# Power-Efficient Resource Allocation for Active STAR-RIS-Aided SWIPT Communication Systems
## 2 System Model and Problem Description
### Problem Description  
Let \(s_{m}\) represent the signal incoming onto the \(m\)-th unit of the active STAR-RIS, where \(m\in\overline{\mathrm{M}}\triangleq 1,2,...,M\). \(t_{m}=(\sqrt{\beta_{m}^{H}}e^{|\theta_{m}^{H}})s_{m}\) and \(r_{m}=(\sqrt{\beta_{m}^{H}}e^{|\theta_{m}^{H}})s_{m}\) represent trans \begin{table}
\begin{tabular}{c c} \hline \hline
**Acronyms** & **Full Name** \\ \hline SWIPT & Simultaneous Wireless Information and Power Transfer \\ AP & Access Point \\ SDR & Semi-Definite Relaxation \\ SCA & Successive Convex Approximation \\ IoT & Internet of Things \\ RIS & Reconfigurable Intelligent Surface \\ NOMA & Non-Orthogonal Multiple Access \\ BS & Base Station \\ AO & Alternating Optimization \\ SINR & Signal-to-Interference-plus-Noise Ratio \\ TS & Time Switching \\ IRs & Information Receivers \\ ERs & Energy Receivers \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Abbreviations used in this paper. Figure 1: Active STAR-IRS-assisted SWIPT system model. mitted and reflected signals on the \(m\)-th unit, respectively, where \(\sqrt{\beta_{m}^{t}}\in[0,\sqrt{\beta_{\max}}]\), \(\sqrt{\beta_{m}^{m}}\in[0,\sqrt{\beta_{\max}}]\), \(\theta_{m}^{t}\in[0,2\pi)\), \(\theta_{m}^{r}\in[0,2\pi)\), and \(\beta_{\max}\geq 1\). The variable \(\beta_{m}^{t}\) and \(\beta_{m}^{r}\) must meet the following conditions:  
\[\beta_{m}^{t}+\beta_{m}^{r}\leq\beta_{\max},\forall m\in\overline{\mathrm{M}} \tag{1}\]  
This paper focuses on the energy splitting (ES) protocol for active STAR-RIS: For ES mode, it is assumed that all units of active STAR-RIS operate in reflection and transmission modes, and the signal on unit is decomposed into an emission signal and reflection signal. The decomposition ratio is \(\beta_{m}^{t}:\beta_{m}^{r}\). \(\Theta_{t}^{\mathrm{ES}}=\mathrm{diag}\Big{(}\sqrt{\beta_{1}^{t}}e^{i\theta_{1 }^{t}},\sqrt{\beta_{2}^{t}}e^{i\theta_{2}^{t}},...,\sqrt{\beta_{M}^{t}}e^{i \theta_{M}^{t}}\Big{)}\) and \(\Theta_{r}^{\mathrm{ES}}=\mathrm{diag}\Big{(}\sqrt{\beta_{1}^{r}}e^{i\theta _{1}^{t}},\sqrt{\beta_{2}^{r}}e^{i\theta_{2}^{r}},...,\sqrt{\beta_{M}^{r}}e^{ i\theta_{M}^{r}}\Big{)}\) represent the emission and reflection coefficient array, respectively. Let \(x_{i}^{I}\), \(x_{j}^{E}\) denote the signals carrying information and carrying energy, respectively, where \(i\in K_{I}\), and \(j\in K_{E}\). The accepted signal at the \(i\)-th IRs is as follows:  
\[y_{i}^{\mathrm{ES}}=f_{t,i}^{H}P\Theta_{t}^{\mathrm{ES}}G(\sum_{i\in K_{I}}w_ {i}x_{i}^{I}+\sum_{j\in K_{E}}v_{j}x_{j}^{E})+f_{t,i}^{H}Pn_{RIS}+n_{i} \tag{2}\]  
where \(w_{i}\in\mathbb{C}^{N\times 1}\), \(v_{j}\in\mathbb{C}^{N\times 1}\) are the precoding vectors of the \(i\)-th IRs and the \(j\)-th ERs, respectively. For information signals \(x_{i}^{I}\), there are \(x_{i}^{I}\sim CN(0,1),E\big{(}\Big{|}x_{j}^{E}\Big{|}_{2}\big{)}=1\) and \(\forall i\in K_{I},\forall j\in K_{E}\). In Equation (2), there are \(n_{i}\sim CN(0,\sigma_{i}^{2})\), \(P\triangleq\{P_{1},P_{2},...,P_{M}\}^{H}\), and \(n_{RIS}\sim CN(0,\sigma_{RIS}^{2})\). Since energy beams are pseudo-random signals that do not carry any information, their waveforms are assumed to be known at the AP and at each information receiver prior to data transmission, and it is assumed in this paper that the interference they cause can be cancelled out at each information receiver.