## 3 Outage Probability of RIS-ONOMA
### Probability of Outage of RIS-SR-ONOMA  
Figure 1 depicts the block diagram of RIS-SR-ONOMA. It is assumed that user 1 is far from BS, whereas user 2 is nearer to BS. Here RIS consists of \(N\) passive reflecting elements that forms SR scenario, where symmetrical reflecting elements are assigned to _WU_ and _SU_ (\(N_{1}=N_{2}=N/2\)) irrespective of channel conditions. RIS is intended to generate any phase shift between 0 and 360 degrees. However, because of hardware limitations, it functions like an ON-OFF switch. It only generates a small number of discrete phase shifts. Therefore, in many real-world scenarios, the implementation of intelligent RIS assisted transmission is impractical. We cannot accomplish diversity gain in blind transmission since phase compensation is not performed at the RIS elements; instead, only array gain can be attained. BS and RIS communicate using one channel with a channel coefficient \(u_{i}=a_{i}e^{\Phi_{i}}\) and RIS and UE communicate via the other channel with a channel coefficient \(v_{i}=b_{i}e^{\Psi_{i}}\). Both \(u_{i}\in\mathbb{C}N(0,\;1)\) and \(v_{i}\in\mathbb{C}N(0,\;1)\) follow Rayleigh distribution [27]. Figure 1: Pictorial representation of RIS-SR-ONOMA system. To implement the NOMA coding principle, BS combines two independent signals using the superposition coding method, and then transmits this combined signal to the users which is given by [28]  
\[x_{s} =\sqrt{\alpha_{1}E_{s}}x_{1}+\sqrt{\alpha_{2}E_{s}}x_{2} \tag{1}\]  
Here \(\alpha_{1}\) and \(\alpha_{2}\) are the power allocation factors assigned to user 1 and 2 respectively. In ONOMA \(\alpha_{2}<\alpha_{1}\) and the sum of power allocation factors must be equal to unity which is given as  
\[\alpha_{1}+\alpha_{2}=1 \tag{2}\]  
Intended symbols for user 1 and 2 are denoted by \(x_{1}\) and \(x_{2}\) respectively. \(E_{s}\) represents average transmit power of the symbols \(x_{1}\) and \(x_{2}\). The received signal \(y_{1}\) at user 1 is given by  
\[y_{1}=\left[\sum_{i=1}^{N}u_{i}v_{i}\right]x_{s}+n_{1} \tag{3}\]  
Here \(n_{1}\) is the complex Gaussian noise added at user 1 with variance \(N_{0}\). Let \(L=\sum_{i=1}^{N}u_{i}v_{i}\) is the overall gain of the dual-hop channel. As per the central limit theorem (CLT), \(L\) follows complex Gaussian distribution \(L\in\mathbb{N}(0,\,N_{1})\)[27]. After substituting (1) in (3), the received signal at user 1 is given as,  
\[y_{1} =L\sqrt{\alpha_{1}E_{s}}x_{1}+L\sqrt{\alpha_{2}E_{s}}x_{2}+n_{1} \tag{4}\]  
The user with the maximum power i.e., user 1 regards the signal from user 2 as interference and recovers its signal without going through any SIC process. The signal-to-interference plus noise ratio (SINR) of user 1 is  
\[\Upsilon_{1}^{x_{1}} =\frac{|L|^{2}\alpha_{1}\varrho_{s}}{|L|^{2}\alpha_{2}\varrho_{s }+1} \tag{5}\]  
where \(\varrho_{s}=\frac{E_{s}}{N_{0}}\) represents received SNR. The capacity of user \(1\) is given by  
\[C_{1}^{x_{1}}=log_{2}\big{(}1+\Upsilon_{1}^{x_{1}}\big{)} \tag{6}\]  
Similarly, the received signal \(y_{2}\) at user 2 is given as,  
\[y_{2}=\left[\sum_{i=\frac{N}{2}+1}^{N}u_{i}v_{i}\right]x_{s}+n_{2} \tag{7}\]  
Here \(n_{2}\) is the complex Gaussian noise added at user 2 with variance \(N_{0}\). Let \(M=\sum_{i=\frac{N}{2}+1}^{N}u_{i}v_{i}\) is the overall gain of the two-hop channel. As per the CLT, \(M\) follows complex Gaussian distribution \(M\in\mathbb{N}(0,\,N_{2})\)[27]. After substituting (1) in (7), the received signal at user 2 is given as,  
\[y_{2}=M\sqrt{\alpha_{1}E_{s}}x_{1}+M\sqrt{\alpha_{2}E_{s}}x_{2}+n_{2} \tag{8}\]  
At user 2, user 1's signal is detected first assuming user 2's signal as interference. The SINR of decoding user 1's signal at user 2 is given by  
\[\Upsilon_{2}^{x_{1}}=\frac{|M|^{2}\alpha_{1}\varrho_{s}}{|M|^{2}\alpha_{2} \varrho_{s}+1} \tag{9}\]  
After the estimation process of \(x_{1}\), its impact is removed from \(y_{2}\) through SIC.