An Efficient Analytical Evaluation of the Electromagnetic Cross-Correlation Green's Function in MIMO Systems
11 An Efficient Analytical Evaluation of theElectromagnetic Cross-Correlation Green’s Functionin MIMO Systems
Debdeep Sarkar,
Member, IEEE , Said Mikki,
Senior Member, IEEE , Yahia Antar,
Life Fellow, IEEE
Abstract —In this paper, we completely eliminate all numericalintegrations needed to compute the far-field envelope cross-correlation (ECC) in multiple-input-multiple-output (MIMO)systems by deriving accurate and efficient analytical expressionsfor the frequency-domain cross-correlation Green’s functions(CGF), the most fundamental electromagnetic kernel neededfor understanding and estimating spatial correlation metrics inmultiple-antenna configurations. The analytical CGF is derivedfor the most general three-dimensional case, which can be usedfor fast CGF-based correlation matrix calculations in MIMOsystems valid for arbitrary locations and relative polarizationsof the constituent elements.
Index Terms —MIMO arrays, Cross-correlation Green’s Func-tion, Infinitesimal Dipole Model (IDM).
I. I
NTRODUCTION
Fifth generation (5G) wireless networks aiming at highdata-rate ( > Gbps) and efficient interference suppressionbetween multiple users in ultra-dense networks (UDNs), de-ploy large antennas arrays/massive multiple-input-multiple-output (MIMO) systems as key enabling technology [1]-[6]. One crucial aspect of such multiple-antenna systems inMIMO transceivers is the latter’s “spatial correlation matrix”,which accounts for mutual interaction (conventionally, onlyfar-field is taken into account) between the constituent antennaelement-pairs [7]-[11]. Despite the pivotal role of this spatialcorrelation matrix in shaping the overall channel matrix andconsequent capacity/interference-suppression issues (see [12]-[15] for details), the impact of pure antenna effects and variouscore electromagnetic aspects in MIMO channel modelling areoften not emphasized adequately.Traditionally, this antenna spatial correlation performance isdetermined from a formula involving the radiation patterns ofindividual elements [10], which makes it very cumbersometo perform antenna current level optimization aiming at adesired diversity performance. To relate the antenna correlationdirectly to the radiating antenna current distribution (i.e.,essentially bypassing altogether the original far-field pattern Debdeep Sarkar (Corresponding Author) and Yahia Antar are with theRoyal Military College, PO Box 17000, Station Forces Kingston, Ontario,K7K 7B4, Canada (Emails: [email protected] ; [email protected]). Said Mikki is with University of New Haven, West Haven, Connecticut,300 Boston Post Rd, 06516, USA (Email: [email protected]). c (cid:13) route), the concept of cross-correlation Green’s functions(CGFs) was first introduced in [16],[17] and later elaboratedfor antenna design applications [30], [18]. A preliminary stepof this CGF-based correlation calculation is to construct asuitable infinitesimal dipole model (IDM) for the radiatingMIMO antenna current distribution (see [19]-[28] for detailedtheory and applications of IDM). The next step is to employsuitable CGFs in order to compute individual ID-pair inter-actions systematically and then combine them togethoer toconstruct the global correlation matrices of the system [16].Application of CGFs to realize high diversity gain MIMOantenna arrays as well as dual-polarized massive MIMOsystems has been reported in recent past [29]-[32]. By efficientintegration of the CGFs with finite-difference-time-domain(FDTD) computational paradigm, one can also perform wide-band time-domain correlation analysis for arbitrary antennas[33]-[34]. The CGFs are also extended to deal with radiatorsinvolving both electric/magnetic current sources [35]. Possibleapplication of the CGF methodology for near-field stochasticsystems [36] and antenna directivity analysis [37] are alsobeing actively explored presently.However, calculation of the CGF tensor components re-quires numerical integration involving elevation ( θ ) and az-imuth ( φ ) angle dependent terms in the argument of com-plex exponential functions [16], [33]. Therefore, it becomesextremely difficult to efficiently embed these CGFs in fastoptimization routines aiming at finding optimum current dis-tributions for desired diversity performance. This necessitatesa robust analytical evaluation scheme for the determination ofthe CGF tensor components and the possibility of achievingthis was in fact already suggested in [16]. Although somespecialized approximation formulas of the time-domain CGFswere attempted in [38], IDM-synthesis of MIMO antennasstrictly requires frequency-domain CGFs, and the analyticalevaluation of the latter at a very general level has been so faran open problem. Mitigating this shortcoming in the literaturewill be the main contribution of the present work.In this paper, we first employ a series-expansion approach toapproximate the complex exponential functions in CGF tensors(some preliminary ideas were briefly suggested [39]). Next, bydeploying carefully selected mathematical properties enjoyedby the Beta and Gamma functions, we analytically evaluate thefull angular space integration involving oscillatory terms. Inthis way, analytical expressions of CGF tensors are presentedhere for the most general three-dimensional case. a r X i v : . [ phy s i c s . c l a ss - ph ] F e b II. A
NALYTICAL
CGF D
ETERMINATION IN O NE /T WO /T HREE D IMENSIONAL A RRAYS
A. Review of the CGF Tensor
In standard MIMO literature, the complex correlation co-efficient ρ between the far-field patterns E = E ( θ, φ ) and E = E ( θ, φ ) , respectively generated by complexcurrent distributions J = J ( r (cid:48) ) and J ( r (cid:48)(cid:48) ) , is traditionallycalculated by [10] ρ = (cid:82) π [ E · E ∗ ] d Ω (cid:113)(cid:82) π [ E · E ∗ ] d Ω (cid:82) π [ E · E ∗ ] d Ω , (1)where d Ω is the solid angle element given by d Ω = sin θdθdφ .That is, only the radiation fields appear in the original defi-nition. This makes the process of evaluating ρ and desingingoptimum antennas for spatial diversity applications very chal-lenging since the 3D computation of the far-field pattern isdemanding. Moreover, the geometrical details of the radiator,e.g., shape, orientations, excitations, do not directly manifestthemselves in the far field. For those reasons, in [16] ρ isexpressed directly in terms of J and J as: ρ = (cid:82) d r (cid:48) (cid:82) d r (cid:48)(cid:48) J · ¯C · J ∗ (cid:113)(cid:2)(cid:82) d r (cid:48) (cid:82) d r (cid:48)(cid:48) J · ¯C · J ∗ (cid:3) (cid:2)(cid:82) d r (cid:48) (cid:82) d r (cid:48)(cid:48) J · ¯C · J ∗ (cid:3) , (2)where all integrals are performed over the entire antenna radi-ating surface (the support of the current distribution functions J ( r ) and J ( r ) . Here, ¯C = ¯C ( r (cid:48) , r (cid:48)(cid:48) ) stands for the CGFtensor in uniform propagation environment given by [16]: ¯C ( r (cid:48) , r (cid:48)(cid:48) ) = (cid:90) π (cid:90) π (cid:2) ¯I − ˆ r ˆ r (cid:3) e j k · ( r (cid:48) − r (cid:48)(cid:48) ) sin θdθdφ. (3)The quantity ¯I is the unit dyad, while r (cid:48) and r (cid:48)(cid:48) denote thespatial dependencies of J and J , respectively. Moreover, k = k ˆ r , with k = 2 π/λ ( λ = operating wavelength) and ˆ r being the radial unit-vector in spherical coordinate systemgiven by ˆ r ( θ, ϕ ) = ˆ r (Ω) := ˆ x cos ϕ sin θ + ˆ y sin ϕ sin θ + ˆ z cos θ. (4)The nine components C pq (where p = x, y, z and q = x, y, z )of the CGF tensor ¯C ( r (cid:48) , r (cid:48)(cid:48) ) in (3) can be derived after using(4) and elementary dyadic arithmetic rule. The results areexpressed as: C pq = (cid:90) π (cid:90) π f pq exp [ jkr d ] dθdφ, (5)where r d = ˆ r · ( r (cid:48) − r (cid:48)(cid:48) ) = x d sin θ cos φ + y d sin θ sin φ + z d cos θ , with x d = x (cid:48) − x (cid:48)(cid:48) , y d = y (cid:48) − y (cid:48)(cid:48) and z d = z (cid:48) − z (cid:48)(cid:48) and values of f pq = f pq ( θ, φ ) are defined as [16], [33]: f xx = sin θ (cid:0) − sin θ cos φ (cid:1) ,f yy = sin θ (cid:0) − sin θ sin φ (cid:1) ,f zz = sin θ, (6) f xy = f yx = − sin θ, cos φ sin φ,f yz = f zy = − sin θ cos θ sin φ,f zx = f xz = − sin θ cos θ cos φ. (7) Consequently, the CGF-based technique completely eliminatesthe requirement of going through the conventional pattern-based route of (1) by focusing instead on the total radiatingantenna current distribution, i.e., only current points reflectingthe radiator excitation and geometry are needed, and these areconsiderably smaller in number than the far-field points [16],[33], [34]. Clearly, still all components of the CGF involvetwo-dimensional angular integration operations over the entiresphere. In general, for every position pair r (cid:48) , r (cid:48)(cid:48) , these integralsmust be computed again. Therefore, for large antenna arraysthe net number of numerical computations becomes large. B. General Idea of CGF Tensor Approximation
While evaluation of C pq via (5) requires computing anangular-space integration of f pq exp ( jkr d ) , it was pointedout in [16] that CGF might be computed or at least well-approximated analytically. Various approaches may be pursuedhere. For example, it is possible to expand the integrand ofevery integral into orthogonal functions then evaluate angularintegrations using exact orthogonality relations. However, theorthogonal expansion itself often requires numerical integra-tions to obtain the needed Fourier coefficients (the weightof every orthogonal function) and hence it may not leadto efficient algorithm. Moreover, because each of the nineintegrals entering into the determination of the full × dyad ¯C may involve a distinct angular function in the integrand,performing an orthogonal function expansion here becomescumbersome and very tedious.Another idea is to simplify (5) by making use of a Bessel’sfunction utilizing the Jacobi-Anger expansion [42]. However,an alternative approach is proposed in this paper, where wesimply utilize the familiar Taylor series expansion of theexponential functions exp [ jkr d ] = ∞ (cid:88) n =0 ( jkr d ) n n ! , (8)after which C pq in (5) can be approximated by truncating intofinite number of N terms giving C pq ≈ N (cid:88) n =0 ( jk ) n I pq,n , (9)where, I pq,n = 1 n ! (cid:90) π (cid:90) π f pq ( r d . ) n dθdφ. (10)The subsequent sections will build fast and robust algorithmfor the evaluation of the quantity I pq,n for various infinitesimaldipole (ID) array configurations (one/two/three dimensionaltopologies will be considered).We also note at this juncture that the formal limit N → ∞ in (9) will give the exact C pq value as obtained from (5).However, soon we will discover that opting for N → ∞ is notrequired in our correlation matrix computation algorithm. Theproposed method then provides a tradeoff between exactnessand computational efficiency. While the use of N makes ourevaluation less exact than using complete orthogonal functionseries expansion, nevertheless the final algorithm turns out to be very efficient for reasonably finite values of N , while itavoids the mathematical complexity of the angular sphericalfunction approach. Fig. 1. Variation of | ρ | with respect to normalized inter-element spacing p = d/λ , for: (a) two y -directed IDs, and (b) two z -directed IDs, placed along y -axis separated y d = d (schematics shown in inset). We perform numericalcalculation using the exact formula (5) and the series-approximation formula(9) for N = 20 , , , . C. Analytical CGF Estimation for One-dimensional LinearDipole Arrays
To begin with, let us consider the linear/one-dimensionalarray configuration that consists of infinitesimal dipoles (IDs)strictly placed along the y -axis, but yet are allowed to havearbitrary polarization and inter-element spacing. Since x d = z d = 0 for this case, we have r d = y d sin θ sin φ . Applyingthis r d in (11), the reduced expression for I pq,n is obtained asfollows: I pq,n = ( y d ) n n ! G pq,n , (11)where, G pq,n = (cid:90) π (cid:90) π f pq (sin θ sin φ ) n dθdφ. (12)Now the natural question that arises here is this: how canone decide the proper value n = N in (9) needed to efficiently compute the far-field correlation of given ID pairwith acceptable accuracy? To answer this question, we probe further into C pq , by ex-pressing the inter-element spacing y d in terms of the operatingwavelength λ . Writing y d = pλ and using k = 2 π/λ , one canput C pq via (9), (11) and (12) in the following form C pq ≈ N (cid:88) n =0 j n G pq,n (cid:20) (2 πp ) n n ! (cid:21) . (13)At this point, we use the well known Stirling’s formula for n ! ,which is very accurate for large values of n [40]-[41]: n ! = √ πn (cid:16) ne (cid:17) n . (14)This further reduces (13) to: C pq ≈ N (cid:88) n =0 j n (cid:18) G pq,n √ πn (cid:19) (cid:18) πepn (cid:19) n . (15)Note that, the factor G pq,n / √ πn in (15) decreases asymp-totically with n , and is ignored for the time-being. By carefulobservation of the next term involving n -th power of (2 πep/n ) and using πe ≈ . , we provide a rule-of-thumb todetermine the necessary n = N for a given value of p requiredto truncate the series in (9) yet while not compromisingcorrelation calculation accuracy: N > p. (16)For the example p = 5 , i.e. if the two IDs are placed λ apart, one should need approximately 85 terms to accuratelydetermine C pq from (9). A numerical verification of this rule-of-thumb formula (16) can be found in Fig. 1(a) and Fig. 1(b),where variations of | ρ | with respect to inter-element spacing p = d/λ (where y d = d ) are shown respectively for a y -directed and z -directed ID pair, placed along the y -axis. Onecan observe from Fig. 1(a) and Fig. 1(b) that for d/λ > N/ ,the | ρ | value using (9) quickly deviates from that determinedvia the exact formula (5). Observing Fig. 1(a) and Fig. 1(b),it can be said that for inter-element spacing > λ , spatialcorrelation magnitude | ρ | is sufficiently small ( | ρ | < . ), andmay be ignored for practical application purpose. This fact willalso come in handy in formulating a general spatial correlationdetermination algorithm in the subsequent section.The next challenge is to determine G pq,n analytically, there-fore completely eliminating the need for numerical integrationroutines, as emphasized before. We demonstrate the derivationfor G zz,n to start with. Using f zz from (6) in (12), we obtain: G zz,n = (cid:90) π (cid:90) π f zz (sin θ sin φ ) n dθdφ = (cid:20)(cid:90) π sin n +3 θdθ (cid:21) (cid:20)(cid:90) π sin n φdφ (cid:21) . (17)Note that, n can be either odd or even. For odd values of n ,the integral with φ -dependent term vanishes, i.e. we have: (cid:90) π sin n φdφ = 0 , (18) Therefore, G zz,n = 0 for odd values of n . On the other hand,for even values of n , we have: (cid:90) π sin n φdφ = 2 B (cid:18) , n (cid:19) = 2Γ (cid:0) (cid:1) Γ (cid:0) n + 1 (cid:1) Γ (cid:0) n + 1 (cid:1) = 2 √ π ( n )! (cid:20) n !2 n ( n )! √ π (cid:21) = π n − n ! (cid:2) ( n )! (cid:3) , (19)with B and Γ standing for Beta and Gamma functions respec-tively [42]. Also, when n is even, n + 3 is odd , yielding: (cid:90) π sin n +3 θdθ = B (cid:18) , n (cid:19) = Γ (cid:0) (cid:1) Γ (cid:0) n + 2 (cid:1) Γ (cid:0) n + (cid:1) = √ π ( n (cid:20) n +4 ( n + 2)! √ π ( n + 4)! (cid:21) = (cid:2) ( n )! (cid:3) n ! (cid:16) n (cid:17) (cid:34) n +4 (cid:0) n + 2 (cid:1) (cid:0) n + 1 (cid:1) ( n + 4)( n + 3)( n + 2)( n + 1) (cid:35) = (cid:2) ( n )! (cid:3) n ! (cid:16) n (cid:17) (cid:20) n +2 ( n + 3)( n + 1) (cid:21) (20)Therefore, using (17), (19) and (20), we obtain for evenvalues of n : G zz,n = (cid:16) n (cid:17) W n , (21)where, W n = 8 π ( n + 1)( n + 3) . (22) W n is a “general weighing factor” for this one-dimensionalID array scenario, which would soon prove to be readilyapplicable for the more general three dimensional case. Theseveral relevant formulas used to simplify the integrations arecollected in the Appendix.The rest of the derivations for G pq,n follows a similar route,and consequently will not be elaborately shown here. Withthe help of a symbolic computer package (e.g., the symbolictoolbox of MATLAB or Mathematica), further verification ofthe detailed derived expressions for G pq,n were conducted bythe authors. The following general observations can be drawnfrom the results:1) The following condition holds always true: G pq,n = 0 for odd values of n for all p, q . (23)This is very significant, since it literally halves thenumber of integrations to be solved.2) The coefficients for mutually orthogonal ID pairs allvanish, i.e. for all values of n : G xy,n = G yx,n = G yz,n = G zy,n = G xz,n = G zx,n = 0 . (24)3) The coefficients for the ID pairs orthogonal to theplacement axes (i.e. for x -directed or z -directed IDpairs) are identical, and can be expressed as: G xx,n = G zz,n = (cid:16) n (cid:17) G yy,n . (25)where G yy,n = W n . These analytical results and the various details about thebehaviour of various terms indexed by n will be fully exploitedin what follows to build efficient and robust cross-correlationcomputation algorithms for massive MIMO. D. Analytical CGF Estimation for Two-dimensional PlanarDipole Arrays
In the last section, we considered that placement of IDs isrestricted along y -axis, i.e. x d = z d = 0 , which significantlysimplified the scenario. Next, let us take the case of a two-dimensional/planar array of IDs placed in the yz -plane. Sincehere x d = 0 , we have r d = y d sin θ sin φ + z d cos θ . Bydeploying the binomial series to expand r d and following somealgebraic manipulations, we get: ( r d ) n = ( y d sin θ sin φ + z d cos θ ) n = n (cid:88) m =0 n ! m !( n − m )! y md z n − md sin m θ cos n − m θ sin m φ. (26)Therefore, following (10), the expression for I pq,n becomes: I pq,n = 1 n ! n (cid:88) m =0 n ! m !( n − m )! V npq,m y md z n − md , (27)where, V npq,m = (cid:90) π (cid:90) π f pq sin m θ cos n − m θ sin m φdθdφ. (28)To solve for V npq,m (which will finally lead to I pq,n ) we need tocarefully choose f pq expressions from (6) and apply suitableproperties of Beta and Gamma functions. We demonstrate thesolution for I zz,n here. V nzz,m = (cid:20)(cid:90) π sin m +3 θ cos n − m θdθ (cid:21) (cid:20)(cid:90) π sin m φdφ (cid:21) (29)Note that, the integral with φ vanishes for odd values of m .Similar to the 1D case, we have for even m , we have: (cid:90) π sin m φdφ = 2 B (cid:18) , m (cid:19) = π m − m ! (cid:2) ( m )! (cid:3) . (30)Now, we consider the two scenarios of n . When n is odd with m being even , both the quantities m + 3 and n − m are odd .Therefore using the fact that cosine function cos θ is odd withrespect to π/ we have: (cid:90) π sin m +3 θ cos n − m θdθ = 0 . (31)Once again, we have I zz,n = 0 for odd values of n . On theother hand, when n is even with m also being even , m + 3 is odd while n − m is even . Therefore, (cid:90) π sin m +3 θ cos n − m θdθ = B (cid:18) n − m , m (cid:19) = Γ (cid:0) n − m + (cid:1) Γ (cid:0) m + 2 (cid:1) Γ (cid:0) n +42 + (cid:1) = (cid:16) m (cid:17) ! (cid:20) ( n − m )!2 n − m ( n − m )! (cid:21) (cid:20) n +4 ( n + 2)!( n + 4)! (cid:21) . (32) Using (32) and (30) in (28), we obtain: V nzz,m = 2 π (cid:20) m !( n − m )! n ! (cid:21) (cid:20) ( n )!( n − m )!( m )! (cid:21) × (cid:34) (cid:0) m + 1 (cid:1) (cid:0) n + 2 (cid:1) (cid:0) n + 1 (cid:1) ( n + 4)( n + 3)( n + 2)( n + 1) (cid:35) = 8 π (cid:0) m + 1 (cid:1) ( n + 3)( n + 1) (cid:20) m !( n − m )! n ! (cid:21) (cid:20) ( n )!( n − m )!( m )! (cid:21) . (33)When V nzz,m is substituted in (27) and the expression for W n is recognized from (22), the expression for I zz,n becomes: I zz,n = W n n ! n (cid:88) m =0 (cid:34) (cid:0) m + 1 (cid:1) ( n )!( n − m )!( m )! (cid:35) y md z n − md , (34)At this point, we notice that I zz,n is actually a sum of twoseries-summations as follows: n (cid:88) m =0 (cid:20) ( n )!( n − m )!( m )! (cid:21) y md z n − md = ( y d + z d ) n , (35) n (cid:88) m =0 (cid:20) ( n )!( n − m )!( m − (cid:21) y md z n − md = y d (cid:16) n (cid:17) n (cid:88) m =0 (cid:20) ( n − n − m )!( m − (cid:21) y m − d z n − md = n y d ( y d + z d ) n − . (36)After substituting these series summation values in (34), thefinal expression for I zz,n reduces to: I zz,n = W n n ! (cid:0) y d + z d (cid:1) ( n − ) (cid:104)(cid:16) n (cid:17) y d + z d (cid:105) . (37)In a similar fashion, the expressions for other I pq,n for even values of n can be derived as follows: I xx,n = W n n ! (cid:16) n (cid:17) (cid:0) y d + z d (cid:1) ( n − , (38) I yy,n = W n n ! (cid:0) y d + z d (cid:1) ( n − ) (cid:104) y d + (cid:16) n (cid:17) z d (cid:105) , (39) I yz,n = I zy,n = − W n n ! (cid:16) n (cid:17) ( y d z d ) (cid:0) y d + z d (cid:1) ( n − ) (40) I xy,n = I yx,n = I xz,n = I zx,n = 0 . (41)Note that, the condition I pq,n = 0 for odd values of n holds true. Furthermore, (41) suggests that the IDs orientedorthogonal to the plane of arrangement (i.e. x -directed IDs)do not have any correlation with the IDs oriented along theplane of arrangement (i.e. y -directed or z -directed IDs). E. Analytical CGF Estimation for Three-Dimensional DipoleArrays: Generalized Case
Finally we consider the most general case of three-dimensional arrays with no restrictions imposed on the dipolelocations, i.e. in general, x d (cid:54) = y d (cid:54) = z d (cid:54) = 0 . Here, it turnsout we have to deal with a trinomial expansion or successivebinomial expansions of ( r d ) n where r d = x d sin θ cos φ + y d sin θ sin φ + z d cos θ. (42) Therefore, the analytical integrations needed to evaluate I pq,n (see (10)) become slightly more complicated. Performingintegration both by-hand using the properties of Beta andGamma functions as before, we determine the followinggeneral formula for I pq,n for even values of n : I xx,n = W n n ! (cid:104) x d + (cid:16) n (cid:17) y d + (cid:16) n (cid:17) z d (cid:105) d n − , (43) I yy,n = W n n ! (cid:104)(cid:16) n (cid:17) x d + y d + (cid:16) n (cid:17) z d (cid:105) d n − , (44) I zz,n = W n n ! (cid:104)(cid:16) n (cid:17) x d + (cid:16) n (cid:17) y d + z d (cid:105) d n − , (45) I xy,n = I yx,n = − W n n ! (cid:104) nx d y d (cid:105) d n − , (46) I yz,n = I zy,n = − W n n ! (cid:104) ny d z d (cid:105) d n − , (47) I zx,n = I xz,n = − W n n ! (cid:104) nx d z d (cid:105) d n − , (48)where, d = (cid:113) x d + y d + z d (49)However the results are further validated by use of the sym-bolic toolbox in MATLAB (see appendix). It is observed thatfor odd values of n , I pq,n = 0 . Also note that, the expressionsfor one-dimensional and two-dimensional ID arrays can beeasily computed back from (43)-(48), substituting x d , y d and z d accordingly. Consequently, the results of this subsection (43) - (45) are the most general but we opted for presentingthe one- and two-dimensional cases for convenience since themathematical treatment is considerably more complex in threedimensional arrays while lower-dimensional MIMO systemstend to be more commonly encountered in practice.Now, similar to our approach for the linear array (or 1Dcase), it is crucial to predict the maximum number of terms N needed to truncate the series in (9). With that objective inmind, we start by carefully examining I xx , n , where we notethat for n > : I xx,n = W n n ! (cid:104) x d + (cid:16) n (cid:17) y d + (cid:16) n (cid:17) z d (cid:105) d n − < W n n ! (cid:16) n (cid:17) (cid:2) x d + y d + z d (cid:3) d n − . (50)Using d from (49), following the same procedure for all all p = x, y, z , and applying the Stirling’s approximation (14),the upper-bound of I pp,n can be expressed as: I pp,n | U.B. = W n (cid:16) n (cid:17) d n n ! = W n (cid:18) n + 22 √ πn (cid:19) (cid:18) epλn (cid:19) n , (51)where d = pλ . Therefore, when this I pp,n is used in (9)containing the ( jk ) n term where k = 2 π/λ , we would obtaina term (2 πep/n ) n , very much similar to (15). Therefore, itcan be deduced that the maximum N -value for the three-dimensional array situation also follows the same guidelinegiven by (16). III. C
ONCLUSION
The present paper analytically estimates the frequency-domain CGF tensor by employing an accurate series expan-sion, followed by detailed integration of functions involvingtrigonemetric expressions of elevation ( θ ) and azimuth ( φ )angles. Expressions for coefficients I pq,n are formulated forthe general three-dimensional case (see (43)-(45)), whichenables very efficient CGF-tensor calculation by simply us-ing the knowledge of inter-element spacing ( x d , y d and z d )between the ID-pairs, along with their relative polarizations.We systematically demonstrate the effect of number of terms N needed to truncate the series used for approximating theCGFs (see (9)), and proposed an effective equation (16) forestimating N from spacing between the IDs.A PPENDIX
While deriving the I pq,n expressions for the one/two/threedimensional cases, we encountered several definite integrations(over the full θ - φ space), involving arbitrary powers of trigono-metric functions sin θ , cos θ , sin φ and cos φ . To obtainedclosed form solutions of these integrations, we utilized thewell-known Beta functions B ( m, n ) having the mathematicalform [42] B ( m, n ) = (cid:90) z m − (1 − z ) n − dz. (52)The connection between the Beta functions and the integralsinvolving trigonometric functions is established using B ( m + 1 , n + 1) = 2 (cid:90) π cos m +1 θ sin n +1 θdθ. (53)Now, the analytical evaluation of Beta functions is performedby invoking the Gamma functions via the relation [42] B ( m, n ) = Γ( m )Γ( n )Γ( m + n ) , (54)where, Γ ( n + 1) = n ! and Γ (cid:18) n + 12 (cid:19) = (2 n )!2 n n ! √ π. (55)Next, we provide solutions for the four general classes ofdefinite integrals involving sin θ and cos θ , that are encoun-tered during the derivations. Class-I: Power of sin odd, Power of cos even: (cid:90) π sin m +1 θ cos n θdθ = B (cid:18) n + 12 , m + 1 (cid:19) . (56) (cid:90) π sin m +1 θ cos n θdθ = 0 . (57) Class-II: Power of sin odd, Power of cos odd: (cid:90) π sin m +1 θ cos n +1 θdθ = (cid:90) π sin m +1 θ cos n +1 θdθ = 0 . (58) Class-III: Power of sin even, Power of cos odd: (cid:90) π sin m θ cos n +1 θdθ = (cid:90) π sin m θ cos n +1 θdθ = 0 . (59) Class-IV: Power of sin even, Power of cos even: (cid:90) π sin m θ cos n θdθ = 12 (cid:90) π sin m θ cos n θdθ = B (cid:18) n + 12 , m + 12 (cid:19) . (60)R EFERENCES[1] J. R. Hampton,
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