# RIS-Assisted High-Speed Communications with Time-Varying Distance-Dependent Rician Channels
## 5 Numerical Evaluations and Discussion
### Ergodic Spectral Efficiency  
Figure 3 shows that the maximized ergodic SE of the proposed RIS-aided HSC system \(\mathrm{SE^{upper}}(t)\) in (29) increases from 5.41 bit/s/Hz to 11.79 bit/s/Hz, when the vehicle moving distances are 100 m and 250 m, respectively. This is because \(d_{0}(t)\) and \(d_{mn}^{\mathrm{MR}}(t)\) are the smallest when the moving distance of the vehicle is 250 m. Then, the SE decreases to 5.41 bit/s/Hz again at 400 m moving distance, which is as expected since the channel gains \(A_{0}(t)\) and \(A_{mn}(t)\) and Rician factors are all distance-dependent. Therefore, the ergodic SE and the transmission distances (i.e., \(d_{0}(t)\), \(d_{\mathrm{BS}}^{mn}\) and \(d_{mn}^{\mathrm{MR}}(t)\)) are inversely proportional. Besides, for ease of comparison, we also provide the case without phase optimization, i.e., \(\phi_{mn}(t)\in\mathcal{U}(0,2\pi)\) where \(\mathcal{U}\) is the uniform distribution. It is shown that before optimization, the ergodic SE of the proposed system is almost the same as the scenario without the RIS. This is because the power of the received signal increases only proportional to \(M\), as (33) shows. Thus, obtaining the optimal phase shift \(\phi_{mn}^{\mathrm{opt}}(t)\) in (23) is crucial for the proposed system. Moreover, note that the analytical and simulated results of the ergodic SE match well, which validates Proposition 1. Therefore, we use the analytical upper bound SE expression in (29) in the rest of this section.  
Figure 3: Ergodic SE for one vehicle pass with the distance-dependent Rician factors.  
Figure 2: Top view of the proposed HSC system for numerical evaluations in this section.  
In order to validate Remarks 2, 6 and 7, we consider the MR position as \([-5000+vt,15,20]^{\text{T}}\) m. It can be observed from Figure 4 that when the vehicle is far from the BS and the RIS, i.e., the vehicle moving distance is equal to or smaller (larger) than 1500 m (3500 m), the channel \(h(t)\) is the same as the Rayleigh channel. However, when the \(d_{0}(t)\) and \(d_{min}^{\text{MR}}(t)\) are small enough, i.e., the vehicle moving distance range is from 2350 m to 2650 m, \(h(t)\) changes to the LoS channel. This result reveals that fixed Rician factors are not practical when describing the practical channel in the HSC system.  
Figure 5 shows the ergodic SE of the cases with different environmental factors \(\varrho\) and \(\iota\). It can be observed that the ergodic SE of the cases with \(\varrho=13\), \(\iota=0.03\)[20] and \(\varrho=0\), \(\iota=0\) are only the same when the vehicle moving distance is 1975 m. Other two of the same points appear in 2175 m and 2350 m, if \(\varrho=5\), \(\iota=0\) and \(\varrho=10\), \(\iota=0\)[19]. This result reveals that there is a clear difference between distance-dependent and fixed Rician factors. Therefore, the simulation results in Ref. [19] could be improved since it only considered the fixed Rician factors.  
Figure 4: Ergodic SE for one vehicle pass with distance-dependent Rician factor (\(\varrho=13\), \(t=0.03\)) [20], LoS Rician factor (\(\varrho=10^{2}\), \(\iota=0\)), and NLoS Rician factor (\(\varrho=-10^{2}\), \(\iota=0\)).  
Figure 5: Ergodic SE for one vehicle pass with the distance-dependent Rician factor (\(\varrho=13\), \(\iota=0.03\)) [20] and fixed Rician factors (\(\varrho=0\), \(\iota=0\); \(\varrho=5\), \(\iota=0\); \(\varrho=10\), \(\iota=0\)[19]). The moving distance of the vehicle is from 1800 m to 2400 m.  
It can be observed from Figure 6 that from 2450 m to 2550 m, the case with dynamic Rician factors has the highest ergodic SE, which reveals that when the vehicle is near the BS and the RIS, the channel is almost the same as the LoS channel, rather than still with fixed Rician factors (e.g., \(\varrho=10\), \(\iota=0\)[19]). From Figures 5 and 6, it can be observed that the SE performance is overestimated (underestimated) when the transmission distance increases (decreases) in Ref. [19].