## 4 Proposed Scheme
### Finding the Power Coefficients to Maximize the SINR  
To maximize the ratio in (13), the numerator is maximized and the denominator is minimized. Equation (13) can also be written as:  
\[\gamma_{list}=\frac{a_{il}}{\sum_{v=1}^{i-1}a_{vl}}\ \forall\ i\in\{2,3, \ldots,n\} \tag{17}\] \[\gamma_{list}=\frac{a_{il}}{1-\sum_{x=i+1}^{n}a_{xl}}\ \forall\ i\in\{2,3, \ldots,n\}. \tag{18}\]  
For the lowest channel gain user in a cluster \(i=n\), putting \(i=n\) in (18), we get:  
\[\gamma_{nlsat}=\frac{a_{nl}}{1-a_{nl}}. \tag{19}\]  
If \(a_{nl}\) is maximized in (19), then \(1-a_{nl}\) is minimized and the ratio, i.e., \(\gamma_{nl}\) is maximized. Similarly, for \(i=n-1\) (18) becomes,  
\[\gamma_{n-1,l}=\frac{a_{n-1,l}}{1-a_{nl}-a_{n-1,l}}. \tag{20}\]  
In (20), \(a_{nl}\) is already maximized and \(a_{n-1,l}\) needs to be maximized. From the (19) and (20), it can be concluded that, if we maximize the power coefficients in the order from \(a_{n,l},a_{n-1,l},a_{n-2,l},\ldots,a_{1,l}\), i.e., the lowest channel gain to second-lowest channel gain to the successive higher channel gain and so on, then it can be assured to achieve the maximum SINRs for all the users of an RIS-assisted NOMA cluster.  
For performing the SIC process successfully at the receiver, it is important to distribute the total cluster power such that the lowest channel gain user must have more than 50% of total cluster power. Similarly, second lowest channel gain user must have more than 25% of the total cluster power and so on for successive lower channel gain users [11]. In the first iteration of power coefficient allocation, if minimum 50% of cluster power is allocated to \(D_{nl}\) (\(a_{nl}=\frac{1}{2}\)), 25% of cluster power is allocated to \(D_{n-1,l}\) (\(a_{n-1,l}=\frac{1}{4}\)), 12.5% of cluster power is allocated to \(D_{n-2,l}\) (\(a_{n-2,l}=\frac{1}{8}\)) and so on. Then remaining power in the fraction after allocation till the highest channel gain user would be  
\[1-\frac{1}{2}-\frac{1}{4}-\frac{1}{8}-...-\frac{1}{2^{n}}=\frac{1}{2^{n}}.\]  
Now, this remaining power needs to be redistributed among the \(n\) users of the cluster in the second iteration. If half of the remaining power is allocated to lowest channel gain user, i.e., \(\frac{1}{2}\times\frac{1}{2^{n}}\), then \(a_{n,l}=\frac{1}{2}+\frac{1}{2^{n+1}}\). After this allocation, remaining power becomes \(\frac{1}{2^{n}}-\frac{1}{2^{n+1}}=\frac{1}{2^{n+1}}\). Half of this remaining power is distributed to the next lowest channel gain user (\(D_{n-2,l}\)). The process of allocating half of the remaining power is repeated till the remaining power is negligible as compared to \(P_{Toll}\), where \(P_{Toll}\) is minimum detectable power at the NOMA receiver. If \(\frac{P_{Toll}}{P_{T}|k_{\min}|^{2}}<\frac{1}{2^{2n}}\), then the geometric progression series for calculating the power coefficient can be given as:  
\[a_{n}=\frac{1}{2}+\frac{1}{2^{(n+1)}}+\frac{1}{2^{(2n+1)}}+\frac{1}{2^{(3n+1 )}}... \tag{21}\]  
\[a_{n-1}=\frac{1}{2^{2}}+\frac{1}{2^{(n+2)}}+\frac{1}{2^{(2n+2)}}+\frac{1}{2^{( 3n+2)}}... \tag{22}\]\[\begin{split}\ldots\ldots\ldots\\ a_{1}=\frac{1}{2^{n}}+\frac{1}{2^{(2n)}}+\frac{1}{2^{(3n)}}+\frac{1}{2^{(4n )}}\ldots\end{split} \tag{23}\]  
So, if \(\frac{P_{Tull}}{P_{T}|h_{t_{min}}|^{2}}<\frac{1}{2^{2n}}\) and \(\rho_{i}|h_{i\|}g_{ii}|^{2}\geq 1\) then, the generalized expression for \(i\)th user's power coefficient calculation, when cluster size is equal to \(n\), can be written as:  
\[a_{i}=\frac{1}{2^{(n-i+1)}}+\frac{1}{2^{(2n-i+1)}}+\frac{1}{2^{(3n-i+1)}}+....\forall i\in[1,n]. \tag{24}\]  
For \(n\geq 3\) the first five terms of the series can be taken into consideration and subsequent terms may be neglected because they do not contribute significantly in the magnitude of the power coefficient.