# RIS-Assisted High-Speed Communications with Time-Varying Distance-Dependent Rician Channels
## 4 Performance Analysis
### Ergodic Spectral Efficiency  
**Proposition 1**.: _Consider \(P_{t}\) and \(\sigma^{2}\) are respectively the transmitted and the additional noise power, and \(h(t)\) is the channel for the proposed model. Then, the ergodic SE of the RIS-assisted HSC system at time \(t\) is upper bounded by_  
\[\text{SE}(t)=\mathbb{E}\bigg{\{}\log_{2}\bigg{(}1+\frac{P_{t}}{\sigma^{2}}|h( t)|^{2}\bigg{)}\bigg{\}}\leq\text{SE}^{\text{upper}}(t)=\log_{2}\bigg{(}1+ \frac{P_{t}}{\sigma^{2}}\mathbb{E}\big{\{}|h(t)|^{2}\big{\}}\bigg{)}, \tag{29}\]  
_where_  
\[\begin{split}\mathbb{E}\big{\{}|h(t)|^{2}\big{\}}& =\big{(}K_{1}^{2}(t)+K_{2}^{2}(t)\big{)}A_{0}^{2}(t)+K_{3}^{2}(t)G _{1}^{2}\bigg{(}\sum_{m=1,n=1}^{\sqrt{M}\sqrt{M}}A_{mn}(t)\bigg{)}^{2}+\\ &\big{(}K_{3}^{2}(t)K_{4}^{2}(t)+K_{4}^{2}(t)G_{1}^{2}+K_{4}^{2}( t)G_{2}^{2}\big{)}\sum_{m=1,n=1}^{\sqrt{M},\sqrt{M}}A_{mn}^{2}(t)+\\ & 2K_{1}(t)K_{3}(t)G_{1}A_{0}(t)\sum_{m=1,n=1}^{\sqrt{M},\sqrt{M}}A_ {mn}(t)\end{split} \tag{30}\]  
_with \(K_{1}(t)=\sqrt{\frac{\kappa_{d}(t)}{\kappa_{d}(t)+1}}\), \(K_{2}(t)=\sqrt{\frac{1}{\kappa_{d}(t)+1}}\), \(K_{3}(t)=\sqrt{\frac{\kappa_{\text{BS}}(t)}{\kappa_{\text{BS}}(t)+1}}\), \(K_{4}(t)=\sqrt{\frac{1}{\kappa_{\text{BS}}(t)+1}}\), \(G_{1}=\sqrt{\frac{\rho_{\text{RBS}}+1}{\rho_{\text{RBS}}+1}}\), \(G_{2}=\sqrt{\frac{1}{\rho_{\text{RBS}}+1}}\), and \(A_{mn}(t)=A_{\text{BS}}^{mn}A_{mn}^{\text{BS}}(t)\)._  
Proof.: See Appendix A.  
**Remark 5**.: _If we assume that the gain for all elements is the same, i.e., \(A_{nm}(t)=A_{1}(t)\). This happens when the distance between the BS, the RIS, and the MR is far greater than the size of the RIS [1, 2]. Consequently, (30) becomes_  
\[\begin{split}\mathbb{E}\big{\{}|h(t)|^{2}\big{\}}& \approx(K_{1}^{2}(t)+K_{2}^{2}(t))A_{0}^{2}(t)+2 MK_{1}(t)K_{3}(t)G_{1}A_{0}( t)A_{1}(t)+\\ &\bigg{(}M^{2}K_{3}^{2}(t)G_{1}^{2}+M\big{(}K_{3}^{2}(t)K_{4}^{2}( t)+K_{4}^{2}(t)G_{1}^{2}+K_{4}^{2}(t)G_{2}^{2}\big{)}\bigg{)}A_{1}^{2}(t).\end{split} \tag{31}\]  
**Remark 6**.: _If \(K_{1}(t)=K_{3}(t)=G_{1}=0\), then (30) becomes_  
\[\mathbb{E}\big{\{}|h(t)|^{2}\big{\}}=K_{2}^{2}(t)A_{0}^{2}(t)+K_{4}^{2}(t)G_{2 }^{2}\sum_{m=1,n=1}^{\sqrt{M}\sqrt{M}}A_{nm}^{2}(t). \tag{32}\]  
_And accordingly, (31) becomes_  
\[\mathbb{E}\big{\{}|h(t)|^{2}\big{\}}\approx K_{2}^{2}(t)A_{0}^{2}(t)+MK_{4}^ {2}(t)G_{2}^{2}A_{1}^{2}(t). \tag{33}\]  
We observe that in this case, \(h(t)\) is under Rayleigh fading, and \(\mathbb{E}\{|h(t)|^{2}\}\) is proportional to the number of elements of the RIS, \(M\). This often happens when the vehicle is far from the BS and the RIS. Besides, (33) is not related to the phase shift \(\phi_{m}^{\rm opt}(t)\) of the RIS.  
**Remark 7**.: _If \(K_{2}(t)=K_{4}(t)=G_{2}=0\), then (30) becomes_  
\[\begin{split}\mathbb{E}\big{\{}|h(t)|^{2}\big{\}}& =K_{1}^{2}(t)A_{0}^{2}(t)+K_{3}^{2}(t)G_{1}^{2}\bigg{(}\sum_{m=1,n =1}^{\sqrt{M}\sqrt{M}}A_{nm}(t)\bigg{)}^{2}+\\ & 2K_{1}(t)K_{3}(t)G_{1}A_{0}(t)\sum_{m=1,n=1}^{\sqrt{M}\sqrt{M}}A_{ nm}(t),\end{split} \tag{34}\]  
_and (31) becomes_  
\[\mathbb{E}\big{\{}|h(t)|^{2}\big{\}}\approx K_{1}^{2}(t)A_{0}^{2}(t)+M^{2}K_{ 3}^{2}(t)G_{1}^{2}A_{1}^{2}(t)+2 MK_{1}(t)K_{3}(t)G_{1}A_{0}(t)A_{1}(t). \tag{35}\]  
It can be observed that in this case, \(h(t)\) only contains LoS components, and \(\mathbb{E}\big{\{}|h(t)|^{2}\big{\}}\) increases proportional to \(M\) and \(M^{2}\). This often appears when the vehicle approaches the BS and the RIS. Besides, it is noteworthy that if the direct link is blocked, we have \(\mathbb{E}\big{\{}|h(t)|^{2}\big{\}}=A_{1}^{2}(t)M^{2}K_{3}^{2}(t)G_{1}^{2}\), which increases proportional to \(M^{2}\), i.e., the _square power law_[15]. Therefore, the gain of the reflected beam from the RIS to the vehicle is more important than that of the direct link because, in practical scenarios, the cascaded link is usually the only channel available, and the square power law can bring a more promising channel gain.