## 2 System Model and Problem Formulation
### Channel and System Models for 5G NOMA-Based Networks
From this diagram, it is noticed that the UC is essential for a multi-carrier NOMA system in which the clustering of users can ensure efficient power allocation and fulfill the SIC constraints. Furthermore, since the channel realization of every user is distinct on different subcarriers, the UC strategy is crucial to ensure that the channel diversity of all users can be fully explored and exploited to enable efficient clustering. Figure 1: SIC operation performed by \(N\) users sharing a specific frequency band in a NOMA-based 5G system. In the NOMA system where N users are sharing a subcarrier \(m\) in a cluster \(C_{i}\), the signal received at \(U_{n,m}\in C_{i}\) can be formulated as  
\[y_{n,m}=\sqrt{p_{n,m}}g_{n,m}\ x_{n,m}+\sum_{n=1,q\
eq n}^{N}\sqrt{p_{q,m}}g_{q,m }\ x_{q,m}+\eta_{n,m} \tag{2}\]  
where \(x_{n,m}\) represents the transmitted signal by the BS to \(U_{n,m}\) and \(\eta_{n,m}\sim CN\big{(}0,\ \sigma_{i}^{2}\big{)}\) represents additive white Gaussian noise (AWGN). In order to enable the successful implementation of SIC, power must be allocated to all users within a cluster following the SIC conditions [14]. Based on the power allocation strategy proposed in [14], the channel gains of the users clustered together are first sorted in a descending order \(g_{1,m}>g_{2,m}>\cdots>g_{N,m}\), the powers are then assigned to all users to satisfy the SIC constraints. The received signal-to-interference-plus-noise ratio (SINR) at the receiver side for \(U_{n,m},\ \forall n=1,\ 2,\ 3,\ \ldots,\ N\) can be expressed as  
\[\gamma_{n,m}=\frac{p_{n,m}G_{n,m}}{1+\sum_{q=1}^{n-1}p_{q,m}G_{q,m}},\ \forall n=1,\ 2,\ 3,\ \ldots,\ N \tag{3}\]  
where \(G_{n,m}=|g_{n,m}|^{2}/\sigma_{i}^{2}\) represents the normalized channel gain of the user \(n\) on subcarrier \(m\). From (3), the achievable throughput for \(U_{n,m},\ \forall n=1,\ 2,\ 3,\ \ldots,\ N\) can be written as  
\[R_{n,m}=Blog_{2}\Bigg{[}1+\frac{p_{n,m}G_{n,m}}{\sum_{q=1}^{n-1}p_{q,m}G_{q,m} +1}\Bigg{]},\ \forall n=1,\ 2,\ 3,\ \ldots,\ N \tag{4}\]  
Based on the Shannon equation expressed in (4), the achievable throughput of a cluster \(i\) can be denoted as  
\[R_{i}=\sum_{n=1}^{N}\sum_{m=1}^{M}R_{n,m}=\sum_{n=1}^{N}\sum_{m=1}^{M}Blog_{2} \Bigg{[}1+\frac{p_{n,m}G_{n,m}}{\sum_{q=1}^{n-1}p_{q,m}G_{q,m}+1}\Bigg{]} \tag{5}\]  
Unlike the work in [14] which rigidly pre-defines the number of clusters in the UC of the NOMA system, the UC proposed in this paper fully explores the channel diversity among all users to form clusters, which leads to an unknown number of clusters mainly determined by the diversity of the channel gains of all users. If there are C clusters formed in the NOMA system, the total system throughput of the NOMA network can be denoted as  
\[R=\sum_{i=1}^{C}R_{i}=\sum_{i=1}^{C}\sum_{n=1}^{N}\sum_{m=1}^{M}Blog_{2} \Bigg{[}1+\frac{p_{n,m}G_{n,m}}{\sum_{q=1}^{n-1}p_{q,m}G_{q,m}+1}\Bigg{]} \tag{6}\]  
In this paper, the UC problem is formulated as a throughput maximization problem in which the BS performs efficient UC and power allocation to maximize total system throughput in (6) subject to various constraints. For convenience, we tabulate the complete list of notations adopted in this paper in Table 1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Description** \\ \hline \(K\) & Number of users \\ \(M\) & Number of subcarriers \\ \(B\) & Bandwidth \\ \(n\) & User index \\ \(m\) & Subcarrier index \\ \(p_{n,m}\) & Transmit power of user \(n\) on subcarrier \(m\) \\ \(g_{n,m}\) & Channel gain of user \(n\) on subcarrier \(m\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of notations and mathematical symbols.