# Joint Beam-Forming Optimization for Active-RIS-Assisted Internet-of-Things Networks with SWIPT
## 2 System Models and Problem Modeling
At this point, the base station (BS) uses the available time and frequency resources to transmit signals to all single users, and the energy power received by the user is divided into two parts ID and EH. We set \(\rho_{k}\in(0,1)\) as the PS ratio, where part \(\rho_{k}\) is used for ID, so the decoded signal from user \(k\) can be described as  
\[y_{k}^{\mathrm{ID}}=\sqrt{\rho_{k}}y_{k}+n_{k}, \tag{6}\]  
where \(n_{k}\sim\mathcal{CN}\big{(}0,\delta_{k}^{2}\big{)}\) is the additional noise generated by the signal processing circuitry of the single user \(k\) when performing information decoding. Part \(1-\rho_{k}\) is used for EH, so the energy-harvesting signal of user \(k\) can be expressed as  
\[y_{k}^{EH}=\sqrt{1-\rho_{k}}y_{k}. \tag{7}\]  
Based on this, the signal-to-noise ratio at a single user \(k\) can be expressed as  
\[\mathrm{SINR}_{k}=\frac{\left|\mathrm{h}_{k}^{H}\mathrm{w}_{k}\right|^{2}}{ \sum\limits_{ \begin{subarray}{c}i=1\\ i\
eq k\end{subarray} }^{K}\left|\mathrm{h}_{k}^{H}\mathrm{w}_{i}\right|^{2}+ \sigma_{v}^{2}\|\mathrm{h}_{r,k}^{H}\mathrm{P}\Theta\|^{2}+\sigma_{k}^{2}+ \frac{\delta_{k}^{2}}{\rho_{k}}},\forall k. \tag{8}\]  
For the energy-harvesting (EH) circuit, this paper adopts the linear model from the literature [34], whose linear input power is denoted by \(P_{k}^{L}\). The energy harvested at user \(k\) can then be described as  
\[P_{k}^{L}=\eta_{k}(1-\rho_{k})\left(\sum_{i=1}^{K}\left|\mathrm{h}_{k}^{H} \mathrm{w}_{i}\right|^{2}+\sigma_{v}^{2}\|\mathrm{h}_{r,k}^{H}\mathrm{P} \Theta\|^{2}\right),\forall k, \tag{9}\]  
where \(\eta_{k}\in[0,1]\) represents the energy conversion efficiency of user \(k\). In this paper, we assume that the energy conversion efficiency of all users is 100%, i.e., \(\eta_{k}=1\). After modeling the signal and energy, the sum of energy harvesting at the user end is maximized by co-optimizing the base station beam-former, the PS ratio, and the RIS phase shift/amplification factor, a problem that can be described as  
\[(\mathrm{P}1):\underset{\mathrm{w}_{k},\mathrm{P},\rho_{k}}{\text{maximize}} \ f_{1}=\sum_{k=1}^{K}P_{k}^{L}, \tag{10}\]\[\text{s.t. }\frac{\left|\mathrm{h}_{k}^{H}\mathrm{w}_{k}\right|^{2}}{ \sum\limits_{ \begin{subarray}{c}i=1\\ i\
eq k\end{subarray} }^{K}\left|\mathrm{h}_{k}^{H}\mathrm{w}_{i}\right|^{2}+ \sigma_{v}^{2}\|\mathrm{h}_{r,k}^{H}\mathrm{P}\Theta\|^{2}+\sigma_{k}^{2}+ \frac{\delta_{k}^{2}}{\rho_{k}}}\geqslant\gamma_{k}, \tag{11}\] \[\sum\limits_{i=1}^{K}\|P\Theta G\mathrm{w}_{i}\|^{2}+\sigma_{v}^{ 2}\|P\Theta\|^{2}\leqslant p_{\text{max}},\] (12) \[\sum\limits_{k=1}^{K}\|\mathrm{w}_{k}\|_{2}^{2}\leqslant P_{B}, \forall k,\] (13) \[0<\rho_{k}<1,\forall k,. \tag{14}\]  
where (10) is the sum of energy collected by the user, and the optimization objective in this paper is to maximize it. Equation (11) is the signal-to-dry noise ratio constraint for user \(k\), and \(\gamma_{k}\) is the minimum signal-to-dry noise ratio threshold required for user \(k\). Equation (12) denotes the reflected power constraint of the A-RIS, and \(p_{\text{max}}\) is the maximum reflected power threshold of the A-RIS. Equation (13) denotes the radiant power constraint of the BS, and \(P_{B}\) is the maximum radiant power threshold of the BS. Equation (14) represents the PS factor scaling constraint. Since the variables are entangled with each other and strongly coupled, the problem (P1) is non-convex and cannot be solved directly using the software CVX, so the next step is to turn the problem (P1) into a convex problem.