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3D Mobile-to-Mobile Wireless Channel Model Prasad T. Samarasinghe, Tharaka A. Lamahewa, Thushara D. Abhayapala and Rodney A. Kennedy Applied Signal Processing Group, Research School of Information Sciences and Engineering, ANU College of Engineering and Computer Science, The Australian National University, Canberra ACT 0200, Australia {prasad.samarasinghe,tharaka.lamahewa,thushara.abhayapala,rodney.kennedy}@anu.edu.au In this paper we introduce a novel three-dimensional (3D) M2M channel model based on free-space wave propagation where the scattering environment surrounding the transmitter and the receiver is characterized using a random scattering gain, which is a function of angle of departure (AOD) and angle of arrival (AOA). When compared with other models discussed above, this channel model can be used for any 3D scattering environment whereas the other models are restricted. Based on this channel model we derive the temporal correlation function for a general 3D scattering environment. The temporal correlation function is characterized by the joint angular power distribution at the transmitter and receiver antennas and the speed of transmitter and receiver antennas. As a special case, in this paper we consider separable scattering environments, i.e., the angular power distribution at the receiver is independent of that at the transmitter. In this case, using the von Mises-Fisher distribution as an example for the angular power distribution at each end of the channel, we calculate the temporal correlation function for several fading scenarios and discuss the usefulness of our proposed model. The rest of the paper is organized as follows. In Section II, we present our new 3D M2M channel model. Using this channel model, in Section III we derive the temporal correlation corresponding to a general scattering environment. In Section IV we provide numerical examples to show the strength of the proposed model. Finally, the concluding remarks are given in Section V. Notations: Throughout the paper, the following notations will be used: bold lower letters denote vectors, a unit vector ∗ is represented by x̂ and (·) denotes the complex conjugate operation. The symbol δ(·) denotes the Dirac delta function while E {·} denotes the mathematical expectation. The scalar product √ between vectors x and y is denoted by x · y and i = −1. Abstract—A three-dimensional single-input single-output Mobile-to-Mobile wireless channel model is developed in this paper by considering the underlying physics of free space wave propagation. Based on this channel model, the temporal correlation function for a general three dimensional scattering environment is derived. The temporal correlation function is characterized by the joint angular power distribution at the transmitter and receiver antennas and the speed of transmitter and receiver antennas. Using a von Mises-Fisher distribution as the angular power distribution, the usefulness of the derived temporal correlation function is discussed. I. I NTRODUCTION A communication system in which both the transmitter and receiver are in motion, can be called as a Mobile-toMobile (M2M) communication system. The main applications of M2M systems are in mobile ad-hoc wireless networks and intelligent transport systems. Dedicated Short Range Communication (DSRC) and Wireless Access in a Vehicular Environment (WAVE) are emerging standards for intelligent transport systems focussed on improving traveler safety, efﬁciency and productivity [1]. Understanding of M2M wireless channel and its behavior will support to improve the above applications. In the literature, M2M channels have been modeled in various ways. Akki and Harber [2, 3] were the ﬁrst to propose a statistical channel model for single-input single-output (SISO) M2M Rayleigh fading radio channel under non line of sight conditions. Simulation methods for SISO M2M channels have been proposed in [4–6]. Also, Pätzold in [7] derived a two-dimensional(2D) ray-tracing model for multiple-input multiple-output (MIMO) M2M narrowband channel by considering a geometrical two-ring scattering model. Stüber in [8] proposed a similar 2D ray-tracing narrowband channel model for MIMO M2M channel based on ‘double-ring’ geometrical model. However, since the latter model is independent of the distances between scatters and antenna elements, it is better than the former model. In [9] the narrowband channel model proposed in [10] was extended to a 2D wideband model. In [11] and [12] these two 2D M2M channels models were extended to 3D models by assuming scatterers are placed on cylinders. One shortcoming of these ray-tracing geometrical based channel models is that they cannot be utilized to represent general scattering scenarios. 978-1-4244-5434-1/10/$26.00 ©2010 IEEE II. M OBILE - TO -M OBILE C HANNEL M ODEL Consider the SISO Mobile-to-Mobile channel model depicted in Fig. 1, where the mobile transmitter is located inside a scatterer free sphere of radius rT and the mobile receiver is located inside a scatterer free sphere of radius rR . It is assumed that scatterers are located outside the two scatterfree transmit and receive regions and are in the far-ﬁeld of the receiver and transmitter regions. In Fig. 1 g(φ̂, ϕ̂) represents 30 AusCTW 2010 where we have introduced the shorthand 4 S×S S×S . In this work we assume that the scattering environment is widesense stationary zero-mean and uncorrelated. Therefore, the second-order statistics of the scattering gain function g(φ̂, ϕ̂) can be written as [14] E g(φ̂, ϕ̂)g ∗ (φ̂ , ϕ̂ ) G(φ̂, ϕ̂)δ(φ̂ − φ̂ )δ(ϕ̂ − ϕ̂ ), where G(φ̂, ϕ̂) = E |g(φ̂, ϕ̂)|2 characterizes the joint power spectral density (PSD) or the joint angular power distribution surrounding the transmitter and receiver regions. In this case the temporal correlation function in (4) further simpliﬁes to G(φ̂, ϕ̂)e−ikuτ û·φ̂ eikvτ v̂·ϕ̂ dφ̂dϕ̂. (5) ρ(τ ) = ^ ϕ ^ φ Scatterers ^ ^ g(φ,ϕ ) ^v Receiver u^ (b) Transmitter (c) (a) Fig. 1. A general scattering model for a ﬂat fading M2M wireless communication system S×S From (5) above it can be seen that the temporal correlation function of a mobile-to-mobile fading channel is described largely by the joint angular power distribution G(φ̂, ϕ̂) and the speed of transmitter and receiver. the effective random complex scattering gain function for a signal leaving the transmitter scatter-free region at a direction φ̂ (relative to the transmitter region origin) and entering the receiver scatter-free region from a direction ϕ̂ (relative to the receiver region origin). Suppose the transmitter is moving at constant velocity u in the direction of û and the receiver is moving at constant velocity v in the direction of v̂. Following the derivation of ﬁxed-to-mobile channel model given in [13], the signal received at the mobile receiver at time t can be written as x(t)g(φ̂, ϕ̂)e−iktuû·φ̂ eiktvv̂·ϕ̂ dφ̂dϕ̂ + n(t), y(t) = S×S III. 3-D S CATTERING E NVIRONMENT Using the spherical harmonic expansion of plane waves, the plane wave eikx·ŷ can be expanded in 3D as [15, page 32] eikx·ŷ = n=0 m=−n where x(t) is the baseband transmitted signal, k = 2π/λ is the wave number with λ being the wavelength, n(t) is the additive noise at the receiver. The integration in (1) above is over a unit circle for a two dimensional scattering (or multipath) environment or over a unit sphere for a three dimensional scattering environment. The input-output relationship for the general case then can be written as where 0 ≤ θ ≤ π and 0 ≤ ψ ≤ 2π are respectively the elevation and azimuth angles and Pnm (·) are the associated Legendre functions of the ﬁrst kind. By applying the spherical harmonic expansion (6) in (5), we obtain (2) ρ(τ ) =(4π)2 e−inπ/2 jn (kuτ )Ynm (û) n=0 m=−n p=0 q=−p where The channel temporal correlation, which is the correlation between the channel gain at time t and time t − τ can then be found from p,q βn,m = ρ(τ ) E {h(t)h∗ (t − τ )} . = E g(φ̂, ϕ̂)g ∗ (φ̂ , ϕ̂ ) eiku(−tû·φ̂+(t−τ )û·φ̂ ) n ∞ p ∞ S×S ×e−ikv(−tv̂·ϕ̂+(t−τ )v̂·ϕ̂ ) dφ̂dϕ̂dφ̂ dϕ̂ , (6) and Ynm (·) are the spherical harmonics, which are deﬁned as, 2n + 1 (n − |m|)! |m| (7) Ynm (θ, ψ) P (cos θ)eimψ 4π (n + |m|)! n where, comparing (1) and (2), the underlying time-varying fading channel between the receiver and the transmitter can be written as, g(φ̂, ϕ̂)e−iktuû·φ̂ eiktvv̂·ϕ̂ dφ̂dϕ̂. (3) h(t) = 4 ∗ in 4πjn (kx)Ynm (x̂)Ynm (ŷ), where jn (r) is the spherical function, which is related to the Bessel function by π J 1 (r) jn (r) = 2r n+ 2 (1) y(t) = h(t)x(t) + n(t) n ∞ S2 ×S2 ∗ p,q eipπ/2 jp (kvτ )Ypq (v̂)βn,m , (8) ∗ G(φ̂, ϕ̂)Yn,m (φ̂)Yp,q (ϕ̂)dφ̂dϕ̂ (9) are the scattering environment coefﬁcients which characterize the 3D scattering environment surrounding the receiver and transmitter regions. According to (8), in addition to scattering coefﬁcients, temporal correlation depends on transmitter and receiver velocities (u (4) 31 and v) and time between two received signals (τ ). Simulation results in section IV further explain these relationships. As shown in [16], spherical Bessel functions will exhibit a high pass nature. Therefore, using this property the inﬁnite summations in (8) can be approximated as ρ(τ ) =(4π)2 MT n proposed in the literature for modeling the scattering distributions GT x (φ̂) and GRx (ϕ̂) at the transmitter and the receiver, respectively. Few examples are, isotropic model [18], uniform limited azimuth/elevation model [19] and spherical harmonic model [19]. In this paper we use the 3D von Mises-Fisher model to characterize the scattering power distribution at each end of the channel. In addition, using our temporal correlation function (8), we derive the temporal correlation function for a M2M channel in a 3D isotropic scattering environment. e−inπ/2 jn (kuτ )Ynm (û) n=0 m=−n p MR p=0 q=−p ∗ p,q eipπ/2 jp (kvτ )Ypq (v̂)βn,m , (10) IV. E XAMPLES where NT πerT /λ and NR πerR /λ. In this section we discuss temporal correlation for two example scattering distributions and their simulation results. For both examples we use carrier frequency as 5.9 GHz, symbol duration Ts = 0.0001 seconds and delay τ = 10Ts . A. Special Case: Separable Channels In this section we consider a special case of the joint angular power distribution G(φ̂, ϕ̂) and derive the scattering p,q corresponding to this special environments coefﬁcients βn,m case. When the angle of departure (AOD) φ̂ is independent of the angle of arrival (AOA) ϕ̂, the joint angular power distribution G(φ̂, ϕ̂) can be written as G(φ̂, ϕ̂) = GT x (φ̂)GRx (ϕ̂), where A. 3D Isotropic Scattering Distribution If the waves are transmitting from the transmitter uniformly to all direction in 3D space, i.e., GT x (φ̂) = 1/2π 2 , the scattering environment coefﬁcients at the transmitter is given by (11) GT x (φ̂) = S2 G(φ̂, ϕ̂)dϕ̂, 0,0 βn,m is the angular power distribution at the transmitter and GRx (ϕ̂) = G(φ̂, ϕ̂)dφ̂ is the angular power distribution at the receiver. Fading channels that satisfy (11) are known as separable channels or Kronecker channels. As shown in [14], the separability condition (11) can be assumed when there is a single scattering cluster in the scattering environment. In this case, the scattering environment coefﬁcients are given by where 0,0 βn,m p,q β0,0 √1 , 4π 0, n=m=0 otherwise = √1 , 4π 0, p=q=0 otherwise The temporal correlation function of the 3D M2M channel, in this case, becomes (12) ρ(τ ) = j0 (kuτ )j0 (kvτ ). = = Similarly, if the waves are impinging on the receiver uniformly from all direction in 3D space, i.e., GRx (ϕ̂) = 1/2π 2 , the corresponding scattering environment coefﬁcients at the receiver is given by S2 p,q 0,0 p,q = βn,m β0,0 , βn,m S2 GT x (φ̂)Yn,m (φ̂)dφ̂ (13) (14) Note that in a 2D scattering environment, as shown in [2], the temporal correlation function in a 2D isotropic scattering environment is given by ρ(τ ) = J0 (kuτ )J0 (kvτ ). The magnitude of the time correlation between two received signals after 10 symbol period versus magnitude of transmitter velocity and magnitude of receiver velocity are shown in Fig. 2 and contour plot of the same is shown in Fig. 3, where both transmitter and receiver scattering distributions are isotropic. In this simulation, transmitter and receiver are moving with zero elevation and zero azimuth angles with respect to reference points.The simulation result shows a high degree of correlation (> 0.3) until both transmitter and receiver reach 20 m/s and reduce to zero temporal correlation when transmitter and or receiver are at 25 m/s. are the scattering environment coefﬁcients at the transmitter and p,q ∗ β0,0 = GRx (ϕ̂)Yp,q (ϕ̂)dϕ̂. S2 are the scattering environment coefﬁcients at the receiver. Power distributions are mainly characterized by the mean angle of arrival ϕ0 (or departure φ0 ) and the angular spread σ at the receiver (or transmitter). A number of univariate power distributions in 3D scattering environments1 have been 1 In 2D scattering environments, researchers modeled the scattering environment using azimuthal power distribution models such as von-Mises [17], Laplacian, uniform-limited, truncated Gaussian, etc. 32 and 1 πκR 1 I 1 (κR )Yp,q (ϕ0 ), 2 sinh κR p+ 2 where κT and κR are the non-isotropy factors at the transmitter and receiver regions, respectively. 0.8 0.6 0.4 1 0.2 0.9 60 0 0 10 Absolute value of time correlation | ρ(τ) | Absolute value of time correlation | ρ(τ) | p,q = β0,0 40 20 30 20 40 50 60 0 v (m/s) u (m/s) Fig. 2. Time correlation between two receiver signals after τ = 10 symbol time periods in 3D M2M environment where both transmitter and receiver scattering distributions are isotropic. 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 60 50 0 0 60 40 30 10 20 20 30 40 55 50 10 60 0 v (m/s) 00 50 00 0 0 0 0 00 u (m/s) 45 Fig. 4. Time correlation between two receiver signals after τ = 10 symbol time periods in 3D M2M environment. Where both transmitter and receiver scattering distributions are von Mises-Fisher with κ = 1. 35 00 0 30 0 25 0 v (m/s) 40 0 00 00 20 60 0.4 00 00 15 50 0.4 10 0 .8 5 20 30 u (m/s) 40 50 60 40 Fig. 3. Contour plot of time correlation between two receiver signals after τ = 10 symbol time periods in 3D M2M environment where both transmitter and receiver scattering distributions are isotropic. 30 0.2 v (m/s) 10 0.2 20 0.4 0.6 10 0.2 The von Mises-Fisher distribution in a 3-D scattering environment deﬁned as [20], G(θ) = κ eκ cos(θ−θ0 ) , |θ − θ0 | ≤ π, 4π sinh κ 0.2 0.4 B. von Mises-Fisher distribution 0.8 0 (15) 0 10 20 30 u (m/s) 40 50 60 Fig. 5. Contour plot of time correlation between two receiver signals after τ = 10 symbol time periods in 3D M2M environment where both transmitter and receiver scattering distributions are von Mises-Fisher with κ = 1. where κ ≥ 0 represents the non-isotropy factor of the distribution and Im (·) is the modiﬁed Bessel function of the ﬁrst kind. Note that κ = 0 represents isotropic scattering. As shown p,q 0,0 and β0,0 in [21], the scattering environment coefﬁcients βn,m corresponding to the von Mises-Fisher distribution are given by πκT 1 0,0 ∗ βn,m = I 1 (κT )Yn,m (φ0 ), 2 sinh κT n+ 2 Assuming both transmitter and receiver sides are of von Mises-Fisher distribution and separable channels, the magnitude of temporal correlation between two receiver signals after 10 symbol period is shown in Fig. 4 and contour plot of the same is shown in Fig. 5. In this simulation, transmitter and receiver are moving with zero elevation and zero azimuth angles with respect to reference points. In addition, AOA and AOD also have zero elevation and azimuth angles with 33 respect to reference points. In this situation a high degree of correlation (> 0.3) is shown until both reach 25 m/s, but when compared to isotropic case rate of reduction of temporal correlation is lesser. The same equation and simulations can be used to understand the impact of the eight angles (elevation and azimuth angles of transmitter velocity, receiver velocity, AOA and AOD) to the temporal correlations. One example of different angles is shown in Fig. 6 where transmitter azimuth and elevation angles are 300 and other angles remain at zero. In this situation receiver is reaching to the same temporal correlation at lesser speeds than the all angles zero example. Hence the rate of reduction of temporal correlation is higher with respect to receiver speeds whereas temporal correlation related to transmitter speeds remains same in both cases. [2] A.S. Akki and F. 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The new model could be used as a generalized 3D model than any existing model. The ﬂexibility of our new model is that it can be applied to different scattering distributions as shown in our simulations, where we have analyzed temporal correlation between two receiver signals for isotropic distribution and von Mises-Fisher distribution as examples. R EFERENCES [1] W. Xiang, P. Richardson, and J. Guo, “Introduction and preliminary experimental results of wireless access for vehicular environments (wave) systems,” in Mobile and Ubiquitous Systems: Networking and Services, 2006 Third Annual International Conference on, July 2006, pp. 1–8. 34