## 3 Proposed CG-LIB Algorithm and User Vehicle Localization
### User Vehicle Localization
Based on (34), we demonstrate the proposed double-step iterative algorithm for the user vehicle localization in Algorithm 2. The double-step at each \(\mathbf{\Sigma}_{1}\mathbf{\Sigma}_{2}\mathbf{\Sigma}_{1}\mathbf{\Sigma}_{2}\) iterations is necessary to obtain an exact solution for the \(\alpha\mathbf{I}+\mathbf{\Sigma}_{1}\) and \(\alpha\mathbf{I}+\mathbf{\Sigma}_{2}\) matrices, which strength diagonal properties of the \(\mathbf{\Sigma}_{1}\) and \(\mathbf{\Sigma}_{2}\) matrices. The complexity of the proposed Algorithm 2 is \(\mathcal{O}(N_{R}\text{log}_{2}N_{R})\) instead of the original complexity \(\mathcal{O}(N_{R}^{2})\). ```
1:Input parameters:\(k\), \(\alpha\)
2:Output:\(\rho(\mathbf{\Sigma}(\alpha))\). 3:If \(x^{(k)}\) is a double-step sequence, then \(x^{(\frac{2k+1}{2})}=\mathbf{M}_{1}^{-1}\mathbf{N}_{1}x^{(k)}+\mathbf{M}_{1}^ {-1}b\), and \(x^{(k+1)}=\mathbf{M}_{2}^{-1}\mathbf{N}_{2}x^{(\frac{2k+1}{2})}+\mathbf{M}_{2}^ {-1}b\), where \(\mathbf{\Sigma}\in\mathbb{C}^{N_{R}\times N_{r}},\mathbf{\Sigma}=\mathbf{M}_{q}- \mathbf{N}_{q}(q=1,2)\) and \(\mathbf{\Sigma}x=b\) for all initial vectors \(x^{(0)}\in\mathbb{C}^{N_{u}}\). 4:If \(k=0,1,\ldots,\) then compute \(x^{(k+1)}=\mathbf{\Sigma}(\alpha)x^{(k)}+,(\alpha)b\), where \(\mathbf{\Sigma}(\alpha)=\mathbf{M}_{2}^{-1}\mathbf{N}_{2}\mathbf{M}_{1}^{-1} \mathbf{N}_{1}\), \(\mathbf{\Sigma}_{(}\alpha)=\mathbf{M}_{2}^{-1}(\mathbf{I}+\mathbf{N}_{2}\mathbf{M}_ {1}^{-1})\), and \(\rho(\mathbf{\Sigma}(\alpha))\leq\sigma(\alpha)<1,\forall_{x}>0\). 5:When \(\ratio (SNR) at the receiver is defined as \(\Gamma=\frac{P}{N_{t}\sigma_{2}^{2}}||Y||_{F}^{2}\), where \(\mathrm{Y}=\mathbf{G}\mathbf{\Theta}\mathbf{C}\mathbf{F}_{A}\mathbf{F}_{D}\), \(P=1\) watt, \(\sigma_{2}^{2}=-174+10\text{log}_{10}B=-90\) dBm and the bandwidth \(B=251.188\) MHz [28]. The azimuth angle of departure (AoD) and angle of arrival (AoA) are uniformly distributed in the \((-\pi,\pi)\) and the elevation AoD and AoA are uniformly distributed in the interval \((-\pi/2,\pi/2)\)[5, 41]. Let the coordinates of the transmit intelligent vehicle, the RIS and the user vehicle are \((x_{t},y_{t})=(0,0)\text{m}\), \((x_{R},y_{R})=(110,20)\text{m}\), and \((x_{u},y_{u})=(210,10)\text{m}\), respectively. We set the phase-shifters resolution \(b=2\) bits and the algorithmic parameter \(\tau=10^{-6}\), \(J=100\), \(\xi=0.6\) and \(\alpha=3\)[42] in the computer simulations. For the line-of-sight channel (\(l=0\) case), the RIS-assisted channel vector \(\mathbf{g}\) is given using

\[\mathbf{g}=\sqrt{N_{R}}\hat{\beta}_{0}\varrho_{t}\varrho_{r}\mathbf{a}_{R}( \theta_{t}^{\alpha z},v_{t}^{el}), \tag{42}\]

where \(\varrho_{t}\) and \(\varrho_{r}\) are the transmit and receive antenna gains, \(\hat{\beta}_{0}\sim\mathcal{CN}(0,10^{-0.1 PL(d2)})\) 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]

\[PL(d2)=35.6+22.2\text{log}_{10}(d_{2})+\mathcal{X}, \tag{43}\]

where \(d_{2}\) denotes the distance between the RIS and user vehicle as shown in Figure 1, and \(\mathcal{X}\sim\mathcal{N}(0,\sigma_{\mathcal{X}}^{2})\). The parameter \(\sigma_{\mathcal{X}}\) is set at \(\sigma_{\mathcal{X}}=5.8\) dB, respectively. Similarly, we also can measure the corresponding channel \(\mathbf{C}\) 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.