IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 1 Spherical Harmonic Expansion of Fisher- and 3D Spatial Fading Correlation for Multiple-Antenna Systems Yibeltal F. Alem, Student Member, IEEE, Zubair Khalid, Member, IEEE and Rodney A. Kennedy, Fellow, IEEE

Abstract—This paper considers the 3D spatial fading correla- closed-form expressions for evaluating the SFC function have tion (SFC) resulting from an angle-of-arrival (AoA) distribution been developed in the existing literature (e.g., [10], [16]–[25]). that can be modelled by a mixture of Fisher-Bingham distribu- The elliptic (directional) nature of the Fisher-Bingham tions on the sphere. By deriving a closed-form expression for the spherical harmonic transform for the component Fisher-Bingham distribution offers flexibility in modelling the distribution of distributions, with arbitrary parameter values, we obtain a normalized power or AoA of the multipath components for closed-form expression of the 3D-SFC for the mixture case. The most practical scenarios. Exploiting this fact, a 3D spatial 3D-SFC expression is general and can be used in arbitrary multi- correlation model has been developed in [9], where the dis- antenna array geometries and is demonstrated for the cases of tribution of AoA of the multipath components is modelled by a 2D uniform circular array in the horizontal plane and a 3D regular dodecahedral array. In computational aspects, we use a positive linear sum of Fisher-Bingham distributions, each recursions to compute the spherical harmonic coefficients and with different parameters. Such modelling has been shown give pragmatic guidelines on the truncation size in the series to be useful in a sense that it fits well with the multi-input representations to yield machine precision accuracy results. The multi-output (MIMO) field data and allows the evaluation results are further corroborated through numerical experiments of the SFC as a function of the angular spread, ovalness to demonstrate that the closed-form expressions yield the same results as significantly more computationally expensive numerical parameter, azimuth, elevation, and prior contribution of each integration methods. cluster. Although the SFC function presented in [9] is general in a sense that it is valid for any arbitrary antenna array Index Terms—Fisher-Bingham distribution, spherical har- monic expansion, spatial correlation, MIMO, angle of ar- geometry, it has not been expressed in closed-form and has rival (AoA). been only evaluated using numerical integration techniques, which can be computationally intensive to produce sufficiently accurate results. If the spherical harmonic expansion of the I.INTRODUCTION Fisher-Bingham distribution is given in a closed-form, the The Fisher-Bingham distribution, also known as the Kent SFC function can be computed analytically using the spherical distribution, belongs to the family of spherical distributions harmonic expansion of the distribution of AoA of the multipath in [1]. It has been used in applications components [21]. To the best of our knowledge, the spherical in a wide range of disciplines for modelling and analysing harmonic expansion of Fisher-Bingham distribution has not directional data. These applications include sound source lo- been derived in the existing literature. calization [2], [3], joint set identification [4], modelling protein In the current work, we present the spherical harmonic structures [5], 3D beamforming [6], classification of remote expansion of Fisher-Bingham distribution and a closed-form sensing data [7], modelling the distribution of AoA in wireless expression that enables the analytic computation of the spher- communication [8], [9], to name a few. ical harmonic coefficients. We also address the computational arXiv:1501.04395v1 [cs.IT] 19 Jan 2015 In this work, we focus our attention to the use of Fisher- considerations required to be taken into account in the evalua- Bingham distribution for modelling the distribution of angle tion of the proposed closed-form. Using the proposed spherical of arrival (AoA) (also referred as the distribution of scatterers) harmonic expansion of the Fisher-Bingham distribution, we in wireless communication and computation of spatial fading also formulate the SFC function experienced between two correlation (SFC) experienced between elements of multiple- arbitrary points in 3D-space for the case when the distribution antenna array systems. Modelling the distribution of scatterers of AoA of the multipath components is modelled by a mixture and characterising the spatial correlation of fading channels (positive linear sum) of Fisher-Bingham distributions. The is a key factor in evaluating the performance of wireless SFC presented here is general in a sense that it is expressed as communication systems with multiple antenna elements [10]– a function of arbitrary points in 3D-space and therefore can be [15]. It has been an active area of research for the past two used to compute spatial correlation for any 2D and 3D antenna decades or so and a number of spatial correlation models and array geometries. Through numerical analysis, we also validate The authors are with the Research School of Engineering, College the correctness of the proposed spherical harmonic expansion of Engineering and Computer Science, The Australian National Univer- of Fisher-Bingham distribution and the SFC function. In this sity, Canberra, ACT 2601, Australia. Emails: {yilbeltal.alem, zubair.khalid, paper, our main objective is to employ the proposed spherical rodney.kennedy}@anu.edu.au. This work was supported by the Aus- tralian Research Council’s Discovery Projects funding scheme (Project no. harmonic expansion of the Fisher-Bingham distribution for DP150101011). computing the spatial correlation. However, we expect that IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 2

` the proposed spherical harmonic expansion can be useful in where dm,m0 (·) denotes the Wigner-d function of degree ` and various applications where the Fisher-Bingham distribution is orders m and m0 [26]. used to model and analyse directional data (e.g., [2]–[7]). By completeness of spherical harmonics, any finite energy The remainder of the paper is structured as follows. We function f(xb) on the 2-sphere can be expanded as review the mathematical background related to signals defined ∞ ` on the 2-sphere and spherical harmonics in Section II. In X X m m f(xb) = (f)` Y` (xb), (4) Section III, we define the Fisher-Bingham distribution and `=0 m=−` present its application in modelling 3D spatial correlation. where (f)m is the spherical harmonic coefficient given by We present the spherical harmonic expansion of the Fisher- ` Z Bingham distribution and analytical formula for computing m m (f)` = f(xb)Y` (xb) ds(xb). (5) the spherical harmonic coefficients in Section IV, where we 2 S also address computational issues. We derive a closed-form The signal f is said to be band-limited in the spectral domain expression for the 3D SFC between two arbitrary points in 3D- m at degree L if (f)` = 0 for ` > L. For a real-valued function space when the AoA of an incident signal follows the Fisher- f(x), we also note the relation Bingham probability density function (pdf) in Section V. b −m m m In Section VI, we carry out a numerical validation of the (f)` = (−1) (f)` , (6) proposed results and provide examples of the SFC for uniform which stems from the conjugate symmetry property of spher- circular array (2D) and regular dodecahedron array (3D) ical harmonics [26]. antenna elements. Finally, the concluding remarks are made in Section VII. C. Rotation on the Sphere II.MATHEMATICAL PRELIMINARIES The rotation group SO(3) is characterized by Euler angles A. Signals on the 2-Sphere (ϕ, ϑ, ω) ∈ SO(3), where ϕ ∈ [0, 2π), ϑ ∈ [0, π] and ω ∈ [0, 2π). We define a rotation operator on the sphere We consider complex valued square integrable functions 2 D(ϕ, ϑ, ω) that rotates a function on the sphere, according defined on the 2-sphere, S . The set of such functions forms to 0zyz0 convention, in the sequence of ω rotation around the L2( 2) a Hilbert space, denoted by S , that is is equipped with z-axis, ϑ rotation around the y-axis and ϕ rotation around f g the inner product defined for two functions and defined z-axis. A rotation of a function f(θ, φ) on sphere is given by on S2 as [26] −1 Z (D(ϕ, ϑ, ω)f)(xb) , f(R xb), (7) hf, gi , f(x)g(x) ds(x), (1) 2 b b b S where R is a 3 × 3 real orthogonal unitary matrix, referred which induces a norm kfk hf, fi1/2. Here, (·) as rotation matrix, that corresponds to the rotation operator , D(ϕ, ϑ, ω) and is given by denotes the complex conjugate operation and xb , [sin θ cos θ, sin θ sin φ, cos θ]T ∈ 2 ⊂ 3 represents a point S R R = R (ϕ)R (ϑ)R (ω), on the 2-sphere, where [·]T represents the vector transpose, z y z (8) θ ∈ [0, π] and φ ∈ [0, 2π) denote the co-latitude and longitude where cos ϕ − sin ϕ 0 respectively and ds(xb) = sin θ dθ dφ is the surface measure on the 2-sphere. The functions with finite energy (induced norm) Rz(ϕ) = sin ϕ cos ϕ 0 , are referred as signals on the sphere. 0 0 1  cos ϑ 0 sin ϑ B. Spherical Harmonics Ry(ϑ) =  0 1 0  . Spherical harmonics serve as orthonormal basis functions − sin ϑ 0 cos ϑ for the representation of functions on the sphere and are Here R (ϕ) and R (ϑ) characterize individual rotations by defined for integer degree ` ≥ 0 and integer order |m| ≤ ` as z y ϕ along z-axis and ϑ along y-axis, respectively. m m m m imφ Y` (xb) ≡ Y` (θ, φ) , N` P` (cos θ)e , III.PROBLEM FORMULATION with s 2` + 1 (` − m)! A. Fisher-Bingham Distribution on Sphere N m , (2) ` , 4π (` + m)! Definition 1 (Fisher-Bingham Five-Parameter (FB5) Distribu- tion). The Fisher-Bingham five-parameter (FB5) distribution, m q is the normalization factor such that Y` ,Yp = δ`,pδm,q, also known as the Kent distribution, is a distribution on the where δm,q is the Kronecker delta function: δm,q = 1 for sphere with probability density function (pdf) defined as [1], m m = q and is zero otherwise. P` (·) denotes the associated [5] Legendre polynomial of degree ` and order m [26]. We also 1 T T g(x; κ, µ, β, A) = eκµb xb+xb βAxb, (9) note the following relation for associated Legendre polynomial b b C(κ, βA) s (` + m)! where A is a symmetric matrix of size 3 × 3 given by P m(cos θ) = d` (θ), (3) ` (` − m)! m,0 T T A = (ηb1ηb1 − ηb2ηb2 ). (10) IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 3

Here µb, ηb1 and ηb2 are the unit vectors (orthonormal set) that mixture refers to the positive linear sum of a number of FB5 denote the mean direction (centre), major axis and minor axis distributions, each with different parameters. We defer the of the distribution, respectively, κ ≥ 0 is the concentration formulation of FB5 based correlation model until Section V. parameter that quantifies the spatial concentration of FB5 Although the correlation model proposed in [9] is general in a distribution around its mean and 0 ≤ β ≤ κ/2 is referred to sense that the SFC function can be computed for any arbitrary as the ovalness parameter that is a measure of ellipticity of the antenna geometry, the formulation of SFC function involves distribution. In (9), the term C(κ, β) denotes the normalization the computation of integrals that can only be carried out using constant, which ensures kgk2 = 1 and is given by numerical integration techniques as we highlighted earlier. ∞ If the spherical harmonic expansion of the FB5 distribution X Γ(r + 1/2) C(κ, β) = 2π β2r(κ/2)−2r−1/2I (κ), is given in closed-form, the SFC function can be computed Γ(r + 1) 2r+1/2 r=0 analytically following the approach used in [21]. (11) In this paper, we derive an exact expression to compute where Ir(·) denotes the modified Bessel function of the first the spherical harmonic expansion of the FB5 distribution. kind of order r. Using the spherical harmonic expansion of FB5 distribution, The FB5 distribution is more concentrated and more elliptic we also formulate exact SFC function for the mixture of FB5 for larger values of κ and β, respectively. As an example, the distributions, defining the distribution of AoA of the multipath components. We also analyse the computational considerations FB5 distribution g(xb; κ, µb, β, A) is plotted on the sphere in Fig. 1 for different values of parameters. involved in the evaluation of spherical harmonic expansion of The FB5 distribution belongs to the family of spherical FB5 and distribution and SFC function. distributions in directional statistics and is the analogue of the Euclidean domain bivariate with un- IV. SPHERICAL HARMONIC EXPANSION OF THE FB5 constrained [1], [5]. DISTRIBUTION In this section, we derive an analytic expression for the B. 3D Spatial Fading Correlation (SFC) computation of spherical harmonic coefficients of the FB5 In MIMO systems, the 3D multipath channel impulse re- distribution. For convenience, we determine the spherical har- monic coefficients of the FB5 distribution with mean (centre) sponse for a signal arriving at antenna array is characterized by 0 0 0 the steering vector of the antenna array. For an antenna array µb located on z-axis and major axis ηb1 and minor axis ηb2 3 aligned along x-axis and y-axis respectively, that is, consisting of M antenna elements placed at zp ∈ R , p = 1, 2,...,M, the steering vector, denoted by α(x), is given by 0 T 0 T 0 T b µb = [0 0 1] ηb1 = [1 0 0] ηb2 = [0 1 0] . (14)   ikzp·x α(xb) = α1(xb), α2(xb), . . . αL(xb) , αp(xb) , e b, With the centre, major and minor axes as given in (14), (12) the FB5 distribution, referred to as the standard Fisher- 3 where xb ∈ R denotes a unit vector pointing in the direction Bingham (FB) distribution, has the pdf given by of wave propagation and k = 2π/λ with λ denoting the 1 2 h(x) κ cos θ+β sin θ cos 2φ wavelength of the arriving signal. For any b representing f(xb; κ, β) , e , (15) the pdf of the angles of arrival (AoA) of the multipath C(κ, β) components or the unit-normalized power of a signal received which is related to the FB5 distribution pdf g(xb; κ, β, µb, A) from the direction xb, the 3D SFC function between the p- given in (9) through the rotation operator D(ϕ, ϑ, ω) as q z z th and the -th antenna elements, located at p and q, −1 respectively, with an assumption that signals arriving at the g(xb; κ, β, µb, A) = (D(ϕ, ϑ, ω)f)(xb; κ, β) = f(R xb; κ, β), antenna elements are narrowband, is given by [21] (16) Z where the rotation matrix is related to µb, ηb1, ηb2 (parameters of FB5 distribution) as ρ(zp, zq) , h(x) αp(x) αq(x) ds(x) 2 b b b b S Z R = [η1, η2, µ], (17) ik(zp−zq )·x b b b = h(x) e b ds(x) ≡ ρ(zp − zq), (13) 2 b b S and the Euler angles (ϕ, ϑ, ω) are related to the rotation which indicates that the SFC only depends on zp − zq and is, matrix R through (8). As an example, we rotate each of the therefore, spatially wide-sense stationary. FB5 distribution g(xb; κ, µb, β, A) plotted in Fig. 1 to obtain the standard Fisher-Bingham distribution f(xb; κ, β), plotted C. FB5 Distribution Based Spatial Correlation Model and in Fig. 2, where we have indicated the Euler angles that Problem Under Consideration relate the FB5 distribution and the Fisher-Bingham distribution through (16). The FB5 distribution offers the flexibility, due to its direc- tional nature, to model the distribution of normalized power or AoA of the multipath components for most practical scenarios. A. Spherical Harmonic Expansion of Standard FB distribution Utilizing this capability of FB5 distribution, a 3D spatial Here, we derive a closed-form expression to compute the correlation model for the mixture of FB5 distributions defining spherical harmonic coefficients of the standard FB distribution, the AoA distribution has been developed in [9]. Here the given in (15). Later in this section, noting the relation between IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 4

3.5

3

2.5

2

1.5

1

0.5

0

(a) κ = 25, β = 10 (b) κ = 100, β = 10 (c) κ = 100, β = 49

Fig. 1: The Fisher-Bingham five-parameter (FB5) distribution g(xb; κ, µb, β, A), given in (9), is plotted on the sphere for T T T T parameters κ, β, as indicated and: (a) µb = [0 0 1] , ηb1 = [0 1 0] , minor axis ηb2 = [1 0 0] . (b) µb = [1 0 0] , T T T T T ηb1 = [0 1 0] , minor axis ηb2 = [0 0 1] . (c) µb = [0 1 0] , ηb1 = [1 0 0] , minor axis ηb2 = [0 0 1] .

(a) κ = 25, β = 10, (b) κ = 100, β = 10, (c) κ = 100, β = 49, (ϕ, ϑ, ω) = (π/2, 0, 0). (ϕ, ϑ, ω) = (π/2, π/2, 0). (ϕ, ϑ, ω) = (π/2, π/2, π/2).

Fig. 2: The standard Fisher-Bingham (FB) distribution f(xb; κ, β), given in (15), is plotted on the sphere for the concentration parameter κ and ovalness parameter β, as indicated. Each subfigure is related to the respective subfigure of Fig. 1 through the the relation (16), with Euler angles indicated. standard FB distribution and FB5 distribution given in (16), ical harmonic coefficients, we rewrite (18) explicitly as and the effect of the rotation operation on the spherical m Z π m N` κ cos θ m harmonic coefficients, we determine the coefficients of FB5 (f)` = e P` (cos θ) sin θdθ distribution. C(κ, β) 0 Z 2π m −imφ β sin2 θ cos 2φ The spherical harmonic coefficient, denoted by (f)` , of × e e dφ the standard Fisher-Bingham distribution given in (15) can be 0 m Z π expressed as N` κ cos θ m = e Fm(θ) P` (cos θ) sin θdθ, C(κ, β) 0 where Z 2π m m m Z 2 (f)` , hf( · ; κ, β),Y` i = f(x; κ, β)Y` (x)ds(x). β sin θ cos 2φ−imφ 2 b b b Fm(θ) = e dφ, S (18) 0 Since f(xb; κ, β) is a real function, we only need to compute which we evaluate as 1 the spherical harmonic coefficients for positive orders 0 ≤ ( 2 m ≤ ` for each degree ` ≥ 0. The coefficients for the negative 2πIm/2(β sin θ) m ∈ 0, 2, 4,... Fm(θ) = orders can be readily computed using the conjugate symmetry 0 m ∈ 1, 3, 5, . . .. relation, noted in (6). To derive a closed-form expression for computing the spher- 1Using Mathematica. IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 5

2 By expanding the modified Bessel function Im/2(β sin θ) as Once the Euler angles (ϕ, ϑ, ω) are extracted from the µ, η , η ∞ 2t+m/2 parameters b b1 b2 using (24)–(26), the spherical harmonic X β  (sin θ)4t+m I (β sin2 θ) = , (19) coefficients of the FB5 distribution g(xb; µb, κ, A) given in (9) m/2 2 t!(t + m/2)! t=0 can be computed following the effect of the rotation operation on the spherical harmonic coefficients as [26] the exponential eκ cos θ as [8] m m m ∞ (g)` , hg( · ; µ, κ, A),Y` i = hD(ϕ, ϑ, ω)f, Y` i r π b κ cos θ X 0 ` e = (2n + 1)In+1/2(κ)P (cos θ), (20) 2κ n X ` m0 n=0 = Dm,m0 (ϕ, ϑ, ω)(f)` , (27) 0 and using the relation between associated Legendre poly- m =−` nomial and Wigner-d function given in (3), along with the m0 where (f)` is the spherical harmonic coefficient of degree following expansion of Wigner-d function in terms of complex ` and order m0 of the standard Fisher-Bingham distribution exponentials ` f(xb). In (27), Dm,m0 (ϕ, ϑ, ω) denotes the Wigner-D function 0 ` of degree ` and orders |m|, |m | ≤ ` and is given by [26] ` n−m X ` ` iuθ dm,n(θ) = i du,m(π/2) du,n(π/2) e , (21) ` −imϕ ` −im0ω Dm,m0 (ϕ, ϑ, ω) = e dm,m0 (ϑ)e . (28) u=−` we obtain a closed-form expression for the spherical harmonic C. Computational Considerations coefficient in (18) as Here we discuss the computation of Wigner-d functions, at r ∞ πi−m 2` + 1 X (f)m = (2n + 1)I (κ) × a fixed argument of π/2, which are essentially required for ` C(κ, β) 2κ n+1/2 n=0 the computation of spherical harmonic expansion of standard n ∞ 2t+m/2 FB or FB5 distribution using the proposed formulation (22). X 2 X (β/2) dn (π/2) × Another computational consideration that is addressed here is u,0 Γ(t + 1)Γ(t + m/2 + 1) u=−n t=0 the evaluation of infinite summations over t and n involved in ` the computation of (22). X ` ` 0 du0,0(π/2)du0,m(π/2) G(4t + m + 1, u + u ), 1) Computation of Wigner-d functions: For the computation u0=−` of spherical harmonic coefficients of the standard FB distri- (22) bution using (22), we are required to compute the Wigner- ` where Γ(·) denotes the Gamma function and we have used the d functions at a fixed argument of π/2, that is du,m(π/2), following identity [27, Sec. 3.892] for each |u|, |m| ≤ `. Let D` denote the matrix of size (2` + 1) × (2` + 1) with entries d` (π/2) for |u|, |m| ≤ `. Z π u,m p iqθ The matrix D can be computed for each ` = 1, 2,..., using G(p, q) , (sin θ) e dθ, ` 0 the relation given in [28] that recursively computes D` from πeiqπ/2Γ(p + 2) D . = . (23) `−1 p p+q+2 p−q+2 2) Truncation over n: The infinite sum over n arises 2 (p + 1)Γ( 2 )Γ( 2 ) from (20) where an exponential function of the concentration B. Spherical Harmonic Expansion of FB5 distribution measure κ and the co-latitude angle θ is expanded as an infinite sum of the modified Bessel functions and Legendre We use the relation between standard FB distribution and polynomials. It is well known that the Legendre polynomials FB5 distribution given in (16) to determine the spherical are oscillating functions with a maximum value of one. The harmonic expansion of FB5 distribution. The Euler angles modified Bessel function In+1/2(κ) that makes up the core (ϕ, ϑ, ω) in (16), which characterize the relation between the of the expansion given in (20) decays quickly (and monoton- standard FB distribution and the FB5 distribution, can be ically) to zero as n → ∞ for a given κ. We use this feature obtained from the rotation matrix R formulated in (17) in of the modified Bessel function In+1/2(κ) to truncate the terms of the parameters of FB5 distribution. By comparing summation over n. We plot In+1/2(κ) for different values of n (8) and (17), ϑ is given by and 0 ≤ κ ≤ 100 in Fig. 3, where it is evident that the Bessel −1 function quickly decays to zero. We propose to truncate the ϑ = cos (R3,3), (24) summation over n in (20), or equivalently in (22), at n = N −16 where Ra,b denotes the entry at a-th row and b-th column such that IN+1/2(κ) < 10 (double machine precision). We of the matrix R given in (17). Similarly ϕ ∈ [0, 2π) and approximate the linear relationship between such truncation ω ∈ [0, 2π) can be found by a four-quadrant search satisfying level and the concentration parameter κ, given by

R2,3 R1,3 3 sin ϕ = , cos ϕ = , (25) N = κ + 24, (29) p 2 p 2 1 − (R3,3) 1 − (R3,3) 2 which is also indicated in Fig. 3. This truncation level is found R3,2 −R3,1 sin ω = , cos ω = , (26) to give truncation error less than the machine precision level p1 − (R )2 p1 − (R )2 3,3 3,3 for the values of concentration parameter κ in the range 0 ≤ respectively. κ ≤ 100 (used in practice [8], [9]). We note that the truncation IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 6

100 50 0 90 45 0

80 40 −50 −50 70 35 −100 60 −100 30

50 25 −150 κ β −150 40 20 −200 30 −200 15

20 10 −250 −250 10 5 −300 0 −300 0 0 50 100 150 200 0 20 40 60 80 100 Truncation level, n Truncation level, t

Fig. 3: Surface plot of modified Bessel function In+1/2(κ) for Fig. 4: Surface plot of the term inside the summation on the −16 different values of κ and n. The region In+1/2(κ) < 10 right hand side of (19) for m = 0 and θ = π/2, that is, is approximated by the region below the straight line, n = β2t S(β, t) , t!t! 22t for different values of the ovalness parameter 3 −16 −16 2 κ + 24. Only those values In+1/2(κ) > 10 plotted above β and t. The region R(β, t) < 10 is approximated by the the straight line account for the sum in (20) to make it accurate 36 region below the straight line, t = 25 β + 12. to the machine precision level.

FB distribution. error becomes smaller for large values of κ, indicating that the truncation at level less than N given in (29) may also allow sufficiently accurate computation of spherical harmonic coefficients. Further analysis on establishing the relationship V. 3D SPATIAL FADING CORRELATION FOR MIXTUREOF between the concentration parameter κ and the truncation level FISHER-BINGHAM DISTRIBUTIONS is beyond the scope of current work. 3) Truncation over t: The expansion of modified Bessel In this section, we derive a closed form expression to com- functions of the first kind for a given value of β as given in (19) pute the SFC function for the spatial correlation model based introduces the infinite sum over t. Again, the decaying (non- on a mixture of FB5 distributions defining the distribution of monotonic in this case) characteristics of the term inside the AoA of multipath components [8], [9]. Let h(xb) denotes the summation on right hand side of (19) with the increase in t pdf of the distribution defined as a positive linear sum of W makes it possible to truncate the infinite sum given in (19) that FB5 distributions, each with different parameters, and is given mainly depends on the ovalness parameter β with a minimal by error. We plot the term inside the summation on the right hand 2t W side of (19) for m = 0 and θ = π/2, that is, S(β, t) β X , t!t! 22t h(x) = K g(x; κ , µ , β , A ), (31) for different values of the ovalness parameter β in Fig. 4. b w b w bw w w w=1 We again propose to truncate the summation over t at the truncation level T such that the terms after the truncation where Kw denotes the weight of w-th FB5 distribution with level T are less than 10−16 and do not have impact on the parameters κw, µbw, βw and Aw. We assume that each Kw summation. We approximate the linear relationship between is normalized such that khk2 = 1. For the distribution with the ovalness parameter β and the truncation level T pdf defined in (31), the SFC function, given in (13) has been formulated in [9] and analysed for a uniform circular antenna 36 T = β + 12. (30) array but the integrals involved in the SFC function were 25 numerically computed. Since we are computing coefficients for positive orders 0 ≤ We follow the approach introduced in [21] to derive a m ≤ ` for each degree `, the truncation level given by closed-form 3D SFC function using the proposed closed-form (30), that is obtained for order m = 0 and θ = π/2, for spherical harmonic expansion of the FB5 distribution. Using the summation in (19), is also valid for all positive orders spherical harmonic expansion of plane waves [29]: 0 ≤ m ≤ ` and all θ ∈ [0, π] because the term inside the summation is maximum for m = 0 and θ = π/2 for a given summation variable t. ∞ ` X X We finally note that the truncation levels proposed here ikzp·xb `  m  m e = 4π i j` kkzpk Y` zp/kzpk Y` (xb), allow accurate computation of spherical harmonic coefficients `=0 m=−` to the level of double machine precision. Later in the paper, we validate the correctness of the derived expression for the and expanding the h(xb) in (31), following computation of spherical harmonic coefficients of the standard (4), and employing the orthonormality of spherical harmonics, IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 7

we write the SFC function in (13) as [21] 100 κ = 2.5,β = 0.5 κ = 25,β = 10 ∞ κ = 100,β = 49 X `  ρ(zp − zq) = 4π i j` kkzp − zqk ×

`=0 ) −5

L 10 (

` ǫ X m m zp − zq  (h)` Y` , (32) kzp − zqk m=−` 10−10 where Spatial error,

m m (h)` = hh, Y` i

W ` 10−15 0 X X ` m = KwDm,m0 (ϕw, ϑw, ωw)(f)` , (33) 0 20 40 60 80 100 120 140 w=1 m0=−` Band-limit, L

m0 Fig. 5: Spatial error (L), given in (34), between the standard which is obtained by combining (27) and (31). Here, (f)` denotes the spherical harmonic coefficient of the standard FB distribution given in (15) and the reconstructed standard FB distribution from its coefficients up to the degree L−1. The FB distribution and the Euler angles (ϕw, ϑw, ωw) relate the convergence of the spatial error to zero (machine precision), w-th FB5 distribution g(x; κw, µw, βw, Aw) of the mixture b b as L increases, corroborates the correctness of the derived and the standard FB distribution f(xb) through (16). In the computation of the SFC using the proposed formulation, given spherical harmonic expansion of standard FB distribution. in (32), we note that the summation for ` over first few terms yields sufficient accuracy as higher order Bessel functions decay rapidly to zero for points near each other in space, which quantifies the error between the standard FB distribution as indicated in [21], [30]. We conclude this section with given in (15) and the reconstructed standard FB distribution a note that the SFC function can be analytically computed from its coefficients, computed using the proposed analytic expression (22), up to the degree L − 1. The summation in using the proposed formulation for an arbitrary antenna array 2 geometry and the distribution of AoA modelled by a mixture (34) is averaged over L number of samples of the sampling of FB5 distributions, each with different parameters. In the scheme [31]. We plot the spatial error (L) against band-limit next section, we evaluate the proposed SFC function for L for different parameters of the standard FB distribution in uniform circular array and a 3D regular dodecahedron array Fig. 5, where it is evident that the spatial error converges to of antenna elements. zero (machine precision) as L increases. Consequently, the standard FB distribution reconstructed from its coefficients converges to the formulation of standard FB distribution in VI.EXPERIMENTAL ANALYSIS spatial domain and thus validates the correctness of proposed We conduct numerical experiments to validate the cor- analytic expression. rectness of the proposed analytic expressions, formulated in (22)–(23) and (32)–(33), for the computation of the spherical B. Illustration - SFC Function harmonic coefficients of standard FB distribution and the SFC function for a mixture of FB5 distributions defining the Here, we validate the proposed closed-form expression for distribution of AoA, respectively. For computing the spherical the SFC function through numerical experiments. In our anal- harmonic transform and discretization on the sphere, we em- ysis, we consider both 2D and 3D antenna array geometries in ploy the recently developed optimal-dimensionality sampling the form of uniform circular array (UCA) and regular dodec- scheme on the sphere [31]. Our MATLAB based code to ahedron array (RDA), respectively. The antenna elements of M compute the spherical harmonic coefficients of the standard -element UCA are placed at the following spatial positions FB (or FB5) distribution and the SFC function using the results 2πp 2πp T z = R cos ,R sin , 0 ∈ 3, (35) and/or formulations presented in this paper, is made publicly p M M R available. where R denotes the circular radius of the array. For the RDA, 20 antenna array elements are positioned at the vertices of a A. Accuracy Analysis - Spherical Harmonic Expansion of regular dodecahedron inscribed in a sphere of radius R, as standard FB distribution shown in Fig. 6 for R = 1. We assume that the AoA follows a standard Fisher-Bingham distribution. In order to analyse the accuracy of the proposed analytical Using the proposed closed-form expression in (32), we expression, for the computation of spherical harmonic coeffi- determine the SFC between the second and third UCA an- cients of the standard FB distribution, given in (22), we define tenna elements and plot the magnitude of the SFC function the spatial error as ρ(z2 − z3) in Fig. 7 against the normalized radius R/λ. In L−1 ` the same figure, we also plot the numerically evaluated SFC 1 X X X m m 2 (L) = f(xp) − (f) Y (xp) , (34) function, formulated in (13) and originally proposed in [9], L2 b ` ` b xbp `=0 m=−` which matches with the proposed closed-form expression for IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 8

1 RDA, κ = 2.5, β = 0.5 RDA, κ = 25, β = 0.5 0.9 RDA, κ = 25, β = 10 Numerical

) 0.8 q z

− 0.7 p z 0.5 z ( q ρ 0.6

z 0.5 0 p 0.4

−0.5 0.3

Spatial Correlation 0.2

0.1

0.5 0 0 1 2 3 4 5 0 0.5 0 Spatial Separation R/λ −0.5 −0.5 Fig. 8: Magnitude of the SFC function ρ(zp − zq) between Fig. 6: Regular dodecahedron array (RDA) – 20 antenna antenna elements, placed at vertices shaded in black in Fig. 6, elements positioned at the vertices of a regular dodecahedron of 20-element RDA of radius R. inscribed in a sphere of radius R = 1.

1 3D SFC function between two arbitrary points in 3D-space UCA, κ = 2.5, β = 0.5 UCA, κ = 25, β = 0.5 when the AoA of an incident signal has the Fisher-Bingham 0.9 UCA, κ = 25, β = 10 Numerical distribution. Furthermore, we have validated the correctness of

) 0.8 3

z proposed closed-form expressions for the spherical harmonic

− 0.7 2 coefficients of the Fisher-Bingham distribution and the 3D z ( ρ 0.6 SFC using numerical experiments. We have focussed on the use of Fisher-Bingham distribution for the computation of SFC 0.5 function. However, we believe that the proposed spherical 0.4 harmonic expansion of the Fisher-Bingham distribution has 0.3 a great potential of applicability in various applications for

Spatial Correlation 0.2 directional statistics and data analysis on the sphere.

0.1

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