## 2 Signal and Channel Model
### Channel Model  
Let \(\mathbf{H}_{e}=\mathbf{G}\mathbf{\Theta}\mathbf{C}\) where \(\mathbf{G}\) is an \(N_{u}\times N_{R}\) channel between RIS to the user vehicle as  
\[\mathbf{G}=\gamma_{g}\sum_{l=1}^{L_{g}}\tilde{\beta}_{l}\mathbf{a}_{u}(\phi_{u}^ {l})\mathbf{a}_{R}^{H}(\theta_{t}^{l},\
u_{t}^{l}), \tag{5}\]  
where \(\tilde{\beta}_{l}\) denotes the complex channel gain at \(l\)-th path of \(\mathbf{G}\) channel, \(\phi_{u}^{l}\) represents the angle of arrival associated with the user vehicle, \(\gamma_{g}=\sqrt{N_{u}N_{R}/L_{g}}\), and \(\theta_{t}^{l}\) and \(\upsilon_{t}^{l}\) 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].  
\[\mathbf{a}_{u}(\phi_{u}^{l})=\frac{1}{\sqrt{N_{t}}}\bigg{[}1,\,e^{\frac{(2\pi i \sin(\theta_{t}^{l})}{\lambda}},\dots,e^{\frac{(2\pi(N_{t}-1)\sin(\theta_{t}^ {l})}{\lambda}})\bigg{]}^{T}, \tag{6}\]  
Figure 1: System model is illustrated by RIS-assisted V2V communications.  
Figure 2: Sensors connected autonomous vehicle with WiFi-based mmWave transceiver.  
where the wavelength \(\lambda=c/f_{c}\), \(c\) is the speed of light, \(f_{c}\) is the carrier frequency, and \(d=\lambda/2\) is the antenna spacing. The normalized array response vector for uniform planner array is given by [28]  
\[\begin{split}\mathbf{a}_{R}(\theta_{t}^{l},\
u_{t}^{l})& =\frac{1}{\sqrt{N_{R}}}[1,\,e^{\frac{\left[2 nd}{\lambda}\left((n_{1}-1) \cos{(\
u_{t}^{l})}\sin{(\theta_{t}^{l})}+(n_{2}-1)\sin{(\
u_{t}^{l})}\right) }\right.,\\ &\left.\ldots e^{\frac{2 nd}{\lambda}\left\{(N_{R,y}-1)\cos{(\
u_{ t}^{l})}\sin{(\theta_{t}^{l})}+(N_{R,z}-1)\sin{(\
u_{t}^{l})}\right\}}\right]^{T}, \end{split} \tag{7}\]  
where \(N_{R}=N_{R,y}\times N_{R,z}\), and \(\mathbf{C}\) denotes the \(N_{R}\times N_{t}\) channel between the transmit vehicle and the RIS controller, that is  
\[\mathbf{C}=\gamma_{c}\sum_{l=1}^{L_{c}}\tilde{\gamma}_{l}\mathbf{a}_{R}( \theta_{r}^{l},\
u_{r}^{l})\mathbf{a}_{t}^{H}(\phi_{t}^{l}), \tag{8}\]  
where \(\tilde{\gamma}_{l}\) denotes the complex channel gain at \(l\)-th path of \(\mathbf{C}\) channel, \(\gamma_{c}=\sqrt{N_{l}N_{R}/L_{c}}\), \(\mathbf{a}_{R}(\theta_{r}^{l},\
u_{r}^{l})\) and \(\mathbf{a}_{t}^{H}(\phi_{t}^{l})\) are designed in the same manner of \(\mathbf{G}\) channel and the \(N_{R}\times N_{R}\) RIS element response matrix \(\mathbf{\Theta}\) is given by  
\[\mathbf{\Theta}=\frac{1}{\sqrt{N_{R}}}\mathbf{D}, \tag{9}\]  
where \(\mathbf{D}\) is the diagonal phase matrix as [25],  
\[\mathbf{D}=\text{diag}\{\theta_{1},\ldots,\theta_{N_{R}}\}, \tag{10}\]  
where \(\boldsymbol{\theta}=[\theta_{1},\ldots,\theta_{N_{R}}]^{T}\), \(\theta_{n}=e^{i\varphi_{n}}\), \(n=1,2,\ldots,N_{R}\) and \(\varphi_{n}\) denotes the phase-shift which is given by  
\[\boldsymbol{\varphi}=\begin{bmatrix}\varphi_{1,1}&\cdots&\varphi_{1,N_{R}} \\ \vdots&\ddots&\vdots\\ \varphi_{N_{R},1}&\cdots&\varphi_{N_{R},N_{R}}\end{bmatrix}. \tag{11}\]  
Thus, the effective channel \(\mathbf{H}_{e}\) can be written as  
\[\mathbf{H}_{e}=\sum_{m=1}^{L_{c}}\sum_{l=1}^{L_{y}}\beta_{l}\gamma_{m}\mu_{l, m}\mathbf{a}_{u}(\phi_{R}^{l})\mathbf{a}_{t}^{H}(\theta_{t}^{l}), \tag{12}\]  
where \(\mu_{l,m}=\mathbf{a}_{R}^{H}(\theta_{r}^{l},\
u_{t}^{l})\mathbf{\Theta}\mathbf{ a}_{R}(\theta_{r}^{m},\
u_{r}^{m})\) denotes the passive beamforming gain, \(\beta_{l}=\gamma_{g}\tilde{\beta}_{l}\), \(\gamma_{l}=\gamma_{c}\tilde{\gamma}_{l}\), and \(\mu_{l,m}\) satisfy \(|\gamma_{1}|\geq|\gamma_{2}|,\ldots,\geq|\gamma_{L_{c}}|\) and \(|\beta_{1}|\geq|\beta_{2}|,\ldots,\geq|\beta_{L_{g}}|\).