## 2 System Model
### System Model without Hardware Impairments  
Consider that the direct and cascaded channels are all LoS, and the BS, the \((m,n)\)-th element of the \(i\)-th RIS, and the vehicle are located at \(\mathbf{P}_{\text{BS}}\), \(P_{in,in}\), and \(\mathbf{P}_{\text{V}}(t)\), respectively. Then, the direct link gain is given as [39]  
\[A_{0}(t)=\frac{\lambda_{c}}{4\pi}\frac{\sqrt{G_{\text{BS}}^{\text{V}}(t)G_{ \text{V}}^{\text{BS}}(t)}}{d_{0}(t)}, \tag{3}\]  
Figure 1: A multi-RIS-assisted HSC system. \(K=2\). Note that the real-time phase shift set is sent to RISs through feedback links. \begin{table}
\begin{tabular}{c c c c} \hline
**Notation** & **Description** & **Notation** & **Description** \\ \hline \(P_{\text{BS}}\) & BS location. & \(d_{in,in}^{N}(t)\) & RIS to vehicle \\ \(P_{\text{V}}(t)\) & Vehicle location. & \(\tau_{0}(t)\) & Direct link delay. \\  & The \((m,n)\)-th element & & \\ \(P_{in,in}\), & of the \(i\)-th RIS & \(\tau_{in,in}\), & Cascaded link delay. \\  & location. & & \\ \(A_{0}(t)\) & Direct link gain. & \(\phi_{in,in}\), & RIS phase shift set. \\ \(A_{in,in}\), & Cascaded link gain. & \(\gamma_{in,in}\), & RIS HWI. \\ \(d_{0}(t)\) & BS to vehicle distance. & \(\eta_{t}\) & Transmitter HWI. \\ \(d_{\text{BS}}^{in,in}\) & BS to RIS distance. & \(\eta_{R}\) & Receiver HWI. \\ \hline \end{tabular}
\end{table} 
Table 3: Notations of system model coefficients. where \(G_{\rm BS}^{\rm V}(t)\) is the BS antenna gain when the observation point is the vehicle antenna and \(G_{\rm V}^{\rm BS}(t)\) is the vehicle antenna gain when the observation point is the BS antenna. \(\lambda_{c}\) denotes the wavelength of the signal and \(d_{0}(t)=\|P_{\rm V}(t)-P_{\rm BS}\|\) is the Euclidean distance between the BS and the vehicle at time \(t\). Since the BS and the vehicle antennas are all isotropic, then  
\[A_{0}(t)=\frac{\lambda_{c}}{4\pi d_{0}(t)}. \tag{4}\]  
Moreover, the direct link time delay can be obtained as  
\[\tau_{0}(t)=\frac{d_{0}(t)}{c_{0}}, \tag{5}\]  
where \(c_{0}\) is the speed of light. Therefore, the received signal from direct link at time \(t\) can be obtained as  
\[S_{0}(t)=A_{0}(t)\sqrt{P_{t}}x(t-\tau_{0}(t)), \tag{6}\]  
where \(P_{t}\) is the fixed transmit power and \(x(t)\) is the transmit signal with \(\mathbb{E}\{|x(t)|^{2}\}=1\). Furthermore, the cascaded link gain by the \((m,n)\)-th element of the \(i\)-th RIS at time \(t\) can be obtained as  
\[A_{i_{m},i_{n}}(t)=\frac{\lambda_{c}^{2}}{16\pi^{2}}\frac{\sqrt{G_{i_{m},i_{n }}^{\rm BS}G_{i_{m},i_{n}}^{\rm V}G_{i_{m},i_{n}}^{\rm V}(t)G_{\rm V}^{i_{m},i_ {n}}(t)}}{d_{\rm BS}^{i_{m},i_{n}}^{i_{n}}(t)}, \tag{7}\]  
where \(G_{i_{m},i_{n}}^{\rm BS}\) is the gain of the \((m,n)\)-th element of the \(i\)-th RIS when the observation point is the BS antenna, \(G_{\rm BS}^{i_{m},i_{n}}\) is the gain of BS antenna when the observation point is the \((m,n)\)-th element of the \(i\)-th RIS when the observation point is the vehicle antenna, and \(G_{\rm V}^{i_{m},i_{n}}(t)\) is the gain of the vehicle antenna when the observation point is the \((m,n)\)-th element of the \(i\)-th RIS. Furthermore, \(d_{\rm BS}^{i_{m},i_{n}}=\|P_{i_{m},i_{n}}-P_{\rm BS}\|\) and \(d_{i_{m},i_{n}}^{\rm V}(t)=\|P_{\rm V}(t)-P_{i_{m},i_{n}}\|\) are the distances from the BS to the element and from the element to the vehicle, respectively. Since the RISs and transceiver are assumed to be isotropic in this paper, we have  
\[A_{i_{m},i_{n}}(t)=\frac{\lambda_{c}^{2}}{16\pi^{2}}\cdot\frac{1}{d_{\rm BS}^ {i_{m},i_{n}}^{i_{n}}d_{i_{m},i_{n}}^{\rm V}(t)}. \tag{8}\]  
Furthermore, the cascaded link time delay for the \((m,n)\)-th element of the \(i\)-th RIS can be calculated as  
\[\tau_{i_{m},i_{n}}(t)=\frac{d_{\rm BS}^{i_{m},i_{n}}+d_{i_{m},i_{n}}^{\rm V}(t )}{c_{0}}. \tag{9}\]  
Note that \(d_{\rm BS}^{i_{m},i_{n}}\) is the distance from the BS to the \((m,n)\)-th element of the \(i\)-th RIS. It is fixed, since the BS and all RISs are static.