## 3 Performance Analysis
### EC of Perfect SIC  
In this section, we are going to derive the approximated expression of data rate of MU NOMA system, which is defined as the total rate of all users which are served. The instantaneous data rate based on the [PERSON] theory of \((m,n)\)th user is represented by Equation (16):  
[PERSON]R_{m,n} =\log_{2}\left(1+\frac{P_{\mathbb{S}}a_{m,n}|\mathbf{w}_{m}\mathbf{ h}_{m,n}|^{2}}{\sum\limits_{i=n+1}^{N}P_{\mathbb{S}}a_{m,i}|\mathbf{w}_{m} \mathbf{h}_{m,n}|^{2}+\sigma_{m,n}^{2}}\right),\] \[=\log_{2}\left(1+|\mathbf{w}_{m}\mathbf{h}_{m,n}|^{2}\rho\Big{(} \sum\limits_{i=n+1}^{N}a_{m,i}+a_{m,n}\Big{)}\right)-\log_{2}\left(1+\sum \limits_{i=n+1}^{N}\rho a_{m,i}|\mathbf{w}_{m}\mathbf{h}_{m,n}|^{2}\right), \tag{16}\]  
here \(\rho=\frac{P_{\mathbb{S}}}{\sigma_{m,n}^{2}}\).  
From Equation (16), the EC of the \((m,n)\)th user is calculated as in Equation (17):  
\[\bar{R}_{m,n} =\mathbb{E}\{R_{m,n}\} \tag{17}\] \[=\mathbb{E}\Bigg{\{}\log_{2}\left(1+|\mathbf{w}_{m}\mathbf{h}_{m, n}|^{2}\rho\Big{(}\sum\limits_{i=n+1}^{N}a_{m,i}+a_{m,n}\Big{)}\right)\Bigg{\}}- \mathbb{E}\Bigg{\{}\log_{2}\left(1+\sum\limits_{i=n+1}^{N}\rho a_{m,i}| \mathbf{w}_{m}\mathbf{h}_{m,n}|^{2}\right)\Bigg{\}}.\]  
The first expectation of a random variable \(\mathcal{X}\) is defined by Equation (18):  
\[\mathbb{E}[\log_{2}(1+\mathcal{X})]=\int\limits_{0}^{\infty}\log_{2}(1+x)f_{ \mathcal{X}}(x)dx, \tag{18}\]  
where \(f_{\mathcal{X}}(x)\) is PDF of \(\mathcal{X}\) with \(\mathcal{X}\in\{X,Y\}\) and  
\[X =P_{\mathbb{S}}|\mathbf{w}_{m}\mathbf{h}_{m,n}|^{2},\] \[Y =\sum\limits_{i=n+1}^{N}P_{\mathbb{S}}a_{m,i}|\mathbf{w}_{m} \mathbf{h}_{m,n}|^{2}.\]  
Using the integration-by-parts method we can rewrite Equation (17) as Equation (19):  
\[\bar{R}_{m,n}=\frac{1}{\ln 2}\int\limits_{0}^{\infty}\frac{1-F_{X}(x)}{1+x}dx- \frac{1}{\ln 2}\int\limits_{0}^{\infty}\frac{1-F_{Y}(y)}{1+y}dy, \tag{19}\]  
where the \(F_{X}(x)\) and \(F_{Y}(y)\) are the CDFs of the random variables \(X\) and \(Y\), respectively. From Equation (19) and above suppositions, the EC depicted as Equation (20):  
\[\bar{R}_{m,n}=\sum\limits_{n=1}^{N}\binom{N}{n}\frac{(-1)^{n-1}}{\ln 2}\mathcal{ J}_{1}(x)-\sum\limits_{n=1}^{N}\binom{N}{n}\frac{(-1)^{n-1}}{\ln 2}\int \limits_{0}^{\infty}\frac{1}{1+y}\exp\left(-\frac{y}{\sum\limits_{i=n+1}^{N} \rho a_{m,i}\Omega_{m,n}}\right)dy, \tag{20}\]  
where Equation (21):  
\[\mathcal{J}_{1}(x)=\int\limits_{0}^{\infty}\frac{1}{1+x}\exp\left(-\frac{x}{ \rho\Big{(}\sum\limits_{i=n+1}^{N}a_{m,i}+a_{m,n}\Big{)}\Omega_{m,n}}\right)dx. \tag{21}\]To solve Equation (20), the following equation is applied Equation (3.352.4) [22] in Equation (22):  
\[\int\limits_{0}^{\infty}\frac{1}{x+\alpha}\text{exp}(-\mu x)dx=-\text{exp}(- \alpha\mu)\text{Ei}(-\mu\beta), \tag{22}\]  
with \(\alpha=\beta=1\) and \(\text{Ei}(x)=\int_{-\infty}^{x}\frac{\text{exp}(t)}{x^{2}}dt\) denotes the exponential integral function [22]. And then, the approximated expression of the EC of \((m,n)\)th user is given in Equation (23).  
\[\bar{R}_{m,n}=\sum\limits_{n=1}^{N}\binom{N}{n}\frac{(-1)^{n-1}}{ \ln 2}\exp\left(\frac{1}{\rho\Big{(}\sum\limits_{i=n+1}^{N}a_{m,i}+a_{m,n} \Big{)}\Omega_{m,n}}\right)\text{Ei}\left(-\frac{1}{\rho\Big{(}\sum\limits_{i= n+1}^{N}a_{m,i}+a_{m,n}\Big{)}\Omega_{m,n}}\right)\] \[-\sum\limits_{n=1}^{N}\binom{N}{n}\frac{(-1)^{n-1}}{\ln 2}\exp \left(\frac{1}{\sum\limits_{i=n+1}^{N}\rho a_{m,i}\Omega_{m,n}}\right)\text{Ei }\left(-\frac{1}{\sum\limits_{i=n+1}^{N}\rho a_{m,i}\Omega_{m,n}}\right). \tag{23}\]  
Finally, the EC of the \(m\)th cluster is the sum of EC of all users in this cluster Equation (24):  
\[\overline{R}_{m}=\sum\limits_{n=1}^{N}\overline{R}_{m,n}. \tag{24}\]