# Active RIS-Assisted Uplink NOMA with MADDPG for Remote State Estimation in Wireless Sensor Networks
## 2 The System Model
### The Uplink Communication Model  
After obtaining \(\hat{x}_{n}^{k}\), sensor \(n\) transmits it to the BS as a data packet. The communication between the sensors and the BS is based on an uplink NOMA communication model, supported by an active RIS. In this model, the RIS serves as an enhancement tool for signal propagation, optimizing both the signal quality and coverage. The equivalent baseband time-domain channel in the system is composed of three parts: the channel from the RIS to the BS, from sensor \(n\) to the RIS, and from sensor \(n\) to the BS. These channels are denoted by \(\mathbf{h}^{rb}\in\mathbb{C}^{G\times M}\), \(\mathbf{h}_{n}^{sr}\in\mathbb{C}^{M\times 1}\), and \(\mathbf{h}_{n}^{sb}\in\mathbb{C}^{G\times 1}\), respectively. The reflection coefficient of the RIS element \(m\) is expressed as \(\theta_{m}=\alpha_{m}e^{\mathrm{i}\phi_{m}}\), where \(\alpha_{m}\) represents the reflection amplitude and \(\phi_{m}\in[0,2\pi]\) denotes the phase shift. The RIS beamforming is then captured by the reflection coefficient matrix \(\mathbf{\Theta}\triangleq\mathrm{diag}\{\theta_{1},\theta_{2},\ldots,\theta_ {M}\}\)[2]. Unlike passive RISs, active RISs are equipped with amplifiers that consume additional power, making thermal noise at the active RIS significant. Consequently, the signal received at the BS can be modeled as  
\[\mathbf{y}_{g}^{k}=\sum_{n=1}^{N}b_{n,g}^{k}\sqrt{p_{n}^{k}}\mathbf{h}_{n}^{k }f_{n}^{k}+\mathbf{n}_{g}^{k}, \tag{10}\]  
where \(b_{n,g}^{k}\in\{0,1\}\) represents the binary channel selection for sensor \(n\). The total channel gain \(\mathbf{h}_{n}^{k}\) is denoted as \(\mathbf{h}_{k}^{rb}\mathbf{\Theta}_{k}\mathbf{h}_{n,k}^{sr}+\mathbf{h}_{n,k}^ {sb}\), \(\mathbf{h}_{n}^{k}\in\mathbb{C}^{G\times 1}\). The noise \(\mathbf{n}_{g}^{k}\) is defined as \(\mathbf{h}_{k}^{rb}\mathbf{\Theta}_{k}\mathbf{v}_{k}+\mathbf{n}_{0,g}^{k}\), \(\mathbf{n}_{g}^{k}\in\mathbb{C}^{G\times 1}\). The thermal noise at the RIS is modeled by \(\mathbf{v}_{k}\in\mathbb{C}^{M\times 1}\), which follows a circularly symmetric complex Gaussian distribution, i.e., \(\mathbf{v}_{k}\sim\mathcal{CN}(\mathbf{0}_{M},\sigma_{1}^{2}\mathbf{I}_{M})\). Similarly, the thermal noise at the BS is modeled by \(\mathbf{n}_{0,g}^{k}\in\mathbb{C}^{G\times 1}\), where \(\mathbf{n}_{0,g}^{k}\sim\mathcal{CN}(\mathbf{0}_{G},\sigma_{2}^{2}\mathbf{I}_{ G})\)[2]. Let \(f_{n}^{k}\) represent the data symbol transmitted by sensor \(n\) at time \(k\), while \(p_{n}^{k}\) denotes the corresponding power used for computation offloading. We assume that the transmitted symbols are independent and identically distributed (i.i.d.) as \(f_{n}^{k}\sim\mathcal{CN}(0,1)\). Since it is an uplink NOMA channel, we assume that the channel gains between different sensors decrease, i.e., \(|\mathbf{h}_{1}^{k}|^{2}\geq|\mathbf{h}_{2}^{k}|^{2}\cdots\geq|\mathbf{h}_{n}^ {k}|^{2}\). The channel model comprises both path loss and small-scale fading. For the direct link between the BS and the sensor \(n\), we assume Rayleigh fading. The corresponding channel can be expressed as  
\[\mathbf{h}_{n,k}^{sb}=\sqrt{d_{S,n,k}^{-\alpha_{S}}}\hat{\mathbf{h}}_{n,k}^{ sb}, \tag{11}\]  
where \(d_{S,n,k}\) is the distance between the BS and sensor \(n\), \(\alpha_{S}\) is the path loss exponent, and \(\hat{\mathbf{h}}_{n,k}^{sb}\in\mathbb{C}^{G\times 1}\) represents the small-scale fading channel, modeled as \(\mathcal{CN}(\mathbf{0}_{G},\mathbf{I}_{G})\). The active RIS is deployed at a fixed location, where line-of-sight (LoS) links exist from both the BS to the RIS and from the RIS to sensor \(n\). Therefore, the reflection channel follows Rician fading.