## Appendix A Proofs of Equation (3)  
The known channel gain fading coefficient \(h_{1}\) follows the Generalized K distribution, whose Probability Density Function (PDF) [20] is as follows:  
\[f_{h_{1}}(x)=\frac{4m^{(k+m)/2}x^{k+m-1}}{\Gamma(m)\Gamma(k)\Omega^{(k+m)/2}}K _{\alpha}\left[2\left(\frac{m}{\Omega}\right)^{\frac{1}{2}}x\right]. \tag{1}\]  
In the formula, \(\alpha=k-m\). Among them, the small-scale Nakagami fading number is \(m\), and the shape parameter \(k\) representing the log-normal shadow; the \(\alpha\) kind of modifiedBessel function is \(K_{a}[\cdot]\). In addition, \(\Omega\) is the average power of fading, and without loss of generality \(\Omega\triangleq\frac{E\left(X^{2}\right)}{k}\), we can set \(E\big{(}X^{2}\big{)}=1\).  
By using Formulas (8.4.23.1) and (8.2.2.15) in Reference [14], (A1) can be transformed into  
\[f_{h_{1}}(x)=\frac{2}{x\Gamma(m)\Gamma(k)}G_{0,2}^{2,0}\bigg{[}\frac{mx^{2}}{ \Omega}|_{k,m}^{-}\bigg{]}.\] (A2)  
From Equation (2), we know the relationship between the instantaneous signal-to-noise ratio \(\gamma_{1}\) and the channel fading coefficient \(h_{1}\), which is \(\gamma_{1}={h_{1}}^{2}\overline{\gamma}_{1}/L_{UR}\); furthermore, \({h_{1}}^{2}=\gamma_{1}L_{UR}/\overline{\gamma}_{1}\), from the PDF solving method of random variable functions [21], we can obtain the PDF of the instantaneous signal-to-noise ratio \(\gamma_{1}\) as  
\[\begin{array}{rl}f_{\gamma_{1}}(\gamma_{1})&=f_{h_{1}}(h_{1})\Bigg{|}_{h_{1 }=\sqrt{\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{ \left(\frac{\left(\left\frac{\left(  
By utilizing the properties \(s\Gamma(s)=\Gamma(s+1)\) of the Gamma function, it can be concluded that  
\[\begin{array}{l}G_{1,3}^{2}\Bigg{[}\frac{mKL_{ijk}\cdot\gamma_{1}}{\gamma_{1}} \Bigg{|}\begin{array}{c}1\\ k,m,0\end{array}\Bigg{]}\\ \approx\frac{\Gamma(m-k)(mKL_{ijk})^{k}}{k}\Big{[}\frac{\gamma}{\gamma_{1}} \Big{]}^{k}+\frac{\Gamma(k-m)(mKL_{ijk})^{m}}{m}\Big{[}\frac{\gamma}{\gamma_{1} }\Big{]}^{m}.\end{array} \tag{10}\]  
Ordering \(\Delta_{1}=\frac{\Gamma(m-k)(mkL_{ijk})^{k}}{K(m)\Gamma(k)},\Delta_{2}=\frac{ \Gamma(k-m)(mkL_{ijk})^{m}}{m\Gamma(m)\Gamma(k)}\), that is,  
\[F_{\gamma_{1}}(\gamma)\approx\frac{\Delta_{1}\gamma^{k}}{\overline{\gamma}^{k }}+\frac{\Delta_{2}\gamma^{m}}{\overline{\gamma}^{m}}, \tag{11}\]  
Similarly, the asymptotic expression \(F_{\gamma_{2}}(\gamma)\) can be obtained:  
\[F_{\gamma_{2}}(\gamma)\approx\frac{\Delta_{3}\gamma\frac{\xi^{2}}{\overline{ \gamma}^{2}}}{\overline{\gamma}^{\frac{\xi^{2}}{\tau}}}+\frac{\Delta_{4} \gamma\frac{\xi^{2}}{\tau}}{\overline{\gamma}^{\frac{\xi^{2}}{\tau}}}, \tag{12}\]  
\[\text{where }\Delta_{3}=\frac{\Gamma\Big{(}a-\frac{\xi^{2}}{\tau}\Big{)}}{ \Gamma(a)}\frac{1}{(bA_{0})^{\xi^{2}}},\,\Delta_{4}=\frac{\xi^{2}}{\xi^{2}} \frac{1}{(\frac{\xi^{2}}{\tau}-a)}\frac{1}{(bA_{0})^{\varpi}}.\]  
Therefore, when \(\rho_{r}=\overline{\gamma}_{1}=\overline{\gamma}\rightarrow\infty\), \(F_{\gamma^{DF}}(\gamma)\) it can be asymptotically expressed as  
\[F_{\gamma^{DF}}(\gamma)\rightarrow\frac{\Delta_{1}\gamma^{k}}{\overline{\gamma }^{k}}+\frac{\Delta_{2}\gamma^{m}}{\overline{\gamma}^{m}}+\frac{\Delta_{3} \gamma\frac{\xi^{2}}{\tau}}{\overline{\gamma}^{\xi^{2}/\tau}}+\frac{\Delta_{4} \gamma\frac{\varpi}{\tau}}{\overline{\gamma}^{\text{ac}/\tau}}. \tag{13}\]