## 3 Ergodic Performance Analysis of the Proposed Scheme Using Rayleigh and Rician Fading Channels
### The First Scenario
#### 3.1.3 Upper Bound for the Achievable Rate Using Rician Fading Channels for \(D_{1}\)
Furthermore, Figure 5 showcases the severity of the hardware impairment for ergodic performance. In particular, due to the transceiver hardware impairment \(Y_{S}=Y_{D_{1}}=Y_{D_{2}}\), the ergodic performance saturates for the case of \(Y_{S}=Y_{D_{1}}=Y_{D_{2}}=0.5\) and cannot be further improved even when increasing \(\gamma\) over 30 (dB). It is observed at the low region of \(\gamma\) that different levels of transceiver hardware impairment lead to a small gap of achievable rates, and the performance gap among two users is small as well. The capacity ceiling of both users exists when \(Y_{S}=Y_{D_{1}}=Y_{D_{2}}=0.5\) corresponding to \(\gamma\) = 30 (dB). It is strongly confirmed that the capacity ceiling is determined by the levels of transceiver hardware impairment rather than \(N\), \(\gamma\) and other scenarios. Figure 3: The performance of overall achievable rates by varying the number of IRSs (\(N\)= \(N_{1}\)= \(N_{2}\)= \(N_{3}\)) when changing \(\gamma\) with Rayleigh fading. Figure 4: Impact of power allocation factors \(\chi_{1}\) on overall achievable rates by varying \(\gamma\) with Rayleigh fading. Figure 5: Impact of hardware impairments \(\mathrm{Y}_{S}=\mathrm{Y}_{D_{1}}=\mathrm{Y}_{D_{2}}\) on overall achievable rates by varying \(\gamma\) with Rayleigh fading. Figure 6 considers the distances of the source-IRS and IRS-destination links. If IRS is placed at the middle point between the source and destinations, the curves of the achievable rate exhibit the lowest values. When the IRS moves close to either the source or the destinations, the achievable rate improves significantly for the case of \(\alpha=2.5\). In this experiment, the achievable rate of the three-IRSs case is still better than the other case. We can verify how the setting of IRS affects the achievable rate, as shown in Figure 7. At the value of the number of meta-surface \(N=400\), the achievable rate obtains the maximum value. The gap among two cases of \(\chi_{1}\) could be larger when \(N\) is greater than 150. The important conclusion is that it is not necessary to design too many meta-surfaces at the IRS since the achievable rate saturates in the region where \(N\) is greater than 500. The setting of the channel has slight impacts on the ergodic performance since we see a small gap among the curves in Figure 8. Besides the locations of IRS, the quality of the channel gives slight variations in the performance of IoT users. Our similar result in the double-IRS and three-IRSs cases can be compared with the results in the recent work [30]. This means that low-cost IoT can obtain benefits with the double-IRS approach with little influence from selecting the channel models. The setting of IRS, i.e., the reflection coefficient, has the main effect on the achievable rates for two cases of IRS, as shown in Figure 9. We can see that \(\eta=0.8\) corresponds to a higher achievable rate. The reason is that reflecting the capability of IRS could be better for the case of \(\eta=0.8\), and SNDR can be enhanced along with a corresponding improvement of the achievable rate. This simulation still confirms that the three-IRSs case provides higher ergodic performance. However, increasing IRSs leads to a higher cost of deployment. Similarly, Figure 10 confirms that there is a little influence of Rayleigh and Rician channels on ergodic performance for both IoT users. Figure 6: Impact on the performance of overall achievable rates by varying \(x_{l_{1}}=x_{l_{2}}=x_{l_{3}}\) when changing \(\alpha\) with Rayleigh fading. Figure 8: Impact on the performance of overall achievable rates by varying \(\gamma\) when changing the Rician-K factor. Figure 2:0 The achievable rates versus \(N\)= \(N_{1}\)= \(N_{2}\)= \(N_{3}\) when changing \(\chi_{1}\) with Rician fading. Figure 2:1 Impact on the performance of overall achievable rates of Rayleigh fading and Rician fading by varying \(\gamma\). Figure 2:2 Impact on the performance of overall achievable rates by varying \(\gamma\) when changing \(\eta\) with Rician fading.