RIS-Assisted Hybrid Beamforming and Connected User Vehicle Localization for Millimeter Wave MIMO Systems

A reconfigurable intelligent surface (RIS) is a type of metasurface that can dynamically control the reflection and transmission of electromagnetic waves, such as radio waves, by changing its physical properties. Recently, RISs have played an important role in intelligently reshaping wireless propagation environments to improve the received signal gain as well as spectral efficiency performance. In this paper, we consider a millimeter wave (mmWave) vehicle-to-vehicle (V2V) multiple-input multiple-output (MIMO) system in which, an RIS is deployed to aid downlink V2V data transmission. In particular, the line-of-sight path of the mmWave system is affected by blockages, resulting in higher signaling overhead. Thus, the system performance may suffer due to interruptions caused by static or mobile blockers, such as buildings, trees, vehicles, and pedestrians. In this paper, we propose an RIS-assisted hybrid beamforming scheme for blockage-aware mmWave V2V MIMO systems to increase communication service coverage. First, we propose a conjugate gradient and location-based hybrid beamforming (CG-LHB) algorithm to solve the user sub-rate maximization problem. We then propose a double-step iterative algorithm that utilizes an error covariance matrix splitting method to minimize the effect of location error on the passive beamforming. The proposed algorithms perform quite well when the channel uncertainty is smaller than 10%. Finally, the simulation results validate the proposed CG-LHB algorithm in terms of the RIS-assisted equivalent channel for mmWave V2V MIMO communications.

create an alternative transmission link, which can enhance transmission reliability [32]. The performance of single and multiple RIS-assisted systems has also been considered without a direct path between the transmitter and the receiver in indoor and outdoor propagation environments [33]. In addition, researchers proposed a geometry-based stochastic channel model that supports the movements of transceivers and clusters, and the evolution of clusters was considered in the space domain, where a reflecting coefficients design was based on the minimum path loss [34]. Elsewhere, researchers provided a comprehensive overview of state-of-the-art research on emerging and promising RIS/IRS-aided wireless systems, with an emphasis on signal processing techniques for solving various radio localization, transmission design, and channel estimation issues [35]. A tutorial overview of single and multi-IRS-aided wireless networks has also been provided, with an emphasis on addressing the new and more challenging issues in IRS reflection optimization and channel acquisition design, in [36]. In [37], a multi-IRS-aided system was studied in which, the IRS and base station (BS) are managed by a central processing unit to coordinate data transmission and maximize the weighted sum rate of all cell-edge users by jointly optimizing the BS's transmit beamforming and each IRS's phase shifts, subject to the BS transmit power budget. The authors solved the non-convex and unit modulus constraint optimization using an efficient iterative power allocation algorithm. The achievable secrecy performance of mmWave MIMO systems was studied using RISs in [38], where the authors assumed a smart environment in which an RIS is placed between the source and the legitimate user to enhance the main link. The authors, motivated by the aforementioned observations, studied beamforming optimization for an IRS-aided multi-antenna communication system by incorporating signal distortions caused by hardware impairments [39]. The authors optimized both source's transmit active and passive beamforming to maximize the SNR received at the destination. Elsewhere, the application of an active refracting RIS-enabled transmitter for a secure internet of things network was investigated to enhance secure communication in the considered network and develop an alternating optimization algorithm to optimize the sum secrecy rate by jointly designing the power allocation, transmit beamforming, and the phase shifts of an RIS in [40].
Due to the high beam training costs and the effect of user location errors in vehicular MIMO communication systems, we propose an RIS-assisted location-based hybrid beamforming algorithm as well as a double-step iterative algorithm for minimizing the effect of user vehicle location error, which has not yet been investigated in the literature. The contributions of this study are listed as follows: • Due to blockage awareness, we first develop an RIS-assisted V2V MIMO channel model and demonstrate the geometrical relation between the RIS controller and user vehicle position. By using the distance between the transmitting and receiving vehicle in relation to the distance of an RIS controller, we estimate the path amplitude and phase. • We then design an RIS-assisted low-dimensional equivalent effective channel. In response to a sub-design problem of an original beamformer, we propose a conjugate gradient and location-based hybrid beamforming algorithm. We apply a Karush-Kuhn-Tucker condition and a Lagrangian method to solve an original problem into a sub-optimal problem for developing the conventional hybrid beamforming algorithm, which reduces the user sub-rate maximization problem. The proposed algorithm attains significant spectral efficiency performance. • We next consider a covariance splitting method to reform the error covariance matrix. Since the channel is geometrically contained in location information, we propose a double-step iterative algorithm that minimizes the effect of location error in user vehicles. Finally, the simulation results demonstrate the superiority of the proposed algorithms over their counterparts.
The rest of the paper is organized as follows. The signal and channel model and the geometrical relation between the RIS controller and connected autonomous vehicle position are discussed in Section 2. The proposed hybrid beamforming and user vehicle localization is presented in Section 3. Simulation results are provided in Section 4. The conclusions are offered in Section 5.

Signal Model
Consider an access point that serves a point-to-point (P2P) vehicular MIMO network using mmWave technology, as shown in Figure 1, where a transmit vehicle is operating at mmWave frequency bands with N t transmit antennas. The user vehicle is equipped with N u receive antennas. Let the transmit and user vehicles be autonomous, which are assumed to be fully connected with different modules and sensors as the power module is used as a battery to supply power to the other module. The perception module is recognized for driving environments and detected objects using sensors such as light detection and ranging (LiDAR), radars, cameras, a global positioning system (GPS), and inertial measurement unit (IMU), and other sensors as shown in Figure 2. A WiFi-based mmWave transceiver is also intelligently connected to the transmit and receive vehicles in Figure 2. We consider an IEEE 802.11ad Wi-Fi chips with the number of 8 antennas to communicate at the 60 GHz frequency band.
A blockage scenario is considered in the system model where the direct link between the transmit and user vehicles is blocked owing to the critical propagation conditions. Hence, the RIS is deployed in the transmission model to assist and expand the service coverage of the P2P vehicular MIMO communication. The received signal vector r ∈ C N u ×1 can be modeled as where x ∈ C N t ×1 is the transmitted signal vector, which satisfies E[xx H ] = I, H e is the N u × N t effective channel from the transmitting vehicle to the user vehicle, z ∼ CN (0, σ 2 z I) is the N u × 1 additive white Gaussian noise vector and σ 2 z denotes the noise variance. After applying a hybrid precoding, x is given by where F A ∈ C N t ×N RF is the analog radio frequency (RF) beamforming matrix, F D ∈ C N RF ×N s is the digital precoding matrix, ||F A F D || 2 F ≤ P, P is the transmit power, s ∈ C N s ×1 is the transmitted symbol vector that satisfies E[ss H ] = I, N s ≤ min(N t RF , N u RF ) when N t >> N t RF , N u >> N u RF , N t RF denotes the number of RF chains for the transmit vehicle side, and N u RF denotes the number of RF chains for the user intelligent vehicle side. Setting (2) in (1) and the received signal vectorŝ is given after combininĝ ×N s is the baseband digital combiner, and W A ∈ C N u ×N RF is the analog combiner which follows the similar constraints as F A . If the transmitted signal follows a Gaussian distribution over the RIS-assisted mmWave V2V MIMO channel, then using (3), the achievable spectral efficiency is given by where

Channel Model
Let H e = GΘC where G is an N u × N R channel between RIS to the user vehicle as whereβ l denotes the complex channel gain at l-th path of G channel, φ l u represents the angle of arrival associated with the user vehicle, γ g = N u N R /L g , and θ l t and ν l t denotes the azimuth and elevation angles of arrival and departure at l-th path, respectively, the normalized array response vector for the case of the uniform linear array is given by [2][3][4][5].
where the wavelength λ = c/ f c , c is the speed of light, f c is the carrier frequency, and d = λ/2 is the antenna spacing. The normalized array response vector for uniform planner array is given by [28] where N R = N R,y × N R,z , and C denotes the N R × N t channel between the transmit vehicle and the RIS controller, that is whereγ l denotes the complex channel gain at l-th path of C channel, γ c = are designed in the same manner of G channel and the N R × N R RIS element response matrix Θ is given by where D is the diagonal phase matrix as [25], where ϑ = [ϑ 1 , . . . , ϑ N R ] T , ϑ n = e jϕ n , n = 1, 2, . . . , N R and ϕ n denotes the phase-shift which is given by Thus, the effective channel H e can be written as where µ l,m = a H R (θ l t , ν l t )Θa R (θ m r , ν m r ) denotes the passive beamforming gain, β l = γ gβl , γ l = γ cγl , and µ l,m satisfy |γ 1 | ≥ |γ 2 |, . . . , ≥ |γ L c | and |β 1 | ≥ |β 2 |, . . . , ≥ |β L g |.

Geometrical Relation between the RIS Controller and the Connected Vehicle Position
r y , r z ), and u = (u x , u y , u z ) are the position of the transmitting vehicle, the RIS controller and the user vehicle, respectively in the Cartesian coordinate as shown in Figure 1, and g is the vector of G channel. The g is given by where θ az denotes the azimuth of the angle of departure (AoD) and ν el denotes the elevation of AoD at the RIS side. The geometric relation between the RIS controller and the user vehicle position is measured by and Similarly, we can make a positional relationship between the transmitting vehicle and the RIS controller as and ν el r = arcsin Using (13), the passive beamforming gain µ l,m is computed as where Θ = diag{ϑ} and the operator (·) * denotes the conjugate, and denotes the element-wise product. The optimal phase-shift ϑ opt is given by

Proposed CG-LHB Algorithm
The objective function is to maximize the rate of the user vehicle while satisfying the maximum transmission power constraint. Using (4) and (12), the sub-rate maximization problem is written as where i = 1, 2, . . . , N s . The sub-rate maximization problem (20) is a sub-problem with respect to the digital beamformer F D . To maximize the sub-rate, F D should be optimized to minimize ||F A f D,i || 2 based on the following multi-object optimization problem (20) as where F A , W A and ϑ are fixed. We see from (21), the N s column of F D can be optimized separately according to where ς i denotes an all-zero vector except for the i-th entry being unity. Using the streamspecific error expressions in (22), we apply Karush-Kuhn-Tucker (KKT) conditions and the Lagrangian method for the convex problem as where A = F H A F A denotes the equivalent N s × N s analog beamforming matrix, κ is the Lagrangian multiplier and the corresponding Lagrangian is If ∂ f f D,i , κ)/∂f D,i = 0, then f D,i and κ is given by and Thus, combining (25) and (26), the digital beamforming vector f D,i is given by Since F A is a constant modulus in (27), we optimize the phase of the analog beamformer for transforming a complex problem into an unconstrained optimization problem. Let the beamforming design problem can be written by where F A is the set of the analog beamformer, F opt = V 1 , which is measured by using a singular value decomposition method of H e = UΓV H in [41]. Consider F A into F A,m as where Ω m denotes the N t × N s phase shift matrix, which phases need to be quantized into the nearest point in (2πm/2 b , m = 0, 1, . . . , 2 b − 1), if the phase shifters are of infinite or finite b-bit resolution. Hence, the phase optimization problem can be expressed as where Ω m is converged using a conjugate gradient method, which optimizes along the direction ∆Ω m where ∆Ω m = η m Ξ m and ∆Ω m satisfy ∆Ω m ≤ τ, η m is determined by minimizing f (Ω m+1 ), Ξ m is given by and the ξ is given by Using (29) to (33), we summarize the proposed CG-LHB algorithm in Algorithm 1.

User Vehicle Localization
Let RIS to user channel covariance Σ i = E[g i g H i ], where i = 1, . . . , N s } and the channel vector g = vec(G) is geometrically generated for the different positions as shown in Section 2.3. Thus, the user vehicle localization problem can be computed as a hypothesis testing problem as follows: Hence, the optimal likelihood (ML)-based localization problem is given bŷ where p i (g) is the probability density function under G We can now evaluate the error-probability performance under H i , which is given by where P e,i is the conditional error probability under H i . Due to user mobility and high location errors, (36) contains a high channel state information (CSI) error. To overcome this issue, we analyze the location information generated by GPS. Consider the location of the user vehicle from the GPS represented by where u = [u x , u y , u z ] T denotes the actual location vector of the user vehicle,û = [û x ,û y ,û z ] T is the ego GPS location vector, and ∆u = [∆u x , ∆u y , ∆u z ] T is the localization error vector and ∆u satisfy ||∆u|| 2 ≤ . Based on (13) and (38), we obtain the actual channel vector g that satisfy g(u) =ĝ(u) + ∆g(u), where ∆g ∼ N (0, Σ) and Σ = [σ 2 g,x , σ 2 g,y , σ 2 g,z ] T is approximated as diagonal with σ 2 g,x , σ 2 g,y and σ 2 g,z being the channel variances along the x, y, and z directions, respectively. Hence, by using (39), with the location error bound (36), the CSI error minimization problem can be formulated as It is noted that (40) still suffers high localization errors. This issue can be addressed using a double-step iteration method. Let the error covariance matrix Σ possesses and split two different covariance matrices Σ = Σ 1 + Σ 2 in (34), where Σ 1 = F * Λ 1 F, Σ 2 =F * Λ 2F , Λ 1 and Λ 2 are the diagonal matrices holding the eigenvalues of Σ 1 and Σ 2 , respectively. If the matrices Σ 1 and Σ 2 are positive definite and α is a positive constant, then the iterative covariance matrix Σ(α) is given using by [42] where α denotes the positive constant, (34), we demonstrate the proposed double-step iterative algorithm for the user vehicle localization in Algorithm 2. The double-step at each Σ 1 Σ 2 Σ 1 Σ 2 iterations is necessary to obtain an exact solution for the αI + Σ 1 and αI + Σ 2 matrices, which strength diagonal properties of the Σ 1 and Σ 2 matrices. The complexity of the proposed Algorithm 2 is O(N R log 2 N R ) instead of the original complexity O(N 3 R ).
For the line-of-sight channel (l = 0 case), the RIS-assisted channel vector g is given using where t and r are the transmit and receive antenna gains,β 0 ∼ CN (0, 10 −0.1PL(d2) ) denotes the channel path gain, PL(d2) in the path loss in the RIS to user link and PL(d2) is defined as [27,43,44] PL(d2) = 35.6 + 22.2log 10 (d 2 ) + X , where d 2 denotes the distance between the RIS and user vehicle as shown in Figure 1, and X ∼ N (0, σ 2 X ). The parameter σ X is set at σ X = 5.8 dB, respectively. Similarly, we also can measure the corresponding channel C and path loss PL(d 1 ) according to (42) and (43), and d 1 represents the distance from the transmitting vehicle to the RIS controller. Figure 3 depicts the received signal power gain versus SNR. We see that the gain performance of the received signal power is showing the same in the high SNR case. For the low SNR case, the gain performance of the received signal power is around 0.30 dB as shown in Figure 3. Figure 4 shows the achievable spectral efficiency versus transmit power using user location information. The proposed CG-LHB algorithm provides a significant increase in the spectral efficiency for a 32 × 32 reflecting array compared to the traditional RIS-assisted hybrid beamforming [25,27] and with and without RIS-assisted hybrid beamforming method [3]. The achievable spectral efficiency of the proposed CG-LHB algorithm is about 1.30 bits/s/Hz, the conventional RIS-assisted hybrid beamforming method is around 1.01 bits/s/Hz and the hybrid beamforming without RIS is about 0.90 in Figure 4. For the proposed CG-LHB algorithm case, the achievable spectral efficiency performance is increased by approximately 0.301 bits/s/Hz/user at 1 watt transmitted power values. Figure 5 illustrates the achievable spectral efficiency versus N R by evaluating Algorithm 1. we also plot the spectral efficiency of the conventional hybrid beamforming with the G-HB algorithm [25,45]. The performance of the spectral efficiency is around 10% at N R = 32. Figure 6 illustrates the convergence of the proposed algorithm with two gradient descent methods as in Algorithm 1 at ξ = 0.6. By measurement of ∇ f (Ω) in (32), the computational complexity of optimizing F A is dominated and it takes O(N t N RF N s ) multiplications. It is noted that the total algorithm needs O(N t N RF N s J) multiplications, where J is the number of iterations required to converge with a fixed value of parameter τ and J < 100. Instead of the conventional gradient method (Algorithm 3 in [46]), which required O(N 2 t N 2 RF J) multiplications, the proposed algorithm shows the computationally more efficient.     Analog beamforming optimization Conjugate Gradient-based analog optimization Gradient Descent-based analog optimization Figure 6. Convergence of the proposed CG algorithm−based analog beamforming optimization.
In Figure 7, we plot the error probability versus SNR, which relies on the equivalent effective channel. The effective equivalent channel is geometrically generated for the different object locations. To achieve a better error-probability performance, we applied a channel covariance splitting method and proposed a double-step iterative algorithm. In computer simulation, we evaluate P e as a function of SNR, where SNR leads the location information subject to the channel path loss, which progressively adds extra scatter positions to previous positions. From Figure 6, we observe that the proposed double-step algorithm provides higher accuracy than the conventional iterative algorithm [26] in the case of a multi-antenna user vehicle. Figure 8 compares the average bit-error-rate versus SNR of a RIS-assisted V2V MIMO system at N s = 4, N t = 8 N u = 8, and N R = 256. We consider the proposed double-step iterative and conventional iterative algorithms. We observe that the proposed double-step algorithm significantly outperforms the conventional iterative algorithm. In Algorithm 2, we used α = 3 to fix the spectral of the covariance matrices. We also observe that the average bit-error rate is improved by about 1.01 dB at the average bit-error rate of 0.0001. Figure 9 shows the channel state information (CSI) error bound in terms of location error . We used the location error parameter = 3 m to validate the effect of the location error on the passive beamforming. If the number of N R is increased, the CSI error bound will vary as shown in Figure 9.

Conclusions
In this paper, we proposed RIS-assisted CG-LHB and a user vehicle localization algorithm. Utilizing the CG-LHB algorithm, we were able to significantly improve spectral efficiency in mmWave MIMO systems. In addition, for user vehicle localization, we considered a channel covariance splitting method and proposed a double-step iterative algorithm that reduces the effect of location error on the passive beamforming. We validated the effectiveness of the proposed algorithms using the conventional algorithms. Hence, the proposed RIS-assisted CG-LHB algorithm can be extended further to the machine learning-based beam alignment and location error minimization solution of the connected autonomous vehicles, which will be explored in future studies.