## 3 Communication and Channel Modeling
### Channel Modeling
#### 3.1.2 Position-Aware Directional Beam-Steering  
Based on Remark 5, directional beam-steering can be implemented to improve the channel gain and communication performance. Assuming the localization-aware communication and geometry-based directional phase shift, the channel gain can be obtained in the angular domain as follows:  
\[\mathcal{G}_{CLoS}^{TR}\approx\left(\frac{\lambda}{4\pi}\right)^{2}\sqrt{G_{T}} \sqrt{G_{R}}\ \frac{\overbrace{\left(\mathbf{a}_{inc}^{RIS\dagger}\left(\varphi^{T\_RIS},\ \zeta^{T\_RIS}\right)\Psi^{RIS}\mathbf{a}_{ref}^{RIS}\left(\varphi^{RIS\_R}-, \ \zeta^{RIS\_R}\right)\right)}^{RIS}e^{-\left(\phi^{inc}+\phi^{ref}\ \right)}}{\underbrace{\left\|p^{Tx}-p^{ RIS}\right\|_{2}\left\|p^{RIS}-p^{Rx}\right\|_{2}}} \tag{11}\]  
\[\text{with }\left\{\phi_{inc}^{inc}=\frac{2\pi}{\lambda}\pi^{\top}\left( \varphi^{T\_RIS},\ \zeta^{T\_RIS}\right)\left(p^{RIS}-p^{Tx}\right)\right.\]  
where \(\varphi^{T\_RIS}\) and \(\zeta^{T\_RIS}\) denote azimuth and elevation angles of the _incident_ (_inc_) path, respectively; \(\varphi^{RIS\_R}\) and \(\zeta^{RIS\_R}\) denote azimuth and elevation angles of the _reflection_ (_ref_) path, respectively, which are given based on the geometry information, see Figure 4 and Appendix B; \(\mathbf{a}_{inc}^{RIS\_R}\left(\varphi^{T\_RIS},\ \zeta^{T\_RIS}\right)\in\mathbb{C}^{M\times 1}\) denotes the RIS array response (wave) vector to the incident signal from the transmitter to the RIS and is obtained as follows:  
\[\mathbf{a}_{inc}^{RIS}\left(\varphi^{T\_RIS},\ \zeta^{T\_RIS}\right)=\left[e^{-i \phi_{(m-1)}^{inc}},\ldots,e^{-i\phi_{m}^{inc}},\ldots,e^{-i\phi_{M}^{inc}} \right]^{\top} \tag{12}\]  
where \(\phi_{m}^{inc}\) denotes the imposed phase shift to the signal received at the \(m^{th}\) element of the RIS with respect to the center point of RIS given as follows:  
\[\phi_{m}^{inc}=\frac{2\pi}{\lambda}\mathbf{w}^{\top}\left(\varphi^{T\_RIS}, \ \zeta^{T\_RIS}\right)\left(p_{m}^{RIS}-p^{RIS}\right); \tag{13}\]  
\(\mathbf{w}\left(\varphi^{T\_RIS},\ \zeta^{T\_RIS}\right)\in\mathbb{R}^{3}\) denotes the normalized 3D wave vector of the \(T_{x}-\) RIS path given as follows:  
\[\mathbf{w}\left(\varphi^{T\_RIS},\ \zeta^{T\_RIS}\right)=\begin{bmatrix}\cos \left(\zeta^{T\_RIS}\right)\cos\left(\varphi^{T\_RIS}\right)\\ \cos\left(\zeta^{T\_RIS}\right)\sin\left(\varphi^{T\_RIS}\right)\\ \sin\left(\zeta^{T\_RIS}\right)\end{bmatrix}; \tag{14}\]  
Similarly, the RIS array response to the reflected signal \(\mathbf{a}_{ref}^{RIS\_R}\left(\varphi^{RIS\_R},\ \zeta^{RIS\_R}\right)\) can be achieved using the azimuth and elevation angles of the reflection path. The RIS phase shift design is implemented based on Remark 6.  
**Remark 6**.: _Directional beam-steering is achieved by controlling the phase shifts (i.e., UPA phased array) of the RIS. It has been proven that by discrete Fourier transform (DFT)-based codebook design, the directional beam-steering is achieved if the DFT weight vector is equivalent to the normalized array response vector (i.e., the normalized wave vector) [43]. Therefore, the normalized array response vectors of the BS-RIS path and RIS-UE (e.g., RIS-IoVs) path are used to control the phased array for BS-BF and RIS-PhSh, respectively, to realize directional beam steering._  
In this light, the phase shift matrix can be designed as follows:  
\[\Psi^{RIS}=\mathbf{a}_{inc}^{RIS}\left(\varphi^{T\_RIS},\ \zeta^{T\_RIS} \right)\mathbf{a}_{ref}^{RIS\_R}\left(\varphi^{RIS\_R},\ \zeta^{RIS\_R}\right). \tag{15}\]After using (15), (11) is converted to (9). However, the advantage of using a geometry-based phase shift design (i.e., directional beam-steering) is that it can be effectively implemented through an appropriate codebook design, as highlighted in Remark 6. Additionally, this approach is well-suited for dynamic vehicular networks, where vehicles are in motion. Positioning-aware beam steering can be employed as an alternative to CSI, which is challenging to obtain in such dynamic environments.