# RIS-Assisted High-Speed Communications with Time-Varying Distance-Dependent Rician Channels
## 2 System Model with Time-Varying Distance-Dependent Rician Channels
### Cascaded RIS Link \(h_{\text{RIS}}(t)\) from the BS to the RIS and from RIS to the MR  
Suppose the RIS is a planar surface on a rectangular grid spaced \(dx\) and \(dy\) apart in the three-dimensional Cartesian coordinate system. Assume the size and the geometric center of the RIS in Figure 1. are \(\sqrt{M}\times\sqrt{M}\) and \([0,r,0]^{\mathsf{T}}\), respectively. The center of the \(mn\)-th element is given as  
\[D_{mn}=\left[\Psi(m,dx,\sqrt{M}),\Psi(n,dy,\sqrt{M})+r,0\right]^{\mathsf{T}}, \tag{7}\]  
where \(\Psi(m,dx,\sqrt{M})=dx(m-0.5((\sqrt{M}+1)\ \text{mod}\ 2))\), \(m\in\Omega(\sqrt{M})\) with  
\[\Omega(\sqrt{M})=\bigg{\{}((\sqrt{M}+1)\ \text{mod}\ 2))-\left\lfloor\frac{ \sqrt{M}}{2}\right\rfloor,\ldots,\left\lfloor\frac{\sqrt{M}}{2}\right\rfloor \bigg{\}}, \tag{8}\]  
and \(\Psi(n,dy,\sqrt{M})\) is obtained in the same way as \(\Psi(m,dx,\sqrt{M})\).  
Assume that the BS-RIS and the RIS-MR links, i.e., \(g_{mn}(t)\) and \(h_{mn}(t)\), both have LoS and NLoS parts. Let us consider the LoS part of \(h_{\text{RIS}}(t)\) first. For the \(mn\)-th isotropic element of the RIS, the channel gain of the BS-RIS and the RIS-MR links are given respectively as  
\[A_{\text{BS}}^{mn}=\frac{\lambda}{4\pi d_{\text{BS}}^{mn}} \tag{9}\]  
and  
\[A_{mn}^{\text{MR}}(t)=\frac{\lambda}{4\pi d_{mn}^{\text{MR}}(t)}, \tag{10}\]  
where \(d_{\text{BS}}^{mn}=\|D_{mn}-D_{\text{BS}}\|\) and \(d_{mn}^{\text{MR}}(t)=\|D_{\text{MR}}(t)-D_{mn}\|\). The delay of the \(mn\)-th cascaded link is  
\[\tau_{mn}(t)=\tau_{\text{BS}}^{mn}+\tau_{mn}^{\text{MR}}(t)=\frac{d_{\text{BS }}^{mn}}{c}+\frac{d_{mn}^{\text{MR}}(t)}{c}. \tag{11}\]  
The DS for the \(mn\)-th cascaded link \(h_{mn}(t)\) can be calculated as  
\[\begin{split} f_{mn}(t)&=-f_{c}\frac{d}{dt}\bigg{(} \tau_{mn}^{\text{MR}}(t)+\frac{\phi_{mn}(t)}{2\pi f_{c}}\bigg{)}\\ &=-f_{c}\bigg{(}\frac{v(x_{0}+vt-x_{mn})}{cd_{mn}^{\text{MR}}(t) }+\frac{d}{dt}\big{(}\frac{\phi_{mn}(t)}{2\pi f_{c}}\big{)}\bigg{)},\end{split} \tag{12}\]  
where \(\phi_{mn}(t)\in(0,2\pi]\) is the tunable phase shift of the \(mn\)-th element. We assume the maximal amplitude of 1 for all of the elements.  
Therefore, the \(mn\)-th RIS-MR link, \(h_{mn}(t)\), can be obtained as  
\[h_{mn}(t)=A_{mn}^{\text{MR}}(t)\bigg{(}\sqrt{\frac{\kappa_{mn}^{\text{LoS}}} {\kappa_{mn}^{\text{LoS}}}+1}h_{mn}^{\text{LoS}}(t)+\sqrt{\frac{1}{\kappa_{ mn}^{\text{NLoS}}}+1}h_{mn}^{\text{NLoS}}(t)\bigg{)}, \tag{13}\]  
where  
\[h_{mn}^{\text{LoS}}(t)=\exp\bigg{(}-j2\pi(f_{c}+f_{mn}(t))\tau_{mn}^{\text{ MR}}(t)\bigg{)}, \tag{14}\]  
and \(h_{mn}^{\text{NLoS}}(t)\) is the same as \(h_{d}^{\text{NLoS}}(t)\), \(\kappa_{mn}^{\text{LoS}}\) and \(\kappa_{mn}^{\text{NLoS}}\) respectively denote Rician factors of the LoS and NLoS parts of \(h_{mn}(t)\). Similarly, the \(mn\)-th BS-RIS link, \(g_{mn}(t)\), can be obtained as  
\[g_{mn}(t)=A_{\text{BS}}^{mn}\bigg{(}\sqrt{\frac{\rho_{mn}^{\text{LoS}}}{\rho_{ mn}^{\text{LoS}}+1}}g_{mn}^{\text{LoS}}+\sqrt{\frac{1}{\rho_{mn}^{\text{NLoS}}+1}}g_{ mn}^{\text{NLoS}}(t)\bigg{)}, \tag{15}\]  
where \(g_{mn}^{\text{LoS}}=\exp(-j2\pi f_{c}\tau_{\text{BS}}^{mn})\), \(\rho_{mn}^{\text{LoS}}\) and \(\rho_{mn}^{\text{NLoS}}\) are Rician factors of the LoS and NLoS channels of \(g_{mn}(t)\) respectively, and \(g_{mn}^{\text{NLoS}}(t)\sim\mathcal{CN}(0,1)\). Note that although the \(g_{mn}^{\text{LoS}}\) in (15) is not a function of time \(t\), \(g_{mn}(t)\) is still time-dependent because the NLoS part,i.e., \(g_{mn}^{\text{NLoS}}(t)\), would change due to moving obstacles such as cars and pedestrians. Finally, we have the cascaded link as  
\[h_{\text{RIS}}(t)=\sum_{m=-1,n=1}^{\sqrt{M}_{\text{L}}\sqrt{M}}h_{mn}(t)\exp \big{(}-\jmath\phi_{mn}(t)\big{)}g_{mn}(t). \tag{16}\]