# Active RIS-Assisted Uplink NOMA with MADDPG for Remote State Estimation in Wireless Sensor Networks
## 2 The System Model
### Remote State Estimation  
Each sensor transmits its local state estimation measurement data to the remote estimator (i.e., the BS) via a wireless channel, utilizing uplink NOMA technology enhanced by RIS assistance. Thus, the transmission of \(\hat{\mathbf{x}}_{n}^{k}\) can be characterized by a binary random process \(\left\{\lambda_{n}^{k}\right\}\), where (see Figure 2)  
\[\lambda_{n}^{k}= \begin{cases}1,&\text{if $\hat{\mathbf{x}}_{n}^{k}$ arrives error-free at time $k$},\\ 0,&\text{otherwise (regarded as dropout)}.\end{cases}  \tag{16}\]  
Combining (15) and (16), the probability of the successful transmission of \(\hat{\mathbf{x}}_{n}^{k}\) is expressed as  
\[\mathbb{P}\Big{[}\lambda_{n}^{k}=1\Big{]}=1-\text{SER}_{n}^{k}=1-e^{-\gamma_ {n}^{k}}. \tag{17}\]  
Then, the RSE error covariance is [3]  
\[\mathbf{P}_{n}^{k}= \begin{cases}e^{\alpha}\widetilde{\mathbf{p}}_{n},&\text{ if $\lambda_{n}^{k}=1$},\\ e^{\alpha}u_{n}\Big{(}\mathbf{P}_{n}^{k-1}\Big{)},&\text{if $\lambda_{n}^{k}=0$}, \end{cases}  \tag{18}\]  
where \(\alpha\) is a positive parameter and \(u_{n}(\mathbf{X})\triangleq\mathbf{A}_{n}\mathbf{X}\mathbf{A}_{n}^{T}+ \mathbf{Q}_{n}\) (see Algorithm 1). To quantify the remote estimation quality of sensor \(n\) at time \(k\), we define the trace of the expected RSE covariance for the sensor as follows:  
\[ \begin{split}&\text{$j_{n}^{k}=\text{Tr}\Big{\{}\mathbb{E}[ \mathbf{P}_{n,r}^{k}]\Big{\}}$}\\ &=\text{Tr}\Big{\{}\mathbb{P}\Big{[}\gamma_{n}^{k}=0\Big{]}e^{ \alpha}u_{n}\Big{(}\mathbf{P}_{n}^{k-1}\Big{)}+\mathbb{P}\Big{[}\gamma_{n}^{k }=1\Big{]}e^{\alpha}\widetilde{\mathbf{p}}_{n}\Big{\}}\end{split}  \tag{19}\]  
Based on the preceding system model and corresponding derivations, the optimization objective is to minimize the sum of the expected covariance of the RSE errors, which can be expressed as follows:\[\text{Problem 1:}\min_{\mathbf{\theta}^{k},p_{n}^{k},\,b_{n,g}^{k}}\frac{1}{K} \sum_{k=1}^{K}\sum_{n=1}^{N}J_{n}^{k} \tag{20}\] \[\text{s.t.}\quad p_{n}^{k}\leq p_{\max},\quad n\in\mathcal{N},\] (20a) \[\quad b_{n,g}^{k}\in\{0,1\},\quad n\in\mathcal{N},\,g\in\mathcal{G},\] (20b) \[\quad\sum_{g=1}^{G}b_{n,g}^{k}\leq I,\quad n\in\mathcal{N},\] (20c) \[\quad\sum_{n=1}^{N}p_{n}^{k}\|\mathbf{\Theta}_{k}\mathbf{h}_{n,k}^{sr }\|^{2}+\|\mathbf{\Theta}_{k}\|^{2}\sigma_{1,k}^{2}\leq P_{\text{aRIS}}. \tag{20d}\]  
where \(K\) is the maximum value of the time slot set. \(P_{\text{aFIS}}=\xi(P_{\text{total}}-M(P_{\text{DC}}+P_{c}))\) represents the amplification power of the RIS [2]. Here, \(\xi\) is the amplifier efficiency, \(P_{\text{total}}\) is the total power budget, \(P_{\text{DC}}\) is the DC biasing power consumption, and \(P_{c}\) is the circuit power consumption of the active RIS. Constraint (20a) ensures that the transmission power of sensor \(n\) is within the maximum allowed limit. Constraints (20b) and (20c) enforce that each channel supports at most \(I\) sensors and that each sensor is assigned to only one channel. These two constraints together imply that the total number of sensors \(N\) must not exceed the overall channel capacity, i.e., \(G\times I\geq N\), in order to ensure the feasibility of the sensor-to-channel assignment. For example, if \(N=15\) sensors are to be scheduled using \(G=3\) channels, and each channel can support up to \(I=5\) sensors, then the total capacity is \(G\times I=15\), and the assignment is feasible. However, if \(N=16\), at least one sensor cannot be accommodated, violating the constraints. Therefore, this relationship is essential to guarantee a valid channel allocation. Constraint (20d) limits the amplification power of the active RIS. ```
0:\(\mathbf{A}_{n,i}\mathbf{C}_{n},\mathbf{Q}_{n},\mathbf{R}_{n},\mathbf{w}_{n}^ {k},\mathbf{v}_{n}^{k},\epsilon,k_{\max}\). 0:\(\mathbf{P}_{n}^{k}\). Initialization:\(t=1\), \(\Delta=1\).