## 2 System Model and Problem Formulation  
A MU MISO RSMA system assisted by a PB with SWIPT is considered, as shown in Figure 1. The number of antennas at the BS is \(M\geq 2\), the number of antennas at the PB is \(N\geq 2\), and there are \(K\) single-antenna PS users and \(G\) single-antenna EH users. EH users are exclusively dedicated to harvesting RF energy, whereas PS users are equipped with a PS mechanism, allowing them to partition the incoming RF signal into two components, serving both information decoding and EH based on a designated PS ratio, \(\theta_{k}\). The BS transmits information signals to both harvest energy at the EH module and decode information at the ID module for PS users. The PB, on the other hand, transmits energy-carrying signals used for energy harvesting by EH users and the EH module of PS users. However, these energy signals are considered interference at the ID module of PS users. In one-layer RSMA, the original message intended for the \(k\)th PS user, \(W_{k}\), is separated into a common message, \(W_{k,c}\), and a private message, \(W_{k,p}\). A super-common message is produced by combining all the \(K\) common messages and encoding them into the common stream \(z_{c}^{PS}\), which needs to be decoded by all PS users. The \(K\) private messages are independently encoded into \(K\) private streams \(\left\{z_{k}^{PS}\right\}\) to be decoded by their respective user. At the PB, the energy-carrying signals are represented by \(\left\{z_{g}^{EH}\right\},g=1,\ldots,G\). Therefore, the signals transmitted at the BS and PB after precoding are \(\mathbf{x}_{BS}=\mathbf{p}_{0}z_{c}^{PS}+\sum\limits_{k=1}^{K}\mathbf{p}_{k}z _{k}^{PS}\) and \(\mathbf{x}_{PB}=\sum\limits_{g=1}^{G}\mathbf{e}_{g}z_{g}^{EH}\), respectively, where \(\mathbf{p}_{0}\in\mathbb{C}^{M\times 1}\) and \(\mathbf{p}_{k}\in\mathbb{C}^{M\times 1}\) represent the information beamforming vectors, and \(\mathbf{e}_{g}\in\mathbb{C}^{N\times 1}\) corresponds to the energy beamforming vector for \(z_{c}^{PS}\), \(z_{k}^{PS}\), and \(z_{g}^{EH}\). Figure 2 shows the RSMA transmission scheme in the considered system model. Figure 1: MISO SWIPT system aided by a PB. At the user side, the \(k\)th PS user first decodes the common stream \(z_{c}^{PS}\) with an achievable rate given by  
\[R_{k,0}=\log_{2}\left(1+\frac{\left|\mathbf{q}_{k,PS}^{H}\mathbf{p}_{0}\right|^{ 2}}{\sum\limits_{i=1}^{K}\left|\mathbf{q}_{k,PS}^{H}\mathbf{p}_{i}\right|^{2}+ \sum\limits_{g=1}^{G}\left|\mathbf{h}_{k,PS}^{H}\mathbf{e}_{g}\right|^{2}+ \sigma_{k}^{2}+\frac{\gamma_{k}^{2}}{\theta_{k}}}\right),\,\forall k, \tag{1}\]  
where \(\mathbf{q}_{k,PS}\in\mathbb{C}^{M\times 1}\) represents the channel vector from the BS to the \(k\)th PS user; \(\mathbf{h}_{k,PS}\in\mathbb{C}^{N\times 1}\) is the channel vector from the PB to the \(k\)th PS user; \(\theta_{k}\in(0,1)\) is the PS factor; \(n_{k}\sim\mathcal{CN}\big{(}0,\sigma_{k}^{2}\big{)}\) represents the Gaussian noise at the antenna of the \(k\)th PS user; and \(v_{k}\sim\mathcal{CN}\big{(}0,\delta_{k}^{2}\big{)}\) is the data processing noise at the ID module of the \(k\)th PS user. After decoding \(z_{c}^{PS}\), RSMA employs the SIC procedure to eliminate interference due to the common stream \(z_{c}^{PS}\). The \(k\)th PS can decode its private stream \(z_{k}^{PS}\) with an achievable rate given by  
\[\mathrm{R}_{k}=\log_{2}\left(1+\frac{\left|\mathbf{q}_{k,PS}^{H}\mathbf{p}_{k }\right|^{2}}{\sum\limits_{i=1,j\
eq k}^{K}\left|\mathbf{q}_{k,PS}^{H}\mathbf{ p}_{i}\right|^{2}+\sum\limits_{g=1}^{G}\left|\mathbf{h}_{k,PS}^{H}\mathbf{e}_{g} \right|^{2}+\sigma_{k}^{2}+\frac{\gamma_{k}^{2}}{\theta_{k}}}\right),\, \forall k. \tag{2}\]  
The rate at which \(z_{c}^{PS}\) is transmitted, expressed as \(R_{0}\), must satisfy \(R_{0}\leq\min\{R_{1,0},\ldots,R_{K,0}\}\) because the common stream \(z_{c}^{PS}\) needs to be decoded by all users.