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Geometrical Modeling of Scattering Environment for Highways in Umbrella Cell Based MIMO Communication Systems M. Yaqoob Wani, M. Riaz & Noor M. Khan Wireless Personal Communications An International Journal ISSN 0929-6212 Wireless Pers Commun DOI 10.1007/s11277-018-5666-9 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science+Business Media, LLC, part of Springer Nature. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to selfarchive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Wireless Pers Commun https://doi.org/10.1007/s11277-018-5666-9 Geometrical Modeling of Scattering Environment for Highways in Umbrella Cell Based MIMO Communication Systems M. Yaqoob Wani1 • M. Riaz2 • Noor M. Khan3 Ó Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract In this paper, we develop a three-dimensional (3D) eccentricity-based cylindrical geometrical channel model for nonisotropic multiple-input-multiple-output (MIMO) communication systems under umbrella macrocellular environment. We use elliptic cylindrical geometry to model the scattering phenomenon in streets, canyons and highways. The scattering objects like, high-rise building, trees and vegetation that lie along the roadside premises are modeled by the height of an elliptical cylinder. The proposed channel model targets fast moving vehicles on the highways in an umbrella-cell of cellular communication networks. We assume that both ends of the communication link are equipped with multiple antenna arrays, where, mobile-station antenna height is lower than base-station antenna. Utilizing the proposed MIMO communication channel model, we obtain closed-form expressions for the space–time correlation function among the MIMO antenna elements. The obtained theoretical expressions are plotted and analyzed for different values of channel parameters. Finally, we compare the proposed model with the existing models in the literature and prove that our model can be deduced to the existing two-dimensional and 3D channel models. & M. Yaqoob Wani [email protected] M. Riaz [email protected] Noor M. Khan [email protected] 1 Department of Computer Science and Information Technology (CS&IT), The University of Lahore, Islamabad Campus, Islamabad, Pakistan 2 Department of Electronics and Electrical Systems, The University of Lahore, Islamabad Campus, 6.7 Japan Road, Zone-V, Islamabad 44000, Pakistan 3 Department of Electrical Engineering, Acme Center for Research in Wireless Communications (ARWiC), Capital University of Science and Technology, Kahuta Road, Zone-V, Islamabad 44000, Pakistan 123 Author's personal copy M. Y. Wani et al. Keywords Umbrella cell Correlation function MIMO capacity Channel modeling Cylinder 1 Introduction Wireless communications has shown tremendous growth in the last few decades due to its potential for facilitating communication links between machines, robots, aircrafts, ships and automobiles, etc. One of the major fields of wireless communication, which expanded exponentially in the last two decades, is cellular mobile communications. It is one of the apogee applications of wireless communication systems, where a stagnant BS with a high rise antenna mounted on the top of a structure communicates with the MSs having low elevated antennas located within its vicinity [32]. Because of the prodigious applications of cellular mobile communication, there has been seen a surge expansion in terms of number of subscribers in the last two decade. This exponential increase in the number of mobile subscribers demanding discriminate features like multimedia applications, live video streaming, internet access and other data-hungry value-added applications, forced the system to increase its spectral efficiency, data rate and link performance. This triggered a gigantic need to increase the throughput with improved quality of service (QoS) and large cellular coverage. In cellular mobile communication systems, high capacity with limited frequency spectrum can be achieved by sub-dividing larger cells in to the smaller ones like microcells, picocells, or femtocells [33, 43]. In such a communication system, mobile subscribers access the system using various multiple access techniques like frequency-division multiple access (FDMA), time-division multiple access (TDMA), code-division multiple access (CDMA) and orthogonal frequency-division multiple access (OFDMA) [43, 45]. A mobile subscriber when crosses the boundary of its serving cell and enters into an neighboring cell during progressing call, hand-off takes place [4, 23, 43]. The smaller size of the cell architecture creates most crucial dilemma in the cellular mobile communication, i.e, it increases the occurrence rate of handoff [48]. These frequent handoffs become an imbroglio situation for speedy vehicles on highways across the microcells [34]. To overcome this frequent problem, fast moving mobile vehicles are handed over to an umbrella cell as shown in Fig. 1. In [10, 11, 13, 14, 24, 31, 36, 47], numerous approaches have been presented by the theorists and researchers to handover the fast-moving vehicles to an umbrella cell. In an umbrella cell, a dedicated BS with high powered antenna mounted on high rise structure is used to cover the large area along the highway to serve these high speed mobile users. Therefore, for the beneficial design of umbrella cell, depth knowledge of the scattering Fig. 1 Umbrella-based cellular communication system 123 Author's personal copy Geometrical Modeling of Scattering Environment for Umbrella Cell... channel between the high rise BS and speedy mobile vehicle is extremely important. In this regard, various approaches like deterministic, stochastic and geometrical-based channel modeling approaches have been published in the literature to analyze the statistics of propagation channel [6, 46]. Geometrically-based channel modeling approach is used to model an umbrella cell for the fast-moving vehicles on highways. In this regard, various regular shaped geometrical channel models for SISO F2M environments have been provided in [5, 7, 15, 25, 28, 38, 41]. These traditional SISO systems are venerable to multipath fading effects therefore, they can not meet the QoS and high date-rate requirements in the rich scattering environments. In contrast to it, theoretically and empirically MIMO antenna architecture have been proved to be the promising candidate that can provide huge throughput in F2M and mobile-to-mobile (M2M) communication environments [20, 22]. However, the high correlation among the antenna array elements is an important practical factor that can degrade the performance of MIMO systems [8, 20]. Therefore, a geometrical channel model is required that can be used to predict the performance of MIMO F2M communication systems in such propagation environments. Various 2D [1, 9, 18] and 3D [26, 37] geometrical channels models have been recommended for F2M communication environments utilizing multiple antenna arrays at each end of the communication link. The authors analyzed the models for various channel characteristic like angle-of-arrival (AoA), time-of-arrival (ToA), Doppler spread and crosscorrelations among the MIMO links. These channels may be applicable in some scenario of mobile communication, however, these models are not appropriate to model umbrella cell scattering environment along highways. An elliptical geometrical shape is more suitable to model the scattering environment along the highways. To accommodate the high-rise scattering objects along the highways like buildings, mountains, and trees, are modeled using the elliptical-based cylindrical shape. The elevated walls of the cylinder represent these elevated scatterers. In this research article, we suggest an eccentricity-based geometrical channel model that represent an umbrella cell in macrocellular environment. We assume that signal propagation takes place in azimuth as well as in elevation planes. To attain higher data rate, multiple antenna array system is installed on each end of the communication link. Using the proposed model, we derive expressions for joint and marginal correlation functions among MIMO antenna elements. The rest of the article is as follows: Sect. 2, describes the methodology to obtain the mathematical expressions of correlations functions among the MIMO links. Section 3, system model of the proposed umbrella cell is described. The derivation of system model is given in Sect. 4. Expressions space–time correlation functions is provided in Sect. 5. The discussion of the theoretical results are presented in Sect. 6, and lastly, outcome of the research article is given in Sect. 7. 2 Research Methodology In this research article, we present and derive the mathematical expressions of the space– time correlation functions among multiple antenna elements installed on the BS and MS. The proposed 3D elliptical-based cylindrical channel model portrays an umbrella cell of the macro-cellular mobile communication environment that accommodates the fast moving vehicles on highways. It is assumed the MS is surrounded by scattering objects lying in azimuth and elevation planes in an elliptical-based cylindrical manner. Signals originated from any antenna array element of the BS propagate towards MS in both horizontal and vertical planes. The wavelets are reflected, diffracted and scattered from the scattering 123 Author's personal copy M. Y. Wani et al. objects located in the vicinity of the MS in haphazard manner resulting in multipath propagation. The multipath signals with different amplitudes and phases superimposed constructively or destructively at the receive antenna array elements that result in formation of complex envelope. The statistics of received signal at each antenna element depends on the distribution of scattering objects, transmit and receive correlations among array elements [17]. Using the geometry of the proposed system model depicted in Fig. 2, expression of the diffused components (5) is obtained with the help of some valid assumptions and binomial approximations. The diffused components are further used for the formulation of the joint space–time correlation function (12). Finally, the obtained correlation expressions for transmit, receive and joint correlation functions are simulated and plots are generated in Matlab for various channel parameters. 3 System Model This section presents a detailed description of the system model of the proposed fixed-tomobile (F2M) MIMO elliptical-based cylindrical geometrical channel model for umbrellacell as shown in Fig. 2. In this proposed, channel model the mobile subscriber is supposed At(p) z ψt(p) dnp z δt x y At(p) dnp dqn Ar(q) βr(n) ψr(q) Sr(n) ht dqn δr Ar(q) hr At(p) vr Ar(p) ar γr br π−θr (q) αr(n) x π−θt(p) Ar(p) D At(p) y Fig. 2 The proposed elliptical-based cylindrical channel model for umbrella cell in macrocellular communication environment 123 Author's personal copy Geometrical Modeling of Scattering Environment for Umbrella Cell... to be located at the center of elliptical cylinder holding low elevated antenna array and the base BS is located on the top of tower of height ht , fixed with multiple antenna array structure. The eccentricity r models the azimuth dimensions of the elliptical scattering region around the MS; whereas, the height hr of the elliptical cylinder models its elevation. The BS and MS are equipped with P and Q number of antenna array elements respectively, for the sake of simplicity, we take P ¼ Q ¼ 2, however, the results can be derived for any ðpÞ configuration. The antenna array elements of BS and MS are symbolized as At and AðqÞ r , and spacings between them are denoted by dt and dr , respectively. The MS is assumed to be located at (0, 0, 0) and BS (D, 0, 0), at an arbitrary time instant, in the Cartesian coordinate system, and the azimuth distance between the center of cylinder to base-station is denoted by D. Moreover it is also assumed that the MS is moving with the speed of vr in the direction of cr . The description of other variables used in the proposed geometry are listed in Table 1. 4 Derivation of Channel Parameters The derivation of the space–time correlation among MIMO antenna elements is based on ~ ðp;pÞ the reference system model depicted in Fig. 2. The multipath signals are arrived at At ~ qÞ from Aðq; via striking at the scattering point SrðnÞ that is situated at the surface of elliptical r ðpÞ cylinder. The propagation distance covered by the transmitted signal from At to AðpÞ r and ~ ðqÞ At ~ pq p~q~ to AðrqÞ , are denoted by dmax and dmax , respectively. The phase change due to the pq p~q~ distance traveled by the signal can be written as ð2p=kÞdmax and ð2p=kÞdmax , where 2p=k is called wavenumber. The gain and phase shift due to the collision of the signals with SðnÞ r Table 1 Notations of various channel parameters used in the system model Symbols Description Rr The dynamic radius of the receiver ellipses ðpÞ At , AðqÞ r Antenna array elements at BS and MS, respectively SðnÞ r Represents the n-th scatter located on the surface of elliptic cylinder dt The spacing between p-th and p~-th antenna elements at Tx dr The spacing between q-th and q~-th antenna elements at Rx ðpÞ hðqÞ r ; ht The azimuth angle of q-th receive and p-th transmit antenna element with respect to x-axis, respectively ðpÞ wðqÞ r ; wt The elevation angle of q-th receive and p-th transmit antenna element with respect to x–y plane, respectively vr Velocities of the MS cr The moving direction of the MS ðnÞ aðnÞ r ; br pq dmax p~q~ dmax AoA of multipath in azimuth and elevation plane respectively ar ; br Represents the major and minor axes of ellipse respectively r Represents the eccentricity of the ellipse surrounding the MS The distance between p-th antenna element to q-th antenna element, dqn ? dnp The distance between p~-th antenna element to q~-th antenna element, dqn ~ ? dnp~ 123 Author's personal copy M. Y. Wani et al. pﬃﬃﬃﬃ can be written as 1= N and xn , respectively. Phase change also occur due to the motion of ðnÞ MS and can be expressed as 2pfRmax cosðaðnÞ r cr Þ cos br t, where, fRmax ¼ vr =k represents the maximum Doppler frequency caused by the motion of receiver mobile node. It is also assumed that both antenna element spacing dt ; dr are much smaller than br . The AoA in azimuth plane and elevation plane are assumed to be random variable and independent of each other [3]. Moreover, xn is also supposed to be a random variable that is uniformly distributed over ½p; p and is also independent from AoA. Furthermore, for the ease of derivation of the space–time correlation among MIMO antenna arrays of the proposed channel model, the following valid assumptions are considered: 1. 2. The proposed eccentricity-based channel model is assumed to be a single-bounced. Infinity number of scattering points are uniformly distributed on the surface of the elliptical cylindrical geometry. Uniform power is reflected from all scatterers. The MS is surrounded by stationary scattering points. Each scatterer behaves as an omni-directional source. 3. 4. 5. Finally, with the help of above assumptions, the diffused components of the transmission ðpÞ link from At to AðqÞ r can be expressed as in [2], N 1 X 2p pq ej k dmax þjxn hpq ðtÞ ¼ lim pﬃﬃﬃﬃ N!1 N n¼1 e ðnÞ ð1Þ ðnÞ j2ptvkr cosðar cr Þ cos br ðpÞ pq ¼ dqn þ dnp , is the total distance traveled by the wave-vector from At to AðqÞ The dmax r . The distances dqn and dnp are obtained by solving the geometry of the proposed model as shown in Fig. 2. The 3D polar coordinates of qth and pth transmit receive antenna element ðqÞ ðqÞ ðqÞ antenna element are denoted by dRx ; dRy ; dRz ðqÞ ðpÞ ðpÞ ðpÞ and dTx ; dTy ; dTz , respectively. Where, ðqÞ ðqÞ ðpÞ ðqÞ ðqÞ ðqÞ ðqÞ dRx ¼ dr cos hðqÞ r cos wr ; dRy ¼ dr sin hr cos wr ; dRz ¼ dr sin wr ; dTx ¼ dt ðpÞ cos hðpÞ t ðpÞ ðpÞ ðpÞ ðpÞ cos wðpÞ . t ; dTy ¼ dt sin ht cos wt ; dTz ¼ dt sin wt ðn or nÞ ~ ðn or nÞ ~ ; dRy ; Similarly, the coordinates of nth scatterer in 3D space is denoted by dRx ~ ðn or nÞ dRz ~ or nÞ or nÞ ~ or nÞ ~ Þ, where dRðn or nÞ~ ¼ Rr cos aðn ; dRðn or nÞ~ ¼ Rr sin aðn ; dRðn or nÞ~ ¼ Rr tan bðn . t r r x z y These polar coordinates depend upon the orientation and configuration of the antenna arrays. Moreover, the dynamic radius Rr of the ellipse around the receiver can be written in terms of minor br and major ar axes is given in [42] as, qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2Þ Rr ¼ ar br = a2r sin2 ar þ b2r cos2 ar Using the distance formula and binomial approximation the approximated distances can be expressed as, dqn Rr cos bðnÞ r cos aðnÞ r 123 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 þ xÞ 1 þ x=2 if ðx 1Þ, ðnÞ ðqÞ ðqÞ dr sin wðqÞ r sin br dr cos hr cos wr cos bðnÞ r dr sin hðqÞ r cos wðqÞ r sin aðnÞ r cos bðnÞ r ð3Þ Author's personal copy Geometrical Modeling of Scattering Environment for Umbrella Cell... d2t h2 R2 sec2 bðnÞ dt Rr tan bðnÞ r r sin wt þ t þ r 2D 2D 2D D ðpÞ ðpÞ dt Rr cos aðnÞ ðpÞ ðnÞ r cos ht cos wt þ dt cos wðpÞ t cos ht Rr cos ar D ðpÞ ðpÞ ðpÞ ðnÞ ðnÞ ht Rr sin hðpÞ cos w h R sin a d R sin a t r t r r r sin ht cos wt t t þ D D D dnp D þ ð4Þ Analogously, the distances dqn ~ and dnp~ can be also obtained using the same geometry. Moreover, the constant phase shift caused by the time independent terms D; d2t =2D; h2t =2D can be set to zero without loss of generality. Furthermore, for the ease of understanding and avoiding complexity, Eq. (1) can be rewritten as, N ðnÞ ðnÞ 1 X Grt ej2ptfRmax cosðaR cr Þ cos br hpq ðtÞ lim pﬃﬃﬃﬃ N!1 N n¼1 ð5Þ 2p where Grt ¼ ej k ðdqn þdnp Þ . 5 Derivation of the Correlation function The normalized space–time correlation function between diffused channel coefficients hpq ðtÞ and hp~q~ðtÞ for the proposed 3D MIMO F2M non-isotropic scattering propagation environment can be obtained using the following relation given in [2, 51], Efhpq ðtÞhp~q~ðt þ sÞg Rpq;p~q~ðsÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Efjhpq ðtÞjg2 þ Efjhp~q~ðtÞjg2 ð6Þ where Efg is the statistical expectation operator and can be applied to all random variables, ðÞ symbolizes the complex conjugate operation and p; p~ 2 {1,...,P}, q; q~ 2 {1,...,Q}. Using Eqs. 5 and 6 the expression for the space–time correlation function can be written as, N ðnÞ ðnÞ 1X Grt Grt ej2psfRmax cosðaR cr Þ cos br N!1 N n¼1 Rpq;p~q~ðdr ; dt ; sÞ ¼ lim ð7Þ In F2M multipath propagation environments, the scattering objects present along the roadside premises are the sources of the energy efficient multipath. Therefore, the distribution of the AoA of multipath depends on the distribution of scattering objects that are lying in the close vicinity of MSs. For the ease of derivation of mathematical expression of space–time correlation function, it is assumed that infinity scattering objects are present around the MS, that implies the discrete scattering distributions may be transformed to a continuous scattering distributions that in turn forces us to change the discrete random ðnÞ variables aðnÞ r and br into continuous random variables ar and br , respectively. Furthermore, we assume that azimuth and elevation angles are independent of each other; therefore, the joint PDF f ðar ; br Þ of AoA at MS can be written in product form as f ðar Þf ðbr Þ, where, f ðar Þ and f ðbr Þ are the PDF’s of AoA in azimuth plane and elevation plane respectively. Therefore, the Eq. (7) can be written in integration form as, 123 Author's personal copy M. Y. Wani et al. Rpq;p~q~ðdr ; dt ; sÞ Z Z bRm bRm j2p k e p e2jpsðfTmax cosðar cr Þ cos br Þ p ðq;qÞ ~ dRx cos ar cos br þ ðp;pÞ ~ Ty D Rr d ~ ðp;pÞ Rr d Tx D cos ar e j2p k ~ ðp;pÞ sin ar þj2p k ~ ðq;qÞ dRz sin br þ Rr dTz ðq;qÞ ~ dRy sin ar cos br tan br D ð8Þ ! R2 sec2 b þ r 2D r e f ðar Þf ðbr Þdar dbr ~ ðq;qÞ where dRx ~ ðp;pÞ ðpÞ ðqÞ ~ ðqÞ ~ ðq;qÞ ðqÞ ¼ dRx dRx ; dRy ~ ðpÞ ~ ðp;pÞ ~ ðqÞ ~ ðq;qÞ ¼ dRy dRy ; dRz ðpÞ ðqÞ ~ ðqÞ ~ ðp;pÞ ¼ dRz dRz ; dTx ðpÞ ~ ðpÞ ¼ dTx dTx ; ~ ðpÞ dTy ¼ dTy dTy ; dTz ¼ dTz dTz ; and bRm is the maximum elevation AoA of the scatterers causes at the MS. To describe isotropic and non-isotropic scattering environments, various distributions have been proposed in the literature for AoA/AoD [2, 29, 44]. In urban or suburban areas the physical layouts of the streets, canyons and highways in the azimuth plane resemble elliptical shapes. Thus the roadside scatterers can be modeled nonisotropically distributed. Hence, eccentricity-based modified von Mises distribution proposed in [49], is more appropriate for such nonisotropic scattering environments to model azimuth AoA/AoD. The expression of PDF of azimuth AoA/AoD at MS is, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 pðar Þ ¼ 2pI01ð2 Þ er cos ar , where, r ¼ 1 b2r =a2r , the azimuth angle ar 2 ½p; p and I0 ðÞ r is the zeroth-order modified Bessel function of the first kind. Similarly, different scattering distributions have been also proposed in the literature for elevation AoA [2, 12]. In addition, Parsons et al. in [39] proposed PDF of elevation AoA f ðbr Þ ¼ 4bp cos rm 0 p br 2 brm , 0 where, the absolute values of elevation angles brm lie in the range 0 brm 20 [50]. Z bR Z p m 1 p p br Rpq;p~q~ðdr ; dt ; sÞ ¼ cos 2pI0 ð2r Þ bRm p 4bRm 2 bRm 2 er cos ar e2jpsðfRmax ðcos ar cos cr cos br þsin ar sin cr cos br ÞÞ ðp;pÞ ~ e e e j2p k ~ ðq;qÞ dRx cos ar cos br þ Rr d j2p k ~ ðq;qÞ dRy sin ar cos br þ ðq;qÞ ~ dRz ðp;pÞ ~ Ty D cos ar sin br þ ð9Þ Rr d j2p k Tx D sin ar ðp;pÞ ~ Rr d tan br R2 sec2 bðnÞ Tz þ r 2D r D dar dbr The integration in (9) lacks the closed-form solution for the space–time correlation functions among the MIMO antenna elements. Hence, we integrate this equation numerically by using small angle approximation, i.e., sin br br , cos br 1, sec2 br ¼ 1 þ br =2. This assumption of small elevation angles is valid for the proposed model because the distance between the MS and BS is much larger than the antenna heights. By incorporating these approximations, we get the following expression, 123 Author's personal copy Geometrical Modeling of Scattering Environment for Umbrella Cell... Rpq;p~q~ðdr ; dt ; sÞ 1 2pI0 ð2r Þ j2p k e bRm Z p e2jpsðfTmax ðcos ar cos cr þsin ar sin cr ÞÞ p bRm ðq;qÞ ~ dRx j2p k e Z ~ ðp;pÞ Rr d cos ar þ DTx ðp;pÞ ~ Rr d br R2 b ðq;qÞ ~ r r dRz br þ TDz þ 4D cos ar ~ ðq;qÞ j2p k dRy sin ar þ e ðp;pÞ ~ Rr d Ty D sin ar ð10Þ p p br dar dbr cos 4bRm 2 bRm 2 er cos ar The above integrations for variables ar and br can be written separately as, Z p 1 e2jpsðfRmax ðcos ar cos cr þsin ar sin cr ÞÞ Rpq;p~q~ðdr ; dt ; sÞ 2pI0 ð2r Þ p ~ ðp;pÞ Rr d ~ ðp;pÞ Ty ðq;qÞ ~ j2p e j2p k ðq;qÞ ~ dRx þ Rr d Tx D þ2r cos ar ~ ðp;pÞ R2 Tz þ4Dr D e k dRy þ D sin ar dar ð11Þ p p br dbr cos 4bRm 2 bRm bRm ~ ðp;pÞ ðp;pÞ ~ Rr dTy Rr dTx ðq;qÞ ~ ðq;qÞ ~ j2p j2p 2 2jpsfRmax cos cr þ r ; x2 ¼ k dRy þ D By substituting, x1 ¼ k dRx þ D ðp;pÞ ~ Rr dTz ~ ðq;qÞ R2r j2p 2jpsfRmax sin cr ; x3 ¼ k dRz þ D þ 4D in (11), we get the following expression, Z j2pbr k bRm e ~ ðq;qÞ dRz þ Rr d Z p 1 Rpq;p~q~ðdr ; dt ; sÞ ex1 cos ar þx2 sin ar dar 2pI0 ð2r Þ p Z bR m p br x3 br p dbr e cos 4bRm 2 bRm bRm ð12Þ Equation (12) can be written in simplified form by using Bessel function notations given in [21], pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Io x21 þ x22 cosð2p b x3 Þ k Rm h i ð13Þ Rpq;p~q~ðdr ; dt ; sÞ I0 ð2r Þ 1 ð4bRm x3 Þ2 k 6 Result Description In this section, we present the theoretical results of the space–time correlation function and elaborate the impact of various channel parameters on the correlations among MIMO antenna elements. It is observed that the derived expression (13) is the function of various model parameters and physical dimensions of the propagation channel. The values of the parameters used in plotting different curves of space–time correlations are mentioned in the caption of each plot. A normalized sampling period (step size) fRmax Ts ¼ 0:01, is used for plotting the curves. Various 2D and 3D plots of the correlations among MIMO antenna elements are presented for discussion and observation. 123 Author's personal copy M. Y. Wani et al. Receive Correlation 1500 1000 500 0 200 150 100 Eleva tion a 50 ngle of tra nsmit 0 -50 anten na, ψ T -100 -150 [in de gree] -200 -200 -150 -100 -50 0 50 nna, ψ R ive ante of rece ion angle 100 150 200 e] [in degre Elevat Fig. 3 3D space time correlation function with respect elevation angles of transmit and receive antenna arrays The correlations among MIMO antenna elements shown in Figs. 3, 4, 5 and 6 are plotted against the orientation of antenna arrays in the azimuth and elevation planes. The plot depicted in Fig. 3, shows clearly that the correlation is maximum when the elevation angles of any antennas is set at 0 or 180 , and the correlations show a decreasing trend with the increase in elevation angle of any the antenna from 0 to 90 . The plot is generated using channel parameters dr ¼ dt ¼ k=2; br ¼ 20 ; ar ¼ p=4; wr ¼ 30 ; hr ¼ ht ¼ 2p=3; ar ¼ 500 m; br ¼ 200 m. Therefore, for the design of an umbrella cell in cellular mobile communication, antenna orientation has significant impact on the system performance. The proposed elliptical cylindrical channel model is generic in nature and can be reduced to cylindrical model proposed by Feng et al. in [16] by keeping eccentricity r ¼ 0 and distance D = 5 km and other channel parameters are dr ¼ dt ¼ k=2; br ¼ 20 ; ar ¼ p=4; hr ¼ ht ¼ 2p=3; ar ¼ br ¼ 100 m. The joint correlation graph is plotted against elevation angles of transmit and receive antenna arrays, shown in Fig. 4. Resultant plot is well matched with the correlation plot given in [16]. This implies that the channel model given in [16] is the special case of our proposed channel model. Similarly, 3D plot shown in Fig. 5, of joint space–time correlation function is taken against azimuth AoA for the channel parameters dr ¼ dt ¼ k=2; br ¼ 20 ; wr ¼ 30 ; hr ¼ ht ¼ 2p=3; ar ¼ br ¼ 100 m, and its 2D slices are plotted in Fig. 6a, b. It is observed from the plots that the correlation among the MIMO antenna array elements are dependent on the PDF of AoA (ar ) in azimuth plane of the nonisotropic scattering environments; a similar observation was reported in [30, 40]. The antenna spacing is another most important system parameter that has significant effect on the correlations among the MIMO antenna elements [19, 27]. The graph shown in Fig. 7, demonstrates the effect of antenna spacing on antenna correlations. It is seen as the spacing is increased, the correlation shows a decreasing trend and vice versa. Hence, keeping the antenna elements at appropriate spacing, maximum throughput can be achieved from the MIMO systems. Moreover, the proposed 3D eccentricity based channel model is transformed into 3D circular based geometrical channel model given in [35], by making eccentricity equal to 123 Author's personal copy Geometrical Modeling of Scattering Environment for Umbrella Cell... 90 80 Receive Correlation 70 60 50 40 30 20 200 10 100 0 0 -200 -150 -100 -50 0 -100 50 Elevation angle of receive antenna, ψR [in degrees] 100 150 it an nsm f tra le o g n on a tenn a, ψ T es] egre [in d -200 200 ti a Elev Fig. 4 Receive correlation with respect to elevation angles of transmit and receive antenna arrays [16] x 10 -3 3 Receive Correlation 2.5 2 1.5 1 0.5 0 1 0.8 0.6 τf 0.4 Rmax 0.2 0 0 50 100 150 250 200 300 350 400 Azimuth angle, α [in degrees] r Fig. 5 Joint space–time correlation function of azimuth AoA and normalized time delay zero. It is observed from Fig. 8 that elliptical geometry shows increased correlation as compared to circular geometry. It thus confirms the statement that elliptical geometry is more appropriate shape to model scattering environments of streets, canyons or highways in umbrella cellular environment; therefore, the results of our proposed channel model are more justified than circular ones to design and analyze MIMO M2M communication links. These curves are plotted for the following channel parameters dt ¼ k=2; br ¼ 20 ; hr ¼ p=6; wr ¼ 30 ; hr ¼ ht ¼ 2p=3; ar ¼ br ¼ 200 m. 123 Author's personal copy M. Y. Wani et al. 3 x 10 -3 3 x 10 -3 τ fRmax = 0 α r = 0o αr = 30 αr = 60 o αr = 90o 2 1.5 τ fRmax = 0.8 1.5 1 0.5 0.5 0.2 0.4 0.6 0.8 τ fRmax = 0.5 2 1 0 0 τ fRmax = 0.1 2.5 Receive Correlation Receive Correlation 2.5 o 0 0 1 τ fRmax 50 100 150 200 250 300 350 Azimuth angle [in degrees] (b) (a) Fig. 6 2D space time correlation function with respect azimuth AoA 0.12 δr = 0.3λ δr = 0.5 λ 0.1 δr = λ Receive Correlation 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized time delay, τ fRmax Fig. 7 Effect of antenna spacing on space–time correlation function 7 Conclusion In this research article, a single bounce 3D eccentricity-based cylindrical geometrical channel model for MIMO F2M propagation environment has been presented. The proposed elliptic cylindrical geometry is rotatable along the horizontal plane about vertical axis and its dimensions are adjustable corresponding to the physical propagation environment. Based on the proposed model, mathematical expressions of joint and marginal space–time correlation functions among the transmitter and receiver antenna array elements have been 123 Author's personal copy Geometrical Modeling of Scattering Environment for Umbrella Cell... 0.12 Elliptical-Based Cylindrical Model Circular-Based Cylindrical Model 0.1 Receive Correlation 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 0 0.5 1 1.5 τ fRmax Fig. 8 Comparison of space–time correlation function of proposed elliptical based cylindrical channel models with the circular based cylindrical channel models [16] formulated. Various curves are generated using the obtained theoretical expressions for the joint and marginal correlation functions for different values of channel parameters. The resultant correlation plots are thoroughly elaborate over the system performance. 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Wireless Personal Communications, 91(2), 509–524. 50. Yamada, Y., Ebine, Y., & Nakajima, N. (1987). Base station/vehicular antenna design techniques employed in high-capacity land mobile communications system. Review of the Electrical Communication Laboratories, 35(2), 115–121. 51. Zajić, A. G., & Stüber, G. L. (2008). Three-dimensional modeling, simulation, and capacity analysis of space–time correlated mobile-to-mobile channels. IEEE Transactions on Vehicular Technology, 57(4), 2042–2054. https://doi.org/10.1109/TVT.2007.912150. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. M. Yaqoob Wani received his B.S. degree in electronics from the University of Kashmir, Srinager, India, in 1991 and M.S. Telecommunication and Networking from the Iqra University, Islamabad Pakistan in 2008. He received his Ph.D. degree in electrical engineering from Capital University of Science and Technology (CUST), Islamabad, Pakistan in 2017. He is with the Acme Center for Research in Wireless Communications (ARWiC) at CUST. Currently, he is also working as Associate Professor/Head of Department Software Engineering with the University of Lahore, Islamabad, Pakistan. His research interests include channel modeling and characterization, cellular mobile communication networks and vehicle-to-vehicle communications. 123 Author's personal copy M. Y. Wani et al. M. Riaz was born in Pakistan in 1977. He received his M.Sc. degree in electronics from Quaid-i-Azam University, Islamabad, Pakistan, in 2002. He did his M.S and Ph.D. degree in electronic engineering from Mohammad Ali Jinnah University (MAJU), Islamabad, Pakistan in 2009 and 2015, respectively. At present, he is with the University of Lahore, Islamabad Campus as Assistant Professor/Head of Department of Electronics and Electrical Systems. He is also with the Acme Center for Research in Wireless Communications (ARWiC) at Capital University of Science and Technology, Islamabad. His research interests include modeling and characterization of cellular and mobileto-mobile communication channels, localization of wireless sensor network, MIMO communication systems, channel equalization and estimation, and vehicle-to-vehicle communications. Noor M. Khan was born in Pakistan in 1973. He received the B.Sc. degree in electrical engineering from the University of Engineering and Technology, Lahore, Pakistan, in 1998 and the Ph.D. degree from the University of New South Wales, Sydney, Australia, in 2007. From 2002 to 2007, he was a casual academic with the University of New South Wales. He is currently Professor with the Capital University of Science and Technology (CUST), Islamabad, Pakistan. His research interests include smart antenna systems, adaptive multiuser detection, mobile-to-mobile communications, wireless sensor networks, channel characterization and estimation, and physical channel modeling for mobile communications. 123