## 2 System Model  
This paper considers a centralized downlink multi-cell NOMA system, as shown in Figure 1. Each cell contains a base station that transmits data to multiple users over wireless channels, and the signals of different users are superimposed through channel and power allocation in the system. At the receiver, users can recover their individual signals by using the serial interference cancelation (SIC) technique. It is assumed that the multi-cell NOMA system consists of \(M\) cells, \(K\) users, and \(N\) orthogonal subchannels. The total channel bandwidth is \(B_{tot}\), so the bandwidth of each subchannel is \(B_{tot}/N\). Concurrently, each subchannel in each cell can be allocated to \(k^{\prime}{}_{n}\) users for multiplexing, where \(k^{\prime}{}_{n}\) denotes the number of users for which subchannel \(n\) is multiplexed. Let \(S^{n}_{m,u}\) denote the transmission signal of the user \(u\) on the subchannel \(n\) of the base station \(m\). The superimposed signal \(S_{m}^{n}\) on the subchannel \(n\) of the cell \(m\) can be expressed as follows:  
\[S_{m}^{n}=\sum\limits_{u=1}^{K_{m}}G_{m,u}^{n}\sqrt{p_{m,u}^{n}}S_{m,u}^{n} \tag{1}\]  
where \(K_{m}\) is the number of users in cell \(m\), \(G\in R^{M\times N\times K}\) is the channel allocation indicator, and \(G_{m,u}^{n}=1\) indicates that the base station \(m\) allocates the subchannel \(n\) to the user \(u\), otherwise \(G_{m,u}^{n}=0\), and \(p_{m,u}^{n}\) is the power that the base station \(m\) allocates to the user \(u\) on the subchannel \(n\). When the superimposed signal is transmitted through the wireless channel to the receiver, the signal received by the user \(u\) of the cell \(m\) on the subchannel \(n\) is represented as follows:  
\[ \begin{split} j_{m,u}^{n}&=\underbrace{G_{m,u}^{n}H_ {m,u}^{n}\sqrt{p_{m,u}^{n}}S_{m,u}^{n}}_{\text{Desired signal}}+ \underbrace{H_{m,u}^{n}\sum\limits_{r=1,r^{\prime}\
eq u}^{K_{m}}G_{m,r}^{n} \sqrt{p_{m,r}^{n}}S_{m,r}^{n}}_{\text{intra-cell interference}}\\ &+\underbrace{\sum\limits_{c\in M,c\
eq m}H_{c,u}^{n}\Bigg{(} \sum\limits_{i=1}^{K_{c}}G_{c,d}^{n}\sqrt{p_{c,d}^{n}}S_{c,d}^{n}\Bigg{)}}_{ \text{inter-cell interference}}+\underbrace{z_{m,u}^{n}}_{\text{AWGN}}\end{split}  \tag{2}\]  
where \(H_{m,u}^{n}\) represents the channel gain between the base station \(m\) and the user \(u\) over the subchannel \(n\). The superimposed signal consists of four components: the desired signal, the intra-cell interference, the inter-cell interference (ICI), and the additive white gaussian noise (AWGN) with zero mean and variance \(\sigma_{n}^{2}\). It is assumed that the user's receiver in this multi-cell NOMA system is equipped with the SIC technique to decode the superimposed signals in the subchannel. The SIC technique is realized by using the channel-to-noise ratio (CNR) of the user \(u\) on the subchannel \(n\) with the base station \(m\), i.e., \(\psi_{m,u}^{n}\), as expressed below:  
\[\psi_{m,u}^{n}=\frac{G_{m,u}^{n}\big{|}H_{m,u}^{n}\big{|}^{2}}{\sigma_{n}^{2} +\sum\limits_{c\in M,c\
eq m}\big{|}H_{c,u}^{n}\big{|}^{2}p_{c}^{n}} \tag{3}\]  
Figure 1: System model. where \(p_{c}^{n}\) is the sum of the power allocated by the base station \(c\) to the users on the subchannel \(n\), i.e., \(p_{c}^{n}=\sum_{d=1}^{K}p_{c,d}^{n}\). Then, the users in the same subchannel are ranked according to the magnitude of their CNRs. According to the fairness principle of the NOMA system, users with higher CNR are allocated to lower power, while users with lower CNR are allocated to higher power. Additionally, users allocated with higher power will regard the signals of users allocated with lower power as interference. Therefore, the SIC can decode the superimposed signal according to the powers of users. In this process, once the SIC decodes a user signal, interference from the signal is eliminated. In this paper, each subchannel is assumed to be multiplexed by two users.