained via Gram-Schmidt orthonormalisation: differential geometry to the analysis of the array manifold v (s) u (s) = k (16a) can be demonstrated using a number of results found in the k v (s) k k k literature. In [8], the fundamental performance capabilities k 1 − H (detection, resolution and estimation error) of an array system vk(s) , uk0 1(s) um(s)uk0 1(s) um(s) (16b) − − − were defined in the context of the array manifold and derived m=1 X
explicitly using the circular approximation of the array mani- In other words, each successive coordinate vector simply fold. The way in which anomalies in the manifold lead to array points in the direction that the hyperplane spanned by the ambiguities was investigated in [9]. A differential geometric previously-defined vectors is moving. perspective was applied to the topic of array design (i.e. the It is worth noting that, under certain circumstances, the array judicious placement of array sensors) in [10]. The sensitivity manifold curve may not span the full N-dimensional complex of array systems to uncertainties in sensor locations, based observation space and so it is not always necessary to define on their location in the overall array geometry, was analysed N frame vectors. Consider, for example, array configurations in [11]. In [12], the task of designing virtual arrays for the where a sensor exists at the array centroid (i.e. the phase purpose of array interpolation was explored. origin). It is clear that that sensor’s response must be constant The study of more sophisticated array systems which incor- for all p and, consequently, the p-curve cannot depart from the porate additional system and channel parameters (such as code (N 1)-dimensional hyperplane associated with this constant division multiple access spreading/scrambling codes, lack of value.− synchronisation, Doppler effects, polarisation parameters and subcarriers) has been introduced using the concept of extended B. The Complex Cartan Matrix array manifolds in [2]. The Cartan matrix provides the curvatures of the curve a(s), which in turn are the curvilinear coordinates of the curve on C. Paper Organisation the moving polyhedron U(s). These curvatures are a means The remainder of this paper is organised as follows. In of assessing the changes in the polyhedron, expressed by the Section II, the basic principles of the proposed theoretical first derivative, U0(s), of U(s), and hence the Cartan matrix, framework are presented. The strictly orthogonal moving C(s), is used to express U0(s) as a function of U(s): frame is established and the complex Cartan matrix is derived. (s) = (s) (s) (17) In Section III, specific properties of hyperhelical manifold U0 U C N N curves are investigated. Explicit expressions are derived for Note that in complex multi-dimensional space, U(s) C × N N ∈ the orthonormal frame vectors and entries of the complex and hence C(s) C × . In particular, C(s) should be skew- Cartan matrix and numerical examples are presented. Finally, Hermitian since this∈ would yield, as a special case, the familiar the paper is concluded in Section IV. skew-symmetric purely real Cartan matrix for curves in RN . It is proven in Appendix A that the complex Cartan matrix II.CURVES IN CN can be written in the form: H In this section, the strictly orthonormal moving frame (or u1 u10 v2 0 0 − k k ···. . moving polyhedron), (s), will be formulated. Using the H . . U  v2 u2 u20 v3 . .  definition of U(s), the exact structure of the complex Cartan k k − k. k . C(s) =  0 v .. .. 0  matrix, C(s), will then be derived.  k 3k   ......   . . . . vN   −H k k   0 0 v u u0  A. The Moving Polyhedron  ··· k N k N N   (18) In order to describe meaningfully the way in which the which is indeed skew-Hermitian: curve a(s) twists through N-dimensional complex space (at H a point s), the moving frame of N orthonormal coordinate C (s) = C(s) (19) − vectors, U(s), at the running point, s: This property is easily proven, since differentiating the identity uH u = 1 leads to: U(s) , [u1(s), u2(s), . . . , uN (s)] (13) k k H H will now be derived, where strict orthonormality implies: uk uk0 + uk0 uk = 0 H Re u u0 = 0 (20) H k k U (s)U(s) = IN (14) ⇒ and so the diagonal entries of C(s) are purely imaginary. N Specifically, in accordance with common practice in R (see, Clearly, the structure of C(s) (Equation 18) shows certain for example, [3, p.159] and [4, p.13]), the unit tangent vector similarities to Cr(s) in the previous framework (Equation 10). is first defined as: The key differences to note are that, firstly, C(s) is generally a u (s) a0(s) (15) smaller matrix ((N N) compared to (d d)) and, secondly, 1 , × × IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL. ?, NO. ?, ???? 4 it can have non-zero, albeit purely imaginary, diagonal entries. curves of planar arrays (where α and β denote cone angles), As a result, in the general case (2 k N 1), the via the concept of the equivalent linear array [5, p.113]. In ≤ ≤ − N frame vector derivative u0 (s) is a linear combination of not such cases, r R denotes the equivalent linear array sensor k ∈ only uk 1(s) and uk+1(s) (with purely real coefficients) locations. − but also uk(s) itself (with the purely imaginary coefficient It will be shown in this section that the manifold curve H uk (s)uk0 (s)). In this way, C(s) completely characterises the traced out by any a (p) with this structure (Equation 24) is fully complex nature of the curve (including the concept of of hyperhelical shape. A number of important properties of inclination angle in the previous framework). This point can hyperhelical manifold curves will then be derived explicitly. now be demonstrated by considering the radius of the circular approximation of the manifold. A. Manifold Vector Derivatives As a preliminary to investigating the differential geometry C. The Radius of the Circular Approximation of the hyperhelical array manifold, a compact differentiation A powerful tool in the analysis of array manifold curves is procedure of the response vector is established. Applying the the circular approximation of the manifold [5, Ch.8]. More chain rule, leads to: specifically, it has been proven that any sufficiently small a˙ (p) a˙ (p) a (s) = = section (arc length) of an array manifold curve can be ap- 0 s˙(p) a˙ (p) k k proximated as a circular arc. Using the previous differential jπr sin p a (p) = geometric framework, it was shown that the inverse of the π r sin p radius of the circular approximation was only equal to the k k = jr a (p) (25) principal curvature, κ1, in the special case of symmetric linear arrays. For general array geometries, however, the inclination where the normalised (equivalent) sensor locations have been angle of the manifold, ζ, needed to be accounted for as defined as: r follows: r (26) , r k k κˆ1(s) = κ1(s) sin ζ(s) (21) Applying the chain rule repeatedly to a(s) in this way leads 2 H 2 a pattern to emerge which is straightforward to prove by = u0 (s) u (s)u0 (s) (22) k 1 k − 1 1 induction: q the proof of which was derived in [5, p.209]. k (k) d a(s) k The corresponding curvature in the complex Cartan matrix a (s) = (jr) a(s) (27) , dsk derived in this paper is v (s) : k 2 k For notational convenience, we define the diagonal matrix H v2(s) = u10 (s) u1 (s)u10 (s) u1(s) R: k k − N N diag(r) R × (28) 2 H 2 R , = u10 (s) u1 (s)u
10 (s) (23) ∈ k k − which allows Equation 27 to be equivalently written as: which provides the desiredq result directly. This demonstrates (k) k k both the equivalence between the two theoretical frameworks a (s) = j R a(s) (29) as well as the directness and convenience of using the complex This notation will prove useful in investigating the differential Cartan matrix. geometry of the hyperhelical array manifold curve. Note that 2 Tr R2 = r = 1. III.HYPERHELICAL ARRAY MANIFOLD CURVES k k  Having established (in Equations 16 and 18) the new frame- B. Differential Geometry of Hyperhelical Manifold Curves work for investigating the geometry of array manifold curves, It is now possible to investigate the orthonormal basis the important class of curves which have hyperhelical shape U(s) and the elements of the Cartan matrix, C(s), for the can now be studied in detail. In this way, a number of concepts hyperhelical array manifold. Indeed, in Appendix B, it is such as the frame matrix and manifold radii vector can be shown that the (unnormalised) orthogonal frame vector vk(s) rederived in order to compare to the analogous quantities can be written in the form: obtained via the previous framework. New results regarding the eigendecomposition of the complex Cartan matrix are also vk(s) = Xka(s) (30) given, followed by specific numerical examples. where the diagonal matrix Xk+1 can be obtained recursively Consider an array system with a response vector having the for 2 k N 1 as: form: ≤ ≤ − j vk j H a (p) = exp ( jπr cos p + w) (24) Xk+1 = XkR+ k k Xk 1 Tr Xk XkR Xk − v v − − v 3 k kk k 1 k where r and w are constant vectors and p is the variable − k k  (31) parameter of interest. The most commonly seen example of with initial conditions: such a response vector is that of of a linear array (where X1 = jR (32) r = r cos(φ ) and w = 0 ) as a function of azimuth (p = θ). x 0 N H X2 = jX1R j Tr X1 X1R X1 (33) However, this model also applies more generally to φ, α and β −  IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL. ?, NO. ?, ???? 5

Furthermore, it is shown that the entries of the complex Cartan 81 in terms of uk+1(s) can be seen to coincide precisely matrix, C(s), for all k = 1, 2,...,N can be obtained as with [5, Eq. 2.11]. A numerical example demonstrating this follows: equivalence will be given in Section III-F.

H vk = Tr Xk Xk (34) k k D. Frame Matrix and Radii Vector H qj  H uk uk0 = 2 Tr Xk XkR (35) Consider a point s = 0 on the curve, a(s), whose local vk 6 k k orthonormal basis, U(s), is unknown. Given the set of ortho- It is evident from Equations 34 - 35 that the complex Cartan normal vectors, U(s), at s = 0, namely U (0), a continuous matrix is constant (independent of s), confirming that the N N differentiable transformation matrix F(s) C × must exist manifold is of hyperhelical shape. ∈ which relates U(s) to U (0) such that

C. Symmetric Linear Arrays U(s) = U (0) F(s) (41) Before proceeding with our discussion of hyperhelical mani- This transformation matrix F(s) is called the frame matrix, and fold curves, it is interesting to address the special case of linear is a nonsingular matrix which must satisfy the initial condition arrays whose sensors exist in symmetric pairs about the origin. Specifically, it will be shown that, for this class of arrays, the F (0) = IN (42) diagonal entries of the complex Cartan matrix vanish and so For hyperhelical array manifolds (for which the complex N the manifold admits a full representation in R . As a result, Cartan matrix is constant) the differential equation of Equation the Cartan matrix and moving frame will be seen to coincide 17 is reduced to the following first order differential equation: exactly with those obtained under the previous framework. H To prove that the diagonal entries of C vanish (i.e. uk uk0 = U0(s) = U(s)C (43) 0 for all k = 1, 2,...,N), the following property of symmetric linear arrays can be used: which admits a solution of the type: i (s) = (0) expm s (44) Tr R = 0, i odd (36) U U C ∀ { } H where U (0) accounts for initial conditions and expm Therefore, in order to prove that uk uk0 = {·} j H denotes the matrix exponential. The frame matrix of Equation 2 Tr k = 0, it is sufficient to prove that v Xk X R k kk 41 is therefore given by: Xk is a polynomial in R of either strictly even or strictly odd powers. F(s) = expm sC (45) { } Given that X1 = jR, it naturally follows that: which clearly satisfies the condition (0) = N . Since (s) is, H F I F Tr X1 X1R = 0 (37) in general, complex-valued, its properties are subtly different from the analogous frame matrix derived under the previous From Equation 33, this leads to the simplified expression X2 = 2 framework [5, p.29]. Firstly, due to the skew-Hermitian nature jX1R = R and so: − of C, it can be deduced that: Tr H = 0 (38) X2 X2R H 1 F (s) = F( s) = F− (s) (46) − Using X1 = jR (a polynomial of strictly odd power) and 2 and so (s) is unitary: X2 = R (a polynomial of strictly even power) as initial F − conditions in Equation 31 will then lead to the desired result. H F (s)F(s) = IN (47) Specifically, given Xk, a polynomial of strictly odd powers of R and Xk 1, a polynomial of strictly even powers of R, Using Equations 41 and 43, the complex Cartan matrix may − Equation 31 becomes: be written as a function of the frame matrix:

j vk = H (s) (s) (48) Xk+1 = XkR + k k Xk 1 (39) C F F0 vk vk 1 − k k even order − even order and so a similar first order differential equation to Equation which is a polynomial in R|of{z strictly} even powers.| {z } Obviously, 43 relates the frame and Cartan matrices: equivalent reasoning can be applied to the converse case (Xk F0(s) = F(s)C (49) as a polynomial of strictly even powers and Xk 1 a polynomial − T of strictly odd powers). Therefore, it follows in general that: The manifold radii vector, (s) = [ 1(s),..., N (s)] N R R R ∈ H C , is defined as: Tr Xk XkR = 0 (40) H H (s) = U (s)a(s) (50) which confirms that ukuk0 = 0 for all k = 1, 2,...,N. R Using this result with the expressions derived in Appendix A such that its elements constitute the inner products of the reveals an exact equivalence with the previous framework in manifold vector and the coordinate vectors the special case of symmetric linear arrays (which exhibit zero H “inclination angle”). In particular, a rearrangement of Equation k(s) = u (s)a(s) (51) R k IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL. ?, NO. ?, ???? 6 for k = 1,...,N. In the case of a hyperhelical manifold, multiplied by the pure imaginary number, j, and the eigen- k(s) can be seen to be independent of arc length: vectors are given by U(s). R H Note that, from Equation 61, it is possible to deduce that Xka(s) k = a(s) for any positive integer, m > 0: R vk  k k  m H m Tr X∗ = (s)(j ) (s) = { k} (52) C U R U v k kk   From Equation 50, the manifold vector may be written as a H H = U (s)(jR) U(s)U (s) (jR) U(s) ...  function of the strictly orthonormal basis and complex Cartan matrix:  =IN  H m  = U (s)(jR) |U(s){z } (62) a(s) = U(s) (s) R = U (0) expm sC (s) (53) It is worth drawing attention to the fact that Equation 60 is a { }R first order differential equation similar to Equation 43, except Further examination of the recursive Equation 31 allows the here U(s) is pre-multiplied by a constant matrix. The solution following pattern to be observed: to this differential equation takes the form: X1 = jR imag. = j j Tr H real U(s) = expm jsR U(0) (63) X2 X1R X1 X1R X1 { } − v = j + 2 j Tr H imag. X3 v X2R kv kX1 3 X2 X2R X2 Therefore, a pre-multiplying frame matrix, (s), is obtained 2 1 − v2 E k k k k k k X4 = . . . real (as an alternative to post-multiplying F(s)) such that: Indeed, it is straightforward to prove by induction that: U(s) = E(s)U(0) (64) real, k even = (54) Xk ∀ with E(s) expm jsR which has initial condition E(0) = imaginary, k odd , { }  ∀ IN . In fact, it can be seen that E(s) is also the matrix which Using this result in conjunction with Equation 52 leads to: relates the original manifold vector a(0) to any other manifold real, k even vector: k = ∀ (55) R imaginary, k odd  ∀ a(s) = U(s) (s) In other words, consecutive entries of the radii vector toggle R = E(s)U(0) (s) between purely real and purely imaginary. R Finally, the length of the radii vector is given by: = E(s)a(0) (65)

(s) = aH (s)U(s)UH (s)a(s) kR k F. Numerical Examples = √qN (56) For the first example, consider a (non-symmetric) linear which corresponds to the radius of the hypersphere upon which array of four sensors, i.e. N = 4, with sensor positions in half- the manifold curve lies, as might be expected. wavelengths given by r = [ 2.1, 1.1, 0.9, 2.3]T . Assuming x − − φ0 = 0, the transformation matrices, Xn, for n = 1,..., 4, E. Cartan Matrix Eigendecomposition and the Cartan matrix, C, are given by: We begin by differentiating Equation 16a and substituting j0.6134, 0, 0, 0 for vk as follows: − 0, j0.3213, 0, 0 ka0(s) X1 =  −  u (s) = X (57) 0, 0, +j0.2629, 0 k0 v k kk  0, 0, 0, +j0.6718 a(s)   = j Xk (58)  0.4115, 0, 0, 0  R v − k kk 0, 0.1217, 0, 0 X2 =  −  = jRuk(s) (59) 0, 0, 0.0540, 0 −  0, 0, 0, 0.4128 Collecting all uk0 (s) for k = 1, 2,...,N therefore gives:  −  +j0.0682, 0, 0, 0 U0(s) = jRU(s) (60) 0, j0.1229, 0, 0 X3 =  −  Combining Equations 43 and 60 leads to: 0, 0, +j0.1351, 0  0, 0, 0, j0.0494 j (s) = (s)  −  RU U C 0.0589, 0, 0, 0  H C = jU (s)RU(s) (61) 0, 0.2181, 0, 0 ⇒ = X4  0, − 0, 0.2158, 0  which reveals that the eigenvalues of C are equal to the −  0, 0, 0, 0.0339 normalised sensor positions of the (equivalent) linear array     IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL. ?, NO. ?, ???? 7

0, 0.6006, 0, 0, 0, 0, 0, 0 +0.6006, − 0, 0.2140, 0, 0, 0, 0, 0  0, +0.2140, − 0, 0.3534, 0, 0, 0, 0  −  0, 0, +0.3534, 0, 0.2357, 0, 0, 0  r =  −  (67) C  0, 0, 0, +0.2357, 0, 0.5588, 0, 0   −   0, 0, 0, 0, +0.5588, 0, 0.1209, 0   −   0, 0, 0, 0, 0, +0.1209, 0, 0.2935  −   0, 0, 0, 0, 0, 0, +0.2935, 0      and matrix to be purely real, the framework proposed in this paper +j0.0574, 0.5979, 0, 0 incorporated a complex, skew-Hermitian Cartan matrix. In this +0.5979, +−j0.0185, 0.2011, 0 way, a strictly orthogonal moving frame has been established, = C  0, +0.2011, −j0.0314, 0.3143  which facilitates a more convenient and direct method of − −  0, 0, +0.3143, j0.0445 analysis.  −   (66) A number of specific features of the proposed framework Using the traditional framework, the purely real Cartan matrix have been highlighted and compared to the traditional ap- of the same array is the 8 8 real matrix shown in Equation proach (with particular attention given to hyperhelical array 67, above. × manifold curves). Concepts such as the radius of circular Comparing the results obtained for the two Cartan matrices, approximation, manifold curve radii vector and the frame it is clear that the first curvature of Cr (namely κ1 = 0.6006) matrix were revisited and rederived under the new framework. can be obtained as the magnitude of the first two elements H of C, i.e. κ1 = v2 + u1 u10 = 0.5979 + j0.0574 . k k | | APPENDIX A Subsequent real curvatures κ2, κ3, . . . , κd 1 may be obtained − DERIVATION OF THE COMPLEX CARTAN MATRIX via more complicated expressions involving the lower order entries of the complex Cartan matrix (but not vice versa, since In order to evaluate the elements of the complex Cartan C contains information regarding inclination angle, which is matrix, Equation 16b is first rearranged, following a change absent from Cr). of variables, as: For the second example, consider the symmetric linear array T k rx = [ 2.3, 1.1, 1.1, 2.3] . Assuming φ0 = 0, the complex H Cartan− matrix− of its manifold is given by: uk0 (s) = vk+1(s) + um(s)uk0 (s) um(s) (71) m=1 0, 0.5903, 0, 0 X
− H +0.5903, 0, 0.2069, 0 Given that um(s)uk(s) = 0 for m = 1, . . . , k 1 with 2 = (68) − ≤ C  0, +0.2069, − 0, 0.3297 k N 1, it follows that: − ≤ −  0, 0, +0.3297, 0    d H   um(s)uk(s) = 0 and the radii vector = [0, 1.6939, 0, 1.0633]T . These ds H H R − −  u (s)u0 (s) = u0 (s)u (s) (72) results, and indeed the orthonormal basis U (s), computed at ⇒ m k − m k any azimuth, θ0, are found to coincide exactly with those obtained through the previous curvilinear differential geometry Substituting this result into Equation 71 yields for 2 k N 1: ≤ ≤ framework for the same symmetric array: − k Cr = C (69) H u0 (s) = v (s) u0 (s)u (s) u (s) (73) k k+1 − m k m Uw(s) = U (s) (70) m=1 X
The convergence of the two frameworks for symmetric linear or, by a simple change of variables: arrays, where the manifold dimensionality is reduced to real m only, demonstrates that the two approaches are fundamentally H u0 (s) = v (s) u0 (s)u (s) u (s) (74) correct and that the difference lies in the handling of the purely m m+1 − i m i i=1 imaginary dimensions. X
Substituting this expression for um0 (s) back into Equation 73 IV. CONCLUSION then leads to (note that (s) is dropped for simplicity for the In this paper, a new theoretical framework has been pre- next few equations): sented which provides an alternative to the existing differential k m H geometric approach to analysing array manifold curves. While H u0 = v v u0 u u u u (75) k k+1 − m+1 − i m i k m the traditional approach restricted the entries of the Cartan m=1 " i=1 # X X IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL. ?, NO. ?, ???? 8

This equation can be rewritten as: (where v = 1 has not been deleted for the sake of k 1k k k m consistency as k is increased). Next, it can be seen that the H H H first element of the complex Cartan matrix, uH (s)u (s), is u0 = v v u u + u u0 u u u 1 10 k k+1 − l+1 k l m i i k m l=1 m=1 i=1 also independent of s: X
X X

(76) H j H H Then, noting that: u1 u10 = 2 a (s)X1 X1Ra(s) v1 H k k vl+1uk = 0 iff l = k 1 (77) j H H 6 − = 2 Tr X1 X1R (87) ui uk = 0 iff i = k (78) v1 6 k k  allows Equation 76 to be simplified to: Proceeding to k = 2 and substituting from the above H expressions leads to: uk0 = vk+1 vk uk 1 + uk uk0 uk (79) − k k − H v2(s) = u10 (s) u1 u10 u10 (s) Finally, noting that vk = vk uk, the method
for calculating − k k j j uk0 (s) can now be stated in full.
H = X1R 3 Tr X1 X1R X1 a(s) (88) For k = 1: " v1 − v1 # k k k k H  u0 = u u0 u + v u (80) X2 1 1 1 1 k 2k 2 For 2 k N 1:
and continuing| as before allows{z the second diagonal} entry of ≤ ≤ − the Cartan matrix to be obtained: H uk0 = vk uk 1 + uk uk0 uk + vk+1 uk+1 (81) j − k k − H H u2 u20 = 2 Tr X2 X2R (89) For k = N:
v2 k k  H Proceeding in this manner leads the pattern stated in uN0 = vN uN 1 + uN uN0 uN (82) − k k − Equations 31 and 35 to emerge. Proof by induction is a where Equation 80 is obtained by direct
rearrangement of straightforward extension of the previous discussion. Equation 73 (with k = 1). Meanwhile, Equation 82 follows from the fact that attempting to evaluate v (in Equation N+1 REFERENCES 16b) must yield the zero vector. [1] H. R. Karimi and A. Manikas, “Cone-angle parametrization of the array Using Equations 80 - 82 with Equation 17 leads directly to manifold in DF systems,” J. Franklin Inst. (Eng. Appl. Math.), vol. 335B, C(s) in Equation 18. pp. 375–394, 1998. [2] G. Efstathopoulos and A. Manikas, “Extended Array Manifolds: Func- tions of Array Manifolds,” IEEE Transactions on Signal Processing, APPENDIX B vol. 59, no. 7, pp. 3272 –3287, july 2011. PROPERTIES OF HYPERHELICAL ARRAY MANIFOLDS [3] H. Guggenheimer, Differential Geometry. McGraw Hill (or Dover Edition), 1963 (1977). In order to analyse the differential geometry of the hyperhe- [4] W. Kühnel, Differential Geometry: Curves - Surfaces - Manifolds, lical array manifold curve, the parameters presented in Section 2nd ed. American Mathematical Society, 2005. II are next evaluated for the special array manifold vector [5] A. Manikas, Differential Geometry in Array Processing. Imperial College Press, 2004. structure of Equation 24. Specifically, in order to identify the [6] A. Manikas, A. Sleiman, and I. Dacos, “Manifold Studies of Nonlinear orthonormal basis vectors, uk(s), and elements of the Cartan Antenna Array Geometries,” Signal Processing, IEEE Transactions on, H vol. 49, no. 3, pp. 497 –506, mar 2001. matrix, expressions will be developed for v (s) and u u0 . k k k [7] H. Karimi and A. Manikas, “Manifold of a planar array and its effects on It will be shown in this appendix that, in the case of the accuracy of direction-finding systems,” Radar, Sonar and Navigation, hyperhelical array manifolds, vk(s) can be written in the form: IEE Proceedings -, vol. 143, no. 6, pp. 349 –357, dec 1996. [8] A. Manikas, A. Alexiou, and H. Karimi, “Comparison of the Ultimate vk(s) = Xka(s) (83) Direction-finding Capabilities of a Number of Planar Array Geometries,” IEE Proc. Radar, Sonar, Navig., vol. 144, pp. 321–329, 1997. To begin, we write the first (unnormalised) frame vector as: [9] A. Manikas, C. Proukakis, and V. Lefkaditis, “Investigative study of planar array ambiguities based on "hyperhelical" parameterization,” Signal Processing, IEEE Transactions on, vol. 47, no. 6, pp. 1532 – v1(s) = jR a(s) (84) 1541, jun 1999. X1 [10] N. Dowlut and A. Manikas, “A polynomial rooting approach to super- resolution array design,” Signal Processing, IEEE Transactions on, Clearly, v1(s) is independent|{z} of arc length, s: vol. 48, no. 6, pp. 1559 –1569, jun 2000. k k [11] A. Sleiman and A. Manikas, “The Impact of Sensor Positioning on the H T Array Manifold,” Antennas and Propagation, IEEE Transactions on, v1 = Tr X1 X1 = Tr R R k k { } vol. 51, no. 9, pp. 2227 – 2237, sep 2003. =q 1  q (85) [12] M. Buhren, M. Pesavento, and J. Bohme, “Virtual array design for array interpolation using differential geometry,” in Acoustics, Speech, and and so the first frame vector derivative may be obtained as: Signal Processing, 2004. Proceedings. (ICASSP ’04). IEEE International Conference on, vol. 2, may 2004, pp. ii – 229–32 vol.2. d X1a(s) X1 u10 (s) = = a0(s) ds v1 v1 j k k k k = a(s) (86) v X1R k 1k