The Impact of Fixed and Moving Scatterers on the Statistics of MIMO Vehicle-to-Vehicle Channels

In the last decades the restless evolution of information and communication technologies (ICT) brought to a deep transformation of our habits. The growth of the Internet and the advances in hardware and software implementations modiﬁed our way to communicate and to share information. In this book, an overview of the major issues faced today by researchers in the ﬁeld of radio communications is given through 35 high quality chapters written by specialists working in universities and research centers all over the world. Various aspects will be deeply discussed: channel modeling, beamforming, multiple antennas, cooperative networks, opportunistic scheduling, advanced admission control, handover management, systems performance assessment, routing issues in mobility conditions, localization, web security. Advanced techniques for the radio resource management will be discussed both in single and multiple radio technologies; either in infrastructure, mesh or ad hoc networks.


The Geometrical Street Model
A typical highway propagation environment for V2V communication is presented in Fig. 1. The highway encompasses three lanes used for traffic in the same direction. We can distinguish between two types of scatterers namely fixed scatterers and moving scatterers. The fixed scatterers are represented by the buildings located on both sides of the street, while the moving scatterers are the vehicles in the vicinity of the transmitter MS T and the receiver MS R . In order to be able to develop an appropriate channel model for the propagation scenario presented in Fig. 1, we first need to produce a representative geometrical model for such an environment. Towards this aim, we model each building by a cluster of scatterers located on a straight line on the left or right hand side of the street. A vehicle can be modeled by a cluster of scatterers located on a line as well. The fixed clusters are represented by solid lines while the moving clusters are represented by dashed lines. The geometrical street model encompassing fixed and moving scatterers is illustrated in Fig. 2. The fixed scatterers around the transmitter (receiver) are denoted by S T m (S R n ). The moving scatterers are designated by S M p . The propagation environment encompasses C T (C R ) fixed clusters around the transmitter (receiver) and C M moving clusters. For fixed clusters, the AoD is referred to as α T m , whereas the AoA is denoted by β R n . For moving clusters, the symbols α M p and β M p stand for the AoD and the AoA, respectively. It has to be noted that for fixed clusters the AoD and the AoA are independent since double-bounce scattering is assumed. For moving clusters, the AoD and the AoA are closely related due to the single-bounce scattering assumption. All scatterers belonging to a given moving cluster have the same velocity v S and the same direction of motion φ S . The transmitter and the receiver are moving with velocity v T and v R , respectively. The angle of motion of the transmitter and the receiver w.r.t the x-axis is referred to as φ T and φ R , respectively. Moreover, the transmitter (receiver) is equipped with M T (M R ) antenna elements. The antenna element spacing at the transmitter and the receiver antenna are denoted by δ T and δ R , respectively. The angle γ T (γ R ) describes the tilt angle of the transmit (receive) antenna array.

The Reference Model
Starting from the geometrical model shown in Fig. 2, we derive a reference model for the MIMO V2V channel. First, we consider the case where we have one moving cluster and two fixed clusters: one cluster is near to the transmitter and the other cluster is close to the www.intechopen.com  receiver. The total number of fixed scatterers around the transmitter (receiver) is denoted by M (N), while the number of moving scatterers is referred to as P. The complex channel gain g kl (t) describing the link between the lth transmit antenna element A T l (l = 1, 2, . . . , M T ) and the kth receive antenna element A R k (k = 1, 2, . . . , M R ) of the underlying M T × M R MIMO V2V channel model can be expressed as g kl (t) = g F kl (t) + g M kl (t). The term g F kl (t) stands for the channel gain due to double-scattering from the fixed clusters. The channel gain caused by the moving cluster is denoted by g M kl (t). We assume that the line-of-sight component is obstructed. Next, we derive analytical expressions of the channel gains g F kl (t) and g M kl (t).

The Channel Gain Due to Fixed Scatterers
The plane wave emitted from the lth transmit antenna element A T l travels over the scatterers S T m and S R n before impinging on the kth receive antenna element A R k . Based on the geometrical model in Fig. 2, the channel gain, due to double-scattering from fixed clusters, g F kl (t) can be www.intechopen.com

Radio Communications 54
written as where c mn and θ mn stand for the joint gain and the joint phase shift resulting from the interaction with the fixed scatterers S T m and S R n . The joint channel gain can be written as c mn = 1/ √ MN, while the joint phase shift can be expressed as θ mn = (θ m + θ n ) mod 2π, where mod stands for the modulo operation. The terms θ m and θ n are the phase shifts associated with the scatterers S T m and S R n , respectively. It has to be noted that θ m , θ n , and θ m,n are independent identically distributed (i.i.d.) random variables uniformly distributed over [0, 2π). The second phase term in (1), k T m · r T , is related to the transmitter movement. The symbol k T m denotes the wave vector pointing in the propagation direction of the mth transmitted plane wave, and r T is the spatial translation vector of the transmitter. The scalar product k T m · r T can be expressed as where f T max =v T /λ stands for the maximum Doppler frequency associated with the mobility of the transmitter. The symbol λ denotes the wavelength. The third phase term in (1), k R n · r R , is caused by the receiver movement. The symbol k R n denotes the wave vector pointing in the propagation direction of the nth received plane wave, and r R is the spatial translation vector of the receiver. The scalar product k R n · r R can be written as where f R max =v R /λ stands for the maximum Doppler frequency due to the receiver movement. The term k 0 d mn in (1) is associated with the total travelled distance and can be expressed as where D lm denotes the distance from the lth transmit antenna element A T l to the scatterer S T m . The symbol D mn stands for the distance between the scatterers S T m and S R n . The term D nk denotes the distance from the scatterer S R n to the kth receive antenna element A R k . The distances D lm and D nk can be approximated as where D T m denotes the distance from the transmitter to the scatterer S T m and D R n corresponds to the distance from the receiver to the scatterer S R n . After substituting (2)-(6) in (1) the channel gain caused by double-scattering from fixed clusters can be expressed as www.intechopen.com The Impact of Fixed and Moving Scatterers on the Statistics of MIMO Vehicle-to-Vehicle Channels It has to be mentioned that the envelope |g F kl (t)| follows a double Rayleigh distribution since double-bounce scattering is assumed (Salo et al., 2006).

The Channel Gain Due to Moving Scatterers
The plane wave emitted from the lth transmit antenna element A T l travels over the scatterer S M p before impinging on the kth receive antenna element A R k . Based on the geometrical model in Fig. 2, the channel gain g M kl (t) of the moving cluster can be expressed as where c p and θ p represent the gain and the phase shift resulting from the interaction with the moving scatterer S M p , respectively. The channel gain is given by c p = 1/ √ P, while the phase shifts θ p are i.i.d. random variables uniformly distributed over [0, 2π). The phase changes k T p · r T and k R p · r R are associated with the movement of the receiver and the transmitter, respectively, and can be written as The spatial translation r S of the moving scatterer S M p influences the wave emitted from the transmitter resulting in a phase change k T p · r S . Moreover, the scatterer S M p interacts with the wave reflected to the receiver resulting in a phase change k R p · r S . These phase changes can be expressed as where f S max =v S /λ is referred to as the maximum Doppler frequency caused by the moving cluster. Recall that all scatterers S M p belonging to the moving cluster have the same speed v S . The phase change resulting from the total travelled distance d p can be expressed as www.intechopen.com

Radio Communications 56
with where D T p and D R p denote the distances from the scatterer S M p to the transmitter and the receiver, respectively. After substituting (14)-(20) in (13) the channel gain due to the moving cluster can be written as It has to be noted that the AoD α M p and the AoA β M p are dependent since single-bounce scattering is assumed. The exact relationship between the AoD and the AoA can be found in (Chelli & Pätzold, 2007). The envelope |g M kl (t)| follows a Rayleigh distribution due to the single-bounce scattering assumption.

The Multiple-Cluster Channel Gain
The channel gain g kl (t) has been derived assuming a scattering environment with two fixed clusters and one moving cluster. However, in real environment, one can find several buildings and several vehicles near to the mobile transmitter and receiver. Therefore, it is of interest to derive an expression for the channel gain in a multiple-cluster case z kl (t). The environment encompasses C T (C R ) fixed clusters around the transmitter (receiver) and C M moving clusters. We added the subscripts (·) c T , (·) c R , and (·) c M to all affected symbols to distinguish between the fixed clusters around the transmitter, the fixed clusters around receiver, and the moving clusters, respectively. The fixed cluster c T has a limited length L c T  In a multiple-cluster scenario the channel gain describing the link A T l -A R k can be expressed as where z F kl (t) is the received diffuse component due to double-scattering from all fixed clusters. The term z M kl (t) is the received diffuse component caused by single-scattering from all moving clusters. The channel gains z F kl (t) and z M kl (t) can be written as where w c T , w c R , and w c M are positive constants representing the weighting factors of the clusters c T , c R , and c M , respectively. We impose the boundary condition ∑ to normalize the mean power of z kl (t) to unity.

Correlation Properties
In this section, we derive analytical expressions for the correlation functions of the proposed MIMO V2V channel model, such as the 3D space-time CCF, the temporal ACF, and the 2D space CCF. The 3D space-time CCF ρ kl,k ′ l ′ (δ T , δ R , τ) can be expressed as where (·) * denotes the complex conjugation and E{·} stands for the expectation operator. The term ρ F kl,k ′ l ′ ,c T ,c R (δ T , δ R , τ) represents the 3D space-time CCF due to double-scattering from the clusters c T and c R . This correlation function can be written as www.intechopen.com

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and are the transmit and the receive correlation functions, respectively, and The distributions of the AoD α T c T and the AoA β R c R are denoted by p α T The temporal ACF r z kl (τ) of the channel gain z kl (t) is defined as r z kl (τ) := E{z * kl (t)z kl (t + τ)} (Papoulis & Pillai, 2002). The temporal ACF r z kl (τ) can be deduced from the 3D space-time CCF ρ kl,k ′ l ′ (δ T , δ R , τ) by setting the antenna element spacings δ T and δ R to zero, i.e., by setting τ to zero, i.e.,

Numerical Results
In this section, we confirm the validity of the analytical expressions presented in the previous section by simulations making use of the sum-of-cisoids method. The simulation models for moving and fixed scatterers are designed using the modified method of equal area (MMEA) proposed in (Gutiérrez & Pätzold, 2007). In order to model all fixed clusters, 50 cisoids are used for the simulation model. The same number of cisoids is used to model all moving clusters. The propagation environment contains six moving clusters: three clusters are located on the right side of the transmitter and the receiver, while the remaining clusters lie on the left side. Each moving cluster has a length of 5 m and is separated by a distance of 45 m from its neighbour clusters. The distance between the transmitter and the moving scatterers located on the left and the right side is set to 3 m. For the fixed clusters, we consider a propagation environment encompassing three clusters on each side of the transmitter. Each cluster has a length of 2 m and is separated by a distance of 34 m from its neighbour clusters. The same number of fixed clusters is considered around the receiver. The distance between the transmitter and the fixed scatterers on the left and right side is set to 300 m. The distance between the transmitter and the receiver is equal to 100 m. The transmitter and the receiver have a speed of 50 km/h and equal angles of motion φ T = φ R = 0. The transmitter and the receiver antenna tilts γ T and γ R are set to π/2. We consider the case of non-isotropic scattering conditions.  Fig. 3. We study the influence of the speed of the vehicles in the vicinity of the transmitter and www.intechopen.com

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the receiver on the channel behaviour. The term v S in Fig. 3 denotes the speed of the vehicles on the left and right side. From Fig. 3, we can notice that as the speed v S decreases, the coherence time of the channel increases. It is well known that the coherence time indicates whether we are facing a fast or a slow fading. As the speed of the vehicles relatively to the transmitter and the receiver decreases the channel changes more slowly. A good fitting between the simulation results and the theoretical results can be observed in Fig. 3. In Fig. 4  numerical results obtained for the 2D space CCF ρ M 11,22 (δ T , δ R ) caused by moving scatterers. It could be seen from Fig. 4 that the cross-correlation function decreases as we increase the antenna spacings δ T and δ R . However, the decay of the 2D space CCF ρ M 11,22 (δ T , δ R ) is faster along the δ R direction. Hence, a small antenna spacing at the receiver side guarantees a diversity gain, but at the receiver side, we need a larger spacing to get non-correlated channels. In

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The Impact of Fixed and Moving Scatterers on the Statistics of MIMO Vehicle-to-Vehicle Channels 61 Fig. 5, we illustrate the numerical results for the transmit correlation function ρ F T (δ T , τ) of the reference model resulting from fixed scatterers. The obtained results are confirmed by simulation in Fig. 6. Similar results have been found for the receive correlation function ρ F R (δ R , τ) associated with fixed scatterers since the same setting for the scatterers is considered around the transmitter and the receiver. Limited space prevents us from including these results.

Conclusion
In this chapter, we have presented a narrowband MIMO V2V channel model, where both the impact of fixed and moving scatterers were taken into account. Double-bounce scattering is assumed for the fixed scatterers, while single-bounce scattering is considered for the moving scatterers. For reasons of brevity, we have restricted our investigations to non-line-of-sight situations. A reference model has been derived starting from the geometrical street model. The statistical properties of the proposed channel model have been studied. We have provided analytical expressions for the 3D space-time CCF, the temporal ACF, and the 2D space CCF. Supported by our analysis, we are convinced that the effect of moving scatterers on the statistics of V2V MIMO channels cannot be neglected. The investigation of the impact of moving scatterers have revealed that as the speed of the vehicles in the vicinity of the transmitter and the receiver decreases, the channel coherence time increases. The proposed channel model is suitable for a highway environment under congestion conditions. In such conditions, the low relative speed of the vehicles in the vicinity of the transmitter and the receiver allows us to consider that the AoD and the AoA seen from moving clusters are non-time-variant during a sufficiently large period of time. Actually, if the AoD and the AoA are time-variant, the channel model becomes non-stationary. The latter aspect will be investigated in future work.