## 4 Numerical Results
### Numerical Results  
#### 4.2.1 EE Performance with Different \(R_{min}\)  
The minimum achievable rate \(R_{min}\) can represent the effectiveness of the communication system. According to Formula (48), \(R_{min}\) can affect the starting iterative value of \(\mathbf{N}\). Figure 4 illustrates the EE versus \(N\) with different \(R_{min}\).  
As can be seen from Figure 4, EE shows a trend of increasing first and then decreasing monotonically with the increasing of N, itmeans that EE(N) is a unimodal function, which is consistent with the proof in the paper. With the different \(R_{min}\), the iterative range \(N~{}=~{}\big{\{}N\in\mathbb{N}^{+}|\underline{N}\leq N\leq\overline{N}\big{\}}\) changes. When \(R_{min}<33.99\) Mbps, the iterative range of \(N\) is independent of \(R_{min}\). This is because \(\min\{N\in\mathbb{N}^{+}|R(N)\geq R_{min}\}\leq\frac{\sigma\sqrt{\pi}}{\pi H_{1 }\sqrt{\pi}}\), and \(\frac{1}{H_{1}\sqrt{\delta}}\leq\frac{\sigma\sqrt{\pi}}{\pi H_{1}\sqrt{\pi}}\), so the starting iterative point of \(N\) is \(\underline{N}=min\Big{\{}N\in\mathbb{N}^{+}|N\geq\frac{\sigma\sqrt{\pi}}{\pi H _{1}\sqrt{\pi}}\Big{\}}\). When \(R_{min}\geq 33.99\) Mbps, \(\min\{N\in\mathbb{N}^{+}|R(N)\geq R_{min}\}\geq\frac{\sigma\sqrt{\pi}}{\pi H _{1}\sqrt{\pi}}\geq\frac{1}{H_{1}\sqrt{\delta}}\), \(\underline{N}=\min\{N\in\mathbb{N}^{+}|R(N)\geq R_{min}\}\).  
When \(33.99\) Mbps \(\leq R_{min}\leq 86.31\) Mbps, \(EE(N)\) increases first and then decreases monotonically. The binary search method proposed can be used to find \(N_{opt}\). When \(86.31\) Mbps \(\leq R_{min}\leq R\big{(}\overline{N}\big{)}\), such as \(R_{min}=100\) Mbps in the Figure 4, \(EE(N)\) decreases monotonically, so \(N_{opt}=\underline{N}\) and \(EE_{max}(N)=EE(\underline{N})\). When \(R_{min}>R\big{(}\overline{N}\big{)}\), the achievable rate cannot meet the requirements and \(N_{opt}\) does not exist.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(z_{m}\) & 0.62 m \\ \((w_{m},h_{m})\) & (0.01, 0.01) m \\ \(\Phi_{1/2}\) & 60 deg. \\ \(\Psi_{C}\) & 35 deg. \\ \(A_{d}\) & 1.0 cm\({}^{2}\) \\ \(T_{s}\) & 1.0 \\ \(\mu\) & 1.5 \\ \(\omega\) & 0.44 W/A \\ \(\rho\) & 0.8 \\ \(\eta\) & 0.54 A/W \\ \(\zeta\) & 1.2 \\ B & 20 MHz \\ \hline \hline \end{tabular}
\end{table}
Table 2: Main simulation parameters.  
Figure 4: EE versus \(N\) with different \(R_{min}\).  
Since the changing of \(R_{min}\) will changes the iterative range of \(N\), the calculation amount will also be different when finding \(N_{opt}\). Table 3 gives \(N_{opt}\) and iterations with different \(R_{min}\).  
As can be seen from the Table 3, when \(R_{min}=40,\ 50,\ 60,\ 70,\ 80\) Mbps, \(N_{opt}=168\). This is consistent with the previous analysis. Compared to the total number of mirrors in the IRS \(N_{max}=3600\), \(EE(N)\) can be maximized only using \(4.67\%\) of the total number of mirrors in IRS after optimization. The BSCI algorithm needs 12 iterations to find \(N_{opt}\), which reduces the amount of computation by \(10^{5}\) orders of magnitude, compared with the Bubble Sort method. When \(R_{min}=90,\ 100\) Mbps, \(EE(N)\) decreases monotonically and \(N_{opt}\) increases gradually. The BSCI algorithm only needs one iteration to find \(N_{opt}\). However, the Bubble Sort method still needs a lot of computation. Therefore, the BSCI algorithm is more efficient.