advertisement

Characterization of 3D Elliptical Spatial Channel Model for MIMO Mobile-toMobile Communication Environment M. Yaqoob Wani & Noor M. Khan Wireless Personal Communications An International Journal ISSN 0929-6212 Wireless Pers Commun DOI 10.1007/s11277-017-4479-6 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media New York. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive 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 DOI 10.1007/s11277-017-4479-6 Characterization of 3D Elliptical Spatial Channel Model for MIMO Mobile-to-Mobile Communication Environment M. Yaqoob Wani1 • Noor M. Khan1 Springer Science+Business Media New York 2017 Abstract In this paper, we develop three dimensional (3D) elliptical cylindrical geometrical channel model for multiple-input–multiple-output mobile-to-mobile communication environments. It is assumed that both the mobile nodes are surrounded by uniformly distributed infinite number of scatterers sprinkled over the surfaces of an elliptical-based cylindrical shapes. The mobile nodes are located at the centers of the bottom surfaces of elliptical cylinders and both the mobile nodes are equipped with low-elevated multiple antenna arrays. The proposed model is designed for urban areas, where mostly the mobile subscribers reside and are on the move. This model takes into account the effect of multiple antenna array attributes, roadside infrastructure, the dimensions of the propagation medium, transmit–receiver distance and the velocity of mobile nodes. Using the proposed channel model, expressions for the joint and marginal cross correlation functions are derived for non-isotropic scattering environments. The derived expression are simulated for various parameters to verify their effect on the antenna correlations. The obtained correlation graph is compared with measured data that confirms a close agreement with it. Finally, by changing various parameters of the proposed channel model, some existing 2D and 3D channel models are deduced. Keywords Mobile-to-mobile channels Correlation function MIMO channels M2M Wireless channel modeling Elliptical geometry Antenna correlations & M. Yaqoob Wani [email protected] Noor M. Khan [email protected] 1 Acme Center for Research in Wireless Communications (ARWiC), Department of Electrical Engineering, Capital University of Science and Technology, Kahuta Road, Zone-V, Islamabad 44000, Pakistan 123 Author's personal copy M. Y. Wani, N. M. Khan 1 Introduction High capacity wireless links with better quality of service (QoS) are the fundamental demands of all mobile subscribers in every aspect of the life. Therefore, providing high capacity data links to mobile subscribers with limited resources is a constant driving force in the research arena. Mobile-to-mobile (M2M) communication technology give rise to various innovative advancements over a short span of time in vehicular, railway and defense sectors to provide audio, data, live video streaming, teleconferencing and other value added applications without any centralized static base station (BS) [9, 43]. However, hazardous situation for these high speed, communicating nodes can arise when they are on the move in scattering-rich urban areas [49]. In such propagation environments, the channel characteristics are unpredictable and multipath components cause the received signal in deep fading. Therefore, traditional single-input–single-output (SISO) communication link may not provide the required capacity with QoS in such communication scenarios. Recently, the rich scattering environment has been recognized feasible situation for the high data-rate applications, if multiple antenna arrays are employed instead of a single antenna structure at both ends of the wireless link [20]. Multiple-input and multiple-output (MIMO) systems, therefore, exploit this rich scattering environment in a constructive manner and provide spatial multiplexing that can guarantee astonishing increase in throughput [13, 20, 22, 24, 48]. Promised large capacity gain of a MIMO system can be achieved contingent upon a good understanding of spatial characteristics of the radio fading channel [24]. In this regard, various techniques are adopted to model the wireless channel to estimate its statistics on the basis of communication scenario and the distribution of the scattering objects in the propagation environments. Mostly, communication channels are modeled in the literature using empirical, deterministic, stochastic and geometry-based stochastic channel modeling (GBSCM) approaches [2, 8, 29]. Empirical channel models are based on experimental measurements and observations for a particular communication scenario. Hence, these models are valid only for specific sites under particular circumstances [18]. The deterministic channel modeling approach is applicable when the transmitter, receiver, scattering objects are static and other channel characteristics are known. The stochastic channel modeling (SCM) approach is based on the PDFs of various channel parameters and is also site specific [12]. Since, in Mobile-to-mobile (M2M) communication environment, both communicating nodes are assumed to be on the move; therefore, previously discussed channel modeling approaches are incongruous to model M2M propagation channels. Such non-stationary propagation environments can be modeled perfectly by GBSCM approaches under the assumption of quasi-stationary scattering [28]. GBSCM approach is based on fundamental laws of wave propagation and on the physical geometry of propagation environment [16]. In addition, geometrical channel models may be utilized for various propagation scenarios just by adjusting the model input parameters [25]. On the basis of physical dimensions of propagation environments, probability distribution functions of various parameters of the M2M wireless channel may be derived geometrically. Various 2D and 3D versions of GBSCM’s, have been proposed in the literature for fixed-to-mobile (F2M) and M2M communication scenarios with both SISO and MIMO antenna structures. In case of F2M framework, mostly researchers have derived mathematical expressions for PDFs of angle-of-arrival (AoA) and time-of-arrival (ToA), and space-time correlation functions among MIMO elements [1, 19, 25, 30, 31, 34, 35]. Besides F2M, Akki and Haber proposed geometrical channel model for SISO and M2M land communication channels and developed closed-form 123 Author's personal copy Characterization of 3D Elliptical Spatial Channel Model… expressions for some statistical characteristics of propagation channel [6, 7]. Authors in [10, 11, 14, 16, 38–39, 41, 54] have extended precursory research by assuming the scattering objects along circular or elliptical loci with isotropic and non-isotropic distributions for both SISO and MIMO environments. The authors have derived expressions for different propagation parameters like AoA, ToA, Doppler spread, level crossing rate, average fade duration and space-time correlation among MIMO antenna elements. In the abovementioned 2D geometrical channel models, signal propagation has been considered only along the azimuth plane, ignoring the elevation plane. In the previously reported geometrical channel models, the effect of azimuth angle of wavelets were considered for the channel statistics. Hence, 2D geometrical channel models may be appropriate in some of the rural areas, but may not decorous for streets, canyons, urban or metropolitan areas. Because in such areas the source of communication is predominantly with scattered waves that are diffracted, reflected down the streets or canyons by the edges of the surrounding infrastructure. Therefore, geometrical channel models that consider the effect of elevation angle of the wavelets are suitable models for such urban environments. The aspect of elevation angle of the wavelets is introduced in [15, 32, 33, 52, 55, 57, 58] by presenting 3D geometrical channel models for SISO and MIMO M2M communication systems, where the authors have assumed that mobile stations are located at the centers of cylinders, spheres, semi-spheres or ellipsoids. In these proposed 3D channel models, authors have ignored the propagation-distance traveled by the wavelets from the scatterer located in the transmitter scattering-region to the scatterer located in the receiver scattering-region. The authors derived closed-form expressions for Doppler spread, PSD and joint space-time correlation functions among MIMO coefficients while assuming non-isotropic scattering environments. Moreover, the authors in [45] have derived the transmit, receive antenna correlations as separate identity under the context of Kronecker model while ignoring the effect of distances between transmitter and receiver on the antenna correlations. These models may be favorable in such urban areas where the mobile nodes reside in close proximity to each other and the scattering objects surround the mobile nodes in a cylindrical manner. Whereas, in reality the physical layout of streets, canyons, deep-cut railway tracks and highways are mostly narrow in width and longer in length that have close resemblance to the elliptical shape than circular ones. High-rise buildings, vegetation and other infrastructure present along the roadside premises are the sources of multipaths in the azimuth and elevation planes. Understanding this usefulness of elliptical shape, Riaz et al. [46, 47] proposed an ellipsoid geometrical channel model for SISO M2M environments, where they derived expressions for the PDFs of ToA and AoA both in azimuth and elevation planes. The authors assumed that the signals are equally likely from all directions of the ellipsoid, which contradicts the realistic propagation environment. Because the probability of AoA of scattered signals from the sky-top of urban areas is almost zero, Ahmed et al. [5] proposed modified geometrical channel model of [46, 47] with top surface open and derived PDFs for AoA and AoD. These proposed M2M communications models can be made more beneficial for high data rate applications if they are equipped with MIMO systems. This motivates us to propose a geometrical channel model for MIMO M2M communication environments that can relegate all the above-mentioned impairments of previously suggested channel models. In this paper, we propose an eccentricity-based elliptical cylindrical channel model for MIMO M2M communication environments. We derive expressions for the marginal and joint correlation functions among the antenna elements. These correlation functions are further simulated to analyze the impact of different system parameters on the correlations among antenna-array element. Furthermore, different 2D and 3D existing channel models 123 Author's personal copy M. Y. Wani, N. M. Khan are deduced, by changing the parameters of proposed geometry. The correlation curves of proposed geometry are compared with the results of these previously published geometrical channel models. Finally, simulation result of space-time correlation function is compared with measurement campaign. Rest of the paper is organized as follows: Sect. 2 describes the geometry of the proposed eccentricity-based elliptical cylindrical channel model for MIMO M2M. Section 3 presents the derivations of the reference model. Section 4 presents derivation of the closeform expressions of correlation functions for the proposed geometry. Section 5 provides the comprehensive discussion of the simulation results. Finally, Sect. 6 provides some concluding remarks. 2 System Model In this section, we present the system architecture of the proposed elliptical cylindrical geometrically-based channel model for MIMO M2M communication environment as shown in Fig. 1. In this proposed channel model, transmitting and receiving mobile stations (MSs) are denoted by MSt and MSr respectively. These MSs are assumed to be located at the centers bottom surface of the elliptical cylinders, having major axes at and ar and minor axes bt and br with eccentricities t and r , respectively. The mobile nodes are moving independently with the velocities of vt and vr making angles at and ar with the xaxis. The center to center distance between the two elliptical cylindrical is represented by d (such that d at þ ar ) and to avoid the channel may not experience the keyhole behavior the distance should not be greater than 4Rt Rr Hr =ðkðTr 1ÞðHt 1ÞÞ [21]. The azimuth plane dimension of the physical propagation channel around the MSs are adjusted in the system model by eccentricities t and r of the ellipses. Whereas, the surrounding scatter hight around transmitter and receiver are represented by ht and hr respectively. The scatters z AT z ST(m) dpm (p) ψT(p) dmn AR(q) βT(m) ψR(q) dp∼m CT ∼ vR vT γT βR(m) dnq∼ CR ∼ AT(p) AT(p) Rt SR(n) dnq αT(m) AR(q)’ ∼ AT(p) AR(q) γR Rr C’R C’T π−θT αR(n) ∼ AR(q) π−θR x d y Fig. 1 Proposed 3D elliptical channel model for MIMO Mobile-to-Mobile channels with Ht ¼ Hr ¼ 2 antenna elements 123 Author's personal copy Characterization of 3D Elliptical Spatial Channel Model… present in the close vicinity of mobile nodes are assumed to be uniformly distributed on the surfaces of elliptical cylinders. Moreover, the elliptical cylinders are rotatable congruous to the directions of motion of the MSs such that their major axes at and ar make angles at (or ar ) respectively, with the x-axis. Mobile nodes are equipped with low hight antennas arrays with configuration Ht Hr , where Ht and Hr are the number of antennas mounted on MSt and MSr , respectively. For simplicity, we take Ht = Hr = 2. However, the results can be derived for any configuration. The transmit and receive antenna array elements are denoted ðpÞ by At and ArðqÞ , and the distance between the antenna array elements are denoted by dt and dr , which are very small as compared to the minor axes of the surrounding ellipses respectively. The description of the other parameters involved in the proposed geometrical model are narrated in Table 1. 3 Derivation of the Reference Model The derivations of the various characteristic of the MIMO M2M fading channel are based on the reference model depicted in Fig. 1. It can be observed that a signal that is transðpÞ ðpÞ mitted from the transmit antenna array element At , first strikes at the scatterer St present on the surface of elliptical cylindrical surrounding the transmitter node and then travels Table 1 Definitions of the channel parameters used in the system model Symbols Description d The distance between center to center of elliptical cylinders surrounding MSt and MSr Rt ; Rr The dynamic radius of the transmitter and receiver ellipses, respectively at ; ar The major axes of the transmitter and receiver ellipses, respectively bt ; br The minor axes of the transmitter and receiver ellipses, respectively ht ; hr The hight of the scatterers at transmit and receive elliptical cylinder, respectively Ht ; Hr Number of antenna array elements at transmitter and receiver, respectively t ; r The eccentricities of the transmitter and receiver ellipses, respectively dt ðp; p~Þ The spacing between pth and p~th antenna elements at Tx dr ðq; q~Þ The spacing between qth and q~th antenna elements at Tx ðqÞ hðpÞ t ; hr The azimuth angle of pth transmit and qth receive antenna element (relative to x-axis), respectively ðqÞ wðpÞ t ; wr The elevation angle of pth transmit and qth receive antenna element (relative to x–y plane), respectively vt ; vr The velocities of the Tx and Rx, respectively ct ; cr The moving directions of the Tx and Rx, respectively ðmÞ at ; aðnÞ r The azimuth angles of departure (AAoD) and the azimuth angles of arrival (AAoA), respectively ðnÞ bðmÞ t ; br The elevation angle of departure (EAoD) and the elevation angle of arrival (EAoA), respectively ðpÞ ðmÞ ðpÞ ~ ðmÞ ðmÞ ðqÞ ðqÞ ~ , d St ; SðnÞ , d SðnÞ The distances d At ; St , d At ; St , and d SðnÞ r r ; Ar r ; Ar dpm dpm ~ dmn dnq dnq~ pq dmax dpm þ dmn þ dnq p~q~ dmax dpm ~ þ dmn þ dnq~ 123 Author's personal copy M. Y. Wani, N. M. Khan towards the scatter SðqÞ located on the surface of elliptical cylindrical surrounding the r receiver node, and then finally reaches the receiving antenna array element AðqÞ r . During ðpÞ ðpÞ ~ ~ propagation, the distances covered by the wavelets from At to AðqÞ to AðrqÞ , are r and At pq p~q~ denoted by dmax and dmax , respectively. Therefore, the phase change due to the distance pq p~q~ traveled by the wavelet can be written as ð2p=kÞdmax and ð2p=kÞdmax , where 2p=k is called wavenumber. Moreover, the joint gain and phase shift due to the collision of the wavelet pﬃﬃﬃﬃﬃﬃﬃﬃ ðpÞ ðqÞ with St and St cab be expressed as 1= MN and /mn respectively. Other sources of phase shift in M2M communication which is due to the motion of transmitter and receiver, ðmÞ ðnÞ cos bðnÞ can be expressed as 2pfTmax cos at ct cos bðmÞ t and 2pf cos a c Rmax r r t r t. Where, fTmax ¼ vt =k and fRmax ¼ vR =k represents the maximum Doppler frequency caused by the motion of mobile nodes. Moreover, for the ease of derivations of the different expressions of the proposed channel model, following valid assumptions are considered 1. 2. 3. 4. 5. Each multipath component of the propagating signal undergoes two bounces while traveling from the transmitter mobile node to the receiver mobile node. Infinite number of scatterers are uniformly distributed on the elliptical cylindrical surfaces with uniformly distributed phases. Equal distributed power is reflected from all the scatterers. All the waves reaching at the receiver antenna elements are equal in power. The scatterers are fixed and MSs are quasi-stationary for a short period of time. Finally, with the help of above assumptions the diffused components of the transmission ðpÞ link from At to AðqÞ r can be expressed as, M X N 2p 1 X ej k ðdpm þdmn þdnq Þþj/mn hpq ðtÞ ¼ lim pﬃﬃﬃﬃﬃﬃﬃﬃ M;N!1 MN m¼1 n¼1 e ðmÞ ðmÞ j2ptfTmax cosðat ct Þ cos bt e ð1Þ ðnÞ ðnÞ j2ptfRmax cosðar cr Þ cos br The expressions of the distances dpm , dpm ~ , dnq and dnq~ are obtained by solving the geometry of the proposed model as shown in Fig. 1. It is observed that these distances are the functions of random angles of azimuth and elevation of transmit antenna arrays, receive antenna arrays and scatter. The 3D polar coordinates of pth transmit and qth receive antenna elements in 3D space are denoted by ðpÞ ðpÞ ðpÞ ðpÞ dTx ; dTy ; dTz ðpÞ and ðqÞ ðqÞ ðqÞ ðpÞ ðpÞ dRx ; dRy ; dRz , where dTx ¼ dðAt ; ðpÞ ðpÞ ðqÞ ðpÞ ðpÞ ðpÞ ðpÞ Ct Þ cos hðpÞ t cos wt , dTy ¼ dðAt ; Ct Þ sin ht cos wt , dTz ¼ dðAt ; Ct Þ sin wt , dRx ¼ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðqÞ ðpÞ dðAðpÞ dRy ¼ dðAðpÞ r ; Cr Þ cos hr cos wr , r ; Cr Þ sin hr cos wr ; dRz ¼ dðAr ; Cr Þ sin wr . Similarly, the coordinates in 3D space are denoted by of mth and nth scatterer ðm=mÞ ~ ðm=mÞ ~ ðm=mÞ ~ ðn=nÞ ~ ðn=nÞ ~ ðn=nÞ ~ ðm=mÞ ~ ðm=mÞ ~ ðm=mÞ ~ dAx ; dAy ; dAz ¼ Rt cos at , d Ay ¼ and dAx ; dAy ; dAz , where dAx ~ ðm=mÞ Rt sin at ~ ðm=mÞ , dAz nÞ ~ Rr tan bðn= . t ~ ðn=nÞ ~ mÞ ¼ Rt tan bðm= , dAx t ~ ðn=nÞ nÞ ~ ¼ d þ Rr cos aðn= , dAy r ~ ðn=nÞ nÞ ~ ¼ Rr sin aðn= , dRz r ¼ These polar coordinates depend upon the orientation and configuration of the antenna arrays. The dynamic radii Rr and Rt of the transmit and receive ellipses can be written qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ in terms of minor ai and major bi axis as, ai bi = a2i sin2 ai þ b2i cos2 ai [44]. Using the pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ distance formula, and binomial approximation ð1 þ xÞ 1 þ x=2 if ðx 1Þ, the approximated distances can be expressed as, 123 Author's personal copy Characterization of 3D Elliptical Spatial Channel Model… dp=p;m ~ Rt cosbðmÞ t ðpÞ ðpÞ ðmÞ ðmÞ ðpÞ ðmÞ ðpÞ d At ;Ct sinwðpÞ t sinbt d At ;Ct coswt cosbt cos at ht ð2Þ dn;q=q~ Rr cosbðnÞ r ðqÞ ðnÞ ðqÞ ðnÞ ðqÞ ðpÞ ðnÞ d AðpÞ r ;Cr sinwr sinbr d Ar ;Cr coswr cosbr cos ar hr ð3Þ ðpÞ ðqÞ The propagation path length, dmn , from St to St is greater than maxðdpm ;dnq Þ and contributes significantly in the phase shift to the received signal. It is the function of ðmÞ ðnÞ ðnÞ ðqÞ ðpÞ random angles bðmÞ and dynamic radii of the ellipses and can be t ;at ;br ar , br , bt obtained by solving the geometry as shown in Fig. 2. With the help of approximations used in (2) and (3), the simplified form of dmn can be expressed as, ðmÞ ðnÞ 2 2 R R cos a a r t t r R R ðmÞ Rt cos at þ Rr cos aðnÞ dmn d þ r þ t r 2d 2d d ð4Þ 2 2 ðqÞ ðpÞ ðqÞ Rt tan bðpÞ R tan b r t r Rt tan bt Rr tan br þ : þ 2d 2d d ðmÞ Rr Rt cos aðnÞ r at ðmÞ dmn d Rt cos at þ Rr cos aðnÞ r d ð5Þ ðpÞ ðqÞ 2 2 ðpÞ 2 2 ðqÞ Rt sec bt Rr sec br Rt tan bt Rr tan br þ : þ 2d 2d d Substituting values of dpm , dnq and dmn in (1), we get, M X N 1 X ap;m bn;q cp;q ej/mn hpq ðtÞ ¼ lim pﬃﬃﬃﬃﬃﬃﬃﬃ M;N!1 MN m¼1 n¼1 ðmÞ at ct ej2p½fTmax cosð Þ ðmÞ cos bt þfRmax cosð ðnÞ ar cr ð6Þ Þ ðnÞ cos br t : St(m) dmn z dpm Sr(n) x ht dqn y At(p) at βt Rt Ar(q) αt d (0,0,0) ar bt hr βr Rr (d,0,0) αr x br Fig. 2 2D view of proposed elliptical channel model for MIMO M2M channels 123 Author's personal copy M. Y. Wani, N. M. Khan where jp k d 2þ ap;m ¼ e 2Rt ðmÞ cos b t ðmÞ ðmÞ 2p j2pd p cos at ej k ðRt cos at Þ e k Tx ðmÞ j2pdT p sin bt þ e k jp k z p j2p k dT sin at e y ðmÞ cos bt ðpÞ ðRt sec br Þ2 2d 2Rr d 2þ ðnÞ cos br bn;q ¼ e ðmÞ ðmÞ cos bt ðnÞ ðnÞ 2p j2pd q cos ar þ ej k ðRr cos ar Þ e k Rx ðnÞ ðnÞ cos br e ðnÞ q j2p k dR sin ar cos br y 2 ðnÞ q j2p k dR sin br þ e jp k cp;q ¼ e z ðRr sec bðqÞ r Þ 2d ðpÞ ðqÞ R R cos aðnÞ aðmÞ r t Rt tan b Rr tan br r t t þ d d ð Þ The proposed geometrical can be also extended for Ht Hr antenna arrays structure. The ðp=qÞ ðp=qÞ ðp=qÞ ðp=qÞ polar coordinates of antenna elements is denoted by ðdTx =Rx ; dTx =Rx ; dTx =Rx Þ where, dTx =Rx ¼ ðp=qÞ ðp=qÞ ðp=qÞ dðt=rÞ 0:5Hðt=rÞ :5 pðqÞ cos ht=r cos wt=r ; dTx =Rx ¼ dðt=rÞ 0:5Hðt=rÞ :5 pðqÞ ðp=qÞ ðp=qÞ ðp=qÞ ðp=qÞ sin ht=r cos wt=r ; dTx =Rx ¼ dðt=rÞ 0:5Hðt=rÞ :5 pðqÞ sin wt=r : The parameter p; p~ 2 f1; . . .; Ht g and q; q~ takes values from f1; . . .; Hr g. The coordinates for p~ element and q~ element can be obtained by replacing dðt=rÞ with dðt=rÞ . Where, dðt=rÞ is the spacing between two adjacent elements of transmitter (receiver) antenna arrays. 4 Derivation of Space-Time Correlation Function of the Proposed Model The normalized space-time correlation function between diffused channel coefficients hpq ðtÞ and hp~q~ðtÞ for the proposed 3D model can be found using the following relation, E½hpq ðtÞhp~q~ðt þ sÞ Rpq;p~q~½s ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E½jhpq ðtÞj2 þ E½jhp~q~ðtÞj2 ð7Þ where, E½ is the statistical expectation operator and can be applied only to all random variables and ðÞ symbolizes as the complex conjugate operation. Using (6) and (7), the space-time correlation function can be formulated as, M X N h 1 X E ap;m bn;q cp;q ap;m ~ bn;q~cp~q~ M;N!1 MN m¼1 n¼1 Rpq;p~q~½s ¼ lim e ðmÞ ðmÞ ðnÞ ðnÞ 2jpsðfTmax cosðat ct Þ cos bt þfRmax cosðar cr Þ cos br ð8Þ i Þ ; It is assumed that infinite number of scattering objects reside around each mobile station, which implies that the scattering distributions may be transformed from discrete to conðmÞ tinuous that in turn forces to change the discrete random variables (e.g., at , bðmÞ t ) into continuous random variables (at , bt ). Furthermore, we assume that azimuth and elevation angles are independent of each other, therefore, f ðat ; bt Þ and f ðar ; br Þ can be written in product form as f ðat Þf ðbt Þ and f ðar Þf ðbr Þ, respectively. Hence, for the continuous time random variables, the above equation can be written in integration form as, 123 Author's personal copy Characterization of 3D Elliptical Spatial Channel Model… Rpq;p~q~ðsÞ Z Z bRm bRm j2p ek bTm bTm ðp;pÞ ~ dTx Z Z p p p e2jpsðfTmax cosðat ct Þ cos bt Þ p ðp;pÞ ~ cos at cos bt þdTz sin bt þ ðRt sec bt Þ2 2d e2jpsðfRmax cosðar cr Þ cos br Þ e ~ j2p ðp;pÞ e k dRx sin ar cos br j2p k ~ ðp;pÞ dTy ~ j2p ðp;pÞ dTy ek sin at cos bt ~ ðq;qÞ sin at cos bt þdRz sin br þ ðRr sec br Þ2 2d ð9Þ f ðat Þf ðbt Þf ðar Þf ðbr Þdat dbt dar dbr where bRm , bTm represents the maximum elevation angles of the scatters present around ~ ðp;pÞ transmit and receiver mobile nodes and dTx ~ ðp;pÞ ðpÞ ~ ðpÞ ~ ðq;qÞ ðqÞ ~ ðqÞ ~ ðq;qÞ ðqÞ ðpÞ ~ ðpÞ ~ ðp;pÞ ¼ dTx dTx , dTy ~ ðqÞ ~ ðq;qÞ ðqÞ ðpÞ ~ ðpÞ ¼ dTy dTy , ~ ðqÞ ¼ dTz dTz , dRx ¼ dRx dRx , dRy ¼ dRy dRy , dRz ¼ dRz dRz . Furthermore, different scattering distribution have been proposed in the literature for isotropic and non-isotropic environments [30]. In urban areas streets, canyons and highways are more likely non-isotropic environments. So, in this proposed research work, we have used eccentricity-based (i.e., i ) modified von Mises distribution for azimuth AoA/ AoD [50]. The PDF of azimuth AoA/AoD at each MS can be written as, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 f ðai Þ ¼ 2pI01ð2 Þ ei cos ai ; i ¼ t; r, where, i ¼ 1 b2i =a2i , the azimuth angle ai 2 ðp; pÞ dTz i and I0 ðÞ is the zeroth-order modified Bessel function of the first kind. Similarly, It has been observed from the experiments that the elevation AoA of incoming signals ranges from 0 to 20 [51] and the distance between two mobile stations is much larger than each of their antenna heights; therefore, using the small angle approximation sin bi bi , cos bi 1, sec2i b ¼ 1 þ bi =2 and substituting PDF of elevation AoA/AoD, i.e., f ðbi Þ ¼ p p bi as proposed in [36], where, the absolute values of elevation angles 4b cos 2 b im im (bim ; i 2 T; R ) lies in the range 00 bim 200 , in Eq. (9) we get, Z p Z p 1 exp ½c cos a þ d sin a da exp ½cr cos ar þ dr sin ar dar Rpq;p~q~½s 2 t t t t t 4p Io ð2t ÞpIo ð2r Þ p p 2 2 Z Z bRm bRm bTm e j2p k ðp;pÞ ~ R dTz þ 4dt bt e j2p k ðq;qÞ ~ R dRz þ 4dr br bT m p pbt p pbr cos cos dbt dbr 4bTm 2bTm 4bRn 2bRn ð10Þ where ~ ðp;pÞ j2pdTx 2jpsfTmax cos ct þ 2t ; k ðp;pÞ ~ j2pdTy 2jpsfTmax sin ct ; dt ¼ k ðq;qÞ ~ j2pdRz cr ¼ 2jpsfRmax cos cr þ 2r ; k ~ ðq;qÞ j2pdRw dr ¼ 2jpsfRmax sin cr : k ct ¼ 123 Author's personal copy M. Y. Wani, N. M. Khan Equation (10) can be further simplified by introducing trigonometric transformation and pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Rp the equality p exp ½ci cos ai þ di sin ai dai ¼ I0 ð2p c2i þ di2 Þ [26]. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z j2p ðp;pÞ~ R2t dTz bt þ 4d bt Io ð2p c2t þ dt2 Þ bTm p pbt k e Rpq;p~q~½s cos dbt 4b 2b 2pIo ð2t Þ bTm Tm Tm ð11Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z 2 Io ð2p c2r þ dr2 Þ bRm p pbr j2pk dRzðq;qÞ~ br þR4dr br e cos dbr 2bRn 2pIo ð2r Þ bRm 4bRn It is seen that the joint correlation function in (11) is the product of transmit and receive correlation functions, that can be written as, Rpq;p~q~½s qð/t ; bt ; dt ; sÞqð/r ; br ; dr ; sÞ ð12Þ where pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z j2p ðp;pÞ~ R2t dTz bt þ 4d bt 2pIo ð c2t þ dt2 Þ bTm p pbt k e qð/t ; bt ; dt ; sÞ cos dbt 2 2bTm 2pIo ðt Þ bTm 4bTm ð13Þ Integrating (13) with respect to bt , and introduce approximations for the complex error function as in [4], the simplified expression can be written as, ðp;pÞ~ 1 0 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d2 k2 cos pð4dTz dþRt ÞbTm 2dk C 2pIo c2t þ dt2 B B C ð14Þ qð/t ; bt ; dt ; sÞ @ A 2 ðp; pÞ ~ 2 2 2 2 2 2pIo ðt Þ d k ðRt þ 4dTz dÞ bTm 0 1 pbTm R2t pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B ðp;pÞ ~ 2p Io 2p c2t þ dt2 Bcos k bTm dTz þ 2dk C C qð/t ; bTm ; dt ; sÞ B 2 C ðp;pÞ ~ @ A 2pIo ð2t Þ 2 4b d R b 1 tdkTm þ Tmk Tz 0 1 pbRm R2r pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B ~ ðq;qÞ 2p Io 2p c2r þ dr2 Bcos k bRm dRz þ 2dk C C qð/r ; bRm ; dr ; sÞ B 2 C ~ ðq;qÞ @ A 2pIo ð2r Þ 4bRm dRz R2r bRm 1 dk þ k ð15Þ ð16Þ The closed-form expression for the receive space-time correlation function can be obtained by replacing the index t with index r, and the joint space-time correlation function (12) can be written as 0 1 2 pb R ðq; qÞ ~ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B Rm r 2p Io 2p c2t þ dt2 Io 2p c2r þ dr2 Bcos k bRm dRz þ 2dk C C Rpq;p~q~½s B 2 C ðq;qÞ ~ @ A 2pIo ð2t Þ 2pIo ð2r Þ 4bRm dRz R2r bRm 1 dk þ k 0 1 ð17Þ pbTm R2t ~ ðp;pÞ 2p Bcos k bTm dTz þ 2dk C B C B 2 C ~ ðp;pÞ @ A 2 4b d R b 1 tdkTm þ Tmk Tz 123 Author's personal copy Characterization of 3D Elliptical Spatial Channel Model… The existing 3D cylindrical channel model [56] can become the spacial case of our proposed geometrical channel model. By using the assumption that is the radii are much smaller than the distance d (i.e., maxfRr ; Rt g d) moreover,to avoid keyhole behavior of the wireless channel the distance d\4Rt Rr Lq =ðkðLp 1ÞðLq 1ÞÞ. Using this assumption the expression (17) can be reduced to space-time correlation function (34) of [56] for isotropic propagation environments. ðq;qÞ ~ ðp;pÞ ~ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2p cos 2p b d T Tz Io 2p c2t þ dt2 Io 2p c2r þ dr2 cos k bRm dRz m k Rpq;p~q~½s 2 2 ð18Þ ðq;qÞ ~ ðp;pÞ ~ 2pIo ð2t Þ 2pIo ð2r Þ 4bRm dRz 4bTm dTz 1 1 k k When mobile stations reside on such highways where the low elevated scattering objects are non isotropically distributed around transmitter and receive. The probability of elevation plane waves cab be negligible. In that case the elevation angles bRm ; bTm can be illuminated in the proposed space- time correlation functions (17). The resultant expression is the space-time correlation function given by Wani et al. [50], which is the special case of the proposed channel model. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Io 2p c2t þ dt2 Io 2p c2r þ dr2 ð19Þ Rpq;p~q~½s 2pIo ð2t Þ 2pIo ð2r Þ The expression (19) can be further explored for the space-time correlation functions in isotropic scattering environments, that is proposed by Pätzold et al. [39] by keeping eccentricity r ; t ¼ 1. 5 Results and Description In this section, we describe the theoretical results obtained from the derived correlation functions (19) and its impact on the MIMO channel capacity. It can be observed that the derived joint correlation expressions (19) is the function of various parameters that are linked with physical MIMO system and the propagation channel environment. For the ease of discussion our focus is mainly on the receive correlation function (16), the receive correlation can be directly obtained by replacing subscript index t with r. In all simulations, a normalized sampling period fRmax Ts ¼ :01, is used (where fRmax ; fTmax are the maximum Doppler frequencies and Ts is the sampling period). Furthermore, the orientation angle of the antenna arrays, elevation angle of scatterers and other parameters are mentioned on the respective plots for each simulation. Different 2D and 3D plots of the correlations among MIMO channel coefficients are obtained for observation and discussion. The simulation plots of space-time correlation among receive antenna array elements is shown in Figs. 3 and 4. It is observed from the graph that the correlation decays rapidly in both space and temporal domains by increasing distance (dr ) between antenna array elements and normalized time delay. The temporal correlation is dependent on the velocity vr , AoA/AoD of multipaths, carrier frequency fc and the speed of light c. Therefore, for the design of MIMO M2M communication system in the rich scattering environment, antenna spacing, and velocity of the mobile stations are the constraints that have significant impact on the system performance. Similarly, the correlations among receive antenna elements are obtained by varying the eccentricity (r ) of the receiver ellipse as shown in Fig. 5. It is observed from the graph that 123 Author's personal copy M. Y. Wani, N. M. Khan [ Correlation ] 1 0.8 0.6 0.4 0.2 0 0 1 τ fRmax 2 3 0 2 Fig. 3 3D space-time correlation function qr ðdr ; sÞ of ðbr ¼ 20 ; ar ¼ 60 ; wr ¼ 30 ; hr ¼ 60 ; ar ¼ 100 m; br ¼ 50 mÞ 8 6 4 σr / λ 22 MIMO M2M channel, 1 δr=λ δr=λ/2 Receive Correlation 0.8 δr=λ/4 δr=λ/5 0.6 0.4 0.2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 τ fRmax Fig. 4 Effect of spacing between antenna elements on correlation function. The curves are obtained using the parameter ðbr ¼ 20 ; ar ¼ 60 ; wr ¼ 30 ; hr ¼ 60 ; ar ¼ 100 m; br ¼ 50 mÞ the correlation has a decreasing trend as of eccentricity is decreased from 0.9 (elliptical channel model) to 0 (circle model). This indicates that the capacity is degraded when the mobile units resides in the narrow streets or canyons, similar findings were observed by Abouda et al. [3]. Therefore, for the design and implementation of MIMO M2M communication system for such propagation environments, the simulation results based on elliptical channel model are more appropriate than circular channel model. Correlation among antenna elements is also evaluated for different relative velocities of the mobile nodes. It is discerned from the simulation results as shown in Fig. 6, that the correlation has a decreasing trend with the increasing relative velocity of the mobile nodes. Similar observations were reported in [17, 23]; however, increasing velocity leads to other severe constraints in the wireless channel that degrade the system throughput. 123 Author's personal copy Characterization of 3D Elliptical Spatial Channel Model… Fig. 5 3D space-time correlation function with respect to eccentricity, ðdr ¼ 0:5k; br ¼ 20 ; ar ¼ 60 ; wr ¼ 30 ; hr ¼ 60 1 Vr = 120km/h Vr = 100km/h Vr = 90km/h Vr = 60km/h Receive correlation 0.8 0.6 0.4 0.2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 τ fRmax Fig. 6 2D space-time correlation function for different velocities, ðdr ¼ 0:5k; br ¼ 20 ; ar ¼ 60 ; wr ¼ 30 ; hr ¼ 60 ; ar ¼ 100 m; br ¼ 50 mÞ The proposed channel model is transformed into existing 3D circular-based cylindrical geometrical channel model [56] by adjusting ai ¼ bi where, i ¼ r; t. The comparison of correlation functions of the deduced 3D geometrical channel model and the proposed model is depicted in Fig. 7. It is observed from the correlation curves that the space-time correlations using circular-based channel model is lower as compared to elliptical-based channel model. Since, the streets and canyons can be modeled more appropriately exploiting elliptical shape as discussed earlier in introductory section. Therefore, the theoretical results, based on the elliptical geometry can predict more appropriately the achievable capacity of MIMO M2M communication systems in streets and canyons. By adjusting the parameters of the proposed model the correlation curves are validated by comparing with the measurement results obtained from experimental campaigns carried 123 Author's personal copy M. Y. Wani, N. M. Khan 1 Circular-based Cylindrical Model Elliptical-based Cylindrical Model 0.8 Receive Correlation 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 τ fRmax Fig. 7 Comparison of space-time correlation function of circular-based cylindrical model with the proposed channel model, ðdr ¼ 0:5k; ar ¼ 60 ; wr ¼ 30 ; hr ¼ 60 ; br ¼ 20 Þ Space−time correlation 1 Measurement Proposed 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 10 λ [m] Fig. 8 Comparison of space-time correlation function with the measurement results out for COST2100 channel model in the outdoor communication environment [59]. It is observed from Fig. 8 that the proposed model is in close agreement with the measurement data. Furthermore, the proposed model is more dynamic in nature, i.e. various other existing geometrical channel models can become the special cases of it. By adjusting the appropriate values for ai ; bi ; wi ; bi ; di ; ði ¼ t; rÞ parameters in Eq. (17) of the proposed channel model. The 2D and 3D geometrical channel models listed and depicted in Table 2 can be deduced, 123 Communication scenarios MIMO M2M MIMO M2M MIMO M2M MIMO M2M Scattering model Proposed Zajić et al. [56] Pätzold et al. [39], Stüber et al. [27] and Zajić et al. [53] Wani et al. [50] Table 2 Generalization of proposed geometrical channel model 3D elliptical-based cylindrical model 3D circular-based cylindrical model 2D circular model 2D elliptical model ar ¼ br , at ¼ bt at ¼ bt , ar ¼ br , bt ¼ br ¼ 0o at 6¼ bt , ar 6¼ br , bt ¼ br ¼ 0 Respective scattering models – Corresponding substitutions y AT(p) Rt AT(p) CT ψT(p) γT vT ST(m) ∼ AT(p) ∼ dp∼m αT(m) βT(m) dpm C’T π−θTAT(p) z y AT(p) Rt AT(p) CT ψT(p) γT vT ST(m) ∼ AT(p) ∼ dp∼m αT(m) βT(m) dpm d C’T π−θTAT(p) z dmn d dnq SR(n) γR ∼ ∼ AR(q) AR(q) π−θR αR(n) βR(m) dnq∼ z Rr C’R vR CR ψR(q) dmn AR(q)’ AR(q) x AR(q)’ AR(q) SR(n) γR ∼ ∼ AR(q) AR(q) π−θR αR(n) βR(m) dnq∼ z Rr C’R vR CR ψR(q) dnq x Geometry of scattering regions Author's personal copy Characterization of 3D Elliptical Spatial Channel Model… 123 Communication scenarios SISO M2M SISO M2M SISO F2M F2M MIMO Scattering model Baltzis et al. [11] Paul et al. [40] Baltzis et al. [30] Abidi et al. [42] Table 2 continued Respective scattering models 2D elliptical model 2D circular model 2D model 2D circular model Corresponding substitutions at ¼ 6 bt , ar 6¼ br , bt ¼ br ¼ 0 , dr ¼ dt ¼ 0 at ¼ bt , ar ¼ br , dt ¼ dr ¼ bt ¼ br ¼ 0 ar ¼ br , at ¼ bt ¼ 0, bt ¼ br ¼ dt ¼ dr ¼ fTmax ¼ 0o at ¼ bt , ar ¼ br ¼ 0, bt ¼ br ¼ 0o Geometry of scattering regions Author's personal copy M. Y. Wani, N. M. Khan 123 Author's personal copy Characterization of 3D Elliptical Spatial Channel Model… 6 Conclusion In this article, 3D eccentricity based elliptical cylindrical geometrical model for MIMO M2M communication scenario is proposed. The geometry of the proposed model is rotatable along x–y plane and its dimensions are adjustable in all axes corresponds to the physical propagation scenario. Based on the proposed model, joint and marginal PDFs of correlation functions among transmitter and receiver antenna array elements is formulated. The expression of correlation function is simulated by changing various parameters of the proposed channel model and obtained correlation results are meticulously described. Correlation results obtained using proposed model show close agreement with the measurement data. Moreover, the proposed model is versatile in nature, by varying the system parameters different existing 2D and 3D geometrical channel models are obtained. References 1. Abdi, A., & Kaveh, M. (2002). A space-time correlation model for multielement antenna systems in mobile fading channels. IEEE Journal on Selected Areas in communications, 20(3), 550–560. 2. Abhayawardhana, V., Wassell, I., Crosby, D., Sellars, M., & Brown, M. (2005). Comparison of empirical propagation path loss models for fixed wireless access systems. In Proceedings of the IEEE Vehicular Technology Conference, vol. 1 (pp. 73–77). IEEE. 3. Abouda, A. A. (2007). Characterization of MIMO channel capacity in urban microcellular environment. Ph.D. Thesis, Electrical and Communications Engineering, Helsinki University of Technology. 4. Abrarov, S., & Quine, B. (2014). Accurate approximations for the complex error function with small imaginary argument. ArXiv preprint arXiv:1411.1024 5. Ahmed, A., Nawaz, S. J., & Gulfam, S. M. (2015). A 3D propagation model for emerging land mobile radio cellular environments. PLoS ONE, 10(8), e0132,555. 6. Akki, A. S. (1994). Statistical properties of mobile-to-mobile land communication channels. IEEE Transactions on Vehicular Technology, 43(4), 826–831. doi:10.1109/25.330143. 7. Akki, A. S., & Haber, F. (1986). A statistical model of mobile-to-mobile land communication channel. IEEE Transactions on Vehicular Technology, 35(1), 2–7. doi:10.1109/T-VT.1986.24062. 8. Asplund, H., Glazunov, A. A., Molisch, A. F., Pedersen, K. I., & Steinbauer, M. (2006). The COST 259 directional channel model-part II: Macrocells. IEEE Transactions on Wireless Communications, 5(12), 3434–3450. 9. Avazov, N., & Pätzold, M. (2012). Design of wideband MIMO car-to-car channel models based on the geometrical street scattering model. Modelling and Simulation in Engineering, 2012, 1–5. 10. Baltzis, K. B. (2011). A simplified geometric channel model for mobile-to-mobile communications. Radioengineering, 20(4), 961–967. 11. Batool, T., Matthias, P., et al. (2011). A geometrical three-ring-based model for MIMO mobile-tomobile fading channels in cooperative networks. EURASIP Journal on Advances in Signal Processing, 2011, 1–13. 12. Bello, P. (1963). Characterization of randomly time-variant linear channels. IEEE Transactions on Communications Systems, 11(4), 360–393. 13. Blum, R. S., & Winters, J. H. (2002). On optimum MIMO with antenna selection. In Proceedings of the IEEE International Conference on Communications (ICC) 2002 (Vol. 1, pp. 386–390). 14. Chelli, A., & Patzöld, M. (2007). A MIMO mobile-to-mobile channel model derived from a geometric street scattering model. In Proceedings of the Wireless Communication Systems (ISWCS) (pp. 792–797). 15. Chen, J., & Pratt, T. G. (2012). A three-dimensional geometry-based statistical model of 2 2 dualpolarized MIMO mobile-to-mobile wideband channels. Modelling and Simulation in Engineering, 2012, 6. 16. Cheng, X., Wang, C. X., Laurenson, D. I., Salous, S., & Vasilakos, A. V. (2009). An adaptive geometrybased stochastic model for non-isotropic MIMO mobile-to-mobile channels. IEEE Transactions on Wireless Communications, 8(9), 4824–4835. 17. Chu, M. J., & Stark, W. E. (2000). Effect of mobile velocity on communications in fading channels. IEEE Transactions on Vehicular Technology, 49(1), 202–210. 123 Author's personal copy M. Y. Wani, N. M. Khan 18. Crane, R. K. (1980). Prediction of attenuation by rain. IEEE Transactions on Communications, 28(9), 1717–1733. 19. Ertel, R. B., & Reed, J. H. (1999). Angle and time of arrival statistics for circular and elliptical scattering models. IEEE Journal on Selected areas in Communications, 17(11), 1829–1840. 20. Foschini, G. J., & Gans, M. J. (1998). On limits of wireless communications in a fading environment when using multiple antennas. Wireless Personal Communications, 6(3), 311–335. 21. Gesbert, D., Bolcskei, H., Gore, D. A., & Paulraj, A. J. (2002). Outdoor MIMO wireless channels: Models and performance prediction. IEEE Transactions on Communications, 50(12), 1926–1934. 22. Goldsmith, A., Jafar, S. A., Jindal, N., & Vishwanath, S. (2003). Capacity limits of MIMO channels. IEEE Journal on Selected Areas in Communications, 21(5), 684–702. 23. Haibin, L., Liangjun, X., Jibing, Y., & Linhua, Z. (2013). Modeling and analysis on mobile-to-mobile cascade channel of amplify-and-forward two-way relay networks. In Proceedings of the National Doctoral Academic Forum on Information and Communications Technology (pp. 1–6). IET. 24. Holter, B. (2001). On the capacity of the MIMO channel: A tutorial introduction. In Proceedings of the IEEE Norwegian Symposium on Signal Processing (pp. 167–172). 25. Janaswamy, R. (2002). Angle of arrival statistics for a 3D spheroid model. IEEE Transactions on Vehicular Technology, 51(5), 1242–1247. 26. Jeffrey, A., & Zwillinger, D. (2007). Table of integrals, series, and products. Cambridge: Academic Press. 27. Kang, H., Stuber, G., Pratt, T. G., & Ingram, M. A. (2004). Studies on the capacity of MIMO systems in mobile-to-mobile environment. In Wireless Communications and Networking Conference, 2004. WCNC. 2004 IEEE (Vol. 1, pp. 363–368). IEEE. 28. Karedal, J., Tufvesson, F., Czink, N., Paier, A., Dumard, C., Zemen, T., Mecklenbräuker, C. F., & Molisch, A. F. (2009). Measurement-based modeling of vehicle-to-vehicle MIMO channels. In Proceedings of the IEEE International Conference on Communications (pp. 1–6). IEEE. 29. Khan, N. M. (2006). Modeling and characterization of multipath fading channels in cellular mobile communication systems. Ph.D. Thesis, University of New South Wales (UNSW) Sydney, Australia. 30. Khan, N. M., Simsim, M. T., & Rapajic, P. B. (2008). A generalized model for the spatial characteristics of the cellular mobile channel. IEEE Transactions on Vehicular Technology, 57(1), 22–37. 31. Liberti, J. C., & Rappaport, T. S. (1996). A geometrically based model for line-of-sight multipath radio channels. In Proceedings of the IEEE Vehicular Technology Conference (Vol. 2, pp. 844–848). 32. Michailidis, E. T., Theofilakos, P., & Kanatas, A. G. (2012). A 3D model for mimo mobile-to-mobile amplify-and-forward relay fading channels. In Proceedings of the European Conference on Antennas Propagation (EUCAP) (pp. 2073–2077). IEEE. 33. Narrainen, J., Besnier, P., & Ibambe, M. G. (2016). A geometry-based stochastic approach to emulate V2V communications main propagation channel metrics. International Journal of Microwave and Wireless Technologies, 8(03), 455–461. 34. Nawaz, S. J., Qureshi, B. H., & Khan, N. M. (2010). A generalized 3D scattering model for a macrocell environment with a directional antenna at the BS. IEEE Transactions on Vehicular Technology, 59(7), 3193–3204. doi:10.1109/TVT.2010.2050015. 35. Olenko, A. Y., Wong, K. T., & Hui-On Ng, E. (2006). Analytically derived uplink/downlink TOA and 2-D-DOA distributions with scatterers in a 3-D hemispheroid surrounding the mobile. IEEE Transactions on Antennas and Propagation, 54(9), 2446–2454. 36. Parsons, J., & Turkmani, A. (1991). Characterisation of mobile radio signals: Model description. In Proceedings of the IEE Communications, Speech and Vision (Vol. 138, pp. 549–556). IET. 37. Patzöld, M., Hogstad, B. O., & Youssef, N. (2008). Modeling, analysis, and simulation of MIMO mobile-to-mobile fading channels. IEEE Transactions on Wireless Communications, 7(2), 510–520. 38. Pätzold, M., & Hogstad, B. O. (2008). A wideband MIMO channel model derived from the geometric elliptical scattering model. Wireless Communications and Mobile Computing, 8(5), 597–605. 39. Patzold, M., Hogstad, B. O., & Youssef, N. (2008). Modeling, analysis, and simulation of MIMO mobile-to-mobile fading channels. IEEE Transactions on Wireless Communications, 7(2), 510–520. 40. Paul, B. S., Hasan, A., Madheshiya, H., & Bhattacharjee, R. (2009). Time and angle of arrival statistics of mobile-to-mobile communication channel employing circular scattering model. IETE Journal of Research, 55(6), 275–281. 41. Pätzold, M., Hogstad, B. O., Youssef, N., & Kim, D. (2005). A MIMO mobile-to-mobile channel model: Part I-the reference model. In Proceedings of the IEEE 16th International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC) (Vol. 1, pp. 573–578). 42. Petrus, P., Reed, J. H., & Rappaport, T. S. (2002). Geometrical-based statistical macrocell channel model for mobile environments. IEEE Trans. Commun., 50(3), 495–502. doi:10.1109/26.990911. 123 Author's personal copy Characterization of 3D Elliptical Spatial Channel Model… 43. Qu, F., Wang, F. Y., & Yang, L. (2010). Intelligent transportation spaces: Vehicles, traffic, communications, and beyond. IEEE Communications Magazine, 48(11), 136–142. 44. Rade, L., & Westergren, B. (2013). Mathematics handbook for science and engineering. Berlin: Springer. 45. Raghavan, V., Kotecha, J. H., & Sayeed, A. M. (2010). Why does the kronecker model result in misleading capacity estimates? IEEE Transactions on Information Theory, 56(10), 4843–4864. 46. Riaz, M., Khan, N. M., & Nawaz, S. J. (2015). A generalized 3D scattering channel model for spatiotemporal statistics in mobile-to-mobile communication environment. IEEE Transactions on Vehicular Technology, 64(10), 4399–4410. 47. Riaz, M., Nawaz, S. J., & Khan, N. M. (2013). 3D ellipsoidal model for mobile-to-mobile radio propagation environments. Wireless Personal Communications, 72(4), 2465–2479. 48. Telatar, E. (1999). Capacity of multi-antenna Gaussian channels. European Transactions on Telecommunications, 10(6), 585–595. 49. Wang, L., Qi, X., Xiao, J., Wu, K., Hamdi, M., & Zhang, Q. (2016). Exploring smart pilot for wireless rate adaptation. IEEE Transactions on Wireless Communications, 15(7), 4571–4582. 50. Wani, M. Y., Riaz, M., & Khan, N. M. (2016). Modeling and characterization of mimo mobile-tomobile communication channels using elliptical scattering geometry. Wireless Personal Communications, 91(2), 509–524. 51. 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. 52. Yuan, Y., Cheng, X., Wang, C. X., Laurenson, D. I., Ge, X., & Zhao, F. (2010). Space-time correlation properties of a 3D two-sphere model for non-isotropic MIMO mobile-to-mobile channels. In Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM) (pp. 1–5). IEEE. 53. Zajić, A. G., & Stüber, G. L. (2006). Space-time correlated MIMO mobile-to-mobile channels. In Proceedings of the IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (pp. 1–5). 54. Zajić, A. G., & Stüber, G. L. (2008). Space-time correlated mobile-to-mobile channels: Modelling and simulation. IEEE Transactions on Vehicular Technology, 57(2), 715–726. 55. 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. 56. Zajić, A. G., & Stüber, G. L. (2008). Three-dimensional modeling, simulation, and capacity analysis of space-time correlated mobile-to-mobile channels. IEEE Trans. Veh. Technol., 57(4), 2042–2054. doi:10.1109/TVT.2007.912150. 57. Zajic, A. G., & Stuber, G. L. (2009). Three-dimensional modeling and simulation of wideband MIMO mobile-to-mobile channels. IEEE Transactions on Wireless Communications, 8(3), 1260–1275. 58. Zhou, J., Jiang, H., & Kikuchi, H. (2015). Generalised three-dimensional scattering channel model and its effects on compact multiple-input and multiple-output antenna receiving systems. IET Communications, 9(18), 2177–2187. 59. Zhu, M., Eriksson, G., & Tufvesson, F. (2013). The COST 2100 channel model: Parameterization and validation based on outdoor MIMO measurements at 300 MHz. IEEE Transactions on Wireless Communications, 12(2), 888–897. 123 Author's personal copy M. Y. Wani, N. M. Khan Mohd Yaqoob Wani received his B.Sc. degree in electronics from the University of Kashmir, Srinager, India, in 1991 and MS Telecommunication and Networking from the Iqra University, Islamabad Pakistan in 2008. At present, he pursues his Ph.D. degree in electrical engineering at Capital University of Science and Technology (CUST), Islamabad, Pakistan. He is currently with the Acme Center for Research in Wireless Communications (ARWiC) at CUST. Currently, he is also working as Assistant Professor with the University of Lahore, Islamabad, Pakistan. His research interests include channel modeling and characterization, cellular mobile communication networks. 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