# Performance Analysis of a Cognitive RIS-NOMA in Wireless Sensor Network
## 2 The System Model  
In this paper, we consider the cognitive RIS-aided NOMA communications network shown in Figure 1, which operates over the spectrum of a primary network where a secondary source (\(\mathcal{S}\)) and \(M\) secondary user \(\{\mathcal{SU}_{m}\}_{m=1}^{M}\) are assisted by a RIS (\(\mathcal{R}\)) with \(N\) reflective elements, the \(K\) primary user \(\{\mathcal{PD}_{k}\}_{k=1}^{K}\) is in the area of interest of a primary source (\(\mathcal{PS}\)). Furthermore, all PN and SN nodes are equipped with a single antenna. We designate \(h_{\mathcal{SR}_{n}},n\in\{1,\ldots,N\}\) as the channel from \(\mathcal{S}\) to \(n\)-th reflective elements of \(\mathcal{R}\), while \(h_{\mathcal{SR}_{n}\mathcal{U}_{m}}\) denotes the channel from the \(n\)-th reflective elements of \(\mathcal{R}\) to \(\mathcal{SU}_{m}\); \(h_{\mathcal{SU}_{m}}\) denotes the channel from \(\mathcal{S}\) to \(\mathcal{SU}_{m}\), \(h_{\mathcal{SD}_{k}}\) denotes the channel from \(\mathcal{S}\) to \(\mathcal{PD}_{k}\), \(h_{\mathcal{PU}_{m}}\) denotes the channel from \(\mathcal{PS}\) to \(\mathcal{SU}_{m}\). In our paper, the CSI of \(\mathcal{PS}\) is not available in SN [44]. Moreover, all primary transmitter interference signals are treated as AWGN with \(\mathcal{CN}(0,\zeta\sigma^{2})\), where \(\zeta\) denotes the scaling coefficient. All channels follow Rayleigh fading. Without loss generality, the channel from \(\mathcal{S}\) to \(M\) SU and the cascade channel from \(\mathcal{S}\) to \(\mathcal{R}\) and \(\mathcal{R}\) to \(M\) SU are ordered as \(\left|\mathbf{h}_{\mathcal{RU}_{1}}^{H}\Theta\mathbf{h}_{\mathcal{SR}}+h_{ \mathcal{SU}_{1}}\right|^{2}\leqslant\cdots\leqslant\left|\mathbf{h}_{ \mathcal{RU}_{M}}^{H}\Theta\mathbf{h}_{\mathcal{SR}}+h_{\mathcal{SU}_{M}} \right|^{2}\)[33], where \(\mathbf{h}_{\mathcal{SR}_{n}}=\left[h_{\mathcal{SR}_{1}},\ldots,h_{\mathcal{ SR}_{N}}\right]\), \(\mathbf{h}_{\mathcal{RU}_{1}}=\left[h_{\mathcal{R}_{1}\mathcal{U}_{1}},\ldots,h_{ \mathcal{R}_{N}\mathcal{U}_{1}}\right]\), and \(\Theta=\text{diag}\big{(}\pi_{1}e^{\beta_{1}},\ldots,\)\(\alpha_{N}e^{\beta_{N}}\big{)}\), with \(\alpha_{n}=1\), \(\forall n\) and \(\theta_{n}\in[0,2\pi)\) being the amplitude coefficient and the phase shift, respectively, of the \(n\)-th reflective elements of \(\mathcal{R}\). In the CR model, \(\mathcal{S}\) must dynamically modify its transmit power in accordance with the maximum allowable interference level, \(P_{th}\), in order to prevent the interference that an SU transmission causes for the PU. As a result, transmit power at \(\mathcal{S}\) according to [42], is  
\[P_{\mathcal{S}}=\min\Bigg{(}\bar{P}_{\mathcal{S}},\frac{P_{th}}{\max\limits_{ k=1,\ldots,K}\left|h_{\mathcal{SD}_{k}}\right|^{2}}\Bigg{)} \tag{1}\]where \(\bar{P}_{S}\) denotes the maximum transmit power at S. Next, \(\mathcal{S}\) sends superposed signals to the \(M\) SU assisted by the RIS. Thus, the received signal at \(\mathcal{SU}_{m}\) is  
\[y_{\mathcal{SU}_{m}}=\left(\mathbf{h}_{\mathcal{SU}_{m}}^{H}\Theta\mathbf{h}_{ \mathcal{SR}}+h_{\mathcal{SU}_{m}}\right)\sum_{i=0}^{M}\sqrt{P_{S}\beta_{i}}x_ {i}+h_{\mathcal{PU}_{m}}+n_{\mathcal{SU}_{m}} \tag{2}\]  
where \(\beta_{i}\) denotes the power allocation with \(\beta_{1}\geqslant\ldots\geqslant\beta_{M}\), and \(\sum_{i=0}^{M}\beta_{i}=1\), with \(x_{i}\) denoting the message of \(i\)-th \(\mathcal{SU}\), and \(n_{\mathcal{SU}_{m}}\) denoting AWGN with \(\mathcal{CN}(0,\sigma^{2})\). Following NOMA procedures [30], the signal-to-interference plus noise ratio (SINR) at the \(m\)-th SU to detect information of \(p\)-th SU (\(m>p\)) is given by  
\[\text{SINR}_{m\to p}=\frac{\mathcal{G}_{m}^{2}P_{S}\beta_{p}}{\mathcal{G}_{m}^ {2}P_{S}\sum_{i=p+1}^{M}\beta_{i}+\xi\sigma^{2}+\sigma^{2}} \tag{3}\]  
where \(\mathcal{G}_{m}=|G_{m}+h_{\mathcal{SU}_{m}}|\), and \(G_{m}=|\sum_{i=1}^{N}h_{\mathcal{SR}_{m}}h_{\mathcal{R}_{m}d_{m}}e^{\beta_{m}}|\).