Spatial Correlation from Multipath with 3D Power Distributions having Rotational Symmetry

Transcription

1 Spatia Correation from Mutipath with 3D Power Distributions having Rotationa Symmetry Rodney A. Kennedy, Zubair Khaid and Yibeta F. Aem Research Schoo of Engineering, Coege of Engineering and Computer Science, The Austraian Nationa University, Canberra, ACT 0200, Austraia Emai: Abstract In this paper we give a genera expression for the 3D spatia correation experienced between two sensors in 3D-space for the cass of normaized power distributions representing farfied mutipath sources having a rotationa symmetry about their mean direction axis. A genera expansion for the 3D spatia correation is presented and interpreted in terms of an associated eigenfunction equation. This enabes us to deveop cosed-form coefficient expressions for the spatia correation for a number of distributions such as the Gauss-Weierstrass kerne based distribution and the previous known resuts for the von Mises-Fisher power distribution. Anaytica resuts generated fuy account for the effect of varying in the reative orientation between the sensors in 3D and the power distribution mean direction which can be arbitrariy oriented. The resuts provide information on pacement of sensors to reduce correation effects. Index Terms wireess channe, spatia correation, mutipath, von Mises distribution, von Mises-Fisher distribution, Gauss- Weierstrass distribution, spherica harmonics. I. INTRODUCTION Genera expressions for the spatia correation between two sensors for arbitrary mutipath power distributions were given in [1] for both the 2D case which restricts sensors and mutipath to the horizonta pane and the unrestricted 3D case. The 3D case permits mutipath to arrive with different eevations and azimuths, and the sensors to be at different heights and orientations. In the 2D case, cosed-form expressions were given for the coefficients of number of mutipath distributions: uniform, uniform azimuth imited [2], power of cosine [3], von Mises [4], truncated gaussian [5], truncated Lapacian [6], etc., in a spatia correation expansion which converges rapidy. These 2D mutipath distributions have the property that they are symmetric about their mean direction of arriva. For other distributions one can resort to numerica integration methods to compute the spatia correation but there is a cear preference for anaytica resuts where the effects of parameters is reveaed. For the 3D case, which is our focus in this paper aso, a genera expression was given in [1] which identified the spherica harmonic coefficients of the mutipath power distribution with the coefficients used in an expansion for the spatia This work was supported under the Austraian Research Counci s Discovery Projects funding scheme project no. DP correation. Then to obtain a cosed-form coefficient spatia correation expansion one ony needs to find expressions for the power distribution which admit suitabe expansions in spherica harmonics or Legendre poynomias. In [1] two exampes were given of such a procedure, the first was the we-known omnidirectiona case which eads to the famous sinc spatia correation ρz 1 z 2 sin k z 1 z 2 k z 1 z 2 where k 2π/λ, λ is the waveength and z 1 z 2 is the eucidean distance between spatia points z 1 and z 2. Because of the zero crossings, this formua can be seen as a theoretica basis for the haf waveength, λ/2, sensor spacing rue-ofthumb. The other exampe given in [1] was for mutipath that was uniformy distributed over a imited range of eevations, which is somewhat contrived and unikey to have practica significance. More recenty, it was shown that the 3D von Mises-Fisher distribution admits a suitabe cosed-form expansion in spherica harmonics in the form of a recursion [7] and in cosed-form using haf-integer-order modified Besse functions of the first kind in [8]. This atter case is a direct 3D anaog of the 2D von Mises distribution case deveoped in [1], which uses integer-order modified Besse functions of the first kind. The von Mises-Fisher distribution has rotationa symmetry about its mean direction which can have both azimuth and eevation components. This rotationa symmetry is the key attribute that permits simpification and we expoit this to deveop resuts for distributions more genera than the von Mises-Fisher distribution. Simiary we make no restriction on the pacement of the sensors in 3D space. It is this type of distribution which we target so as to deveop anaytica resuts where the effect of various parameters can be reveaed. In this paper, we deveop a genera expression for the spatia correation for 3D distributions having rotationa symmetry about some axis incuding as a specia case the von Mises- Fisher distribution. Firsty, the genera expression reveas features and therefore insights in common to a cases. Then for a number of specia cases, refecting distributions more reaistic in nature, we provide cosed-form expressions for the /13/$ Crown

2 spatia correation from which it is possibe to characterize convergence. II. ROTATIONALLY SYMMETRIC DISTRIBUTIONS A distribution f x; µ on the 2-sphere, denoted, with random unit vector x with rotationa symmetry about some axis µ, see Fig. 1, can be written in terms of a rea-vaued univariate function as foows fz, defined on z [, +1] f x; µ f x µ. 1 If η denotes the north poe, then a symmetric function f x; η f x η is caed azimuthay symmetric. The requirements to be a vaid distribution with equivaent conditions on the univariate function are: and non-negativity: f x; µ 0, x fz 0, z [, +1]. 2 and normaization f x; µ ds x 1 2π +1 fz dz 1 3 where ds x sin θ dθ dϕ is the uniform surface measure on the 2-sphere see Appendix A. The equivaence in 3 can be proven with change of variabes and the substitution z x µ. Exampe: The von Mises-Fisher distribution on 2-sphere can be defined in terms of the univariate function κ expκ z f κ z, z [, +1], 4 sinh κ eading to f x; µ, κ f κ x µ κ expκ x µ sinh κ where µ is the mean direction, κ 0 is the concentration parameter, x is a random unit vector on the 2-sphere, and the remaining terms serve to normaize the distribution. III. SPHERICAL HARMONIC EXPANSION Appendix A gives the definition of spherica harmonics, inner product, and reated concepts used beow. The objective wi be to provide the spherica harmonic coefficients and spherica harmonic expansion of a rotationay symmetric function on the 2-sphere with axis of symmetry µ, and this task is simpified by the addition theorem of the spherica harmonics [9], [10] m 5 xy m µ P x µ. 6 So we expand the univariate function f in terms of Legendre poynomias with adjustments to directy expoit 6 and simpify ater deveopment: fz λ P z, 7 0 µ x η ϕ z θ x xr, θ, ϕ R3 r x x xθ, ϕ Fig. 1: Coordinate system showing a point in 3D space x, a unit vector x, which ies on the 2-sphere,, and the spherica poar coordinate system with θ [0, π] the co-atitude, ϕ [0, 2π the ongitude and r x the radius. where the eigenvaues are the terminoogy wi be justified ater λ 2π +1 fzp z dz. 8 Since fz is rea-vaued then λ is aso rea-vaued but can positive, negative or zero. With 0 in 8 then we recover 3. This means the normaization is equivaent to λ 0 1. So combining 6, 8 into 7 we can infer the spherica harmonic expansion of any symmetric distribution f x; µ f x µ λ µ x, 0 m which is of the genera form of a spherica harmonic expansion f x; µ Y m x, 9 0 m f µ m where the spherica harmonic coefficients are f ; µ, Y m λ µ. 10 f µ m So in summary, given a distribution f x; µ with axis of symmetry in some direction µ then the formua to determine the spherica harmonic coefficients is 10 which requires us to compute the eigenvaues 8 from the associated univariate function fz. A. Properties and Exampes Exampe 1: von Mises-Fisher Distribution: The eigenvaues for the von Mises-Fisher distribution are λ κ 2π +1 f κ zp z dz. 11 where the univariate function is given in 4. This expression can be computed recursivey as shown in [7] λ +1 κ λ κ λ κ, 1, 2,.... κ y

3 with initia vaues λ 0 κ 1 and λ 1 κ coth κ 1/κ, eading to { λ κ } { 0 1, coth κ 1 κ, κ2 3κ coth κ + 3 } κ 2,.... Further, it can be estabished that λ κ > 0, for a 0, 1, 2,..., and for a κ 0 [10, p.394]. In [8], [11], [12] it was shown that these coefficients can aso be expressed as λ κ I +1/2κ πκ I +1/2 κ I 1/2 κ 2 sinh κ where I +1/2 is a haf-integer-order modified Besse function of the first kind. With the above, the spherica harmonic coefficients are f µ,κ m f ; µ, κ, Y m I +1/2 κ I 1/2 κ and the spherica harmonic expansion is κ expκ x µ I +1/2 κ sinh κ I 1/2 κ 0 m Y m µ, 12 µ x. Exampe 2: Azimuthay Symmetric Distribution: Aigning the axis of symmetry with the z-axis or north poe means setting µ η [0, 0, 1]. Spherica harmonics for m 0 aso have rotationa symmetry about the z-axis and these are the ony terms that are required in the spherica harmonic expansion as we now show. We use the simpe identity using the definitions given in Appendix A: Then f η m η f ; η, δ m,0. λ η λ and so the spherica harmonic expansion is f x; η f x η Y m x 0 m 0 m 0 f η m λ λ Y 0 x δ m,0 δ m,0 Y m x which can aso be expressed in terms of Legendre poynomias equivaent to 7 with z x η. Exampe 3: Gauss-Weierstrass Distribution: A specia case of this azimuthay symmetric distribution is the Gauss- Weierstrass kerne [13] with concentration parameter κ f x; η, κ e +1/2κ Y 0 x 0 where we see λ κ e +1/2κ and this is approximates the von Mises-Fisher distribution with mean direction η in the imit κ 1, see [7]. If we use a genera mean direction µ, instead of η, the rotated Gauss-Weierstrass kerne becomes f x; µ, κ e +1/2κ µ x. 0 m The Gauss-Weierstrass kerne spatia expression does not simpify to a known cosed-form spatia expression. However, it does have a simpe spectra or eigenvaue characterization. Spatiay they are unimoda with adjustabe width, and when they are narrower they get asymptoticay coser to the von Mises-Fisher distribution [7]. B. Integra Equation, Eigenvaues and Eigenfunctions The underying eigen-structure of the probem under study expains why rotationay symmetric functions ead to an eegant soution and aso expain why it is unikey the methods can be easiy extended to more genera asymmetric functions. Theorem 1: Define the integra operator F as foows Fh x F x, ŷhŷ dsŷ with kerne F x, ŷ f x; ŷ λ x ŷ. 13 Then 0 m FY m x λ Y m x 14 which reveas the λ, 8, are the eigenvaues corresponding to eigenfunctions Y m x of integra operator F. Further, with the finite energy condition +1 f 2 z dz <, 15 the operator F is compact and sef adjoint. Proof: From 10, f ; µ, Y m λ Y m µ, is the same as fŷ; µy mŷ dsŷ λ Y m µ. Then from the definitions given in Appendix A x m Y m x, and from symmetry fŷ; µ f µ; ŷ we get 14. Sef-adjointness of operator F foows from Hermitian symmetry, due to f being rea-vaued, fŷ; µ fŷ; µ f µ; ŷ F F. Then from [10, p.357], the finite energy condition 15 can be shown to be equivaent to finite energy sum square of the eigenvaues summed with mutipicity by the Parseva reation for Legendre poynomias. This impies the kerne 13 is Hibert-Schmidt and, therefore, operator F is compact. This justifies seeking a spherica harmonic representation for rotationay symmetric distributions f x; µ in the first pace, Section III, because they are eigenfunctions. Simiary, making the coefficients identica to the eigenvaues makes the deveopment cogent.

4 IV. SPATIAL CORRELATION A. Rotationay Symmetric Distribution Case We review the deveopment in [1]. The spatia correation between two narrowband fixed wavenumber k compexvaued signas s 1 t, s 2 t at points z 1, z 2 R 3 is given by ρz 1, z 2 E{s 1t s 2 t} E{s 1 t s 1 t} 16 where E{ } denotes expectation over the random compex gains from the mutipath scattering [1]. Then with a power distribution f x; µ, normaized according to 3, representing the average power density as a function of direction x in the farfied of the two points z 1, z 2 R 3 then the spatia correation, 16, becomes spatiay wide-sense stationary in the form ρz 1 z 2 ; µ f x; µe ikz1 z2 x ds x. 17 The wide-sense stationarity does not depend on the rotationa symmetry of fŷ; µ, and the above resut is quite genera. Then by empoying a spherica harmonic expansion of pane waves of the continuous superposition in 17 one arrives at ρz 1 z 2 ; µ i j k z1 z 2 0 m f µ m Y m z1 z 2, 18 z 1 z 2 where f µ m are the spherica harmonic coefficients of the distribution f ; µ, given in 10. B. Spatia Correation Expansion We specify the resuts in [1] for 3D scattering exhibiting rotationa symmetry about some axis of symmetry µ expressed in the context of the eigen-probem. We gather our main theoretica findings into a sef-contained statement. Theorem 2: Let the mutipath be defined by the normaized power distribution with rotationa symmetry f x; µ f x µ where x is the direction, µ is the mean direction, and f is a non-negative, rea univariate function with domain [, +1]. Then the spatia correation between points z 1, z 2 R 3 depends ony on z z 1 z 2, is spatiay widesense stationary, and is given by ρz; µ j i 1 λ P ẑ µ j, 19 where z R 3, ẑ z/ z, z is the Eucidean distance, λ 2π +1 fzp z dz, 0, 1,..., 8 and fz λ P z, 7 0 noting that λ 0 1. Proof: Using 10 and the addition theorem 6, 18 becomes ρz; µ i j λ µ ẑ, 0 m i λ P ẑ µ j. 0 which can be written as in 19 which separates out the 0 term. Comments 1 Note that the 0 term is a spherica Besse function equivaent to a sinc function: sin j 0 λ sin 2π z /λ. 2π z This term, common to a spatia correation expansions because the power distribution is nonnegative there is aways a positive DC term which when normaized is unity, is the 3D omnidirectiona contribution to spatia correation, corresponding to λ 0 1, and has zero crossings at λ/2 spacings except at zero where the correation is unity as shown in Fig. 2. This is equivaent to the von Mises-Fisher distribution case with κ 0. 2 The reative orientation of the vector joining the two sensors and the mean direction axis of rotationa symmetry of the power distribution is ceary deineated in the expression 19 with the appearance of the term ẑ µ. This is the ony way that µ enters the formua. 3 The shape of the power distribution, 7, is defined by the λ which are given by 8. C. von Mises-Fisher Distribution For f x; µ, κ in 5 the univariate functions is given by f µ z in 4 which has eigenvaues λ κ given by 11. Then the spatia correation 19 is given by ρz; µ, κ 2+1 i I +1/2κ I 1/2 κ P ẑ µ j This expression can be compared with the structuray simiar f x; µ, κ I +1/2κ I 1/2 κ P x µ, 21 0 which foows from 1, 7. This is potted in Fig. 3, in the form f κ cos θ, for a range of κ. Corresponding to the highighted cases in Fig. 3 we have the spatia correation 20 potted in Fig. 4, for κ 1, 2, 4, 8 and Fig. 2 for κ 0. The tota number of terms in the expansion required to yied the accuracy shown in the pots was L 1, 6, 7, 10, 13, 18, meaning the sum is over 0, 1,..., L,

5 spatia correation ρz; µ, κ Fig. 2: Magnitude of the spatia correation ρz; µ, κ 0 for the von Mises-Fisher distribution with concentration parameter κ 0, which is the 3D omnidirectiona case. This correation is independent of ẑ µ and where z is the vector between the two sensors and µ is the mean direction. univariate distribution function fκcos θ κ 16 κ 8 κ 4 κ 1 κ π/4 π/2 3π/4 π ange θ cos ẑ µ Fig. 3: Von Mises-Fisher univariate distribution functions, f κ cos θ for κ 0, 1, 4, 8, 16 intermediate vaues increment by 1 potted against θ cos ẑ µ. for κ 0, 1, 2, 4, 8, 16, respectivey. The pots show the effect due to the effective width of the distribution around the mean direction. Naturay as κ gets smaer the width gets arger and vice versa. Aso by varying the ẑ µ we see the fu effect of varying in the reative orientation between the sensors z z 1 z 2 and the distribution mean direction µ. These resuts are fuy anaytica and not simuated which woud be very time consuming to generate accurate data. They are quantitative and fuy consistent with quaitative expectations. Note that the term j, which enters the spatia correation expression 20 but is absent in the distribution expansion 21, serves to speed convergence in the foowing sense which reies on a resut deveoped for the Besse functions in [14], [15]. For the spherica Besse functions the resut says that for πe z /λ the j contribution is very sma effectivey truncating the series [16]. However, the remainder of the series coefficient aso becomes smaer as and an effective truncation can have few terms than that determined by j aone. D. Lebedev Distribution Non-negative rea-vaued functions on [, +1] which have a simpe Legendre poynomia expansion ead to cosed-form 3D distributions and thereby cosed-form spatia correation expansions. An exampe is modifying a function given in [17], and [18, ], and deveoped in [19] fz; µ, η η η η 12π 8π 1 1 x µ 2 P ẑ µ, η [0, 6], which, therefore, has a spatia correation expansion ρz; µ, η j 0 + η 1 i P ẑ µ j where η [0, 6] ensures the distribution is non-negative. Further cosed-form spatia correation expansions can be inferred from the resuts in [19] as we discuss in the concusions. V. CONCLUSIONS Prior to this paper the von Mises distribution had been presented as a versatie correation function in the 2D context [4] aong with other distributions that aso had a mirror symmetry property [5], [6]. In the 3D case, cosed-form spatia correation function having a power distribution with rotationa symmetry property were imited to two simpe cases considered in [1] and the von Mises-Fisher distribution in [8]. This paper has given a genera expansion for the 3D spatia correation for the cass of normaized power distributions representing farfied mutipath sources having a rotationa symmetry about their mean direction. The main theoretica resut, Theorem 2, aso gives good insight into the interpay between the positioning of the sensors and the distribution mean direction. In Theorem 1, we have fuy accounted for the spherica harmonic eigenfunction and eigenvaue structure of the spatia correation probem for power distributions with rotationa symmetry property. Finding other suitabe univariate functions with cosedform coefficient expansions has an interesting reationship with finding cosed-form reproducing kernes on the 2-sphere [19], [20]. Functions which work as the basis for constructing cosed-form reproducing kernes aso mosty work as univariate functions for the purpose of defining cosed-form mutipath power distributions. We gave a few exampes in this paper but others given in [19] aso work as aternatives. In the current probem it is the spatia function that needs to be non-negative as it represents a power/probabiity distribution but the spectra description expressed through the eigenvaues can be positive or negative. In contrast, for the reproducing

6 ẑ µ1 spatia correation ρz; µ, κ 1 10 ẑ µ0 ẑ µ1 spatia correation ρz; µ, κ 4 10 ẑ µ0 a κ 1 b κ ẑ µ ẑ µ1 spatia correation ρz; µ, κ 8 10 ẑ µ0 spatia correation ρz; µ, κ ẑ µ0 c κ 8 d κ 16 Fig. 4: Magnitude of the spatia correation ρz; µ, κ for the von Mises-Fisher distribution with concentration parameter κ and for a range of vaues of ẑ µ {0, 0.1, 0.2,..., 1.0} and where z is the vector between the two sensors and µ is the mean direction. The abeed curves are broadside: ẑ µ 0 and end-fire: ẑ µ 1. kerne case the eigenvaues need to be positive and the spatia function, being the kerne itsef, need not be a positive function but predominanty so. The effectiveness of synthesizing nonunimoda power distributions [8] from a reproducing kerne perspective is yet to be determined. APPENDIX A SPHERICAL HARMONICS Define the 2-sphere by {x R 3 : x 1} and the inner product of two functions whose domain is the 2-sphere f, g f xg x ds x, 22 where x sin θ cos ϕ, sin θ sin ϕ, cos θ R 3 and ds x sin θ dθ dϕ is the uniform surface measure satisfying ds x. Finite energy functions are those that satisfy the bounded induced norm condition f L 2 f f, f 1/2 <. In this work we require the spherica harmonics. They are defined through Y m m! θ, ϕ + m! P m cos θe imϕ Y m x, where {0, 1,...} is the degree, m {, + 1,..., } is the order, the associated Legendre functions are P m and satisfy z m 2 1 z 2 m/2 d+m! dz +m z2 1, m {0, 1,..., } P m m m! z + m! P m z, m {0, 1,..., }, which enabes the determination of the spherica harmonics for m < 0. For m 0 the associated Legendre functions reduce to the standard Legendre poynomias which are denoted P z.

7 For each f L 2 we have expansion in the spherica harmonics f x f m Y m x, 0 m where the spherica harmonic coefficients are f m f,. The equaity in the expansion is understood in the sense of convergence in the mean [10]. REFERENCES [1] P. D. Tea, T. D. Abhayapaa, and R. A. Kennedy, Spatia correation for genera distributions of scatterers, IEEE Signa Process. Lett., vo. 9, no. 10, pp , Oct [2] 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 , Nov [3] R. G. Vaughan, Pattern transation and rotation in uncorreated source distributions for mutipe beam antenna design, IEEE Trans. Antennas Propag., vo. 46, no. 7, pp , Ju [4] A. Abdi and M. Kaveh, A versatie spatio-tempora correation function for mobie fading channes with non-isotropic scattering, in Proc. Tenth IEEE Workshop on Statistica Signa and Array Processing, Pocono Manor, PA, Aug. 2000, pp [5] M. Kakan and R. H. Carke, Prediction of the space-frequency correation function for base station diversity reception, IEEE Trans. Veh. Techno., vo. 46, no. 1, pp , Feb [6] K. I. Pedersen, P. E. Mogensen, and B. H. Feury, Power azimuth spectrum in outdoor environments, Eectron. Lett., vo. 33, no. 18, pp , [7] K.-I. Seon, Smoothing of a-sky survey map with Fisher-von Mises function, J. Korean Phys. Soc., vo. 48, no. 3, pp , Mar [8] K. Mammasis and R. W. Stewart, Spherica statistics and spatia correation for mutieement antenna systems, EURASIP J. Wire. Commun., vo. 2010, Dec. 2010, artice ID [9] D. Coton and R. Kress, Inverse Acoustic and Eectromagnetic Scattering Theory, 3rd ed. New York, NY: Springer, [10] R. A. Kennedy and P. Sadeghi, Hibert Space Methods in Signa Processing. Cambridge, UK: Cambridge University Press, Mar [11] P. H. Leong, T. A. Lamahewa, and T. D. Abhayapaa, Framework to cacuate eve-crossing rate and average fade duration in two-dimensiona and three-dimensiona scattering environments, IET Commun., vo. 6, no. 15, pp , [12] M. Abramowitz and I. Stegun, Handbook of Mathematica Functions. New York, NY: Dover Pubishing Inc., [13] T. Büow, Spherica diffusion for 3D surface smoothing, IEEE Trans. Pattern Ana. Mach. Inte., vo. 26, no. 12, pp , Dec [14] H. M. Jones, R. A. Kennedy, and T. D. Abhayapaa, On dimensionaity of mutipath fieds: Spatia extent and richness, in Proc. IEEE Int. Conf. Acoustics, Speech, and Signa Processing, ICASSP 2002, vo. 3, Orando, FL, May 2002, pp [15] R. A. Kennedy, P. Sadeghi, T. D. Abhayapaa, and H. M. Jones, Intrinsic imits of dimensionaity and richness in random mutipath fieds, IEEE Trans. Signa Process., vo. 55, no. 6, pp , Jun [16] W. Zhang, T. D. Abhayapaa, R. A. Kennedy, and R. Duraiswami, Insights into head reated transfer function: Spatia dimensionaity and continuous representation, J. Acoust. Soc. Am., vo. 127, no. 4, pp , Apr [17] N. N. Lebedev, Specia Functions and Their Appications. New York, NY: Dover Pubications, [18] I. Gradshteyn and I. Ryzhik, Tabe of Integras, Series, and Products, 7th ed. Academic Press, [19] R. A. Kennedy, P. Sadeghi, Z. Khaid, and J. D. McEwen, Cassification and construction of cosed-form kernes for signa representation on the 2-sphere, in Proc. SPIE 8858, Waveets and Sparsity XV, San Diego, CA, Aug [20] J. Cui and W. Freeden, Equidistribution on the sphere, SIAM J. Sci. Comput., vo. 18, no. 2, pp , Mar