## 2 System Model  
This paper proposes a downlink NOMA network consisting a source node equipped with \(K\) antenna and two downlink users, namely user \(1\) (\(U_{1}\)) and user \(2\) (\(U_{2}\)), as in Figure 1. We assume that each user is affected by a random number of randomly distributed interferer nodes. We assume that the information links, i.e., source-user links, experience independent flat Rayleigh fading. Furthermore, the interference links are assumed to experience independent flat Nakagami-\(m\) fading. In the proposed model, the source node delivers an unit energy superimposed signal \(\sqrt{\theta_{1}}x_{1}+\sqrt{\theta_{2}}x_{2}\) through a selected \(k\)-th antenna to both users, in which \(\theta_{n}\) is the power allocation for the \(n\)-th user, \(n\in\{1,2\}\), where \(\theta_{1}+\theta_{2}=1\). Hence, the received signal at each \(U_{n}\) is given by  
\[y_{n}(k)=\sqrt{P_{S}}\frac{h_{n}^{(k)}}{\sqrt{d_{n}^{\eta}}}(\sqrt{\theta_{1} }x_{1}+\sqrt{\theta_{2}}x_{2})+I_{n}, \tag{1}\]  
in which \(P_{S}\) denotes the transmit power, \(h_{n}^{(k)}\) is the channel coefficient between the \(k\)-th antenna and the \(n\)-th user, \(d_{n}\) is the distance from the source to the \(U_{n}\), \(\eta\) is the path-loss exponent and \(I_{n}\) specifies the aggregated interference signals at the user \(n\). Assuming that the interferers in the vicinity of each user are uniformly distributed and follow independent homogeneous Poisson point process \(\Phi_{n}\), and \(\lambda_{n}\) is the density, i.e., average number of the interferers. Therefore, \(I_{n}\) can be formulated as  
\[I_{n}=\sum_{\forall l\in\Phi_{n}}\sqrt{P_{I}}\frac{g_{n}^{(l)}}{\sqrt{r_{n,l}^{ \eta}}}x_{n}^{(l)}\,, \tag{2}\]  
in which \(P_{I}\) is the transmit power of each interferer, \(g_{n}^{(l)}\) and \(r_{n,l}\) specify the channel coefficient and the distance from the \(l\)-th interferer and the \(n\)-th user, and \(x_{n}^{(l)}\) is the transmitted signal from the \(l\)-th interferer. **Remark 1**. : _It is noteworthy from [38] that \(I_{1}\) and \(I_{2}\) might be correlated and causes significant performance loss at the corresponding user. According to [37], such an impact can be mitigated by having the paired users spatially separated. By means of NOMA, \(U_{1}\) and \(U_{2}\) can be a cell-edge paired with a cell-center user, respectively, which in turns ensures that \(U_{1}\) is distant from \(U_{2}\). Hence, we can assume that \(I_{1}\) and \(I_{2}\) are independent._  
Furthermore, we assume that  
\[\frac{|n_{1}^{(k)}|^{2}d_{1}^{-\eta}}{\sum_{\forall l\in\Phi_{1}}|g_{1}^{(l)}| ^{2}r_{1,l}^{-\eta}}<\frac{|n_{2}^{(k)}|^{2}d_{2}^{-\eta}}{\sum_{\forall l\in \Phi_{2}}|g_{2}^{(l)}|^{2}r_{2,l}^{-\eta}}, \tag{3}\]  
without loss of generality, thus for ensuring fairness, the source node allocates more power for \(U_{1}\), i.e., \(\theta_{1}\geq\theta_{2}\). The achievable rate at the first user to decode the desired signal while being assisted by the \(k\)-th antenna is formulated by  
\[R_{1}(k)=\log_{2}\left(1+\frac{\theta_{1}X_{1,k}}{\theta_{2}X_{1,k}+Z_{1}} \right), \tag{4}\]  
where \(X_{n,k}=\hat{\gamma}|h_{n}^{(k)}|^{2}d_{n}^{-\eta}\) with \(\hat{\gamma}=P_{S}/P_{I}\) specifies the average signal-to-interference (SIR) ratio, and \(Z_{n}=\sum_{\forall l\in\Phi_{n}}|g_{n}^{(l)}|^{2}r_{n,l}^{-\eta}\). At user \(U_{2}\), due to superior allocated power, successive interference cancellation (SIC) is carried out to detect \(x_{2}\). First, \(U_{2}\) decodes \(U_{1}\)'s signal while treating its own signal as an additional interference, thus the instantaneous achievable rate for detecting \(x_{1}\) is given as  
\[R_{1\to 2}(k)=\log_{2}\left(1+\frac{\theta_{1}X_{2,k}}{\theta_{2}X_{2,k}+Z_{2}} \right). \tag{5}\]  
After that, \(x_{1}\) is then cancelled by means of SIC to decode \(x_{2}\) with the following received rate  
\[R_{2}(k)=\log_{2}\left(1+\frac{\theta_{2}X_{2,k}}{Z_{2}}\right). \tag{6}\]  
Figure 1: An illustration of the proposed NOMA system in interference-limited environment.