# Joint Beam-Forming Optimization for Active-RIS-Assisted Internet-of-Things Networks with SWIPT
## 2 System Models and Problem Modeling  
In this paper, we study a multi-user, multi-input, single-output PS-SWIPT network model in which RIS is introduced as an aid, as can be seen in Figure 1, which consists of a BS, RIS, and \(k\) single-input channels of users, where the BS has M input channels and the RIS has N reflector units. The set of users is defined as \(\mathcal{K}=\{1,\cdots,K\}(\forall k\in\mathcal{K})\). The set of array cells for RIS is \(\mathcal{N}=\{1,\cdots,N\}(\forall n\in\mathcal{N})\). Let \(\mathrm{G}\in\mathbb{C}^{N\times M}\) denote the flat fading channel gain of the BS-RIS, \(\mathrm{h}_{b,k}\in\mathbb{C}^{M\times 1}\) denote the flat fading channel gain of the BS-the user \(k\), and \(\mathrm{h}_{r,k}\in\mathbb{C}^{N\times 1}\) denote the flat fading channel gain of the RIS-user \(k\). All channels experience quasi-static flat gradient fading that remains constant over several symbols [33]. Then the signal transmitted from the BS can be described as  
\[\mathrm{x}=\sum_{i=1}^{K}\mathrm{w}_{i}\mathrm{s}_{i}, \tag{1}\]  
where \(\mathrm{w}_{i}\in\mathbb{C}^{M\times 1},\forall i\in\mathcal{K}\) represents the beam-former and \(\mathrm{s}_{i}\) denotes the information symbol of user \(i\) satisfying \(\mathbb{E}\Big{[}\abs{\mathrm{s}_{i}}^{2}\Big{]}=1,i\in\mathcal{K}\). In this way, the reflected signal of a P-RIS with N reflective elements can be written as  
\[\mathrm{r}=\Theta\mathrm{x}, \tag{2}\]  
where \(\Theta=\mathrm{diag}\big{(}e^{\abs{\mathrm{s}_{1}}},e^{\abs{\mathrm{s}_{2}}}, \ldots,e^{\abs{\mathrm{s}_{N}}}\big{)}\), \(\theta_{N}\in[0,2\pi)\) denotes the reflection coefficient matrix of the RIS, and \(\mathrm{x}\) denotes the signal transmitted from the base station (BS). If assisted by an RIS containing N reflective elements, the communication link can be strengthened, resulting in an increase in the energy power available to the user. If the base station (BS) transmits a signal strong enough to propagate through free space directly to a single user \(k\) without being blocked or reflected _en route_, the role of P-RIS becomes negligible due to the \"multipath fading\" effect. To maximize the sum of the energy collected by all users in the communication area in any case, this paper adopts A-RIS to solve this problem. The biggest difference between A-RIS and P-RIS is that A-RIS can amplify the reflected signals, and compared with the traditional relay, A-RIS is smaller in size and more flexible in layout. The output signal of the incident signal after reflection that occurs through the A-RIS can be written as  
\[\mathrm{r}=\mathrm{P}\Theta\mathrm{x}+\mathrm{P}\Theta\mathrm{v}+\
u, \tag{3}\]  
Figure 1: Multi-user MISO A-RIS-assisted SWIPT system. where \(\mathrm{P}=\mathrm{diag}(p_{1},\ldots,p_{N})\in\mathbb{R}^{N\times N}\) represents the A-RIS amplification factor matrix with all elements greater than 1, \(\mathrm{P}\Theta\
u\) represents the dynamic noise at the A-RIS, and \(\
u\) represents the static noise at the A-RIS. In addition, \(\
u\sim\mathcal{CN}\big{(}0_{N},\sigma_{v}^{2}\mathrm{I}_{N}\big{)}\) is only related to the device noise and input noise inherent in the A-RIS, whereas the static noise \(\
u\) is not related to \(\mathrm{P}\) and can therefore be disregarded. In the base station (BS) to the A-RIS channel, the signal received at the A-RIS can be represented as  
\[\widetilde{\mathrm{r}}=\mathrm{P}\Theta\mathrm{G}\sum_{i=1}^{K}\mathrm{w}_{i} \mathrm{s}_{i}+\mathrm{P}\Theta\
u. \tag{4}\]  
In an A-RIS to single user \(k\) channel, the signal received by the user \(k\) can be described as  
\[y_{k}=\sum_{i=1}^{K}\mathrm{h}_{k}^{H}\mathrm{w}_{i}\mathrm{s}_{i}+\mathrm{h} _{k}^{H}\mathrm{P}\Theta\
u+z_{k},\forall k, \tag{5}\]  
where \(\mathrm{h}_{k}^{H}=\mathrm{h}_{r,k}^{H}\mathrm{P}\Theta+\mathrm{h}_{b,k}^{H}\) represents the total channel gain from the base station to user \(k\), and \(z_{k}\sim\mathcal{CN}\big{(}0,\sigma_{k}^{2}\big{)}\) represents the complex Gaussian noise received at user \(k\).