# RIS-Assisted High-Speed Communications with Time-Varying Distance-Dependent Rician Channels
## Appendix A Proof of Proposition 1  
Consider \(\mathbb{E}\{\log_{2}(1+x)\}\leq\log_{2}\{1+\mathbb{E}\{x\}\}\)[21], we then focus on the derivation of \(\mathbb{E}\big{\{}|h(t)|^{2}\big{\}}\). We define \(\Delta_{1}\), \(\Delta_{2}\), \(\Delta_{3}\) and \(\Delta_{4}\) as follows  
Figure 11: Optimized DS for different positions of the BS.  
\[h(t)= \underbrace{A_{0}(t)K_{1}(t)h_{d}^{\text{LoS}}(t)}_{\Delta_{1}}+ \underbrace{\sum_{m=1,n=1}^{\sqrt{M},\sqrt{M}}A_{mn}(t)\exp\big{(}-j\phi_{mn}(t) \big{)}K_{3}(t)h_{mn}^{\text{LoS}}(t)G_{1}g_{mn}^{\text{LoS}}}_{\Delta_{2}}+\] \[\underbrace{A_{0}(t)K_{2}(t)h_{d}^{\text{NLOS}}(t)}_{\Delta_{3}}+ \underbrace{\sum_{m=1,n=1}^{\sqrt{M},\sqrt{M}}A_{mn}(t)\exp\big{(}-j\phi_{mn}(t )\big{)}\bigg{(}K_{3}(t)h_{mn}^{\text{LoS}}(t)G_{2}g_{mn}^{\text{NLoS}}(t)+ \big{.}}_{\Delta_{4}} \tag{10}\]  
Note that \(\Delta_{1}\) and \(\Delta_{2}\) are deterministic, and \(\mathbb{E}\{\Delta_{3}\}=\mathbb{E}\{\Delta_{4}\}=0\). Therefore,  
\[\mathbb{E}\big{\{}|h(t)|^{2}\big{\}} =\mathbb{E}\big{\{}h^{*}(t)h(t)\big{\}} \tag{11}\] \[=\mathbb{E}\big{\{}(\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}) ^{*}\cdot(\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4})\big{\}}\] \[=\mathbb{E}\big{\{}\Delta_{1}^{*}\Delta_{1}+\Delta_{2}^{*}\Delta_ {2}+\Delta_{3}^{*}\Delta_{3}+\Delta_{4}^{*}\Delta_{4}+\Delta_{1}^{*}\Delta_{2} +\Delta_{1}\Delta_{2}^{*}\big{\}},\]  
where \((\cdot)^{*}\) denotes conjugate operation. Firstly, we focus on \(\mathbb{E}\{\Delta_{1}^{*}\Delta_{1}\}\), which can be computed as  
\[\mathbb{E}\big{\{}\Delta_{1}^{*}\Delta_{1}\big{\}}=\mathbb{E}\big{\{}\big{(}A_ {0}(t)K_{1}(t)h_{d}^{\text{LoS}}(t)\big{)}^{*}A_{0}(t)K_{1}(t)h_{d}^{\text{LoS }}(t)\big{\}}. \tag{12}\]  
Recall that \(h_{d}^{\text{LoS}}(t)=\exp\big{(}-j2\pi\big{(}f_{c}+f_{0}(t)\big{)}\tau_{0} \big{)}\), we obtain  
\[\mathbb{E}\big{\{}\Delta_{1}^{*}\Delta_{1}\big{\}}=K_{1}^{2}(t)A_{0}^{2}(t). \tag{13}\]  
We then concentrate on \(\mathbb{E}\big{\{}\Delta_{2}^{*}\Delta_{2}\big{\}}\). Since the optimal phase shift set (23) is achieved, then the phase of the cascaded and direct links are the same, i.e.,  
\[\underbrace{\exp\big{(}-j\phi_{mn}(t)\big{)}h_{mn}^{\text{LoS}}(t)g_{mn}^{ \text{LoS}}}_{\text{The phase of the cascaded link}}=\underbrace{\exp\big{(}-j2\pi \big{(}f_{c}+f_{0}(t)\big{)}\tau_{0}\big{)}}_{\text{The phase of the direct link}}. \tag{14}\]  
Thus, we have  
\[\mathbb{E}\big{\{}\Delta_{2}^{*}\Delta_{2}\big{\}} =\bigg{|}\exp\big{(}-j2\pi\big{(}f_{c}+f_{0}(t)\big{)}\bigg{|}^{2 }\bigg{(}\sum_{m=1,n=1}^{\sqrt{M},\sqrt{M}}A_{mn}(t)K_{3}(t)G_{1}\bigg{)}^{2} \tag{15}\] \[=K_{3}^{2}(t)G_{1}^{2}\bigg{(}\sum_{m=1,n=1}^{\sqrt{M},\sqrt{M}} A_{mn}(t)\bigg{)}^{2}.\]  
Next, we study \(\mathbb{E}\{\Delta_{3}^{*}\Delta_{3}\}\). Recall that \(h_{d}^{\text{NLoS}}(t)\sim\mathcal{CN}(0,1)\), thus \(\mathbb{E}\big{\{}|h_{d}^{\text{NLoS}}(t)|^{2}\big{\}}=1\), then we have  
\[\mathbb{E}\big{\{}\Delta_{3}^{*}\Delta_{3}\big{\}}=A_{0}^{2}(t)K_{2}^{2}(t) \mathbb{E}\big{\{}|h_{d}^{\text{NLoS}}(t)|^{2}\big{\}}=K_{2}^{2}(t)A_{0}^{2}(t). \tag{16}\]  
Fourthly, for \(\mathbb{E}\big{\{}\Delta_{4}^{*}\Delta_{4}\big{\}}\), we notice that \(\mathbb{E}\big{\{}|h_{mn}^{\text{NLoS}}(t)|^{2}\big{\}}=\mathbb{E}\big{\{}|g_{mn }^{\text{NLoS}}(t)|^{2}\big{\}}=1\), \(\mathbb{E}\big{\{}h_{mn}^{\text{NLoS}}(t)\big{\}}=\mathbb{E}\big{\{}g_{mn}^{ \text{NLoS}}(t)\big{\}}=0\), hence \(\mathbb{E}\big{\{}\Delta_{4}^{*}\Delta_{4}\big{\}}\) can be obtained as  
\[\mathbb{E}\big{\{}\Delta_{4}^{*}\Delta_{4}\big{\}}=\big{(}K_{3}^{2}(t)K_{4}^{2}(t )+K_{4}^{2}(t)G_{1}^{2}+K_{4}^{2}(t)G_{2}^{2}\big{)}\sum_{m=1,n=1}^{\sqrt{M}, \sqrt{M}}A_{mn}^{2}(t). \tag{17}\]Finally, since \(\Delta_{1}\) and \(\Delta_{2}\) have the same phase part, then  
\[\mathbb{E}\big{\{}\Delta_{1}^{*}\Delta_{2}\big{\}}=\mathbb{E}\big{\{}\Delta_{1} \Delta_{2}^{*}\big{\}}=A_{0}(t)K_{1}(t)K_{3}(t)G_{1}\sum_{m=1,n=1}^{\sqrt{M}, \sqrt{M}}A_{mn}(t). \tag{30}\]  
Therefore, considering all the results above, we have (30), which completes the proof.