## 3 Proposed CG-LIB Algorithm and User Vehicle Localization
### User Vehicle Localization
The proposed CG-LHB algorithm provides a significant increase in the spectral efficiency for a \(32\times 32\) 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 \(\xi=0.6\). By measurement of \(\
abla f(\mathbf{\Omega})\) in (32), the computational complexity of optimizing \(\mathbf{F}_{A}\) is dominated and it takes \(\mathcal{O}(N_{t}N_{RF}N_{s})\) multiplications. It is noted that the total algorithm needs \(\mathcal{O}(N_{t}N_{RF}N_{s}J)\) multiplications, where \begin{table}
\begin{tabular}{c c} Total number of transmit vehicle antennas & \(N_{t}=8\) \\ Total number of user vehicle antenna & \(N_{t}=8\) \\ Total number of RIS discrete elements & \(N_{R}=32(N_{R,y}=8,N_{R,z}=4)\) \\ Total number of RF chains & \(N_{RF}=4\) \\ Total number of data streams & \(N_{s}=4\) \\ The number of propagation path & \(1\) \\ Wavelength & \(\lambda=1\) cm \\ Light speed & \(c=3\times 10^{8}\,\text{m}/\,\text{s}\) \\ RIS elements distance & \(d=0.5\) cm \\ Carrier frequency & \(f_{c}=28\,\text{GHz}\) \\ RIS dimensions & \(32\times 32\) \\ Noise PSD & \(N_{0}=-174\,\text{dBm}/\,\text{Hz}\) \\ Noise variance & \(\sigma_{2}^{2}=N_{0}+10\text{log}_{10}B\) \\ \end{tabular}
\end{table} 
Table 1: Simulation Parameters. 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 \(\alpha=3\) 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 \(\epsilon\). We used the location error parameter \(\epsilon=3\) 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 \(\epsilon\) will vary as shown in Figure 9.

Figure 6: Convergence of the proposed CG algorithm–based analog beamforming optimization. Figure 7: The error–probability performance of the V2V communications at 20 dB SNR. ## 5 Conclusions

In this paper, we proposed RIS-assisted CG-LHB and a user vehicle localization algorithm.