## 3 Ergodic Rate Analysis  
In this section, an analysis of the ergodic rates is presented. Due to the modelling of non-ideal factors, it is very difficult to obtain a closed-form expression for this problem. We obtain an accurate approximation using the theory of mathematical derivations. The ergodic rate for user \(n\) can be expressed as \[C_{n}=\mathbb{E}\log_{2}(1+\mathrm{SINR}_{n,n}),\quad n\in N \tag{17}\]  
**Theorem 1. **_The expression for the ergodic rate for user \(n\) can be derived as_  
\[ \begin{split} C_{n}\approx&\sum_{s=0}^{n-1}\frac{N!(- )^{s}(N-n+s+1)^{-1}}{(n-1)!(N-n)!\ln 2}\binom{n-1}{s}\\ &\times\left[I\bigg{(}\frac{\theta_{n,2}}{\theta_{n,1}},\frac{ \theta_{n,3}}{\theta_{n,1}},\frac{\theta_{n,4}}{\theta_{n,1}}\bigg{)}-I \bigg{(}\frac{\theta_{n,2}}{\theta_{n,1}},\frac{\theta_{n,3}}{\theta_{n,1}}, \frac{\theta_{n,4}}{\theta_{n,1}}\bigg{)}\right]\end{split}  \tag{18}\]  
_where \(\theta_{n,1}=(b_{n}+\alpha a_{n})\Gamma_{1}\beta_{\mathrm{RU}}\phi_{\mathrm{SR}}\), \(\bar{\theta}_{n,1}=b_{n}\Gamma_{1}\beta_{\mathrm{RU}}\phi_{\mathrm{SR}}\), \(\theta_{n,2}=\beta_{\mathrm{RU}}\Gamma_{2}\), \(\theta_{n,3}=\beta_{\mathrm{SR}}\Gamma_{3}(N-n+s+1)\) and \(\theta_{n,A}=\Gamma_{4}(N-n+s+1)\). The function \(I(\cdot,\cdot,\cdot)\) in (18) is given by_  
\[ \begin{split} I(p,q,r)&\approx\mathrm{e}^{\frac{r} {2}}E_{1}\Big{(}\frac{r}{q}\Big{)}+\mathrm{e}^{q}E_{1}(q)\\ &+\mathrm{e}^{p}E_{1}(p)\,\sum_{\lambda\in\Lambda}\Bigg{\{}h_{ \lambda}\mathrm{e}^{-k_{\lambda}q}\bigg{[}\frac{x_{\lambda}^{2}}{2}+\sqrt{x_{ \lambda}}J_{1}(-2\sqrt{x_{\lambda}})\bigg{]}\Bigg{\}}\\ &+\sum_{\lambda\in\Lambda}\Bigg{\{}h_{\lambda}\mathrm{e}^{-k_{ \lambda}q}\bigg{[}\frac{x^{2}}{2p}-\sum_{v=2}^{V}\sum_{u=1}^{v-1}\frac{(u-1)!(- )^{v-u}x^{v}}{v!(v-1)!p^{u}}\bigg{]}\Bigg{\}},\end{split}  \tag{19}\]  
_where \(\Lambda\), \(h_{\lambda}\), and \(k_{\lambda}\) are defined in (24), and \(x_{\lambda}=k_{\lambda}(pq-r)\), \(J_{1}(\cdot)\) denotes the first-order Bessel function of the first kind._  
Proof. : Through knowledge of order statistics [28], it can be seen that \(|\hat{h}_{n}|^{2}\) has a density function  
\[ \begin{split} f_{n}(x)&=\frac{N!\beta_{\mathrm{RU} }^{-1}}{(n-1)!(N-n)!}\left(\mathrm{e}^{-\frac{1}{P_{\mathrm{RU}}}x}\right)^{N -n+1}\left(1-\mathrm{e}^{-\frac{1}{P_{\mathrm{RU}}}x}\right)^{n-1}\\ &=\frac{N!\beta_{\mathrm{RU}}^{-1}}{(n-1)!(N-n)!}\sum_{s=0}^{n-1} \binom{n-1}{s}(-)^{s}\mathrm{e}^{-\frac{N-n+s+1}{P_{\mathrm{RU}}}x}\mathbf{1 }_{(x\geq 0)},\end{split}  \tag{20}\]  
where \(\mathbf{1}_{(\cdot)}\) denotes the indicator function. Using (15), (17), (20), we obtain  
\[ \begin{split} C_{n}=&\int_{0}^{\infty}\int_{0}^{ \infty}\log_{2}\left(1+\frac{\alpha a_{n}\Gamma_{1}xy}{b_{n}\Gamma_{1}xy+ \Gamma_{2}x+\Gamma_{3}y+\Gamma_{4}}\right)\\ &\times f_{n}(x)\beta_{\mathrm{SR}}^{-1}\mathrm{e}^{-\frac{1}{P_ {\mathrm{SR}}}y}\,\mathrm{d}x\mathrm{d}y.\end{split}  \tag{21}\]  
Changing variable \(x\mapsto(N-n+s+1)\beta_{\mathrm{RU}}^{-1}x\) and \(y\mapsto\beta_{\mathrm{SR}}^{-1}y\), (20) can be expressed as  
\[ \begin{split} C_{n}=&\sum_{s=0}^{n-1}\frac{N!(-)^{s }(N-n+s+1)^{-1}}{(n-1)!(N-n)!\ln 2}\binom{n-1}{s}\\ &\times\int_{0}^{\infty}\int_{0}^{\infty}\ln\left(\frac{\theta_{n,1}xy+\theta_{n,2}x+\theta_{n,3}y+\theta_{n,4}}{\bar{\theta}_{n,1}xy+\theta_{n,2}x+\theta_{n,3}y+\theta_{n,A}}\right)\mathrm{e}^{-x-y}\,\mathrm{d}x\mathrm{d} y.\end{split}  \tag{22}\]  
We could carry on calculating the integral in (22) as follows:  
\[ \begin{split} I(p,q,r)&=\int_{0}^{\infty}\int_{0}^{ \infty}\ln(xy+px+qy+r)\mathrm{e}^{-x-y}\,\mathrm{d}x\mathrm{d}y-\ln r\\ &=\int_{0}^{+\infty}\bigg{[}\ln\Bigl{(}\frac{q}{r}y+1\Bigr{)}+ \mathrm{e}^{\frac{q+r}{q+r}}E_{1}\Bigl{(}\frac{qy+r}{y+p}\Bigr{)}\bigg{]} \mathrm{e}^{-y}\,\mathrm{d}y\\ &\approx\mathrm{e}^{\frac{r}{q}}E_{1}\Bigl{(}\frac{r}{q}\Bigr{)}+ \int_{0}^{+\infty}\sum_{\lambda\in\Lambda}h_{\lambda}\mathrm{e}^{-k_{\lambda} \frac{q+r}{q+r}}\mathrm{e}^{-y}\,\mathrm{d}y\end{split}