# Energy-Efficient RIS-Enabled SISO-OFDMA Communication via Lower Bound Optimization
## 2 System Model and Problem Formulation  
### System Model  
The depicted system model can be observed in Figure 1, illustrating the interaction between a single-antenna base station and \(K\) single-antenna users. These entities establish communication through the intermediary of a RIS comprising \(M\) elements. \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \(H_{T,n}\) & The channel from the transmitter to the RIS \\ \hline \(h^{k}_{R,n}\) & The channel from \(k\)th receiver to the RIS \\ \hline \(\varrho_{k}\) & Set of subcarriers assigned to the \(k\)th user \\ \hline \(\rho_{k,n}\) & Subcarrier \(n\) assigned to the \(k\)th user \\ \hline \(p_{k,n}\) & Transmit power assigned to the \(k\)th user on subcarrier \(n\) \\ \hline \(R\) & Total data rate of the system, where \(R=\sum_{k=1}^{K}\sum_{n=1}^{N}r_{k,n}\) \\ \hline \(\mathbf{\alpha}_{k,\text{RIS}}\) & Hypothetical static circuit power allocated to the \(k_{\text{th}}\) UE and \(n_{\text{th}}\) RIS element \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notations used. The channel model is based on an OFDM with \(N\) subcarriers. The channels from the transmitter and the \(k\)th user to the RIS are defined as \(\mathbf{H}_{T,n}\in\mathbb{C}^{M\times 1}\) and \(\mathbf{h}_{R,n}^{k}\in\mathbb{C}^{M\times 1}\), respectively, where \(n=1,2,\ldots,N\) and \(k=1,2,\ldots,K\). \(x_{n}^{k}\) is denoted as the transmitting signal from the transmitter to the \(k\)th user over the \(n\)th subcarrier. In particular, the direct line-of-sight (LOS) link between the transmitter and the receiver is blocked (i.e., occlusion due to buildings). Thus, the received signal at the \(k\)th receiver is given as  
\[y_{n}^{k}=((\mathbf{h}_{R,n}^{k})^{T}\mathbf{\Phi}_{n}^{k}\mathbf{H}_{T,n})p_{k,n}x_{n}^{k} +w_{n}^{k},\]  
where \(\mathbf{\Phi}_{n}^{k}=\text{diag}[\phi_{1},\phi_{1},\dots,\phi_{M}]\in\mathbb{C}^{ M\times M}\) is the RIS configuration matrix describing the phase shift effect of the RIS on the incident signal. Note that the amplitude of the incident signal does not change, which means \(\phi_{m}=e^{i\theta_{m}}\) for any \(m=1,2,\ldots,M\) and \(\theta_{m}\in[0,2\pi]\). Furthermore, \(w_{n}^{k}\sim\mathcal{CN}(0,\sigma_{n}^{2})\) denotes the additive white Gaussian noise (AWGN) at the receivers. The signal-to-noise ratio (SNR) perceived by the \(k\)th user on the \(n\)th subcarrier is formulated as follows:  
\[\text{SNR}_{k,n}=\frac{\left|(\mathbf{h}_{R,n}^{k})^{T}\mathbf{\Phi}_{n}^{k}\mathbf{H}_{T, n}\right|p_{k,n}}{N_{0}W}. \tag{1}\]  
In this context, \(N_{0}=\sigma^{2}\) represents the variance of the AWGN noise. The total bandwidth, \(B\), is divided into \(N\) subcarriers, each with a bandwidth of \(W=\frac{B}{N}\). Next, the maximum achievable data rate of the \(k\)th user on the \(n\)th subcarrier, \(r_{k,n}\), and the overall rate of the system, R, can be expressed as:  
\[r_{k,n}=W\log_{2}(1+\text{SNR}_{k,n}) \tag{2}\]  
\[\text{R}=\sum_{k=1}^{K}\sum_{n=1}^{N}\rho_{k,n}\log_{2}(1+\text{SNR}_{k,n}). \tag{3}\]  
Also, \(\rho_{k,n}\in\{0,1\}\) indicates whether or not the \(n\)th subcarrier is assigned to the \(k\)th user. A feasible subcarrier assignment indictor matrix, \(\mathbf{\rho}=\left[\rho_{k,n}\right]_{K\times N}\), should satisfy:  
Figure 1: The considered RIS-based multi-user SISO-OFDMA system. \[ \begin{split}\mathbf{\rho}\in\varrho=\{[\rho_{k,n}]_{K\times N}|\sum_{k\in \mathcal{K}}\rho_{k,n}\leq 1,\forall n\in\mathcal{N};\\ \rho_{k,n}\in\{0,1\},\forall k\in\mathcal{K},n\in\mathcal{N}\}, \end{split}  \tag{4}\]  
where \(\mathcal{N}=\{1,2,\cdots,N\}\) and \(\mathcal{K}=\{1,2,\cdots,K\}\) represent the sets of all subcarriers and all UEs, respectively. To minimize interference between users, each subcarrier is assigned exclusively to one user at a time.