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Modeling and Characterization of MIMO Mobile-to-Mobile Communication Channels Using Elliptical Scattering Geometry M. Yaqoob Wani, M. Riaz & Noor M. Khan Wireless Personal Communications An International Journal ISSN 0929-6212 Volume 91 Number 2 Wireless Pers Commun (2016) 91:509-524 DOI 10.1007/s11277-016-3473-8 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 (2016) 91:509–524 DOI 10.1007/s11277-016-3473-8 Modeling and Characterization of MIMO Mobile-toMobile Communication Channels Using Elliptical Scattering Geometry M. Yaqoob Wani1 • M. Riaz1 • Noor M. Khan1 Published online: 4 July 2016 Springer Science+Business Media New York 2016 Abstract In this paper, we develop a two-dimensional (2-D) elliptical geometrical scattering model for multiple-input multiple-output (MIMO) mobile-to-mobile (M2M) communication channels. The elliptical geometry is supposed to be an appropriate approach to model streets and canyons in M2M communication environment. We assume that both mobile stations (MSs) are located at the centers of ellipses and are surrounded by uniformly distributed scatterers present on the elliptical loci. The equal spacing between two consecutive scatterers on the elliptical loci forces the angle-of-arrival (AoA) or angle-ofdeparture (AoD) distribution at either of the MSs to be non-isotropic. We provide an empirical model for such a non-isotropic AoA and compare its results with the numerical curves of the elliptical geometry which results in excellent agreement. Utilizing the nonisotropic AoA and the proposed geometrical model, we derive closed-form expressions for the marginal and joint correlation function of the channel coefficients. We provide various plots to analyze the correlation among the diffused components of M2M MIMO communication link. Furthermore, a comparison of the correlation curves obtained from the mathematical expression of the proposed model is carried out with the existing results in the literature. In order to validate the proposed model, the elliptical geometrical shape is transformed into circular one. The resulting comparative analysis verifies that the circular geometrical models are the special cases of our proposed model. Keywords Correlation function MIMO Channel modeling Elliptical Scattering & M. Yaqoob Wani [email protected] M. Riaz [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, 44000 Islamabad, Pakistan 123 Author's personal copy 510 M. Y. Wani et al. 1 Introduction Continuous demands for high data rates in all aspect of wireless communication systems are becoming a constant driving force in today’s research arena. Situations become even more demanding for mobile-to-mobile (M2M) communication networks where both ends of the links are surrounded by huge number scattering objects. Such a rich scattering environment can offer feasible situation for the higher data rate demands using multipleinput and multiple-output (MIMO). A MIMO system composed of multiple antennas at its both ends can benefit from this rich scattering environment providing spatial multiplexing that can guarantee maximum possible data rates [11, 12, 25]. This is only possible if we can have good understanding of spatial characteristics of radio fading channel and can be able to observe correlations among various elements of the multiple antennas. Communication channels are modeled in the literature exploiting deterministic, stochastic and geometry-based stochastic channel modelings [4, 16]. Deterministic channel modeling approach is employed when locations of the transmitter, receiver and scatterers are known. In M2M communication environment both the communicating nodes are mobile and their locations are not fixed, therefore deterministic channel modeling approach is not appropriate [24]. The stochastic channel modeling approach is based on probability distribution functions (pdfs) which give insight about the statistical behavior of the channel and is an empirical approach to analyze the parameters of the channel in a stochastic manner [7]. Since the vibrant behavior of the M2M wireless channel makes the propagation channel non-stationary, stochastic modeling approach is incapable of modeling the impact of physical scattering phenomenon in such a dynamic propagation environment. However, this issue can be resolved through assuming scattering environment quasi-stationary and using geometry-based characteristics of the channel. Exploiting the geometrical channel models, mathematical expressions for various pdfs and statistical parameters of M2M wireless channel may be derived. Understanding the importance of the geometrical stochastic channel modeling, various geometrical channel models have been proposed in the literature for M2M communication environment. Akki and Haber were the first researchers who proposed a fundamental channel model for M2M communication channels [3]. Using their model, they derived expressions for the pdfs of the time-correlations in channel coefficients. Using these correlations, the authors obtained the power spectral density. Patzöld et al. [19] extended the work in [3] and presented a frequency non-selective circular two-ring model for MIMO M2M communication environment. Assuming infinite scatterers around the mobile stations (MSs), the authors derived expressions for transmit and receive correlation functions and provided a framework for the channel capacity. In [26], under the assumption of wide sense stationary (WSS) scattering, closed-form expressions for the time-autocorrelation function and Doppler spectrum are provided for M2M communication environment. In [18], a general analytical solution was provided for the three-dimensional (3D) space–time cross-correlation function (CCF). It was proved that the 3D CCF is the product of two dimensional (2D) space–time correlation functions of M2M communication channel. Furthermore, the model was simulated for isotropic and non-isotropic environments. Another concept that scatterers are not located exactly at the boundary of the circular rings but at varying distances from the center of the rings, was presented in the form of two-erose-ring model in [27]. The authors also derived formulas for the complex envelope of the diffused antenna components. However, the empirical results were presented on the basis of simulation for 123 Author's personal copy Modeling and Characterization of MIMO Mobile-to-Mobile… 511 the space–time correlation function assuming an isotropic scattering environment. The work in [27] was extended by Riaz et al. [23] for the derivation of mathematical expressions for the time-autocorrelation function and Doppler Spectrum. In [20, 21], geometrical channel models have been proposed by employing circular disc and circular strip shapes for M2M scattering environment. The authors derived closed-form expressions for the pdfs of angle-of-arrival (AoA) and time-of-arrival (ToA) assuming uniform scatterers distribution around MSs. In the previous discussion, two-ring models and two-disc models for single-input single-output (SISO) and MIMO links in M2M communication environment are presented. Such channel models can become good foundations for future research but may not be realistic. The realistic approach to model the streets and canyons for M2M communication scenario is, however the elliptical shape based channel modeling proposed in [5, 6]. In [6], an elliptical scattering model for SISO M2M communication environment is proposed. The authors assumed that scatterers are uniformly distributed within the elliptical regions around the MSs. Using their model, the authors derived expressions for the pdf of AoA and ToA and analyzed angular spread and delay variations. The same statistical parameters are also analyzed by proposing a more realistic and flexible channel model in [5], where the authors introduced adjustable scatterer-free regions in the previously proposed elliptical channel model. These elliptical channel models are better than circular models. However, these models can be made more beneficial if multiple antenna arrays are introduced in these models. Furthermore, in the two-ring circular channel models for M2M communication environment, isotopic scattering environment is considered. However, in [18], non-isotropic scattering environment is taken into consideration without using any geometry. Street canyons in urban areas, highways with road-side vegetation in suburban and rural areas and deep-cut railway tracks in hilly areas are the demanding candidates for nonisotropic physical channel modeling in MIMO M2M communication links. The layout of such propagation environments resembles more closely with elliptical shape than circular. Therefore, such environment can be modeled more realistically with elliptical geometrical shape. Moreover, elliptical geometrical shapes have been strongly recommended in the literature [11] to be used in environment-specific vehicular channel modeling. Therefore, the majority multipath components are expected from roadside scatterers, which benefits a MIMO system [11, 12, 25]. Utilizing the multiple antenna array at both sides of the mobile communication link increases system capacity and QoS dramatically [11, 14]. This motivates us to propose a channel model that addresses all the above-mentioned benefits in this paper. We propose two-elliptical eccentricity-based channel model for MIMO M2M communication environment and derive mathematical expressions for the pdf of AOA/AOD which are further verified through simulation results. Moreover marginal correlation functions at transmitter and receiver antennas for M2M communication channel are derived which are then used to develop joint transmit-receive correlation functions. The rest of the paper is organized as follows: Sect. 2 describes the system model of the proposed geometrical channel model and pdf of AoA is presented in Sect. 3. Expressions for transmit and receive correlation functions are provided in Sect. 4. Discussion of the theoretical and simulation results are presented in Sect. 5. Finally, conclusion of the paper is given in Sect. 6. 123 Author's personal copy 512 M. Y. Wani et al. q p St 1 At bt at t dtp dpq Sr Vr dqr Vt φt βt αt 1 Ar αr φr δr βr br 2 2 Ar At ar d Fig. 1 Proposed elliptical channel model for MIMO mobile-to-mobile communication environment 2 System Model In this section, we present the system architecture of the proposed geometrically-based two-elliptical channel model for MIMO M2M communication environment as shown in Fig. 1. Each of the transmitting and receiving MSs is denoted by Mm , where the subscript m ¼ t or r to denote Mt and Mr , respectively. The MSs are located at the centers of the ellipses having major axes at and ar and minor axes bt and br with eccentricities et and er, respectively. The MSs are separated by a distance d such that d ðat þ ar Þ and are surrounded by uniformly distributed scatterers present on the boundaries of the ellipses. The ellipses are made rotatable with 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. Both MSs, being at the same hight are equipped with multiple antennas with configuration Pt Qr , where Pt and Qr are the number of antennas mounted on Mt and Mr , respectively. For simplicity, we take Pt = Qr = 2. However, the results can be derived for ðpÞ any configuration. The transmit and receive antenna elements are denoted by At and AðqÞ r and the separations between antenna elements on Mt and Mr are denoted by dt and dr , ðpÞ respectively. It is assumed that dt bt and dr br . The symbols St and SðqÞ r in Fig. 1 denote the pth and qth scattering objects around Mt and Mr , respectively. Moreover, the Mt and Mr are moving with velocities vt and vr making angles at and ar with the x-axis. Angle-of-departure (AoD) and AoA of the pth and qth multipath signals are designated as /tðpÞ and /ðqÞ r , respectively. The tilt angles of the transmit and receive antennas are denoted by bt and br. The proposed two-elliptical model is based on the following general assumptions, 1. 2. 3. 4. 5. Double bounce scattering model is considered between communicating MSs. Infinite number of scatterers are uniformly distributed on the boundaries of ellipses. Power is equally reflected from all the scatterers. Each scatterer behaves as an isotropic antenna. The scatterers are fixed and MSs are quasi-stationary for a short period of time. 123 Author's personal copy Modeling and Characterization of MIMO Mobile-to-Mobile… 513 3 Distribution of AoA/AoD For the ease of derivation, in most of the geometrical channel modeling approaches, uniform distribution of scattering objects is assumed thereby relating the scatterer distribution with the AoA. A relationship between arc-lengths and the arriving angles of the multipath signals at the center of the proposed geometry has been presented in the literature for circular scattering models [10]. However, such relationship between the two parameters does not exist for the elliptical geometry. In this section, we find a connection between the arc-lengths with the distribution of AoA of the multipath signals numerically as well as theoretically for the proposed elliptical geometry where scatterers are located at its boundary. Researchers usually use different scattering distributions like Gaussian, Laplacian, Uniform and von Mises distributions for the pdf of AoA/AoD in their proposed geometric channel models. The physical scattering environment around the MSs can either be isotropic or non-isotropic like in circular or elliptical models, respectively. The von Mises probability density function (pdf) is considered matching well in fitting the measured results of azimuth dispersion in mobile radio channels [22]. Abidi et al. in [1] introduced von Mises distribution for non-isotropic scattering environments which is given as, pðaÞ ¼ 1 exp½j cosða lÞ; 2pIo ðjÞ ð1Þ where Io ð:Þ is the modified Bessel function of first kind of order zero, l 2 ðp; pÞ represents the mean angle and j controls the spread of scatters around the mean l. Above mathematical expression of the von Mises provided convenience to express closed-form solutions for correlations functions and other coefficients of the wireless channel. Taking this advantage, authors in [17, 30] derived closed-form expressions of the correlation function among different MIMO channel coefficients for non-isotropic scenarios. In order to obtain the distribution of the AoA of the multipath signals in the proposed scattering model depicted in Fig. 1. One need to concentrate on the non-isotropicity present in the proposed system model. This requires establishing a relationship of any of the abovementioned probability distribution models with dimension (shape and size) of the physical scattering environment that creates non-isotropicity. For the calculation of AoA distribution, we consider an arbitrary elliptical region around the Mr shown in Fig. 2. We create some N point-objects as scattering points on the boundary of the ellipse around either of the MSs of the communication link. The scattering objects are located at points x_1, y_1, (x_2, y_2, (x_3, y_3,..., (xN, yN) in the Cartesian coordinates system. Equal spacing between these points is calculated by dividing the perimeter of the ellipse by N number of scattering points as shown in Fig. 2. Knowing the arc-length and using the coordinates of the scattering points, straight lines are drawn from the center of the ellipse to the coordinates of the scattering objects. Using cosine laws, angle between two consecutive lines (e.g., r1 and r2 ) is measured. Using the same procedure, angle between rest of the pairs of these lines are measured. The resultant angles are then used to draw histogram for different values of the eccentricities er of the ellipse around Mr , as shown in Fig. 3. The figure shows that the pdf of AoA of multipath signals changes to uniform as the geometry of the scattering objects changes from an elliptical to a circular shape. The following empirical formula 123 Author's personal copy 514 M. Y. Wani et al. 15 10 r3 r 5 2 r1 0 r r -5 P P-1 -10 -15 0 5 10 15 20 25 30 35 40 45 50 Fig. 2 Equal distribution of scattering points on the boundary of the ellipse for a specific dimension Probability Distribution Function 5 x 10 -3 ε=0.92 ε=0.80 ε=0.60 ε=0 4.5 4 3.5 3 2.5 2 1.5 50 100 150 200 250 300 350 Angle-of-Arival [Degrees] Fig. 3 Rate of occurrence of the AoA using elliptical scattering loci of various eccentricities around Mt and Mr for the pdf of AoA is also fitted which is validated by the calculated result as shown in Fig. 4. pð/m Þ ¼ 1 2 em cos 2/m 2pIo ð2m Þ where Io ðÞ is the modified bessel function of zeroth order and m ¼ eccentricity of the elliptical loci around Mt or Mr . 123 ð2Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 b2m =a2m is the Author's personal copy Modeling and Characterization of MIMO Mobile-to-Mobile… Probability Distribution Function 14 x 10 515 -3 Simulation Theoretical 12 10 8 6 4 2 0 50 100 150 200 250 300 350 Angle-of-Arival [Degrees] Fig. 4 Comparison of theoretical results of the proposed empirical expression for the pdf of AoA with the numerical results from the geometry of the system model 4 Derivation of the Correlation Function In this section, we derive an expression for CCF using the proposed elliptical channel model for MIMO M2M communication environment. Let the MSs are located at certain points (i.e., Xo, Yo) in the Cartesian coordinates systems. The generalized equation for each ellipse, having semi-major am and semi-minor bm axes, can be expressed as, 2 ðxm Xo Þ cos /m þ ðym Yo Þ sin /m a2m ð3Þ 2 ðxm Xo Þ sin /m þ ðym Yo Þ cos /m þ ¼1 b2m The dynamic radius, r, of the ellipse varies from its minimum to maximum according to the length of minor and major axes of the ellipse, respectively. An expression for this dynamic radius is given by, am bm r ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 am sin /m þ b2m cos2 /m ð4Þ The dynamic radius of the ellipse given in (4) is plotted against AoA/AoD shown in Fig. 5. This dynamic radius can be linked with the distribution of the scattering points lying on the boundary of the ellipse. From these results, it can be concluded that when scatterers are uniformly distributed on the boundary of the ellipse, of any arbitrary eccentricity, the AoA emerges as shown in Fig. 3. In the proposed geometry depicted in Fig. 1, it can be observed that a signal that is ðpÞ ðpÞ transmitted from the transmit antenna element At , first strikes at the scatterer St of the scattering ellipse surrounding the transmitter and then travels to the scatterer SrðqÞ of the 123 Author's personal copy 516 M. Y. Wani et al. 50 Radius of ellipse [meters] 45 40 35 30 25 20 15 10 0 50 100 150 200 250 300 350 φ [in degrees] Fig. 5 Radius of ellipse w.r.t. AoA scattering ellipse surrounding the receiver, and finally reaches the receiving antenna element AðqÞ r . Mathematically, this propagation length of the signal can be expressed as, D ¼ dtp þ dpq þ dqr ð5Þ ðpÞ where dtp = distance from transmitter Mt to the scattering point St , dpq = distance from ðpÞ St scattering point to the scattering point SrðqÞ , dqr = distance from scattering point SðqÞ r to the receiver Mr : As each of the MS is moving with velocity vt (or vr ) causing maximum Doppler ftmax ¼ vt =k (or frmax ¼ vr =k), exploiting the proposed geometrical model, an expression ð1Þ for the channel of the communication link from At to Að1Þ r can be described as in [18], P;Q ðpÞ ðqÞ 1 X fpq ej½2pðfT þfr Þtþhr þhpq þho h11 ðtÞ ¼ lim pﬃﬃﬃﬃﬃﬃﬃ P;Q!1 PQ p;q¼1 fpq ¼ xp yq zpq , where ðpÞ j2p k ðrt cos /t rr ðqÞ cos /r Þ ðpÞ ft ðpÞ xp ¼ ejpðdt =kÞ cos ð/t bt Þ cosð/ðpÞ t , frðqÞ ð6Þ ðpÞ yq ¼ ejpðdr =kÞ cos ð/r br Þ , zpq ¼ cosð/ðqÞ r e , ¼ ftmax at Þ, ¼ frmax ar Þ, ho ¼ 2p k D, ðr þ r Þ, where h is the constant phase that depends on specific orientations of hr ¼ 2p t r r k the ellipses. Similarly, ho is also a constant phase as interspacing between the MSs is fixed. Due to the constant behavior of these phases, there will be no effect on the statistics of the proposed model, therefore these phases can be neglected. Using the proposed geometrical channel model, one can find the other diffused components i:e:; h12 ðtÞ; ð1Þ h21 ðtÞ and h22 ðtÞ . Space–time correlation function between the transmission links At ð2Þ ð2Þ Að1Þ r and At Ar can be expressed as in [18], q11;22 ðdT ; dr ; sÞ ¼ Efh11 ðtÞh22 ðt þ sÞg 123 ð7Þ Author's personal copy Modeling and Characterization of MIMO Mobile-to-Mobile… 517 where the operator Efg is known as the expectation and is applied on all the random ðqÞ phases, AoA and AoD ði:e:; /ðpÞ t ; /r and hpq Þ in the equation. By substituting the values of the major axis xp and minor axis yq of the ellipse and their conjugate transpose in (6) then we get the following expression for the above CCF as, ( Q P X 1 X E xp xpyq yqzpq zpq q11;22 ðdt ; dr ; sÞ ¼ lim P;Q!1 PQ p¼1 q¼1 ) hn oi ðpÞ ðpÞ ðpÞ ðqÞ ðqÞ ðqÞ exp j 2pðft þ fr ft fr Þt þ hpq hpq ðft þ fr Þs ð8Þ ðpÞ ðpÞ Using boundary condition i.e., ft þ frðqÞ ¼ ft in reduced form can be written as, q11;22 ðdt ; dr ; sÞ ¼ þ frðqÞ iff p ¼ p and q ¼ q, it follows that (8) Q P X h i 1 X ðpÞ x2p y2q exp j2pðft þ frðqÞ Þs PQ p¼1 q¼1 ð9Þ ðpÞ In the above equation, xp and ft are the functions of AoD, /tðpÞ , while yq and frðqÞ are the functions of AoA, /ðqÞ r . It has already been assumed that there are infinite number of scatterers that reside around the MSs that is P; Q ! 1. In such a case, the discrete random variables /tðpÞ and /rðqÞ become continuous random variables and take the form /t and /r . These continuous random variables are represented by certain statistical distributions, denoted by pð/t Þ and pð/r Þ, respectively. Power received at the receiver through each diffused component corresponding to differential angles d/t and d/r is proportional to pð/t Þpð/r Þd/td /r . As the AoA, /r , and AoD, /t , are statistically independent random variables, therefore, the joint transmitreceive space–time correlation function (CCF) in (9) can be decomposed and that can be written in the form of the product of transmit and receive correlation functions as, q11;22 ðdt ; dr ; sÞ ¼ qt ðdt ; sÞqr ðdr ; sÞ ð10Þ where the transmit correlation function qt ðdt ; sÞ can be expressed as, qt ðdt ; sÞ ¼ Zp x2p ðdt ; /t Þ exp½j2pft ð/t Þspð/t Þd/t ð11Þ p Putting the values of xp ðdt ; /t Þ and ft ð/t Þ from (6) and pð/t Þ from (2) in (11), we get, 1 qt ðdt ; sÞ ¼ 2pIo ð2t Þ Zp exp½j2pðdt =kÞ cos ð/t bt Þ exp½j2p p ð12Þ 2 ftmax cosð/ðpÞ t at Þs exp½t cos 2/t d/t Expanding the trigonometric functions in the above equation and rearranging the terms, we have, 123 Author's personal copy 518 M. Y. Wani et al. Zp o h n i 1 2 cosð/ qt ðdt ; sÞ ¼ exp 2p jðd =kÞ cosðb Þ jðV =kÞs cosða Þ þ Þ t t t t t t 2pIo ð2t Þ ð13Þ p o i h n exp 2p jðdt =kÞ sinðbt Þ jðVt =kÞs sinðat Þ d/t Substituting, x1 ¼ 2p jðdt =kÞ cos bt jðVt =kÞs cos at þ 2t and x2 ¼ j2pfðdt =kÞ sin bt ðVt =kÞs sin aT g, (13) takes the form, 1 qt ðdt ; sÞ ¼ 2pIo ð2t Þ Zp exp½x1 cos /t þ x2 sin /t d/t ð14Þ p The above equation is integrated by comparing it with eq. (3.338-4) in [13], we get, qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 2 J qt ðdt ; sÞ ¼ ð15Þ o 2p x1 þ x2 2pIo ð2t Þ where Jo ðÞ is the bessel function of zeroth order. Putting back the values of x1 and x2 , (15) can be expressed in simplified form as, qt ðdt ; sÞ ¼ 1 Jo 2p 4t ðdt =kÞ2 ðsftmax Þ2 22t sftmax cos at 2 Io ðt Þ ! 1=2 dt 2 dt þ j2t cos bt þ 2s ftmax cos ðat bt Þ k k ð16Þ Similar equation can be derived for the correlation function at the receiver end. Thus joint correlation function in (10) can be expressed as, q11;22 ðdt ; dr ; sÞ ¼ 1 Jo Io ð2t ÞIo ð2r Þ 2p 4t ðdt =kÞ2 ðsftmax Þ2 22t sftmax cos at 1=2 dt dt þ j22t cosbt þ 2s ftmax cosðat bt Þ k k 2 ðsfrmax Þ ! Jo 2p 4r ðdr =kÞ2 1=2 dr dr 22r sfrmax cosar þ j22r cosbr þ 2s frmax cosðar br Þ k ! k ð17Þ which is a generalized expression for the joint correlation function of MIMO M2M communication links for more realistic channel models which targets environments like streets and canyons. The following remarks can be made about (17): • If elliptical geometrical shapes are replaced with circular ones, the eccentricities of the ellipses around transmitter and receiver i.e., t and r , would approach to zero. In such a case, the scattering environment will become isotropic and the expression for the joint correlation function in (17) reduces to, 123 Author's personal copy Modeling and Characterization of MIMO Mobile-to-Mobile… q11;22 ðdt ; dr ; sÞ ¼ Jo 519 1=2 dt 2p ðdt =kÞ2 ðsftmax Þ2 2s ftmax cos ðat bt Þ k Jo ! 1=2 dr 2p ðdr =kÞ2 ðsfrmax Þ2 2s frmax cos ðar br Þ k ! ð18Þ which is the result presented in [8, 19, 28, 29] • For a SISO case, substituting the spacing between antenna element equal to zero, we get, ð19Þ q11;22 ð0; 0; sÞ ¼ Jo 2pftmax s Jo 2pfrmax s which is the well known result presented in [2, 3] for SISO M2M communication environment. • If the transmitting Mt is supposed to be fixed; the maximum Doppler would occur only due to the motion of the receiving Mr ; then (19) reduces to, ð20Þ qðsÞ ¼ Jo 2pfrmax s which is the well known CCF of the Jake’s model [9, 15]. 5 Results and Description In this section, description of the obtained theoretical results for the derived correlation function is presented. From (17), it is clear the correlation function depends on various parameters like eccentricities of the ellipses, separation between antenna elements, velocities of the MSs and carrier frequency of the arriving signals. It can be observed that Transmit Correlation Function 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 0 0.1 δt/ λ 0.2 0.3 0.4 0 0.1 0.2 0.3 τ f tmax 0.4 0.5 0.6 0.7 0.8 Fig. 6 Correlation function of the non-isotropically arriving signals using MIMO two-elliptical channel model (at = 50 m, bt = 20 m, d = 100 m, fc = 800 MHz, vt = 100 km/h, at = 0, bt = 90, c = 3 9 108 m/s) 123 Author's personal copy Transmit Correlation Function 520 M. Y. Wani et al. 3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5 0 0.5 1 1.5 2 σ /λ 2.5 3 0 1 0.5 2 1.5 2.5 3 3.5 4 τ.ftmax Fig. 7 Correlation function of the non-isotropically arriving signals using MIMO two-ring channel model (at = 50 m, bt = 20 m, d = 100 m, fc = 800 MHz, vt = 100 km/h, at = 0, bt = 90, c = 3 9 108 m/s) 0.25 ε = 0.92 ε = 0.80 ε = 0.60 ε=0 Cross-Correlations Function 0.2 0.15 0.1 0.05 0 -0.05 -0.1 0 1 2 3 4 5 6 Time [Sec] Fig. 8 Comparison of correlation function on the basis of different values of eccentricity (at = 50 m, bt = 20 m, d = 100 m, fc = 800 MHz, vt = 100 km/h, at = 0, bt = 90, c = 3 9 108 m/s) the correlation between antenna elements is maximum at s ¼ 0 and decays with an increasing value of s. It implies that the MIMO capacity will be minimum at the beginning as the correlation is maximum there and will thus show an increasing trend with decreasing values of correlation. Since in the proposed elliptical model, the scattering objects are assumed to be uniformly spaced on the elliptical loci, the AoA distribution at the receiver, Mr would thus be non-isotropic as shown in Fig. 3. For such non-isotropic environment where a non-isotropic AoA distribution emerges from the varying distances of the scattering points from the Mr , expression for the correlation function is derived as given in (17). Various correlation plots with 3-D and 2-D views are generated for the parameters specified in the captions of the figures. The result presented in Fig. 6 shows excellent agreement with the 123 Author's personal copy Modeling and Characterization of MIMO Mobile-to-Mobile… 521 0.25 λ/2 λ/4 λ/8 Transmit Correlations 0.2 0.15 0.1 0.05 0 -0.05 -0.1 0 1 2 3 4 5 6 7 8 9 10 Time [sec] Fig. 9 Effect of spacing between antenna elements on correlation function, (at = 50 m, bt = 20 m, d = 100 m, fc = 800 MHz, vt = 100 km/h, at = 0, bt = 90, c = 3 9 108 m/s) 1.1 Transmit Correlation 1 0.9 0.8 0.7 0.6 0.5 vt =100 km/hr vt =200 km/hr 0.4 vt =300 km/hr 0 10 20 30 40 50 60 70 80 90 100 Temporal Separation, τ [seconds] Fig. 10 Effect of changing velocity on correlations simulation results given in [18]. This is basically the elliptical geometrical shape that forces the correlation functions, qt ðdt ; sÞ, to be having this typical trend. However, the causes of such trend have not be documented in [18]. If we fix the value of the major axis of an ellipse and increase its minor from its minimum value to a value equal to its major axis making eccentricity equal to zero, then the ellipse will be transformed into a circle. In this case, the correlation function, qt ðdt ; sÞ, shows an isotropic behavior of the arriving signals as shown in Fig. 7. This result also perfectly matches the theoretical results of the correlation curves shown in [18]. From the above figures, it is verified that the circular geometrical model is the special case of our proposed elliptical model. The 2-D curves of the correlation function for 123 Author's personal copy 522 M. Y. Wani et al. different values of eccentricities are shown in Fig. 8. It can be seen that the correlation increases with an increase in eccentricity of the ellipse and becomes maximally uncorrelated when the ellipse is transformed into a circle. The effect of spacing between antenna elements, dt , on the correlation function is shown in Fig. 9. The plots are taken for different values of antenna spacing i.e., dt ¼ k=D where, D ¼ 2; 4; 8. The plots show that the correlation is lesser for larger values of antenna spacing and vice versa. It can be seen from (16) and (17) that correlation between antenna elements depends upon velocity of transmitter and receiver. The effect of motion of transmitter is shown in Fig. 10 for different velocities i.e., vt = 100, 200, and 300 km/h. From these curves, we analyze that correlation values are higher for smaller values of velocities. In other words, we say that correlation decreases with the increase in velocity of the MSs. 6 Conclusion In this paper, we have developed a 2-D elliptical geometrical scattering model for MIMO M2M communication channels. The elliptical geometrical shape is supposed to be a more realistic approach than circular geometry to model the streets and canyons. We have assumed that both MSs are located at the centers of ellipses and surrounded by uniformly distributed scatterers present on the elliptical boundaries. Due to being an elliptical boundaries, the distances of the scattering objects from the MSs are not equal which forces the AoA distribution at either of the MSs to be non-isotropic. We have provided an empirical formula for AoA distribution emerged as the result of such non-isotropic arriving signals. We have compared its results with the numerical curves of the elliptical geometry that resulted into an acceptable agreement. Utilizing the non-isotropic AoA and the proposed geometrical model, we have derived a closed-form expression for the joint transmitreceive correlation function. Since the AoA and AoD are statistically independent, therefore, the joint expression for the correlation function can be expressed as a product of transmit and receive correlation functions. Various plots have been provided to analyze the correlation functions among the diffused components of the M2M MIMO communication link. It can be seen from the curves that the correlation depends on spacing between antenna elements, eccentricity of the ellipses, velocities of the MSs and the frequency of the arriving signals. Furthermore, the correlation curves obtained from the mathematical expression of the proposed model have been compared with the existing results in the literature. In order to validate the proposed model, the elliptical geometrical shape has been transformed into a circular one. The resulting comparative analysis verified that the circular geometrical models in [8, 19, 28, 29] are the special cases of our proposed model. References 1. Abdi, A., Barger, J. A., & Kaveh, M. (2002). A parametric model for the distribution of the angle of arrival and the associated correlation function and power spectrum at the mobile station. 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Closed-form expressions for correlation function and power density spectrum in MIMO mobile-to-mobile channels using two-erose-ring model. In Proceedings of the IEEE international conference on information and communication technologies (ICICT) (pp. 1–5). 24. Seidel, S. Y., Rappaport, T. S. (1992). A ray tracing technique to predict path loss and delay spread inside buildings. In Proceedings of the IEEE global telecommunication conference (GLOBECOM) (pp. 649–653). 25. Telatar, E. (1999). Capacity of multi-antenna gaussian channels. European Transactions on Telecommunications, 10(6), 585–595. 26. Vatalaro, F., & Forcella, A. (1997). Doppler spectrum in mobile-to-mobile communications in the presence of three-dimensional multipath scattering. IEEE Transactions on Vehicular Technology, 46(1), 213–219. doi:10.1109/25.554754. 27. Wei, C., Zhiyi, H., Lili, Z. (2007). A reference model for MIMO mobile-to-mobile fading channel. In Proceedings of IEEE international conference on wireless communications, networking and mobile computing (pp. 228–231). 28. Zajić, A. G., Stuber, G. L. (2006). Space–time correlated mimo mobile-to-mobile channels. In Proceedings of IEEE 17th international symposium on personal, indoor and mobile radio communications (pp. 1–5). 29. Zajić, A. G., & Stuber, 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. doi:10.1109/TVT.2007.912150. 123 Author's personal copy 524 M. Y. Wani et al. 30. Ziółkowski, C., & Kelner, J. M. (2015). Geometry-based statistical model for the temporal, spectral, and spatial characteristics of the land mobile channel. Wireless Personal Communications, 83(1), 631–652. Mohd 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. 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. Muhammad Riaz was born in Pakistan in 1977. He received his M.Sc. degree in electronics from Quaid-i-Azam University, Islamabad, Pakistan, in 2002 and M.S. degree in electronic engineering from Mohammad Ali Jinnah University (MAJU), Islamabad, Pakistan in 2009. At present, he pursues his Ph.D. degree in electronic engineering at MAJU, Islamabad, Pakistan. He is currently with the Acme Center for Research in Wireless Communications (ARWiC) at MAJU. His research interests include modeling and characterization of mobile-tomobile communication channels and channel equalization. Noor M. Khan received his B.Sc. degree in electrical engineering from the University of Engineering and Technology (UET), Lahore, Pakistan, in 1998 and Ph.D. degree in electrical engineering from the University of New South Wales (UNSW), Sydney, Australia in 2006. He held several positions in WorldCall, NISTE, PTCL, UNSW, GIK Institute of Engineering Sciences and Technology, and Mohammad Ali Jinnah University, Pakistan from 1998 to 2015. Currently, he is working as Professor with the Capital University of Science and Technology (CUST), Islamabad, Pakistan. He has served as Chair and Co-Chair of the Technical Program Committees of IEEE International Conference on Emerging Technologies (ICET2012) in 2012 and IEEE International Multi-topic Conference (INMIC-2009) in 2009, respectively. He has been awarded Research Productivity Award (RPA) by the Pakistan Council for Science and Technology (PCST) for the years 2011 and 2012. His research interests include channel modeling and characterization, wireless sensor networks, cellular mobile communication networks and ground-to-air communication systems. 123