## 3 Performance Analysis
### EC of Imperfect SIC  
From the instantaneous SINR given in Equation (25), we have the instantaneous capacity of the \((m,n)\)th user is represented by Equation (30):  
\[R_{m,n}^{impSIC}=\log_{2}\Big{(}1+\gamma_{m,n}^{impSIC}\Big{)}= \log_{2}\left(1+\frac{P_{\text{S}}a_{m,n}|\textbf{h}_{m,n}\textbf {w}_{m}|^{2}}{\Big{(}\sum\limits_{i=n+1}^{N}P_{\text{S}}a_{m,i}+\Delta_{k} \Big{)}|\textbf{h}_{m,n}\textbf{w}_{m}|^{2}+\sigma_{m,n}^{2}}\right),\] \[= \log_{2}\left(\frac{\Bigg{[}\Big{(}\sum\limits_{i=n+1}^{N}P_{ \text{S}}a_{m,i}+\Delta_{k}\Big{)}+P_{\text{S}}a_{m,n}\Big{]}\left|\textbf{h}_ {m,n}\textbf{w}_{m}\right|^{2}+\sigma_{m,n}^{2}}{\Big{(}\sum\limits_{i=n+1}^{ N}P_{\text{S}}a_{m,i}+\Delta_{k}\Big{)}|\textbf{h}_{m,n}\textbf{w}_{m}|^{2}+ \sigma_{m,n}^{2}}\right), \tag{30}\]  
where \(\Delta_{k}=\xi\sum_{k=1}^{n-1}a_{k}P_{\text{S}}\). From Equation (30), we can rewrite the EC as Equation (31).  
\[\bar{R}_{m,n}^{impSIC}= \mathbb{E}\Bigg{\{}\log_{2}\Bigg{(}\Big{[}\Big{(}\sum\limits_{i= n+1}^{N}\rho a_{m,i}+\frac{\Delta_{k}}{\sigma_{m,n}^{2}}\Big{)}+\rho a_{m,n} \Big{]}|\textbf{h}_{m,n}\textbf{w}_{m}|^{2}+1\Bigg{)}\Bigg{\}}\] \[-\mathbb{E}\Bigg{\{}\log_{2}\Bigg{(}\Big{(}\sum\limits_{i=n+1}^{ N}\rho a_{m,i}+\frac{\Delta_{k}}{\sigma_{m,n}^{2}}\Big{)}|\textbf{h}_{m,n}\textbf{w}_{m}|^{2 }+1\Bigg{)}\Bigg{\}}. \tag{31}\]  
Let  
\[Z_{1}= \Big{[}\Big{(}\sum\limits_{i=n+1}^{N}\rho a_{m,i}+\frac{\Delta_{k }}{\sigma_{m,n}^{2}}\Big{)}+\rho a_{m,n}\Big{]}|\textbf{h}_{m,n}\textbf{w}_{m}| ^{2},\] \[Z_{2}= \Big{(}\sum\limits_{i=n+1}^{N}\rho a_{m,i}+\frac{\Delta_{k}}{ \sigma_{m,n}^{2}}\Big{)}|\textbf{h}_{m,n}\textbf{w}_{m}|^{2},\]using integration-by-parts method and Equation (18), the \(\hat{R}_{m,n}^{impSIC}\) is rewritten as follows in Equation (32):  
\[\hat{R}_{m,n}^{impSIC}=\underbrace{\frac{1}{\ln 2}\int\limits_{0}^{\infty}\frac{1-F_{ Z_{1}}(z_{1})}{1+z_{1}}dz_{1}}_{Q_{1}}-\underbrace{\frac{1}{\ln 2}\int \limits_{0}^{\infty}\frac{1-F_{Z_{2}}(z_{2})}{1+z_{2}}dz_{2}}_{Q_{2}}. \tag{32}\]  
Note that, according to the Equations (7)-(14) [23] the CDFs of \(Z_{1}\) and \(Z_{2}\) can be given as Equation (33):  
\[F_{Z_{u}}(z)=1-\sum\limits_{n=1}^{N}\binom{N}{n}(-1)^{n-1}\exp\left(-\frac{nz_ {u}}{\Omega_{m,n}}\right), \tag{33}\]  
with \(u\in\{1,2\}\).  
The functions \(Q_{1}\) and \(Q_{2}\) are calculated by replacing the CDF in Equation (33) into Equation (32). In Equations (34) and (35):  
\[Q_{1}=\sum\limits_{n=1}^{N}\binom{N}{n}\frac{(-1)^{n-1}}{\ln 2}\int\limits_{0}^{ \infty}\frac{1}{1+z_{1}}\exp\left(-\frac{z_{1}}{\Xi_{1}\Omega_{m,n}}\right)dz _{1}, \tag{34}\]  
where \(\Xi_{1}=\left[\left(\sum\limits_{i=n+1}^{N}\rho a_{m,i}+\frac{\Delta_{k}}{ \sigma_{m,n}^{2}}\right)+\rho a_{m,n}\right]\).  
\[Q_{2}=\sum\limits_{n=1}^{N}\binom{N}{n}\frac{(-1)^{n-1}}{\ln 2}\int\limits_{0}^{ \infty}\frac{1}{1+z_{2}}\exp\left(-\frac{z_{2}}{\Xi_{2}\Omega_{m,n}}\right)dz _{2} \tag{35}\]  
where \(\Xi_{2}=\left(\sum\
olimits_{i=n+1}^{N}\rho a_{m,i}+\frac{\Delta_{k}}{ \sigma_{m,n}^{2}}\right)\). Using Equation (22), we can obtain \(Q_{1}\) and \(Q_{2}\) as in Equations (36) and (37):  
\[Q_{1}=\sum\limits_{n=1}^{N}\binom{N}{n}\frac{(-1)^{n-1}}{\ln 2}\exp\left(- \frac{1}{\Xi_{1}\Omega_{m,n}}\right)\text{Ei}\left(-\frac{1}{\Xi_{1}\Omega_{ m,n}}\right), \tag{36}\]  
\[Q_{2}=\sum\limits_{n=1}^{N}\binom{N}{n}\frac{(-1)^{n-1}}{\ln 2}\exp\left(- \frac{1}{\Xi_{2}\Omega_{m,n}}\right)\text{Ei}\left(-\frac{1}{\Xi_{2}\Omega_{ m,n}}\right). \tag{37}\]  
Finally, replace \(Q_{1}\) and \(Q_{2}\) into Equation (32), we can obtain the EC of the \((m,n)\)th user, and then the EC of \(m\)th cluster in case of imperfect SIC.  
The result of our paper investigated the performance of the considered NOMA system in derivate the closed-form expressions of outage probability and ergodic capacity of each cluster. Based on the total of users, the number of users in each cluster \(N\), and the number of clusters, \(M\) of the system can be varied. The derived expressions can not apply to another system.