# Multiple-Intelligent Reflective Surfaces (Multi-IRSs)-Based NOMA System
## 2 The Modeled IRS-NOMA System  
The considered scenario is similar to the illustration in Figure 1. A downlink- communication system assisted with \(N\)-elements reflective plane is mode led, with both the source and the destination being considered to be equipped with single antennae. At the destination, the received signal is  
\[y^{u}=\left(h_{SD}^{u}+h_{SR}^{u(T)}\Theta h_{RD}^{u}\right)\sqrt{P_{IRS}^{u}}x^ {u}+n^{u} \tag{1}\]  
where \(\mathbf{h}_{SD}^{u}\in\mathbb{C}^{N}\) denotes the flat fading channel gain between the source in the destination, while \(\mathbf{h}_{SR}^{u}\in\mathbb{C}^{N}\) denotes that between the source and the IRS. In addition, \(\mathbf{h}_{DR}^{u}\in\mathbb{C}^{N}\) and \(\mathbf{x}^{u}\) represent the channel gain between the IRS and the destination and the information signal, respectively. Moreover, \(\Theta=\alpha diag\left(e^{ip_{1}},e^{ip_{2}},.....,e^{ip_{N}}\right)\) is a diagonal matrix and \(\theta_{1},\theta_{2},.....,\theta_{N}\) denotes the phase shift of the IRS elements with \(\alpha\in(0,1]\) standing for a constant reflection coefficient; \(n^{u}\)\(\sim\mathcal{CN}\left(0,\sigma^{2}\right)\) stands for the noise at the receiver. As a milestone, IRS-OMA will be considered in this paper for a comparison against the proposed IRS-NOMA system. The achievable SE of the IRS-OMA system is  
\[SE_{IRS-OMA}^{u}=\log_{2}\left(1+\frac{P_{IRS-OMA}^{u}\left|h_{SD}^{u}\right| }{z^{u}}\right) \tag{2}\]  
where \(P_{IRS-OMA}^{u}\) and \(N\) denote the power assigned to the \(u\)-th user in IRS-OMA system and the total number of reflection elements, respectively. \(z^{u}\sim\mathcal{CN}\left(0,\sigma^{2}\right)\) is the additive white Gaussian noise (AWGN) that has a zero mean and a variance of \(\sigma^{2}\). The considered model assumes that the \(U\)-IRS-NOMA users are scheduled in descending order based on their channel power, then the achievable SE by the \(u\)-th user is  
\[SE_{IRS-NOMA}^{u}=\] \[\log_{2}\left(1+\frac{P_{IRS-NOMA}^{u}\left(h_{SD}^{u}+h_{SR}^{u(T)} \Theta h_{RD}^{u}\right)^{2}}{\xi_{u}+z^{u}}\right) \tag{3}\]  
where \(P_{IRS-NOMA}^{u}\) denotes the power assigned to the \(u\)-th user in IRS-NOMA. \(\xi_{u}=\sum_{\
u=u+1}^{U}P_{IRS-NOMA}^{v}\left(h_{SD}^{u}+h_{SR}^{u(T)} \Theta h_{RD}^{u}\right)^{2}\) represents the effect of other IRS-NOMA users that have a better channel gain than the \(u\)-th user, and they are multiplexed over the same resource, along with the \(u\)-th user. According to the SIC mechanism, each user will consider the other multiplexed peers with higher channel gains as noise, while they cancel the effect of the ones who have lower channel gains before decoding their own signal. Note that \(\xi_{u}=0\) for the \(U\)-th indexed user case. Note that both \(P_{IRS-OMA}^{u}\) and \(P_{IRS-NOMA}^{u}\) are assumed to be allocated equally among all users following \(P_{IRS-OMA}^{u}\)=\(P_{IRS-NOMA}^{u}=\frac{P}{U}\), where \(P\) represents the total transmission power. Knowing that \(h_{SR}^{u(T)}\Theta h_{RD}^{u}=\alpha\sum_{n=1}^{N}e^{i\theta_{n}}\left[h_{SR} ^{u}\right]_{n}\left[h_{RD}^{u}\right]_{n}\), beside the fact that (2) and (3) depend on the channel amplitudes rather than its phase, hence, it is possible to use the notation \(\frac{1}{N}\sum_{n=1}^{N}\left[h_{SR}^{u}\right]_{n}\left[h_{RD}^{u}\right]_{n }=\Gamma_{IRS}^{u}\). The expressions in (2) and (3) could be rewritten as  
\[SE_{IRS-OMA}^{u}=\log_{2}\left(1+\frac{P_{IRS-OMA}^{u}\left(\Gamma_{SD}^{u} \right)^{2}}{z^{u}}\right) \tag{4}\]  
\[SE_{IRS-NOMA}^{u}=\log_{2}\left(1+\frac{P_{IRS-NOMA}^{u}\left(\Gamma_{SD}^{u}+ Na\Gamma_{IRS}^{u}\right)^{2}}{\xi_{u}+z^{u}}\right) \tag{5}\]  
respectively.