CONSIDER an array of N sensors (residing in threedimensional. Investigating Hyperhelical Array Manifold Curves Using the Complex Cartan Matrix

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1 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL.?, NO.?,???? 1 Investigating yperhelical Array Manifold Crves Using the Complex Cartan Matrix Athanassios Manikas, Senior Member, IEEE, arry Commin, Stdent Member, IEEE, and Adham Sleiman Abstract The differential geometry of array manifold crves has been investigated extensively in the literatre, leading to nmeros applications. owever, the existing differential geometric framework restricts the Cartan matrix to be prely real and so the vectors of the moving frame U(s) are fond to be orthogonal only in the wide sense (i.e. only the real part of their inner prodct is eqal to zero). Imaginary components are then acconted for separately sing the concept of the inclination angle of the manifold. The prpose of this paper is therefore to present an alternative theoretical framework which allows the manifold crve in C N to be characterised in a more convenient and direct manner. A continosly differentiable strictly orthonormal basis is established and forms a platform for deriving a generalised complex Cartan matrix with similar properties to those established nder the previos framework. Concepts sch as the radis of circlar approximation, the manifold crve radii vector and the frame matrix are also revisited and rederived nder this new framework. Index Terms Array manifold, differential geometry, array processing. NOTATION a, A Scalar a, A Colmn vector A Matrix ( ) T, ( ) Transpose, congate transpose Absolte vale Eclidean norm of vector a b Element-by-element power exp(a) Element-by-element exponential diag {a} Diagonal matrix whose diagonal entries are a I N (N N) identity matrix 0 N (N 1) vector of zeros Tr { } Matrix trace operator adamard (element-by-element) prodct R The set of real nmbers C The set of complex nmbers I. INTRODUCTION CONSIDER an array of N sensors (residing in threedimensional real space) receiving one or more narrowband plane waves. The response of the array to a signal incident from azimth θ [0, 360 ) and elevation φ ( 90, 90 ) is commonly modelled sing the (N 1) complex array response vector, or manifold vector: a(θ, φ) exp ( [ ] ) r x, r y, r z k (θ, φ) (1) The athors are with the Department of Electrical and Electronic Engineering, Imperial College London, Soth Kensington Camps, London, SW7 2AZ, UK (a.manikas@imperial.ac.k). Manscript received XX XX, 20XX; revised XX XX, 20XX. where the (N 3) real matrix [r x, r y, r z ] denotes the array sensor locations: [r x, r y, r z ] = [r 1, r 2,..., r N ] T R N 3 (2) For the prposes of this paper, it is convenient to define the phase reference at the centroid of the array. In Eqation 1, k(θ, φ) is the wavenmber vector: k(θ, φ) 2π λ (θ,φ) {}}{ [cos(θ) cos(φ), sin(θ) cos(φ), sin(φ)] (3) where λ is the carrier signal wavelength and (θ, φ) is the (3 1) real nit vector pointing from (θ, φ) towards the origin. As is common practice, the notation sed for a(θ, φ) in Eqation 1 ignores dependence pon the constant, known qantities [ r x, r y, r z ] and λ. owever, it is important to note that the manifold vector is by no means restricted to a (θ, φ)-parameterisation. Firstly, this is becase the (θ, φ)- parameterisation is not niqe; other valid directional parameterisations exist (with cone-angle parameterisation particlarly sefl for the analysis of planar arrays [1]). Secondly, many other variable parameters of interest (besides direction) can be incorporated into the response vector modelling. When these additional parameters (sch as delay, Doppler, polarisation, sbcarrier and spreading/scrambling codes) are inclded, the reslting response vector is referred to as an extended manifold vector, the properties of which have been investigated in [2]. Therefore, for the sake of generality, the response vector will simply be denoted as a(p), where the vector p may comprise any or all of the aforementioned variable parameters (and/or others). The central prpose of this paper then lies in exploring the geometrical natre of the mathematical obect which is traced ot by a(p) when p is evalated across the range of all feasible parameter vales (denoted by the parameter space, Ω). This reslting obect is called the array manifold. It completely characterises the array system and is formally defined as: A { a(p), p Ω } (4) In the single-parameter case (i.e. p = p), it can be seen that a(p) traces ot a crve in C N. For two parameters, the manifold is a srface and, similarly, for larger nmbers of parameters, the manifold is some higher dimensional obect. A branch of mathematics dedicated to the investigation of these kinds of (differentiable) manifolds is differential geometry [3,4]. The tools of differential geometry have already been applied extensively in the array signal processing literatre [5]. The main body of this existing research relates to the stdy of manifold crves and srfaces. Althogh the geometrical

2 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL.?, NO.?,???? 2 properties of array manifold srfaces have been investigated directly [6], this is a more complicated task than for crves. To simplify matters, it has been shown that array manifold srfaces can alternatively be considered to consist of families of constant-parameter crves [7]. Therefore, the theoretical framework introdced in this paper will be done so with a focs on array manifold crves. A particlarly important class of array manifold crves is the hyperhelix. yperhelices are especially convenient to analyse since all their crvatres are constant (do not vary from point to point) and may be calclated recrsively. Althogh all linear arrays of isotropic sensors have hyperhelical manifold crves, the sefl properties of the hyperhelix have also been exploited for the analysis of planar arrays sing cone-angle parameterisation and the concept of the eqivalent linear array [5]. Frthermore, hyperhelical manifold crves have been identified in the analysis of extended array manifolds [2]. Therefore, in this paper, particlarly close attention will be paid to the analysis of hyperhelical manifold crves. In order to highlight the motivation of this paper, it is sefl at this point to briefly review the existing differential geometric framework that has been applied to the analysis of array manifold crves. A short smmary of some of its key applications in the literatre will then be given. A. Geometry of Array Manifold Crves: Traditional Approach Given a crve a(p) C N as a fnction of a single real parameter p R (sch as azimth, θ, or elevation, φ), the arc length along the manifold crve, s(p), and its rate of change, ṡ(p), are defined, respectively, as: s(p) p 0 ṡ(p 0 )dp 0 (5) ṡ(p) ds = ȧ(p) (6) dp A sefl featre of parameterising a manifold crve in terms of arc length, s, is that it is an invariant parameter. This means that the tangent vector to the crve (expressed in terms of s): a (s) d ds a(s) = dp ds d dp a(p) = ȧ(p) ȧ(p) is always nit length. (Note that differentiation with respect to s has been denoted by prime and differentiation with respect to p by dot. This convention will be followed throghot this paper). The tangent vector a (s 0 ) provides sefl local information abot the manifold crve in the neighborhood of s 0 (it is a geometric approximation of the first order). owever, in order to bild p a fll (local) characterisation of the manifold crve, it is necessary to attach some additional vectors to the rnning point, s 0. Specifically, an orthonormal moving frame of (local) coordinate vectors can be constrcted. Then, the manner in which this frame twists and trns as the rnning point progresses along the crve will be shown to provide a profondly meaningfl description of the crve. owever, the way in which these frame vectors are chosen is of central importance to this paper. A sbtly different approach will be seen to provide significant new insight. (7) Under the traditional approach, the manifold vector a(p), residing in C N, is instead considered to comprise 2N real components. For this reason, the moving frame, U w (s), consists of p to 2N complex vectors: U w (s) [ w,1 (s), w,2 (s),..., w,d (s) ] (8) where N 1 d 2N (depending on the level of symmetricity exhibited by the sensor array). Together with U w (s), p to 2N 1 non-zero real crvatres can be defined. These allow the motion of the moving frame to be expressed as: U w(s) = U w (s)c r (s) (9) where C r (s) is the prely real Cartan matrix, which contains the crvatres, κ i, according to the following skew-symmetric strctre: 0 κ κ 1 0 κ.. 2. C r (s) =. 0 κ (10) κd κ d 1 0 As a conseqence of constraining the entries of C r (s) to be prely real, it is fond that the moving frame, U w (s), can only be orthonormal in the wide sense (i.e. applying only to the real part): Re { U w (s)u w (s) } = I d (11) As a reslt, it has been proven [5, Eq. 2.23] that this leads to C r (s) taking the form: C r (s) = Re { U w (s)u w(s) } (12) This means that C r (s) does not, in general, niqely describe the shape and size of the manifold crve (since any imaginary components are ignored). Only in the special case of arrays with sensors located symmetrically abot the origin is the manifold seen to admit a representation entirely in R N. In the general case, therefore, imaginary components mst be acconted for separately, sing the concept of the inclination angle of the manifold (defined as the angle between the manifold vector and a certain sbset of the even-nmbered frame vectors [5, p.45]). The prpose of the new theoretical framework presented in this paper is to allow the manifold crve in C N to be characterised in a more convenient and direct manner. The key to doing this lies in reformlating Eqations In particlar, a continosly differentiable strictly orthonormal basis, U(s) C N N, is established and forms a basis for deriving a generalised complex Cartan matrix, C(s), with similar properties to those established nder the previos framework. Concepts sch as the radis of circlar approximation, the manifold crve radii vector and the frame matrix are also revisited and rederived nder this new framework.

3 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL.?, NO.?,???? 3 B. Exemplary Reslts and Motivation The seflness and importance of applying the tools of differential geometry to the analysis of the array manifold can be demonstrated sing a nmber of reslts fond in the literatre. In [8], the fndamental performance capabilities (detection, resoltion and estimation error) of an array system were defined in the context of the array manifold and derived explicitly sing the circlar approximation of the array manifold. The way in which anomalies in the manifold lead to array ambigities was investigated in [9]. A differential geometric perspective was applied to the topic of array design (i.e. the dicios placement of array sensors) in [10]. The sensitivity of array systems to ncertainties in sensor locations, based on their location in the overall array geometry, was analysed in [11]. In [12], the task of designing virtal arrays for the prpose of array interpolation was explored. The stdy of more sophisticated array systems which incorporate additional system and channel parameters (sch as code division mltiple access spreading/scrambling codes, lack of synchronisation, Doppler effects, polarisation parameters and sbcarriers) has been introdced sing the concept of extended array manifolds in [2]. C. Paper Organisation The remainder of this paper is organised as follows. In Section II, the basic principles of the proposed theoretical framework are presented. The strictly orthogonal moving frame is established and the complex Cartan matrix is derived. In Section III, specific properties of hyperhelical manifold crves are investigated. Explicit expressions are derived for the orthonormal frame vectors and entries of the complex Cartan matrix and nmerical examples are presented. Finally, the paper is conclded in Section IV. II. CURVES IN C N In this section, the strictly orthonormal moving frame (or moving polyhedron), U(s), will be formlated. Using the definition of U(s), the exact strctre of the complex Cartan matrix, C(s), will then be derived. A. The Moving Polyhedron In order to describe meaningflly the way in which the crve a(s) twists throgh N-dimensional complex space (at a point s), the moving frame of N orthonormal coordinate vectors, U(s), at the rnning point, s: U(s) [ 1 (s), 2 (s),..., N (s)] (13) will now be derived, where strict orthonormality implies: U (s)u(s) = I N (14) Specifically, in accordance with common practice in R N (see, for example, [3, p.159] and [4, p.13]), the nit tangent vector is first defined as: 1 (s) a (s) (15) and then all sbseqent coordinate vectors (for k 2) are obtained via Gram-Schmidt orthonormalisation: k (s) = v k (s) v k (s) (16a) k 1 v k (s) k 1 (s) ( m (s) k 1 (s)) m (s) (16b) m=1 In other words, each sccessive coordinate vector simply points in the direction that the hyperplane spanned by the previosly-defined vectors is moving. It is worth noting that, nder certain circmstances, the array manifold crve may not span the fll N-dimensional complex observation space and so it is not always necessary to define N frame vectors. Consider, for example, array configrations where a sensor exists at the array centroid (i.e. the phase origin). It is clear that that sensor s response mst be constant for all p and, conseqently, the p-crve cannot depart from the (N 1)-dimensional hyperplane associated with this constant vale. B. The Complex Cartan Matrix The Cartan matrix provides the crvatres of the crve a(s), which in trn are the crvilinear coordinates of the crve on the moving polyhedron U(s). These crvatres are a means of assessing the changes in the polyhedron, expressed by the first derivative, U (s), of U(s), and hence the Cartan matrix, C(s), is sed to express U (s) as a fnction of U(s): U (s) = U(s)C(s) (17) Note that in complex mlti-dimensional space, U(s) C N N and hence C(s) C N N. In particlar, C(s) shold be skew- ermitian since this wold yield, as a special case, the familiar skew-symmetric prely real Cartan matrix for crves in R N. It is proven in Appendix A that the complex Cartan matrix can be written in the form: 1 1 v v v 3... C(s) =. 0 v vn 0 0 v N N N (18) which is indeed skew-ermitian: C (s) = C(s) (19) This property is easily proven, since differentiating the identity k k = 1 leads to: k k + k k = 0 Re { k k} = 0 (20) and so the diagonal entries of C(s) are prely imaginary. Clearly, the strctre of C(s) (Eqation 18) shows certain similarities to C r (s) in the previos framework (Eqation 10). The key differences to note are that, firstly, C(s) is generally a smaller matrix ((N N) compared to (d d)) and, secondly,

4 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL.?, NO.?,???? 4 it can have non-zero, albeit prely imaginary, diagonal entries. As a reslt, in the general case (2 k N 1), the frame vector derivative k (s) is a linear combination of not only k 1 (s) and k+1 (s) (with prely real coefficients) bt also k (s) itself (with the prely imaginary coefficient k (s) k (s)). In this way, C(s) completely characterises the flly complex natre of the crve (inclding the concept of inclination angle in the previos framework). This point can now be demonstrated by considering the radis of the circlar approximation of the manifold. C. The Radis of the Circlar Approximation A powerfl tool in the analysis of array manifold crves is the circlar approximation of the manifold [5, Ch.8]. More specifically, it has been proven that any sfficiently small section (arc length) of an array manifold crve can be approximated as a circlar arc. Using the previos differential geometric framework, it was shown that the inverse of the radis of the circlar approximation was only eqal to the principal crvatre, κ 1, in the special case of symmetric linear arrays. For general array geometries, however, the inclination angle of the manifold, ζ, needed to be acconted for as follows: ˆκ 1 (s) = κ 1 (s) sin ζ(s) (21) = 1 (s) 2 1 (s) 1 (s) 2 (22) the proof of which was derived in [5, p.209]. The corresponding crvatre in the complex Cartan matrix derived in this paper is v 2 (s) : v 2 (s) = 1 (s) ( 1 (s) 1(s) ) 1 (s) = 1 (s) 2 1 (s) 1 (s) 2 (23) which provides the desired reslt directly. This demonstrates both the eqivalence between the two theoretical frameworks as well as the directness and convenience of sing the complex Cartan matrix. III. YPERELICAL ARRAY MANIFOLD CURVES aving established (in Eqations 16 and 18) the new framework for investigating the geometry of array manifold crves, the important class of crves which have hyperhelical shape can now be stdied in detail. In this way, a nmber of concepts sch as the frame matrix and manifold radii vector can be rederived in order to compare to the analogos qantities obtained via the previos framework. New reslts regarding the eigendecomposition of the complex Cartan matrix are also given, followed by specific nmerical examples. Consider an array system with a response vector having the form: a (p) = exp ( πr cos p + w) (24) where r and w are constant vectors and p is the variable parameter of interest. The most commonly seen example of sch a response vector is that of of a linear array (where r = r x cos(φ 0 ) and w = 0 N ) as a fnction of azimth (p = θ). owever, this model also applies more generally to φ, α and β crves of planar arrays (where α and β denote cone angles), via the concept of the eqivalent linear array [5, p.113]. In sch cases, r R N denotes the eqivalent linear array sensor locations. It will be shown in this section that the manifold crve traced ot by any a (p) with this strctre (Eqation 24) is of hyperhelical shape. A nmber of important properties of hyperhelical manifold crves will then be derived explicitly. A. Manifold Vector Derivatives As a preliminary to investigating the differential geometry of the hyperhelical array manifold, a compact differentiation procedre of the response vector is established. Applying the chain rle, leads to: a (s) = ȧ (p) ṡ(p) = ȧ (p) ȧ (p) πr sin p a (p) = π r sin p = r a (p) (25) where the normalised (eqivalent) sensor locations have been defined as: r r (26) r Applying the chain rle repeatedly to a(s) in this way leads a pattern to emerge which is straightforward to prove by indction: a (k) (s) dk a(s) ds k = (r) k a(s) (27) For notational convenience, we define the diagonal matrix R: R diag(r) R N N (28) which allows Eqation 27 to be eqivalently written as: a (k) (s) = k R k a(s) (29) This notation will prove sefl in investigating the differential geometry of the hyperhelical array manifold crve. Note that Tr { R 2} = r 2 = 1. B. Differential Geometry of yperhelical Manifold Crves It is now possible to investigate the orthonormal basis U(s) and the elements of the Cartan matrix, C(s), for the hyperhelical array manifold. Indeed, in Appendix B, it is shown that the (nnormalised) orthogonal frame vector v k (s) can be written in the form: v k (s) = X k a(s) (30) where the diagonal matrix X k+1 can be obtained recrsively for 2 k N 1 as: X k+1 = X kr+ v k vk 1 X k 1 3 Tr { X k X k R } X k with initial conditions: (31) X 1 = R (32) X 2 = X 1 R Tr { X 1 X 1 R } X 1 (33)

5 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL.?, NO.?,???? 5 Frthermore, it is shown that the entries of the complex Cartan matrix, C(s), for all k = 1, 2,..., N can be obtained as follows: = Tr { X k X } k (34) k k = 2 Tr { X k X k R } (35) It is evident from Eqations that the complex Cartan matrix is constant (independent of s), confirming that the manifold is of hyperhelical shape. C. Symmetric Linear Arrays Before proceeding with or discssion of hyperhelical manifold crves, it is interesting to address the special case of linear arrays whose sensors exist in symmetric pairs abot the origin. Specifically, it will be shown that, for this class of arrays, the diagonal entries of the complex Cartan matrix vanish and so the manifold admits a fll representation in R N. As a reslt, the Cartan matrix and moving frame will be seen to coincide exactly with those obtained nder the previos framework. To prove that the diagonal entries of C vanish (i.e. k k = 0 for all k = 1, 2,..., N), the following property of symmetric linear arrays can be sed: Tr { R i} = 0, i odd (36) Therefore, in order to prove that k k = 2 Tr { X k X kr } = 0, it is sfficient to prove that X k is a polynomial in R of either strictly even or strictly odd powers. Given that X 1 = R, it natrally follows that: Tr { X 1 X 1 R } = 0 (37) From Eqation 33, this leads to the simplified expression X 2 = X 1 R = R 2 and so: Tr { X 2 X 2 R } = 0 (38) Using X 1 = R (a polynomial of strictly odd power) and X 2 = R 2 (a polynomial of strictly even power) as initial conditions in Eqation 31 will then lead to the desired reslt. Specifically, given X k, a polynomial of strictly odd powers of R and X k 1, a polynomial of strictly even powers of R, Eqation 31 becomes: X k+1 = X kr even order + vk 1 X k 1 even order (39) which is a polynomial in R of strictly even powers. Obviosly, eqivalent reasoning can be applied to the converse case (X k as a polynomial of strictly even powers and X k 1 a polynomial of strictly odd powers). Therefore, it follows in general that: Tr { X k X k R } = 0 (40) which confirms that k k = 0 for all k = 1, 2,..., N. Using this reslt with the expressions derived in Appendix A reveals an exact eqivalence with the previos framework in the special case of symmetric linear arrays (which exhibit zero inclination angle ). In particlar, a rearrangement of Eqation 81 in terms of k+1 (s) can be seen to coincide precisely with [5, Eq. 2.11]. A nmerical example demonstrating this eqivalence will be given in Section III-F. D. Frame Matrix and Radii Vector Consider a point s 0 on the crve, a(s), whose local orthonormal basis, U(s), is nknown. Given the set of orthonormal vectors, U(s), at s = 0, namely U (0), a continos differentiable transformation matrix F(s) C N N mst exist which relates U(s) to U (0) sch that U(s) = U (0) F(s) (41) This transformation matrix F(s) is called the frame matrix, and is a nonsinglar matrix which mst satisfy the initial condition F (0) = I N (42) For hyperhelical array manifolds (for which the complex Cartan matrix is constant) the differential eqation of Eqation 17 is redced to the following first order differential eqation: which admits a soltion of the type: U (s) = U(s)C (43) U(s) = U (0) expm {sc} (44) where U (0) acconts for initial conditions and expm { } denotes the matrix exponential. The frame matrix of Eqation 41 is therefore given by: F(s) = expm {sc} (45) which clearly satisfies the condition F (0) = I N. Since F(s) is, in general, complex-valed, its properties are sbtly different from the analogos frame matrix derived nder the previos framework [5, p.29]. Firstly, de to the skew-ermitian natre of C, it can be dedced that: and so F(s) is nitary: F (s) = F( s) = F 1 (s) (46) F (s)f(s) = I N (47) Using Eqations 41 and 43, the complex Cartan matrix may be written as a fnction of the frame matrix: C = F (s)f (s) (48) and so a similar first order differential eqation to Eqation 43 relates the frame and Cartan matrices: F (s) = F(s)C (49) The manifold radii vector, R(s) = [R 1 (s),..., R N (s)] T C N, is defined as: R(s) = U (s)a(s) (50) sch that its elements constitte the inner prodcts of the manifold vector and the coordinate vectors R k (s) = k (s)a(s) (51)

6 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL.?, NO.?,???? 6 for k = 1,..., N. In the case of a hyperhelical manifold, R k (s) can be seen to be independent of arc length: ( ) Xk a(s) R k = a(s) = Tr {X k } (52) From Eqation 50, the manifold vector may be written as a fnction of the strictly orthonormal basis and complex Cartan matrix: a(s) = U(s)R(s) = U (0) expm {sc} R(s) (53) Frther examination of the recrsive Eqation 31 allows the following pattern to be observed: X 1 = R X 2 = X 1 R Tr { X 1 X 1 R } X 1 X 3 = v 2 X 2R + v 2 v 1 X 1 Tr { X v X 2 R } X 2 X 4 =... Indeed, it is straightforward to prove by indction that: { real, k even X k = imaginary, k odd imag. real imag. real (54) Using this reslt in connction with Eqation 52 leads to: { real, k even R k = (55) imaginary, k odd In other words, consective entries of the radii vector toggle between prely real and prely imaginary. Finally, the length of the radii vector is given by: R(s) = a (s)u(s)u (s)a(s) = N (56) which corresponds to the radis of the hypersphere pon which the manifold crve lies, as might be expected. E. Cartan Matrix Eigendecomposition We begin by differentiating Eqation 16a and sbstitting for v k as follows: k (s) = X ka (s) (57) = R X ka(s) (58) = R k (s) (59) Collecting all k (s) for k = 1, 2,..., N therefore gives: Combining Eqations 43 and 60 leads to: U (s) = RU(s) (60) RU(s) = U(s)C C = U (s)ru(s) (61) which reveals that the eigenvales of C are eqal to the normalised sensor positions of the (eqivalent) linear array mltiplied by the pre imaginary nmber,, and the eigenvectors are given by U(s). Note that, from Eqation 61, it is possible to dedce that for any positive integer, m > 0: C m = [ U (s) (R) U(s) ] m = U (s) (R) U(s)U (s) (R) U(s)... =I N = U (s) (R) m U(s) (62) It is worth drawing attention to the fact that Eqation 60 is a first order differential eqation similar to Eqation 43, except here U(s) is pre-mltiplied by a constant matrix. The soltion to this differential eqation takes the form: U(s) = expm {sr} U(0) (63) Therefore, a pre-mltiplying frame matrix, E(s), is obtained (as an alternative to post-mltiplying F(s)) sch that: U(s) = E(s)U(0) (64) with E(s) expm {sr} which has initial condition E(0) = I N. In fact, it can be seen that E(s) is also the matrix which relates the original manifold vector a(0) to any other manifold vector: F. Nmerical Examples a(s) = U(s)R(s) = E(s)U(0)R(s) = E(s)a(0) (65) For the first example, consider a (non-symmetric) linear array of for sensors, i.e. N = 4, with sensor positions in halfwavelengths given by r x = [ 2.1, 1.1, 0.9, 2.3] T. Assming φ 0 = 0, the transformation matrices, X n, for n = 1,..., 4, and the Cartan matrix, C, are given by: X 1 = X 2 = , 0, 0, 0 0, , 0, 0 0, 0, , 0 0, 0, 0, , 0, 0, 0 0, , 0, 0 0, 0, , 0 0, 0, 0, , 0, 0, 0 0, , 0, 0 0, 0, , 0 X 3 = 0, 0, 0, , 0, 0, 0 X 4 = 0, , 0, 0 0, 0, , 0 0, 0, 0,

7 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL.?, NO.?,???? 7 0, , 0, 0, 0, 0, 0, , 0, , 0, 0, 0, 0, 0 0, , 0, , 0, 0, 0, 0 C r = 0, 0, , 0, , 0, 0, 0 0, 0, 0, , 0, , 0, 0 0, 0, 0, 0, , 0, , 0 0, 0, 0, 0, 0, , 0, , 0, 0, 0, 0, 0, , 0 (67) and , , 0, 0 C = , , , 0 0, , , , 0, , (66) Using the traditional framework, the prely real Cartan matrix of the same array is the 8 8 real matrix shown in Eqation 67, above. Comparing the reslts obtained for the two Cartan matrices, it is clear that the first crvatre of C r (namely κ 1 = ) can be obtained as the magnitde of the first two elements of C, i.e. κ 1 = v = Sbseqent real crvatres κ 2, κ 3,..., κ d 1 may be obtained via more complicated expressions involving the lower order entries of the complex Cartan matrix (bt not vice versa, since C contains information regarding inclination angle, which is absent from C r ). For the second example, consider the symmetric linear array r x = [ 2.3, 1.1, 1.1, 2.3] T. Assming φ 0 = 0, the complex Cartan matrix of its manifold is given by: 0, , 0, 0 C = , 0, , 0 0, , 0, (68) 0, 0, , 0 and the radii vector R = [0, , 0, ] T. These reslts, and indeed the orthonormal basis U (s), compted at any azimth, θ 0, are fond to coincide exactly with those obtained throgh the previos crvilinear differential geometry framework for the same symmetric array: C r = C (69) U w (s) = U (s) (70) The convergence of the two frameworks for symmetric linear arrays, where the manifold dimensionality is redced to real only, demonstrates that the two approaches are fndamentally correct and that the difference lies in the handling of the prely imaginary dimensions. IV. CONCLUSION In this paper, a new theoretical framework has been presented which provides an alternative to the existing differential geometric approach to analysing array manifold crves. While the traditional approach restricted the entries of the Cartan matrix to be prely real, the framework proposed in this paper incorporated a complex, skew-ermitian Cartan matrix. In this way, a strictly orthogonal moving frame has been established, which facilitates a more convenient and direct method of analysis. A nmber of specific featres of the proposed framework have been highlighted and compared to the traditional approach (with particlar attention given to hyperhelical array manifold crves). Concepts sch as the radis of circlar approximation, manifold crve radii vector and the frame matrix were revisited and rederived nder the new framework. APPENDIX A DERIVATION OF TE COMPLEX CARTAN MATRIX In order to evalate the elements of the complex Cartan matrix, Eqation 16b is first rearranged, following a change of variables, as: k (s) = v k+1 (s) + k ( m (s) k (s)) m (s) (71) m=1 Given that m(s) k (s) = 0 for m = 1,..., k 1 with 2 k N 1, it follows that: d [ ds m (s) k (s) ] = 0 m (s) k (s) = m (s) k (s) (72) Sbstitting this reslt into Eqation 71 yields for 2 k N 1: k (s) = v k+1 (s) k ( m (s) k (s) ) m (s) (73) m=1 or, by a simple change of variables: m (s) = v m+1 (s) m i=1 ( i (s) m (s) ) i (s) (74) Sbstitting this expression for m (s) back into Eqation 73 then leads to (note that (s) is dropped for simplicity for the next few eqations): k = v k+1 [ ] k m v m+1 i m i k m (75) m=1 i=1

8 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING (SPECIAL ISSUE), VOL.?, NO.?,???? 8 This eqation can be rewritten as: k k = v k+1 ( ) v l+1 k l + l=1 Then, noting that: k m m=1 i=1 ( m ) ( ) i i k m (76) v l+1 k 0 iff l = k 1 (77) i k 0 iff i = k (78) allows Eqation 76 to be simplified to: k = v k+1 k 1 + ( k k) k (79) Finally, noting that v k = k, the method for calclating k (s) can now be stated in fll. For k = 1: For 2 k N 1: 1 = ( 1 1) 1 + v 2 2 (80) k = k 1 + ( k k) k + v k+1 k+1 (81) For k = N: N = v N N 1 + ( N N) N (82) where Eqation 80 is obtained by direct rearrangement of Eqation 73 (with k = 1). Meanwhile, Eqation 82 follows from the fact that attempting to evalate v N+1 (in Eqation 16b) mst yield the zero vector. Using Eqations with Eqation 17 leads directly to C(s) in Eqation 18. APPENDIX B PROPERTIES OF YPERELICAL ARRAY MANIFOLDS In order to analyse the differential geometry of the hyperhelical array manifold crve, the parameters presented in Section II are next evalated for the special array manifold vector strctre of Eqation 24. Specifically, in order to identify the orthonormal basis vectors, k (s), and elements of the Cartan matrix, expressions will be developed for v k (s) and k k. It will be shown in this appendix that, in the case of hyperhelical array manifolds, v k (s) can be written in the form: v k (s) = X k a(s) (83) To begin, we write the first (nnormalised) frame vector as: v 1 (s) = R a(s) (84) X 1 Clearly, v 1 (s) is independent of arc length, s: v 1 = Tr { X 1 X } 1 = Tr {R T R} = 1 (85) and so the first frame vector derivative may be obtained as: 1(s) = d X 1 a(s) ds v 1 = X 1 v 1 a (s) = v 1 X 1Ra(s) (86) (where v 1 = 1 has not been deleted for the sake of consistency as k is increased). Next, it can be seen that the first element of the complex Cartan matrix, 1 (s) 1(s), is also independent of s: 1 1 = v 1 2 a (s)x 1 X 1 Ra(s) = v 1 2 Tr { X 1 X 1 R } (87) Proceeding to k = 2 and sbstitting from the above expressions leads to: v 2 (s) = 1 (s) ( 1 1) 1 (s) [ = v 1 X 1R v 1 3 Tr { X 1 X 1 R } ] X 1 a(s) (88) X 2 and contining as before allows the second diagonal entry of the Cartan matrix to be obtained: 2 2 = v 2 2 Tr { X 2 X 2 R } (89) Proceeding in this manner leads the pattern stated in Eqations 31 and 35 to emerge. Proof by indction is a straightforward extension of the previos discssion. REFERENCES [1]. R. Karimi and A. Manikas, Cone-angle parametrization of the array manifold in DF systems, J. Franklin Inst. (Eng. Appl. Math.), vol. 335B, pp , [2] G. Efstathopolos and A. Manikas, Extended Array Manifolds: Fnctions of Array Manifolds, IEEE Transactions on Signal Processing, vol. 59, no. 7, pp , ly [3]. Gggenheimer, Differential Geometry. McGraw ill (or Dover Edition), 1963 (1977). [4] W. Kühnel, Differential Geometry: Crves - Srfaces - Manifolds, 2nd ed. American Mathematical Society, [5] A. Manikas, Differential Geometry in Array Processing. Imperial College Press, [6] A. Manikas, A. Sleiman, and I. Dacos, Manifold Stdies of Nonlinear Antenna Array Geometries, Signal Processing, IEEE Transactions on, vol. 49, no. 3, pp , mar [7]. Karimi and A. Manikas, Manifold of a planar array and its effects on the accracy of direction-finding systems, Radar, Sonar and Navigation, IEE Proceedings -, vol. 143, no. 6, pp , dec [8] A. Manikas, A. Alexio, and. Karimi, Comparison of the Ultimate Direction-finding Capabilities of a Nmber of Planar Array Geometries, IEE Proc. Radar, Sonar, Navig., vol. 144, pp , [9] A. Manikas, C. Prokakis, and V. Lefkaditis, Investigative stdy of planar array ambigities based on "hyperhelical" parameterization, Signal Processing, IEEE Transactions on, vol. 47, no. 6, pp , n [10] N. Dowlt and A. Manikas, A polynomial rooting approach to sperresoltion array design, Signal Processing, IEEE Transactions on, vol. 48, no. 6, pp , n [11] A. Sleiman and A. Manikas, The Impact of Sensor Positioning on the Array Manifold, Antennas and Propagation, IEEE Transactions on, vol. 51, no. 9, pp , sep [12] M. Bhren, M. Pesavento, and J. Bohme, Virtal array design for array interpolation sing differential geometry, in Acostics, Speech, and Signal Processing, Proceedings. (ICASSP 04). IEEE International Conference on, vol. 2, may 2004, pp. ii vol.2.