# Energy-Efficient RIS-Enabled SISO-OFDMA Communication via Lower Bound Optimization
## 4 Simulation Results
Conversely, the performance of the without-RIS scheme remains unaffected by the number of RIS elements in the system, implying the enhancement achievable through RIS in the wireless communication environment. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Values** \\ \hline Bandwidth of the BS B & 180 KHz \\ \hline Maximum transmit power at BS \(P_{\text{max}}\) & 20 dBW \\ \hline Circuit power of the Bs \(P_{B}\) & 39 dBm \\ \hline Power amplifier efficiency at the Bs \(v\) & 0.8 \\ \hline Circuit power of each user \(P_{k}\) & 10 dBm \\ \hline Circuit power of each RIS element \(P_{m}(b)\) & 10 dBm \\ \hline \end{tabular}
\end{table} 
Table 3: Simulation parameters. Figure 3: Average SE versus \(P_{\text{max}}\) for \(R_{\text{min}}=2.5\,\text{bps}/\text{Hz}\), \(\alpha_{\text{k,RIS}}=0.25\), and \(M=4,K=4,N=72\). Remark
* Complexity  
Securing the optimal solution for problem (P1) requires the application of a power allocation algorithm to each feasible subcarrier assignment that meets the constraint in Equation (4) and subsequently selecting the assignment with the highest EE. Its computational complexity depends on the number of optimizing variables, which can be large if the number of subcarriers and/or the number of UEs is large. Nevertheless, the associated complexity proves excessively high, rendering it impractical for real-world scenarios. In contrast, the proposed sub-optimal solution, facilitated by the MDSA algorithm, eliminates the need to optimize a joint and intricate objective in each step. Subcarrier assignment becomes more manageable with this pragmatic approach. This is because, for each UE, the maximum of its individual EE is contingent solely upon its parameters and the subcarriers it will occupy, without reliance on the power adaptation strategies of other UEs. The complexity of the MDSA algorithm for a given \(\boldsymbol{\alpha}_{k,\text{RIS}}\) is roughly \(\mathcal{O}((N_{OL}N)\) times that of the SQP method, where \(N_{OL}\) is the number of iterations in the outer layer, and \(N\) is the total number of subcarriers. The complexity of the SQP algorithm is dependent on the QP subproblem. By using modern interior-point (IP) methods for solving subproblems, a polynomial bound of computational complexity results [29]. Additionally, it is observed that the computational complexity of the PSO algorithm mainly involves the computation of fitness values of \(L\) passive beamformers for \(K\) users. Thus, the computational complexity is \(\mathcal{O}((LK\log(L))\). It is clear that the complexity of the PSO algorithm is only linearly increasing with the number of particles \(L\). * Convergence The iterative alternating algorithm for solving the EE maximization problem (P3) is given in Algorithm 5. Since the updates through SQP and PSO all maximize the objective function at each iteration, the iterations in Algorithm 5 lead to a monotone in  
Figure 4: Average EE versus number of RIS elements, \(M\) for \(R_{\text{min}}=2.5\)\(\text{bps}/\text{Hz}\), \(\alpha_{k,\text{RIS}}=0.25\), \(P_{\text{max}}=20\,\text{dBm}\), and \(K=4,N=72\). crease of the objective function of problem (P3). Since the objective function under the power and amplitude reflection coefficient constraints are bounded, the convergence of the alternating maximization algorithm can be guaranteed with the monotonic convergence theorem [30].