## 1 Introduction
### Problem Formulation for Sound Mathematical Representations  
Consider a downlink communication arrangement with one BS and \(K\) user, where each user has a single antenna. The signal at user \(i\) is received using NOMA transmission and is shown in Equation (1)  
\[y=h_{i}x+n_{i},\ i=1,\ 2,\ \ldots,\ K, \tag{1}\]where \(h_{i}\) signifies the station coefficient, and \(\ n_{i}\ \sim\ N(0,\ N_{0})\) is the additive white Gaussian noise (AWGN) at client \(i\), \(\sum_{i=1}^{K}\sqrt{P_{i}s_{i}}\) is the superposition of \(s_{i}\)'s with power allocation policy \(P=\left\{(P_{1},\ P_{2},\ \ldots,\ P_{K})/\sum_{i=1}^{K}P_{i}=P\right\}\), \(s_{i}\) signifies the data envisioned to communicate to the client \(i\), \(P_{i}\) signifies the power assigned to client \(i\), and P represents the total power constraint. For effortlessness of examination, we accept that \(|h_{1}|\ \geq\ |h_{2}|\ \geq\cdots\ \geq\ |h_{K}|\) and the total bandwidth is normalized to unity in this review. In this paper, we establish the minimal rate constraint \(r^{*}\) In the interest of consumer impartiality. The following condition should be guaranteed by the power allocation policy mathematically as seen in Equation (2):  
\[\underset{i}{min}\ r_{i}\ \geq\ r^{*}, \tag{2}\]  
where \(r_{i}\) Is the user \(i\) possible rate in nats/sec/Hz, which is determined by Equation (3)  
\[r_{i}=ln\left(1+\frac{P_{i}|h_{i}|^{2}}{N_{0}+|h_{i}|^{2}\sum_{j=1}^{i-1}P_{j }}\right) \tag{3}\]  
The summation in the denominator becomes zero for the exceptional case of \(i=1\), and the associated rate changes to as seen in Equation (4):  
\[r_{1}=ln\left(1+\frac{P_{i}|h_{i}|^{2}}{N_{0}}\right) \tag{4}\]  
The fact that the channels are ordered and the user with the stronger channel can decode the messages delivered to the user with the weaker channel indicates that \(r_{i}\) is possible. Consequently, the Equations (5)-(7) that follows is capable of being utilized to address the optimization challenge of enhancing the overall aggregate rate while satisfying the consumer's equitable requirement for NOMA frameworks:  
\[R_{N}\ \triangleq\ \underset{P_{i}}{\max}\sum_{t=1}^{K}r_{i} \tag{5}\]  
\[s.t.\qquad r_{i}=ln\left(1+\frac{P_{i}|h_{i}|^{2}}{N_{0}+|h_{i}|^{2}\sum_{j=1 }^{i-1}P_{j}}\right), \tag{6}\]  
\[\sum_{t=1}^{K}P_{i}\ \leq P, \tag{7}\]  
Conventional OMA structures, which include FDMA or TDMA, allocate duration/ frequency resources quasi-adaptively, giving every customer an established subchannel. To simplify designation, we refer to this kind of OMA as OMA-TYPE I in the present investigation. Therefore, the OMA-TYPE-I optimization challenge is capable of being formulated in the following Equations (8) and (9), presuming that all consumers will receive an equal volume of assets (the duration or bandwidth):  
\[R_{01}\ \triangleq\ \underset{P_{i}}{max}\sum_{t=1}^{K}r_{i} \tag{8}\]  
\[s.t.\qquad r_{i}=\frac{1}{K}ln\left(1+\frac{KP_{i}|h_{i}|^{2}}{N_{0}}\right), \tag{9}\]The optimization problem for jointly designing power and sub-channel allocations will be addressed because some users may encounter poor channel conditions due to significant route loss and random fading caused by sub-channel allocations that are not optimal. When the total time/frequency is divided into \(n\) sub-channels that will be shared orthogonally among \(K\) users, the following Equations (10)-(15) represents the optimization problem:  
\[R_{0X}\ \triangleq\ \underset{P_{i,\
u}}{max}\sum_{S_{i}}\sum_{t=1}^{K}\sum_{n \in S_{i}}r_{i},\ n \tag{10}\]  
\[\underset{i}{min}\ r_{i}\ \geq\ r^{*}, \tag{2}\]0 \[\underset{i}{min}\ r_{i}\ \geq\ r^{*}, \tag{2}\]1 (12) \[\underset{i}{min}\ r_{i}\ \geq\ r^{*}, \tag{2}\]2 (13) \[\underset{i}{min}\ r_{i}\ \geq\ r^{*}, \tag{2}\]3  
\[\underset{i}{min}\ r_{i}\ \geq\ r^{*}, \tag{2}\]4  
\[\underset{i}{min}\ r_{i}\ \geq\ r^{*}, \tag{2}\]5  
The power allotted to and the station coefficient of the client \(i\)'s sub-station, correspondingly, are denoted as \(P_{i},\ n\) and \(h_{i}\).