A MIMO MOBILE-TO-MOBILE CHANNEL MODEL: PART I THE REFERENCE MODEL

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1 A MIMO MOBILE-O-MOBILE CHANNEL MODEL: PA I HE EFEENCE MODEL Matthias Pätzold 1, Bjørn Olav Hogstad 1, Neji Youssef 2, and Dongwoo Kim 3 1 Faculty of Engineering and Science, Agder University College Grooseveien 36, NO-4876 Grimstad, Norway 2 Ecole Supérieure des Communications de unis oute de aoued, Km 3.5, 2083 El Ghazala, unis, unisia 3 Department of Electrical Engineering and Computer Science Hanyang University, 1271 Sa-dong, Ansan, Kyeong-gi, Korea Abstract In this paper, we propose a multiple-input multiple-output (MIMO) channel model for mobile-to-mobile communications. his so-called MIMO mobile-to-mobile channel model is derived from the geometrical two-ring scattering model under the assumption that both the transmitter and the receiver are moving. In Part I of our paper, we focus on the reference model, which is based on the assumption that the transmitter and the receiver are surrounded by an infinite number of local scatterers. From a wave model, the complex channel gains are derived and their statistical properties are studied. General analytical solutions are provided for the three-dimensional (3-D) space-time cross-correlation function (CCF). We show that this 3-D function can be expressed as the product of two 2-D space-time CCFs, which are introduced as the transmit and receive CCF. Especially for the latter CCFs, closed-form expressions and numerical results are presented in case of isotropic scattering. he procedure presented in our paper provides an important framework for designers of future mobile-to-mobile communication systems to verify new transmission concepts under certain propagation conditions. Keywords Mobile fading channels, mobile-to-mobile channels, MIMO channels, multielement antenna systems, spacetime correlation, MIMO channel capacity. I. INODUCION Mobile-to-mobile communications are expected to play an important role in ad hoc networks and intelligent transportation systems, where the communication links must be extremely reliable. o cope with problems faced by the development and performance investigation of future mobile-to-mobile radio transmission systems, a detailed knowledge of the multipath fading channel and its statistical properties is essential. Several papers dealing with the modelling of mobile-to-mobile propagation channels have been appeared in the literature for singleinput single-output (SISO) systems. For such systems, early studies were reported in [1,2], where the statistics of a narrowband mobile-to-mobile fading channel has been investigated by focussing on the characterization of the time-varying channel transfer function. Later, in [3], the temporal correlation properties and the Doppler power spectral density were studied for the case of 3-D scattering environments. In [4], an approach based on rayoptical techniques was used to model the wave propagation. he resulting impulse responses incorporate the complete channel information in the form of time series, which can be used directly for system simulations. Experimental measurements were reported in [5]. here it was suggested that the statistics of certain mobile-tomobile fading channels can appropriately be described by a multiple ayleigh channel model. Experimental results of the propagation characteristics of radio waves at 60 GHz between running vehicles were reported in reference [6], in which also the feasibility of data transmission was demonstrated. he topic of designing simulation models for SISO mobile-to-mobile channels was addressed in [7]. In this paper, we propose an original model for MIMO mobile-to-mobile fading channels. It is well-known from the studies in [8,9] that MIMO channels offer large gains in capacity over SISO channels. We believe that especially this fact, combined with the opportunities opened by large vehicle surfaces on which multielement antennas can be placed, makes MIMO techniques very attractive for mobile-to-mobile communication systems. he scattering environment around the transmitter and receiver is modeled here by the geometrical two-ring scattering model, which is used as a starting point for the derivation of a proper reference model. he analysis of the statistical properties of the reference model is the main objective of the first part of our paper, while Part II [10] focusses on the design of the corresponding simulation model. he rest of this paper is structured as follows. In Section II, we present the geometrical model for a wireless transmission link between two mobile stations. In Section III, we show how a reference model for the path

2 gains can be derived from the geometrical model and provide general expressions for the transmit and receive correlation functions. Section IV presents some numerical results for the special case of isotropic scattering. In Section V, we finally draw the conclusions. II. HE GEOMEICAL WO-ING MODEL v δ α y A (2) A (1) d 1m β 0 d 2m φ (m) S (m) d mn d n2 A (2) S (n) d n1 δ φ (n) 0 v α A (1) β x In this section, we describe briefly the geometrical two-ring model for a narrowband mobile-to-mobile MIMO channel as shown in Fig. 1. For reasons of brevity, we restrict our considerations on a 2 2 antenna configuration, meaning that both the transmitter and the receiver are equipped with only two omnidirectional antennas. Such an elementary antenna configuration can be used to construct many other types of 2-D multielement antenna arrays, including uniform linear arrays, hexagonal arrays, and circular antenna arrays. he mobile transmitter and the mobile receiver are denoted by MS and MS, respectively. For simplicity, it is assumed that there is no line-of-sight path between the two mobile stations. As can be seen from Fig. 1, the local scatterers around the transmitter, denoted by S (m) (m = 1, 2,...), are located on a ring of radius, while the local scatterers S (n) (n = 1, 2,...) around the receiver lie on a separate ring of radius. he symbols φ (m) and φ (m) denote the angles of departure (AOD) and the angles of arrival (AOA), respectively. In the two-ring model, only local scattering is considered, because the contributions of remote scatterers to the total received power can be neglected due to high path loss. It is assumed that the radii and are small in comparison with D, which is the distance between the transmitter and the receiver, i.e., max{, } D. he antenna spacings at the transmitter and the receiver are denoted by δ and δ, respectively. Since the antenna spacings δ and δ are generally small in comparison with the radii and, we might assume that the inequality max{δ, δ } min{, } holds. he tilt angle between the x-axis and the orientation of the antenna array at the transmitter is denoted by β. Analogously, the tilt angle β describes the orientation of the antenna array at the receiver. Moreover, it is assumed that the transmitter (receiver) moves with speed v (v ) in the direction determined by the angle of motion α (α ). he geometrical two-ring model is an appropriate model for describing propagation scenarios in which neither the transmitter nor the receiver is elevated, but both are surrounded by a large number of local scatterers. Such propagation conditions generally occur between vehicles in urban and suburban areas. D Figure 1: he geometrical two-ring model for a 2 2 MIMO channel with local scatterers around a mobile transmitter MS (left) and a mobile receiver MS (right). III. HE EFEENCE MODEL A. Derivation of the eference Model In this section, we derive the reference model for the MIMO mobile-to-mobile channel. he starting point is the geometrical two-ring model shown in Fig. 1 from which we observe that the mth homogeneous plane wave emitted from the first antenna element A (1) of the transmitter travels over the local scatterers S (m) and S (n) before impinging on the first antenna element A (1) of the receiver. It is assumed that all waves reaching the receiver antenna array are equal in power. he reference model is based on the assumption that the numbers of local scatterers, M and N, around the transmitter and the receiver are infinite. Hence, the diffuse component at the first receive antenna A (1) is composed of an infinite number of homogeneous plane waves, each of which carries power that is negligible compared to the total mean power of the diffuse component. Considering the geometrical model in Fig. 1, the diffuse component of the channel describing the link from A (1) to A(1) can be written as h 11 ( r ) = lim M N M,N m,n=1 E mn e j(θmn+ k (m) r k (n) r k0dmn) (1) where E mn and θ mn denote the joint gain and joint phase shift caused by the interaction of the scatterers S (m) and S (n), respectively, k (m) is the wave vector pointing in the propagation direction of the mth transmitted plane wave, and r denotes the spatial translation vector of the transmitter. Analogously, k (n) is the wave vector pointing in the propagation direction of the nth received plane wave and r denotes the spatial translation vector of the receiver. Furthermore, the symbol k 0 denotes the

3 free-space wave number, which is related to the wavelength λ via k 0 = 2π/λ. Finally, D mn is the length of the total distance which a plane wave travels from A (1) to A (1) via the scatterers S(m) and S (n). Each local scatterer causes a gain and a phase shift. It is generally assumed that the gain and the phase shift introduced by a particular scatterer is dependent on the direction of the incoming plane waves. Since D max{, }, the gain E n m and the phase shift θ n m caused by S (n) on the condition that a wave is received from S (m) are approximately identical to those caused when a wave is received from S (m ), where m m. herefore, we may assume that the gain and phase shift caused by S (n) is independent of the location of S(m) and vice versa. Furthermore, since δ, the waves emerging from different transmit antenna elements arrive at a particular scatterer S (m) at approximately the same angle [11]. Similarly, if δ, then we may state that waves emerging from a particular scatterer S (n) arrive at different receive antennas at approximately the same angle. For these reasons, it has been assumed in [12] that for a particular scatterer S (m) (or S (n) ), the gain and phase shift are the same for waves arriving from (or traveling to) different antenna elements. In our model, we assume that each scatterer S (m) on the ring around the transmitter introduces an infinitesimal constant gain E m = 1/ M and a random phase shift θ m. Analogously, on the receiver side, each scatterer S (n) introduces a constant gain E n = 1/ N and a random phase shift θ n. he phase shifts θ m and θ n are assumed to be independent, identically distributed (i.i.d.) random variables with a uniform distribution over the interval [0, 2π). hus, the joint gains E mn and joint phases θ mn in (1) can be expressed as E mn = E m E n = 1/ MN (2) θ mn = (θ m + θ n ) mod 2π (3) where mod denotes the modulo operation. Note that the sum of θ m and θ n modulo 2π results in i.i.d. random variables θ mn, which are also uniformly distributed over [0,2π). r and k (n) r in (1) are due to he phase changes k (m) the motion of the transmitter and receiver, respectively, and can be expressed as (m) k r = 2πf max cos(φ (m) α )t (4) (n) k r = 2πf max cos(φ (n) α )t (5) where f max = v /λ (f max = v /λ) is the maximum Doppler frequency caused by the movement of the transmitter (receiver). In the reference model, the AOD φ (m) and AOA φ (m) are independent random variables determined by the distribution of the local scatterers. Furthermore, the phase change k 0 D mn in (1) is due to the total distance traveled and can be written as k 0 D mn = 2π λ (d 1m + d mn + d n1 ) (6) where d 1m, d mn, and d n1 are the lengths as illustrated in Fig. 1. hese distances can be approximated by using δ, δ, and 1 + x 1 + x/2 (x 1) as follows: d 1m δ 2 cos(φ(m) β ) (7) d mn D + cos φ (n) cos φ (m) (8) d n1 δ 2 cos(φ(n) β ). (9) Finally, after substituting (2) and (4) (6) in (1) and using (7) (9), we may express the diffuse component of the link from A (1) to A(1) approximately as h 11 (t) = lim M N where M,N 1 MN m,n=1 g mn e j[(2π(f(m) +f (n) )t+θmn+θ0)] (10) g mn = a m b n c mn (11) a m b n c mn f (m) f (n) jπ(δ /λ) cos(φ(m) = e β ) (12) jπ(δ/λ) cos(φ(n) = e β) (13) = e j 2π λ ( cos φ(m) cos φ(n) ) (14) = f max cos(φ (m) α ) (15) = f max cos(φ (n) α ) (16) θ 0 = 2π λ ( + D + ). (17) It should be mentioned that the constant phase θ 0 can be set to zero in (10) without loss of generality. From the statistical properties of the diffuse component in (10) it follows that the mean value and the mean power of h 11 (t) are equal to 0 and 1, respectively. Hence, the central limit theorem states that h 11 (t) is a zero-mean complex Gaussian process with unit variance. Consequently, the envelope h 11 (t) is a ayleigh fading process. One can show that the diffuse component h 22 (t) of the link from A (2) to A (2) can be obtained from (10) by replacing a m and b n by their respective complex conjugates a m and b n. Similarly, the diffuse components h 12 (t) and h 21 (t) can directly be obtained from (1) by performing the substitutions a m a m and b n b n, respectively. he four diffuse components h ij (t) (i,j = 1, 2) of the

4 A (j) A(i) link can be combined to the so-called stochastic channel matrix ( ) h11 (t) h H(t) = 12 (t) h 21 (t) h 22 (t) (18) which describes completely the reference model of the proposed two-ring MIMO frequency-nonselective ayleigh fading channel. he capacity C(t) of this channel (in bits/s/hz) is defined as ( C(t) := log 2 [det I 2 + P )] H(t)H H (t) (19) 2N 0 where det( ) denotes the determinant, I 2 is the 2 2 identity matrix, P designates the total transmitted power allocated uniformly to the two antenna elements of the transmitter, N 0 is the noise power, and ( ) H denotes the complex conjugate (Hermitian) transpose operator. B. he Space-ime CCF of the eference Model According to [13], the space-time CCF between the two links A (1) A(1) and A(2) A(2) is defined as the correlation between the diffuse components h 11 (t) and h 22(t), i.e., ρ 11,22 (δ,δ,τ) := E{h 11 (t)h 22(t + τ)}. (20) Here, the expectation operator applies on all random variables: phases {θ mn }, AOD {φ (m) }, and AOA {φ(n) }. Starting from (10) and making use of the fact that h 22 (t) can be obtained from h 11 (t) by performing the substitutions a m a m and b n b n, we can express the 3-D space-time CCF as ρ 11,22 (δ,δ,τ) = lim M N 1 MN M,N m,n=1 E{a 2 mb 2 ne j2π(f(m) +f (n) )τ }. (21) he above expression has been obtained by averaging over the random phases θ mn. Note that a m and f (m) are both functions of the AOD φ (m), while b n and f (n) are depending on the AOA φ (n). If the number of local scatterers approaches infinity (M, N ), then the discrete random variables φ (m) and φ (n) become continuous random variables φ and φ, each of which is characterized by a certain distribution, denoted by p φ (φ ) and p φ (φ ), respectively. he infinitesimal power of the diffuse components corresponding to the differential angles dφ and dφ is proportional to p φ (φ )p φ (φ )dφ dφ. As M and N, this contribution must be equal to 1/(MN), i.e., 1/(MN) = p φ (φ )p φ (φ )dφ dφ. Hence, it follows from (21) that the 3-D space-time CCF of the reference model can be expressed as ρ 11,22 (δ,δ,τ) = ρ (δ,τ) ρ (δ,τ) (22) where ρ (δ,τ) = and ρ (δ,τ) = a 2 (δ,φ )e j2πf (φ )τ p φ (φ )dφ (23) b 2 (δ,φ )e j2πf(φ)τ p φ (φ )dφ (24) are 2-D correlation functions related to the transmit side and the receive side, respectively, and a(δ,φ ) = e jπ(δ /λ) cos(φ β ) (25) b(δ,φ ) = e jπ(δ/λ) cos(φ β) (26) f (φ ) = f max cos(φ α ) (27) f (φ ) = f max cos(φ α ). (28) From (22), we observe that the 3-D space-time CCF ρ 11,22 (δ,δ,τ) can be expressed as the product of the transmit correlation function ρ (δ,τ) and the receive correlation function ρ (δ,τ). his fact was also pointed out for the indoor MIMO channel model proposed in [12]. For the one-ring model, however this property does in general not hold [13,14]. Furthermore, we also observe that ρ 11,22 (δ,δ,τ) is independent of the ring radii (, ) and the distance D between the transmitter and the receiver. he 2-D space CCF ρ(δ,δ ), defined as ρ(δ,δ ) := E{h 11 (t)h 22(t)}, equals the space-time CCF ρ 11,22 (δ,δ,τ) at τ = 0, i.e., ρ(δ,δ ) = ρ 11,22 (δ,δ,0) = ρ (δ,0) ρ (δ,0). (29) he temporal ACF r hij (τ) of the diffuse component h ij (τ) of the link from A (j) to A (i) (i,j = 1,2) is defined by r hij (τ) := E{h ij (t)h ij(t + τ)}. (30) After applying similar techniques as introduced at the beginning of this subsection, we can show that the equation above can be expressed as r hij (τ) = ρ (0,τ) ρ (0,τ) (31) for all i,j {1,2}, where ρ (δ,0) and ρ (δ,0) can be obtained from (23) and (24) as ρ (0,τ) = ρ (0,τ) = e j2πf (φ )τ p φ (φ )dφ (32) e j2πf(φ)τ p φ (φ )dφ (33)

5 respectively. Notice that r hij (τ) equals ρ 11,22 (δ,δ,τ) at δ = δ = 0, i.e., r hij (τ) = ρ 11,22 (0,0,τ) =ρ (0,τ) ρ (0,τ). his result shows that the temporal ACFs of all four diffuse components are identical. It has been mentioned in [14] that this statement holds also for the one-ring model. he reference model described above is basically a theoretical model, which grounds on the assumption that the number of scatterers (N,M) are infinite. Although such a model is unrealizable, it is of central importance for the derivation of stochastic and deterministic simulation models. he design of efficient channel simulators having approximately the same statistical properties as the reference model will be the topic of Part II [10]. reference model results in r hij (τ) = ρ (0,τ) ρ (0,τ) = J 0 (2πf max τ) J 0 (2πf max τ) for all i,j = 1, 2. he observation that the temporal correlation function can be expressed as the product of two Bessel functions has also been made in [1], where a mobile-to-mobile channel with a single transmit and a single receive antenna has been studied. he reference model s transmit correlation function ρ (δ,τ) [see (34)] is illustrated in Fig. 2. Finally, Figure 3 shows the shape of the receive correlation function ρ (δ,τ) determined by (35). IV. NUMEICAL ESULS In this section, we present some numerical results for the special case of isotropic scattering. Here, we concentrate on the transmit and receive correlation functions, which have been identified in Section III as proper statistical quantities to describe the space-time correlation properties of the reference model completely. he following model parameters have been chosen. he antenna tilt angles β and β are defined as β = β = π/2. At the transmit side, the angle of motion α was set to π/4, while the receiver was moving at an angle of α = 0. Identical maximum Doppler frequencies of f max = f max = 91Hz have been assumed, and the wave length λ was set to λ = 0.15m. Under isotropic scattering conditions, where p φ (φ ) = p φ (φ ) = 1/(2π) holds, the transmit and receive correlation functions of the reference model can be expressed in closed form. After solving the integrals in (23) and (24) by using [15, eq. ( )], [15, eq. (3.339)], and [16, eq. (9.6.3)], we can write (δ ) 2 ρ (δ,τ)=j 0 2π + (f max λ τ)2 2g (δ,τ) (34) (δ ) 2 ρ (δ,τ)=j 0 2π + (f max λ τ)2 2g (δ,τ) (35) Figure 2: he transmit correlation functions ρ (δ,τ) of the 2 2 MIMO mobile-to-mobile channel model. where J 0 ( ) denotes the Bessel function of the first kind of order zero and g (δ,τ) = (δ /λ)f max τ cos(α β ) (36) g (δ,τ) = (δ /λ)f max τ cos(α β ). (37) From (34), (35), and (29), it follows that the 2-D space CCF ρ(δ,δ ) can be expressed as the product of two Bessel functions according to ρ(δ,δ ) = ρ (δ,0) ρ (δ,0) = J 0 (2πδ /λ) J 0 (2πδ /λ). Finally, we notice that the temporal ACF r hij (τ) of the Figure 3: he receive correlation function ρ (δ,τ) of the 2 2 MIMO mobile-to-mobile channel model.

6 V. CONCLUSIONS A reference model has been derived for frequencynonselective MIMO ayleigh mobile-to-mobile channels. he starting point of the procedure was the geometrical two-ring scattering model, where both the transmitter and the receiver are moving. he proposed MIMO mobile-to mobile channel model takes not only the Doppler effect caused by two mobile stations into account, but models also accurately the spatial cross-correlation properties between the complex channel gains. General formulas as well as specific closedform expressions have been provided for the transmit and receive space-time CCF. he presented model is an extension of the mobile-to-mobile channel model proposed originally by Akki and Haber with respect to multiple antennas at the transmitter and the receiver. Moreover, it includes the two-ring MIMO channel model introduced by Byers and akawira as a special case when the transmitter is fixed and only the receiver is moving. he proposed MIMO mobile-to-mobile channel model is useful for the design, test and analysis of future MIMO mobileto-mobile systems. Our procedure provides also an important framework for studying the channel capacity of MIMO mobile-to-mobile channels. In Part II [10] of our paper, we show how an ergodic channel simulator can be derived from the non-realizable reference model to enable an efficient simulation of MIMO mobile-to-mobile channels. ACKNOWLEDGMEN he authors would like to express their gratitude to SK elecom and HF, Inc. (Korea), for supporting this work. EFEENCES [6] A. Kato, K. Sato, M. Fujise, and S. Kawakami, Propagation characteristics of 60-GHz millimeter waves for IS inter-vehicle communications, IEICE rans. Commun., vol. E84-B, no. 9, pp , Sept [7] C. S. Patel, G. L. Stüber, and. G. Pratt, Simulation of ayleigh faded mobile-to-mobile communication channels, in Proc. 58th IEEE Veh. echnol. Conf. (VC 03), Orlando, FL, USA, Oct. 2003, pp [8] I. E. elatar, Capacity of multi-antenna Gaussian channels, European rans. elecommun. elated echnol., vol. 10, pp , [9] G. H. Foschini and M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Pers. Commun., vol. 6, pp , [10] B. O. Hogstad, M. Pätzold, N. Youssef, and D. Kim, A MIMO mobile-to-mobile channel model: Part II he simulation model, in Proc. 16th IEEE Int. Symp. on Personal, Indoor and Mobile adio Communications, PIMC 2005, Berlin, Germany, Sept. 2005, accepted for publication. [11] G. J. Byers and F. akawira, he influence of spatial and temporal correlation on the capacity of MIMO channels, in Wireless Communications and Networking 2003, WCNC 2003, Mar. 2003, pp [12] Z. ang and A. S. Mohan, A correlated indoor MIMO channel model, in Canadian Conference on Electrical and Computer Engineering 2003, IEEE CCECE 2003, Venice, Italy, May 2003, vol. 3, pp [13] A. Abdi, J. A. Barger, and M. Kaveh, A parametric model for the distribution of the angle of arrival and the associated correlation function and power spectrum at the mobile station, IEEE rans. Veh. echnol., vol. 51, no. 3, pp , May [14] M. Pätzold and B. O. Hogstad, A space-time channel simulator for MIMO channels based on the geometrical one-ring scattering model, Wireless Communications and Mobile Computing, Special Issue on Multiple-Input Multiple-Output (MIMO) Communications, vol. 4, no. 7, pp , Nov [15] I. S. Gradstein and I. M. yshik, ables of Series, Products, and Integrals, vol. I and II. Frankfurt: Harri Deutsch, 5th edition, [16] M. Abramowitz and I. A. Stegun, Eds., Pocketbook of Mathematical Functions. Frankfurt/Main: Harri Deutsch, [1] A. S. Akki and F. Haber, A statistical model of mobileto-mobile land communication channel, IEEE rans. Veh. echnol., vol. 35, no. 1, pp. 2 7, Feb [2] A. S. Akki, Statistical properties of mobile-to-mobile land communication channels, IEEE rans. Veh. echnol., vol. 43, no. 4, pp , Nov [3] F. Vatalaro and A. Forcella, Doppler spectrum in mobile-tomobile communications in the presence of three-dimensional multipath scattering, IEEE rans. Veh. echnol., vol. 46, no. 1, pp , Feb [4] J. Maurer,. Fügen,. Schäfer, and W. Wiesbeck, A new inter-vehicle communications (IVC) channel model, in IEEE Veh. echnol. Conf. (VC 04), Los Angeles, California, USA, Sept [5] I. Z. Kovacs, P. C. F. Eggers, K. Olesen, and L. G. Petersen, Investigations of outdoor-to-indoor mobile-to-mobile radio communications channels, in Proc. IEEE 56th Veh. echnol. Conf., VC 02, Vancouver BC, Canada, Sept. 2002, pp