## 2 NOMA System Model  
We consider a single-cell NOMA network equipped with a single-input, single-output (SISO) design in which the BS is located at the centre of the cell. The numbers of users and subchannels are defined as \(U\) and \(N\), respectively. Users \(U\) are uniformly distributed within the cell. The BS transmits the superimposed signal for multiple users over \(N\) subchannels, where a subchannel is indexed as \(n\in\{1,2,\ldots,N\}\). The total bandwidth \(B_{Total}\) is divided by the number of subchannels, and the subchannel bandwidth \(B_{sc}\) is divided equally for each subchannel. Moreover, multiple users share the same subchannel in accordance with NOMA concepts, where \(U_{n}\) is the set of users sharing the same subchannel \(n\). The total transmit power distributed over the subchannels is \(P_{Max}\), where \(P_{u,n}\) is the user power, and system power allocation is limited as \(\sum_{n}^{U_{n}}\sum_{u}^{U_{n}}P_{u,n}\leq P_{Max}\). The BS transmits signal \(x_{n}\) on subchannel \(n\), and the transmitted signal is given by:  
\[x_{n}=\sum_{u=1}^{U_{n}}\sqrt{P_{u,n}}\;\widetilde{x}_{u,n}\;, \tag{1}\]where \(P_{u,n}\) is the power allocated to user \(u\) multiplexed on subchannel \(n\), and the modelled symbol is denoted as \(\widetilde{x}_{u,n}\). The signal received by user \(u\) over subchannel \(n\) is described as follows:  
\[\begin{array}{c}y_{u,n}\ =g_{u,n}x_{u,n}+\mathbb{Z}_{u,n}\,\\ y_{u,n}\ =g_{u,n}\sqrt{P_{u,n}}\ \widetilde{x}_{u,n}+g_{u,n}\ \overset{U_{n}}{ =u+1}\ \sqrt{P_{i,n}}\ \widetilde{x}_{i,n}+\mathbb{Z}_{i,n}\,\end{array} \tag{2}\]  
where the channel gain coefficient between the \(u\)th user and the BS is represented as \(g_{u,n}=d_{u,n}^{-1}h_{u,n}\), where \(d_{u,n}^{-1}\) represents the path loss coefficient. The Rayleigh fading channel gain is \(h_{u,n}\). \(d_{u,n}\) is the distance between the \(u\)th user for each channel and the BS. \(\mathbb{Z}_{i,n}\) is the additive white Gaussian noise (AWGN) with a zero-mean complex random variable and \(\sigma_{n}^{2}\) variance is \(\mathbb{Z}_{i,n}\ \sim\mathcal{CN}\big{(}0,\ \sigma_{n}^{2}\big{)}\). The design assumes that the number of users is two times the number of channels (\(U=2N\)). The data rate on a given subchannel for the \(u\)th user is described as follows:  
\[R_{u,\ n}=B_{sc}\log_{2}(1+SINR_{u,n}), \tag{3}\]  
where the signal-to-interference-plus-noise ratio (\(SINR\)) is given as:  
\[SINR_{u,n} =\frac{p_{u,n}|g_{u,n}|^{2}}{\sigma_{n}^{2}+\sum_{i=1}^{l-1}p_{u, n}|g_{u,n}|^{2}}, \tag{4}\] \[=\frac{\frac{p_{u,n}|g_{u,n}|^{2}}{\sigma_{n}^{2}}}{1+\sum_{i=1}^ {l-1}\frac{p_{u,n}|g_{u,n}|^{2}}{\sigma_{n}^{2}}},\] (5) \[=\frac{p_{u,n}G_{u,n}}{1+\sum_{i=1}^{u-1}p_{u,n}G_{u,n}}\, \tag{6}\]  
where \(p_{u,n}\) is the power allocated to the \(u\)th user on the \(n\)th subchannel. \(g_{u,n}\) is the channel response coefficient on the downlink for the \(u\)th user on the \(n\)th subchannel. \(G_{l,n}\) is the channel response normalised by noise (CRNN) for the corresponding user. System design is strictly multiplex, with only two users on each channel. For example, the subchannel \(n\) multiplexes the two users \(U_{1,n}\) and \(U_{2,n}\) with \(|G_{1,n}|^{2}\geq|G_{2,n}|^{2}\), where the power allocation scheme is \(P_{1,n}\leq P_{2,n}\) in accordance with the power domain multiplexing protocol in NOMA [20; 21; 22].  
In our design, we assume that subchannels are sorted such that \(|G_{1,n}|\geq|G_{2,n}|\geq\cdots\geq|G_{U_{n,n}}|\). SIC is applied to each channel, where the intended signal can be detected by each user. The high channel gain user firstly decodes the signal of the low channel gain user, cancels the interference power, and then decodes the intended signal. By contrast, the low channel gain user regards the interference power on the same subchannel as noise. This assumption is common in NOMA based on SIC. Thus, the total throughput is:  
\[R=\sum_{n=1}^{N}\sum_{i=1}^{N_{sc}}B_{sc}\log_{2}(1+SINR_{u,\ n}). \tag{7}\]