Spherical Harmonic Expansion of Fisher-Bingham Distribution and 3D Spatial Fading Correlation for Multiple-Antenna Systems

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION Spherica Harmonic Expansion of Fisher-Bingham Distribution and 3D Spatia Fading Correation for Mutipe-Antenna Systems Yibeta F. Aem, Student Member, IEEE, Zubair Khaid, Member, IEEE and Rodney A. Kennedy, Feow, IEEE arxiv:5.4395v [cs.it] 9 Jan 25 Abstract This paper considers the 3D spatia fading correation (SFC) resuting from an ange-of-arriva (AoA) distribution that can be modeed by a mixture of Fisher-Bingham distributions on the sphere. By deriving a cosed-form expression for the spherica harmonic transform for the component Fisher-Bingham distributions, with arbitrary parameter vaues, we obtain a cosed-form expression of the 3D-SFC for the mixture case. The 3D-SFC expression is genera and can be used in arbitrary mutiantenna array geometries and is demonstrated for the cases of a 2D uniform circuar array in the horizonta pane and a 3D reguar dodecahedra array. In computationa aspects, we use recursions to compute the spherica harmonic coefficients and give pragmatic guideines on the truncation size in the series representations to yied machine precision accuracy resuts. The resuts are further corroborated through numerica experiments to demonstrate that the cosed-form expressions yied the same resuts as significanty more computationay expensive numerica integration methods. Index Terms Fisher-Bingham distribution, spherica harmonic expansion, spatia correation, MIMO, ange of arriva (AoA). I. INTRODUCTION The Fisher-Bingham distribution, aso known as the Kent distribution, beongs to the famiy of spherica distributions in directiona statistics []. It has been used in appications in a wide range of discipines for modeing and anaysing directiona data. These appications incude sound source ocaization [2], [3], joint set identification [4], modeing protein structures [5], 3D beamforming [6], cassification of remote sensing data [7], modeing the distribution of AoA in wireess communication [8], [9], to name a few. In this work, we focus our attention to the use of Fisher- Bingham distribution for modeing the distribution of ange of arriva (AoA) (aso referred as the distribution of scatterers) in wireess communication and computation of spatia fading correation (SFC) experienced between eements of mutipeantenna array systems. Modeing the distribution of scatterers and characterising the spatia correation of fading channes is a key factor in evauating the performance of wireess communication systems with mutipe antenna eements [] [5]. It has been an active area of research for the past two decades or so and a number of spatia correation modes and The authors are with the Research Schoo of Engineering, Coege of Engineering and Computer Science, The Austraian Nationa University, Canberra, ACT 26, Austraia. Emais: {yibeta.aem, zubair.khaid, rodney.kennedy}@anu.edu.au. This work was supported by the Austraian Research Counci s Discovery Projects funding scheme (Project no. DP5). cosed-form expressions for evauating the SFC function have been deveoped in the existing iterature (e.g., [], [6] [25]). The eiptic (directiona) nature of the Fisher-Bingham distribution offers fexibiity in modeing the distribution of normaized power or AoA of the mutipath components for most practica scenarios. Expoiting this fact, a 3D spatia correation mode has been deveoped in [9], where the distribution of AoA of the mutipath components is modeed by a positive inear sum of Fisher-Bingham distributions, each with different parameters. Such modeing has been shown to be usefu in a sense that it fits we with the muti-input muti-output (MIMO) fied data and aows the evauation of the SFC as a function of the anguar spread, ovaness parameter, azimuth, eevation, and prior contribution of each custer. Athough the SFC function presented in [9] is genera in a sense that it is vaid for any arbitrary antenna array geometry, it has not been expressed in cosed-form and has been ony evauated using numerica integration techniques, which can be computationay intensive to produce sufficienty accurate resuts. If the spherica harmonic expansion of the Fisher-Bingham distribution is given in a cosed-form, the SFC function can be computed anayticay using the spherica harmonic expansion of the distribution of AoA of the mutipath components [2]. To the best of our knowedge, the spherica harmonic expansion of Fisher-Bingham distribution has not been derived in the existing iterature. In the current work, we present the spherica harmonic expansion of Fisher-Bingham distribution and a cosed-form expression that enabes the anaytic computation of the spherica harmonic coefficients. We aso address the computationa considerations required to be taken into account in the evauation of the proposed cosed-form. Using the proposed spherica harmonic expansion of the Fisher-Bingham distribution, we aso formuate the SFC function experienced between two arbitrary points in 3D-space for the case when the distribution of AoA of the mutipath components is modeed by a mixture (positive inear sum) of Fisher-Bingham distributions. The SFC presented here is genera in a sense that it is expressed as a function of arbitrary points in 3D-space and therefore can be used to compute spatia correation for any 2D and 3D antenna array geometries. Through numerica anaysis, we aso vaidate the correctness of the proposed spherica harmonic expansion of Fisher-Bingham distribution and the SFC function. In this paper, our main objective is to empoy the proposed spherica harmonic expansion of the Fisher-Bingham distribution for computing the spatia correation. However, we expect that

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 2 the proposed spherica harmonic expansion can be usefu in various appications where the Fisher-Bingham distribution is used to mode and anayse directiona data (e.g., [2] [7]). The remainder of the paper is structured as foows. We review the mathematica background reated to signas defined on the 2-sphere and spherica harmonics in Section II. In Section III, we define the Fisher-Bingham distribution and present its appication in modeing 3D spatia correation. We present the spherica harmonic expansion of the Fisher- Bingham distribution and anaytica formua for computing the spherica harmonic coefficients in Section IV, where we aso address computationa issues. We derive a cosed-form expression for the 3D SFC between two arbitrary points in 3Dspace when the AoA of an incident signa foows the Fisher- Bingham probabiity density function (pdf) in Section V. In Section VI, we carry out a numerica vaidation of the proposed resuts and provide exampes of the SFC for uniform circuar array (2D) and reguar dodecahedron array (3D) antenna eements. Finay, the concuding remarks are made in Section VII. II. MATHEMATICAL PRELIMINARIES A. Signas on the 2-Sphere We consider compex vaued square integrabe functions defined on the 2-sphere, S 2. The set of such functions forms a Hibert space, denoted by L 2 (S 2 ), that is is equipped with the inner product defined for two functions f and g defined on S 2 as [26] f, g f( x)g( x) ds( x), () S 2 which induces a norm f f, f /2. Here, ( ) denotes the compex conjugate operation and x [sin θ cos θ, sin θ sin φ, cos θ] T S 2 R 3 represents a point on the 2-sphere, where [ ] T represents the vector transpose, θ [, π] and φ [, 2π) denote the co-atitude and ongitude respectivey and ds( x) = sin θ dθ dφ is the surface measure on the 2-sphere. The functions with finite energy (induced norm) are referred as signas on the sphere. B. Spherica Harmonics Spherica harmonics serve as orthonorma basis functions for the representation of functions on the sphere and are defined for integer degree and integer order m as Y m ( x) Y m (θ, φ) N m P m (cos θ)e imφ, with N m 2 + ( m)! 4π ( + m)!, (2) is the normaization factor such that Y m, Y p q = δ,p δ m,q, where δ m,q is the Kronecker deta function: δ m,q = for m = q and is zero otherwise. P m ( ) denotes the associated Legendre poynomia of degree and order m [26]. We aso note the foowing reation for associated Legendre poynomia P m ( + m)! (cos θ) = ( m)! d m,(θ), (3) where d m,m ( ) denotes the Wigner-d function of degree and orders m and m [26]. By competeness of spherica harmonics, any finite energy function f( x) on the 2-sphere can be expanded as f( x) = (f) m = m= Y m ( x), (4) where (f) m is the spherica harmonic coefficient given by (f) m = f( x)y m ( x) ds( x). (5) S 2 The signa f is said to be band-imited in the spectra domain at degree L if (f) m = for > L. For a rea-vaued function f( x), we aso note the reation (f) m = ( ) m (f) m, (6) which stems from the conjugate symmetry property of spherica harmonics [26]. C. Rotation on the Sphere The rotation group SO(3) is characterized by Euer anges (ϕ, ϑ, ω) SO(3), where ϕ [, 2π), ϑ [, π] and ω [, 2π). We define a rotation operator on the sphere D(ϕ, ϑ, ω) that rotates a function on the sphere, according to zyz convention, in the sequence of ω rotation around the z-axis, ϑ rotation around the y-axis and ϕ rotation around z-axis. A rotation of a function f(θ, φ) on sphere is given by (D(ϕ, ϑ, ω)f)( x) f(r x), (7) where R is a 3 3 rea orthogona unitary matrix, referred as rotation matrix, that corresponds to the rotation operator D(ϕ, ϑ, ω) and is given by where R = R z (ϕ)r y (ϑ)r z (ω), (8) cos ϕ sin ϕ R z (ϕ) = sin ϕ cos ϕ, cos ϑ sin ϑ R y (ϑ) =. sin ϑ cos ϑ Here R z (ϕ) and R y (ϑ) characterize individua rotations by ϕ aong z-axis and ϑ aong y-axis, respectivey. III. PROBLEM FORMULATION A. Fisher-Bingham Distribution on Sphere Definition (Fisher-Bingham Five-Parameter (FB5) Distribution). The Fisher-Bingham five-parameter (FB5) distribution, aso known as the Kent distribution, is a distribution on the sphere with probabiity density function (pdf) defined as [], [5] g( x; κ, µ, β, A) = eκ µ T x+ x T βa x, (9) C(κ, βa) where A is a symmetric matrix of size 3 3 given by A = ( η η T η 2 η T 2 ). ()

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 3 Here µ, η and η 2 are the unit vectors (orthonorma set) that denote the mean direction (centre), major axis and minor axis of the distribution, respectivey, κ is the concentration parameter that quantifies the spatia concentration of FB5 distribution around its mean and β κ/2 is referred to as the ovaness parameter that is a measure of eipticity of the distribution. In (9), the term C(κ, β) denotes the normaization constant, which ensures g 2 = and is given by Γ(r + /2) C(κ, β) = 2π Γ(r + ) β2r (κ/2) 2r /2 I 2r+/2 (κ), r= () where I r ( ) denotes the modified Besse function of the first kind of order r. The FB5 distribution is more concentrated and more eiptic for arger vaues of κ and β, respectivey. As an exampe, the FB5 distribution g( x; κ, µ, β, A) is potted on the sphere in Fig. for different vaues of parameters. The FB5 distribution beongs to the famiy of spherica distributions in directiona statistics and is the anaogue of the Eucidean domain bivariate norma distribution with unconstrained covariance matrix [], [5]. B. 3D Spatia Fading Correation (SFC) In MIMO systems, the 3D mutipath channe impuse response for a signa arriving at antenna array is characterized by the steering vector of the antenna array. For an antenna array consisting of M antenna eements paced at z p R 3, p =, 2,..., M, the steering vector, denoted by α( x), is given by α( x) = [ α ( x), α 2 ( x),... α L ( x) ], α p ( x) e ikzp x, (2) where x R 3 denotes a unit vector pointing in the direction of wave propagation and k = 2π/λ with λ denoting the waveength of the arriving signa. For any h( x) representing the pdf of the anges of arriva (AoA) of the mutipath components or the unit-normaized power of a signa received from the direction x, the 3D SFC function between the p- th and the q-th antenna eements, ocated at z p and z q, respectivey, with an assumption that signas arriving at the antenna eements are narrowband, is given by [2] ρ(z p, z q ) h( x) α p ( x) α q ( x) ds( x) S 2 = h( x) e ik(zp zq) x ds( x) ρ(z p z q ), (3) S 2 which indicates that the SFC ony depends on z p z q and is, therefore, spatiay wide-sense stationary. C. FB5 Distribution Based Spatia Correation Mode and Probem Under Consideration The FB5 distribution offers the fexibiity, due to its directiona nature, to mode the distribution of normaized power or AoA of the mutipath components for most practica scenarios. Utiizing this capabiity of FB5 distribution, a 3D spatia correation mode for the mixture of FB5 distributions defining the AoA distribution has been deveoped in [9]. Here the mixture refers to the positive inear sum of a number of FB5 distributions, each with different parameters. We defer the formuation of FB5 based correation mode unti Section V. Athough the correation mode proposed in [9] is genera in a sense that the SFC function can be computed for any arbitrary antenna geometry, the formuation of SFC function invoves the computation of integras that can ony be carried out using numerica integration techniques as we highighted earier. If the spherica harmonic expansion of the FB5 distribution is given in cosed-form, the SFC function can be computed anayticay foowing the approach used in [2]. In this paper, we derive an exact expression to compute the spherica harmonic expansion of the FB5 distribution. Using the spherica harmonic expansion of FB5 distribution, we aso formuate exact SFC function for the mixture of FB5 distributions, defining the distribution of AoA of the mutipath components. We aso anayse the computationa considerations invoved in the evauation of spherica harmonic expansion of FB5 and distribution and SFC function. IV. SPHERICAL HARMONIC EXPANSION OF THE FB5 DISTRIBUTION In this section, we derive an anaytic expression for the computation of spherica harmonic coefficients of the FB5 distribution. For convenience, we determine the spherica harmonic coefficients of the FB5 distribution with mean (centre) µ ocated on z-axis and major axis η and minor axis η 2 aigned aong x-axis and y-axis respectivey, that is, µ = [ ] T η = [ ] T η 2 = [ ] T. (4) With the centre, major and minor axes as given in (4), the FB5 distribution, referred to as the standard Fisher- Bingham (FB) distribution, has the pdf given by f( x; κ, β) cos θ+β sin 2 θ cos eκ 2φ, (5) C(κ, β) which is reated to the FB5 distribution pdf g( x; κ, β, µ, A) given in (9) through the rotation operator D(ϕ, ϑ, ω) as g( x; κ, β, µ, A) = (D(ϕ, ϑ, ω)f) ( x; κ, β) = f(r x; κ, β), (6) where the rotation matrix is reated to µ, η, η 2 (parameters of FB5 distribution) as R = [ η, η 2, µ], (7) and the Euer anges (ϕ, ϑ, ω) are reated to the rotation matrix R through (8). As an exampe, we rotate each of the FB5 distribution g( x; κ, µ, β, A) potted in Fig. to obtain the standard Fisher-Bingham distribution f( x; κ, β), potted in Fig. 2, where we have indicated the Euer anges that reate the FB5 distribution and the Fisher-Bingham distribution through (6). A. Spherica Harmonic Expansion of Standard FB distribution Here, we derive a cosed-form expression to compute the spherica harmonic coefficients of the standard FB distribution, given in (5). Later in this section, noting the reation between

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 4 3.5 3 2.5 2.5.5 (a) κ = 25, β = (b) κ =, β = (c) κ =, β = 49 Fig. : The Fisher-Bingham five-parameter (FB5) distribution g( x; κ, µ, β, A), given in (9), is potted on the sphere for parameters κ, β, as indicated and: (a) µ = [ ] T, η = [ ] T, minor axis η 2 = [ ] T. (b) µ = [ ] T, η = [ ] T, minor axis η 2 = [ ] T. (c) µ = [ ] T, η = [ ] T, minor axis η 2 = [ ] T. 3.5 3 2.5 2.5.5 (a) κ = 25, β =, (ϕ, ϑ, ω) = (π/2,, ). (b) κ =, β =, (ϕ, ϑ, ω) = (π/2, π/2, ). (c) κ =, β = 49, (ϕ, ϑ, ω) = (π/2, π/2, π/2). Fig. 2: The standard Fisher-Bingham (FB) distribution f( x; κ, β), given in (5), is potted on the sphere for the concentration parameter κ and ovaness parameter β, as indicated. Each subfigure is reated to the respective subfigure of Fig. through the the reation (6), with Euer anges indicated. standard FB distribution and FB5 distribution given in (6), and the effect of the rotation operation on the spherica harmonic coefficients, we determine the coefficients of FB5 distribution. The spherica harmonic coefficient, denoted by (f) m, of the standard Fisher-Bingham distribution given in (5) can be expressed as (f) m f( ; κ, β), Y m = f( x; κ, β)y m S 2 ( x)ds( x). (8) Since f( x; κ, β) is a rea function, we ony need to compute the spherica harmonic coefficients for positive orders m for each degree. The coefficients for the negative orders can be readiy computed using the conjugate symmetry reation, noted in (6). To derive a cosed-form expression for computing the spherica harmonic coefficients, we rewrite (8) expicity as where (f) m = N m C(κ, β) π 2π = N m C(κ, β) F m (θ) = e κ cos θ P m (cos θ) sin θdθ e imφ e β sin2 θ cos 2φ dφ π 2π e κ cos θ F m (θ) P m e β sin2 θ cos 2φ imφ dφ, (cos θ) sin θdθ, which we evauate as { 2πI m/2 (β sin 2 θ) m, 2, 4,... F m (θ) = m, 3, 5,.... Using Mathematica.

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 5 By expanding the modified Besse function I m/2 (β sin 2 θ) as ( ) 2t+m/2 β I m/2 (β sin 2 (sin θ) 4t+m θ) = 2 t!(t + m/2)!, (9) t= the exponentia e κ cos θ as [8] π e κ cos θ = (2n + )I n+/2 (κ)p 2κ n(cos θ), (2) n= and using the reation between associated Legendre poynomia and Wigner-d function given in (3), aong with the foowing expansion of Wigner-d function in terms of compex exponentias d m,n(θ) = i n m u= d u,m(π/2) d u,n(π/2) e iuθ, (2) we obtain a cosed-form expression for the spherica harmonic coefficient in (8) as (f) m = πi m 2 + (2n + )I n+/2 (κ) C(κ, β) 2κ n= n ( d n u, (π/2) ) 2 (β/2) 2t+m/2 Γ(t + )Γ(t + m/2 + ) u= n u = t= d u,(π/2)d u,m(π/2) G(4t + m +, u + u ), (22) where Γ( ) denotes the Gamma function and we have used the foowing identity [27, Sec. 3.892] G(p, q) = π (sin θ) p e iqθ dθ, πe iqπ/2 Γ(p + 2) 2 p (p + )Γ( p+q+2 2 )Γ( p q+2 2 ). (23) B. Spherica Harmonic Expansion of FB5 distribution We use the reation between standard FB distribution and FB5 distribution given in (6) to determine the spherica harmonic expansion of FB5 distribution. The Euer anges (ϕ, ϑ, ω) in (6), which characterize the reation between the standard FB distribution and the FB5 distribution, can be obtained from the rotation matrix R formuated in (7) in terms of the parameters of FB5 distribution. By comparing (8) and (7), ϑ is given by ϑ = cos (R 3,3 ), (24) where R a,b denotes the entry at a-th row and b-th coumn of the matrix R given in (7). Simiary ϕ [, 2π) and ω [, 2π) can be found by a four-quadrant search satisfying sin ϕ = sin ω = respectivey. R 2,3 (R3,3 ) 2, cos ϕ = R,3 (R3,3 ) 2, (25) R 3,2 (R3,3 ) 2, cos ω = R 3, (R3,3 ) 2, (26) Once the Euer anges (ϕ, ϑ, ω) are extracted from the parameters µ, η, η 2 using (24) (26), the spherica harmonic coefficients of the FB5 distribution g( x; µ, κ, A) given in (9) can be computed foowing the effect of the rotation operation on the spherica harmonic coefficients as [26] (g) m g( ; µ, κ, A), Y m = D(ϕ, ϑ, ω)f, Y m = Dm,m (ϕ, ϑ, ω)(f)m, (27) m = where (f) m is the spherica harmonic coefficient of degree and order m of the standard Fisher-Bingham distribution f( x). In (27), Dm,m (ϕ, ϑ, ω) denotes the Wigner-D function of degree and orders m, m and is given by [26] D m,m (ϕ, ϑ, ω) = e imϕ d m,m (ϑ)e im ω. (28) C. Computationa Considerations Here we discuss the computation of Wigner-d functions, at a fixed argument of π/2, which are essentiay required for the computation of spherica harmonic expansion of standard FB or FB5 distribution using the proposed formuation (22). Another computationa consideration that is addressed here is the evauation of infinite summations over t and n invoved in the computation of (22). ) Computation of Wigner-d functions: For the computation of spherica harmonic coefficients of the standard FB distribution using (22), we are required to compute the Wignerd functions at a fixed argument of π/2, that is d u,m(π/2), for each u, m. Let D denote the matrix of size (2 + ) (2 + ) with entries d u,m(π/2) for u, m. The matrix D can be computed for each =, 2,..., using the reation given in [28] that recursivey computes D from D. 2) Truncation over n: The infinite sum over n arises from (2) where an exponentia function of the concentration measure κ and the co-atitude ange θ is expanded as an infinite sum of the modified Besse functions and Legendre poynomias. It is we known that the Legendre poynomias are osciating functions with a maximum vaue of one. The modified Besse function I n+/2 (κ) that makes up the core of the expansion given in (2) decays quicky (and monotonicay) to zero as n for a given κ. We use this feature of the modified Besse function I n+/2 (κ) to truncate the summation over n. We pot I n+/2 (κ) for different vaues of n and κ in Fig. 3, where it is evident that the Besse function quicky decays to zero. We propose to truncate the summation over n in (2), or equivaenty in (22), at n = N such that I N+/2 (κ) < 6 (doube machine precision). We approximate the inear reationship between such truncation eve and the concentration parameter κ, given by N = 3 2 κ + 24, (29) which is aso indicated in Fig. 3. This truncation eve is found to give truncation error ess than the machine precision eve for the vaues of concentration parameter κ in the range κ (used in practice [8], [9]). We note that the truncation

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 6 9 5 45 8 7 6 5 4 35 3 5 κ 5 4 3 5 2 β 25 2 5 5 2 2 5 5 2 Truncation eve, n 25 3 5 2 4 6 8 Truncation eve, t 25 3 Fig. 3: Surface pot of modified Besse function I n+/2 (κ) for different vaues of κ and n. The region I n+/2 (κ) < 6 is approximated by the region beow the straight ine, n = 3 2 κ + 24. Ony those vaues I n+/2(κ) > 6 potted above the straight ine account for the sum in (2) to make it accurate to the machine precision eve. Fig. 4: Surface pot of the term inside the summation on the right hand side of (9) for m = and θ = π/2, that is, S(β, t) β2t t!t! 2 for different vaues of the ovaness parameter 2t β and t. The region R(β, t) < 6 is approximated by the region beow the straight ine, t = 36 25 β + 2. error becomes smaer for arge vaues of κ, indicating that the truncation at eve ess than N given in (29) may aso aow sufficienty accurate computation of spherica harmonic coefficients. Further anaysis on estabishing the reationship between the concentration parameter κ and the truncation eve is beyond the scope of current work. 3) Truncation over t: The expansion of modified Besse functions of the first kind for a given vaue of β as given in (9) introduces the infinite sum over t. Again, the decaying (nonmonotonic in this case) characteristics of the term inside the summation on right hand side of (9) with the increase in t makes it possibe to truncate the infinite sum given in (9) that mainy depends on the ovaness parameter β with a minima error. We pot the term inside the summation on the right hand side of (9) for m = and θ = π/2, that is, S(β, t) β2t t!t! 2 2t for different vaues of the ovaness parameter β in Fig. 4. We again propose to truncate the summation over t at the truncation eve T such that the terms after the truncation eve T are ess than 6 and do not have impact on the summation. We approximate the inear reationship between the ovaness parameter β and the truncation eve T T = 36 β + 2. (3) 25 Since we are computing coefficients for positive orders m for each degree, the truncation eve given by (3), that is obtained for order m = and θ = π/2, for the summation in (9), is aso vaid for a positive orders m and a θ [, π] because the term inside the summation is maximum for m = and θ = π/2 for a given summation variabe t. We finay note that the truncation eves proposed here aow accurate computation of spherica harmonic coefficients to the eve of doube machine precision. Later in the paper, we vaidate the correctness of the derived expression for the computation of spherica harmonic coefficients of the standard FB distribution. V. 3D SPATIAL FADING CORRELATION FOR MIXTURE OF FISHER-BINGHAM DISTRIBUTIONS In this section, we derive a cosed form expression to compute the SFC function for the spatia correation mode based on a mixture of FB5 distributions defining the distribution of AoA of mutipath components [8], [9]. Let h( x) denotes the pdf of the distribution defined as a positive inear sum of W FB5 distributions, each with different parameters, and is given by h( x) = W K w g( x; κ w, µ w, β w, A w ), (3) w= where K w denotes the weight of w-th FB5 distribution with parameters κ w, µ w, β w and A w. We assume that each K w is normaized such that h 2 =. For the distribution with pdf defined in (3), the SFC function, given in (3) has been formuated in [9] and anaysed for a uniform circuar antenna array but the integras invoved in the SFC function were numericay computed. We foow the approach introduced in [2] to derive a cosed-form 3D SFC function using the proposed cosed-form spherica harmonic expansion of the FB5 distribution. Using spherica harmonic expansion of pane waves [29]: e ikzp x = 4π i ( j k zp ) = Y m m= ( zp / z p ) Y m( x), and expanding the mixture distribution h( x) in (3), foowing (4), and empoying the orthonormaity of spherica harmonics,

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 7 we write the SFC function in (3) as [2] ρ(z p z q ) = 4π i ( j k zp z q ) where m= = ( (h) m Y m zp z ) q, (32) z p z q Spatia error, ǫ(l) 5 κ = 2.5,β =.5 κ = 25,β = κ =,β = 49 (h) m = h, Y m W = K w Dm,m (ϕ w, ϑ w, ω w )(f) m, (33) w= m = which is obtained by combining (27) and (3). Here, (f) m denotes the spherica harmonic coefficient of the standard FB distribution and the Euer anges (ϕ w, ϑ w, ω w ) reate the w-th FB5 distribution g( x; κ w, µ w, β w, A w ) of the mixture and the standard FB distribution f( x) through (6). In the computation of the SFC using the proposed formuation, given in (32), we note that the summation for over first few terms yieds sufficient accuracy as higher order Besse functions decay rapidy to zero for points near each other in space, as indicated in [2], [3]. We concude this section with a note that the SFC function can be anayticay computed using the proposed formuation for an arbitrary antenna array geometry and the distribution of AoA modeed by a mixture of FB5 distributions, each with different parameters. In the next section, we evauate the proposed SFC function for uniform circuar array and a 3D reguar dodecahedron array of antenna eements. VI. EXPERIMENTAL ANALYSIS We conduct numerica experiments to vaidate the correctness of the proposed anaytic expressions, formuated in (22) (23) and (32) (33), for the computation of the spherica harmonic coefficients of standard FB distribution and the SFC function for a mixture of FB5 distributions defining the distribution of AoA, respectivey. For computing the spherica harmonic transform and discretization on the sphere, we empoy the recenty deveoped optima-dimensionaity samping scheme on the sphere [3]. Our MATLAB based code to compute the spherica harmonic coefficients of the standard FB (or FB5) distribution and the SFC function using the resuts and/or formuations presented in this paper, is made pubicy avaiabe. A. Accuracy Anaysis - Spherica Harmonic Expansion of standard FB distribution In order to anayse the accuracy of the proposed anaytica expression, for the computation of spherica harmonic coefficients of the standard FB distribution, given in (22), we define the spatia error as ɛ(l) = f( x L 2 p ) x p L = m= (f) m Y m ( x p ) 2, (34) 5 2 4 6 8 2 4 Band-imit, L Fig. 5: Spatia error ɛ(l), given in (34), between the standard FB distribution given in (5) and the reconstructed standard FB distribution from its coefficients up to the degree L. The convergence of the spatia error to zero (machine precision), as L increases, corroborates the correctness of the derived spherica harmonic expansion of standard FB distribution. which quantifies the error between the standard FB distribution given in (5) and the reconstructed standard FB distribution from its coefficients, computed using the proposed anaytic expression (22), up to the degree L. The summation in (34) is averaged over L 2 number of sampes of the samping scheme [3]. We pot the spatia error ɛ(l) against band-imit L for different parameters of the standard FB distribution in Fig. 5, where it is evident that the spatia error converges to zero (machine precision) as L increases. Consequenty, the standard FB distribution reconstructed from its coefficients converges to the formuation of standard FB distribution in spatia domain and thus vaidates the correctness of proposed anaytic expression. B. Iustration - SFC Function Here, we vaidate the proposed cosed-form expression for the SFC function through numerica experiments. In our anaysis, we consider both 2D and 3D antenna array geometries in the form of uniform circuar array (UCA) and reguar dodecahedron array (RDA), respectivey. The antenna eements of M-eement UCA are paced at the foowing spatia positions z p = [ R cos 2πp M, R sin 2πp M, ] T R 3, (35) where R denotes the circuar radius of the array. For the RDA, 2 antenna array eements are positioned at the vertices of a reguar dodecahedron inscribed in a sphere of radius R, as shown in Fig. 6 for R =. We assume that the AoA foows a standard Fisher-Bingham distribution. Using the proposed cosed-form expression in (32), we determine the SFC between the second and third UCA antenna eements and pot the magnitude of the SFC function ρ(z 2 z 3 ) in Fig. 7 against the normaized radius R/λ. In the same figure, we aso pot the numericay evauated SFC function, formuated in (3) and originay proposed in [9], which matches with the proposed cosed-form expression for

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 8.5.5 z p z q Spatia Correation ρ(zp zq).9.8.7.6.5.4.3.2 RDA, κ = 2.5, β =.5 RDA, κ = 25, β =.5 RDA, κ = 25, β = Numerica.5.5.5 Fig. 6: Reguar dodecahedron array (RDA) 2 antenna eements positioned at the vertices of a reguar dodecahedron inscribed in a sphere of radius R =..5. 2 3 4 5 Spatia Separation R/λ Fig. 8: Magnitude of the SFC function ρ(z p z q ) between antenna eements, paced at vertices shaded in back in Fig. 6, of 2-eement RDA of radius R. Spatia Correation ρ(z2 z3).9.8.7.6.5.4.3.2. UCA, κ = 2.5, β =.5 UCA, κ = 25, β =.5 UCA, κ = 25, β = Numerica 3D SFC function between two arbitrary points in 3D-space when the AoA of an incident signa has the Fisher-Bingham distribution. Furthermore, we have vaidated the correctness of proposed cosed-form expressions for the spherica harmonic coefficients of the Fisher-Bingham distribution and the 3D SFC using numerica experiments. We have focussed on the use of Fisher-Bingham distribution for the computation of SFC function. However, we beieve that the proposed spherica harmonic expansion of the Fisher-Bingham distribution has a great potentia of appicabiity in various appications for directiona statistics and data anaysis on the sphere. 2 3 4 5 Spatia Separation R/λ Fig. 7: Magnitude of the SFC function ρ(z 2 z 3 ) for 6- eement UCA of radius R. the SFC function. We emphasise that numerica evauation of the integras empoys computationay intensive techniques to obtain sufficienty accurate resuts and is therefore time consuming. Simiary, we compute the SFC between two antenna eements positioned at z p and z q on RDA, which are indicated in Fig. 6, and pot the magnitude ρ(z p z q ) Fig. 6, which again matches with the numericay evauated SFC function given in (3) and thus corroborates the correctness of proposed SFC function. VII. CONCLUSIONS In this paper, the spherica harmonic expansion of the Fisher-Bingham distribution and a cosed-form expression that aows the anaytic computation of the spherica harmonic coefficients have been presented. Using the expansion of pane waves in spherica harmonics and the proposed cosedform expression for the spherica harmonic coefficients of the FB5 distribution, we derived an anaytic formua for the REFERENCES [] J. T. Kent, The Fisher-Bingham distribution on the sphere, J. R. Statist. Soc., vo. 44, no., pp. 7 8, Jun. 982. [2] P. Leong and S. Carie, Methods for spherica data anaysis and visuaization, J. Neuro. Metho., vo. 8, no. 2, pp. 9 2, 998. [3] E. H. Langendijk, D. J. Kister, and F. L. Wightman, Sound ocaization in the presence of one or two distracters, J. Acoust. Soc. Am., vo. 9, no. 5, pp. 223 234, 2. [4] D. Pee, W. J. Whiten, and G. J. McLachan, Fitting mixtures of kent distributions to aid in joint set identification, J. American Statistica Asociation, vo. 96, no. 453, pp. 56 63, 2. [5] J. T. Kent and T. Hameryck, Using the Fisher-Bingham distribution in stochastic modes for protein structure, In Barber S, Baxter P, Mardia K, Was R, eds. Quantitative Bioogy, Shape ANaysis and Waveets, Leeds University Press, Leeds, UK, vo. 24, pp. 57 6, 25. [6] C. T. Christou, Beamforming spatiay spread signas with the kent distribution, in Proc. IEEE Int. Conf. on Information Fusion, 28, pp. 7. [7] D. Lunga and O. Ersoy, Kent mixture mode for cassification of remote sensing data on spherica manifods, in Appied Imagery Pattern Recognition Workshop (AIPR), 2 IEEE. IEEE, 2, pp. 7. [8] K. Mammasis and R. W. Stewart, The FB5 distribution and its appication in wireess communications, in Proc. of Int. ITG Workshop on Smart Antennas, pp. 375 38, Feb. 28. [9] K. Mammasis and R. W. Stewart, The Fisher-Bingham spatia correation mode for mutieement antenna systems, IEEE Trans. Veh. Techno., vo. 58, no. 5, pp. 23 236, Jun. 29. [] J. Saz and J. H. Winters, Effect of fading correation on adaptive arrays in digita mobie radio, IEEE Trans. Veh. Techno., vo. 43, no. 4, pp. 49 57, Nov. 994.

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