## 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 \(D_{1}\)  
In this case, it is assumed that line-of-sight (LOS) paths are presented between the links in \(S_{1}D_{i}\), \(S_{1}D_{i}\) and \(S_{1}D_{i}\) and are modeled as Rician fading channels, \(h_{p}=\sqrt{\frac{K_{p}}{K_{p}+1}}\widehat{h}_{p}d_{p}^{-a/2},p\in\{S_{1},I_{ 1}D_{i},S_{1},I_{2}D_{i},S_{1},I_{3},I_{3}D_{i}\}\), where \(\widehat{h}_{p}\) is a fixed-component vector with elements of unit power, and \(K_{p}\) is the Rician K-factor. Proposition 3: _The upper bound for the \(D_{1}\) using the Rician fading channels is given as_  
\[\hat{R}_{\text{upper},\text{D}_{1}}^{(1)}=\text{log}_{2}\Bigg{(}1+ \text{E}\bigg{[}\widetilde{\gamma_{D_{1}}^{(1)}}\bigg{]}\Bigg{)}\] (21) \[=\text{log}_{2}\Bigg{(}1+\frac{\gamma\chi_{1}\big{(}\vartheta_{1}+ 2\vartheta_{2}+\vartheta_{3}\big{)}}{\gamma\chi_{2}\big{(}\vartheta_{1}+2 \vartheta_{2}+\vartheta_{3}\big{)}+\big{(}\text{V}_{2}^{  
\[\begin{split}&\text{E}\bigg{[}\widetilde{\gamma_{D_{1}}^{(2)}}\bigg{]}= \text{E}\Bigg{[}\frac{\gamma\chi_{1}\bigg{(}N_{1}\sqrt{\widetilde{A_{1}^{(1)}} _{1}+N_{2}}\sqrt{\widetilde{A_{2}^{(1)}}_{2}+N_{3}}\sqrt{\widetilde{A_{3}^{(1) }}_{3}}\bigg{)}^{2}}{\gamma\chi_{2}\bigg{(}N_{1}\sqrt{\widetilde{A_{1}^{(1)}} _{1}+N_{2}}\sqrt{\widetilde{\widetilde{A_{2}^{(1)}}_{2}}}+N_{3}\sqrt{ \widetilde{A_{3}^{(1)}}_{3}}\bigg{)}^{2}+\big{(}\mathsf{v}_{5}^{2}+\mathsf{v}_{ 2_{1}}^{2}\big{)}\gamma\bigg{(}N_{1}\sqrt{\widetilde{A_{1}^{(1)}}_{1}+N_{2}} \sqrt{\widetilde{\widetilde{A_{2}^{(1)}}_{2}}}+N_{3}\sqrt{\widetilde{A_{3}^{( 1)}}_{3}}\bigg{)}^{2}+1}\Bigg{]}\\ &=\frac{\gamma\chi_{1}\Gamma_{1}}{\gamma\chi_{2}\Gamma_{1}+ \big{(}\mathsf{v}_{3}^{2}+\mathsf{v}_{2_{1}}^{2}\big{)}\gamma\Gamma_{1}+1}\\ &\leq\frac{\gamma\chi_{1}\big{(}\frac{1}{6}\beta_{1}+\frac{1}{6 }\beta_{3}+\frac{1}{16}\beta_{4}+\frac{1}{6}\beta_{2}+\frac{1}{8}\beta_{5}+ \frac{1}{16}\beta_{6}\big{)}}{\gamma\chi_{2}\big{(}\frac{1}{6}\beta_{1}+\frac{ 1}{16}\beta_{3}+\frac{1}{16}\beta_{4}+\frac{1}{6}\beta_{2}+\frac{1}{8}\beta_{5 }+\frac{1}{8}\beta_{6}\big{)}+\big{(}\mathsf{v}_{5}^{2}+\mathsf{v}_{2_{1}}^{2 }\big{)}\gamma\big{(}\frac{1}{6}\beta_{1}+\frac{1}{16}\beta_{3}+\frac{1}{16} \beta_{4}+\frac{1}{16}\beta_{2}+\frac{1}{8}\beta_{5}+\frac{1}{8}\beta_{6} \big{)}+1}},\\ &\qquad\qquad\qquad\text{in which }\Gamma_{1}=N_{1}^{2}\text{E}\bigg{[} \widetilde{A_{1}^{(1)}}\bigg{]}+2N_{1}N_{2}\text{E}\bigg{[}\sqrt{\widetilde{A_{ 1}^{(1)}}_{1}}\sqrt{\widetilde{A_{2}^{(1)}}_{2}}\bigg{]}+N_{2}^{2}\text{E} \bigg{[}\widetilde{A_{2}^{(1)}}_{2}\bigg{]}+N_{3}^{2}\text{E}\bigg{[} \widetilde{A_{3}^{(1)}}_{3}\bigg{]}\\ &\qquad\qquad\qquad\qquad+\qquad\qquad 2N_{1}N_{3}\text{E}\Bigg{[} \sqrt{\widetilde{A_{1}^{(1)}}_{1}}\sqrt{\widetilde{A_{3}^{(1)}}_{3}}\bigg{]} \qquad\qquad\qquad\qquad\qquad+\qquad 2  
Figure 2: The flowchart of Monte Carlo simulations. Figure 3 demonstrates the ergodic capacity performance as a function of \(N\) and \(\gamma\) for two scenarios. When the IRS elements give additional constructive paths, an enhancement in the SNDR could be affected by the increasing \(N\). It is intuitively seen that a higher average SNR at the source leads to a remarkable gap between the ergodic capacity of the two scenarios. In Figure 4, we confirm the impact of the power allocation factors on the achievable rate. It can be easily seen that more power \(\chi_{1}\) assigned to the first user results in a higher achievable rate while the performance of the second user increases at \(\chi_{1}=0.65\) for the case of \(\gamma\)=20 (dB) and declines remarkably afterward. The number of meta-surfaces in the three-IRSs case plays an important role in enhancing the quality of received signals at destinations. However, more power assigned to the transmit signal to a selected user is a key factor affecting the achievable rate. On the other hand, we can observe that more IRSs served for user \(D_{1}\) can be employed to improve the rate performance significantly if \(\chi_{1}=0.65\) is greater than 0.7.