## 5 Performance Analysis and Optimization
### Average Output SNR and Amount of Fading  
#### 5.2.1 Average Output SNR  
The average output SNR is a useful performance measure serving as an excellent indicator of the overall system's fidelity. The average output SNR of SSC over various Fisher-Snedecor \(\mathcal{F}\) fading channels can be obtained by setting \(q=1\) in (38) (or (43) and (44)); that is, \(\overline{\gamma}_{SCC}=\mu_{\gamma_{SCC}}(1)\).  
To get the optimum switching threshold that maximizes the average output SNR, by differentiating \(\mu_{\gamma_{SCC}}(1)\) in (38) with respect to \(\gamma_{\tau}\) and with some mathematical manipulation, we have:  
\[\frac{\partial}{\partial\gamma_{\tau}}\mu_{\gamma_{\tau_{\tau}}}(1)=B(\gamma_ {\tau})({}_{1}-_{2})+(P/F_{1})(\Phi_{1}(\gamma_{\tau})-\Phi_{2}(\gamma_{\tau }))+\Phi_{2}(\gamma_{\tau}), \tag{61}\]  
where:  
\[B(\cdot)=\frac{f_{\gamma_{2}}(\cdot)F_{1}-f_{\gamma_{1}}(\cdot)F_{2}}{\left(F _{1}+F_{2}\right)^{2}} \tag{62a}\]  
\[\Phi_{i}(\cdot)=\int_{0}^{\infty}\gamma f_{\gamma_{1}\gamma_{2}}(\cdot,\gamma )d\gamma-\gamma_{\tau}f_{\gamma_{i}}(\cdot) \tag{62b}\]  
\[i=\int_{0}^{\infty}\gamma g_{\gamma_{i}}(\gamma)d\gamma+\int_{\gamma_{\tau}} ^{\infty}\gamma f_{\gamma_{i}}(\gamma)d\gamma \tag{62c}\]  
We substitute (1), (13), (19) and (20) into (62), and then set the final results of (61) to zero. For this general case, it is hard to find the analytical solution of the optimum value \(\gamma_{\tau}^{*}\). However, this solution can be evaluated by using the numerical method with the helpof the MATLAB software package or Mathematica tool. For the i.i.d. case, the closed-form expression of the optimum switching threshold for \(\mu_{\gamma_{SCC}}(q)\) can be obtained as:  
\[\gamma_{\tau}^{*}=\bigg{(}\frac{B(m+q,n-q)}{B(m,n)\Lambda^{q}}\bigg{)}^{1/q}, \tag{63}\]  
where \(\mu_{\gamma_{SCC}}(q)\) has the maximum value. When \(q=1\), \(\gamma_{\tau}^{*}=\overline{\tau}\), the maximum average output SNR at the SSC can be written as:  
\[\overline{\tau}_{ssc}=\overline{\tau}\bigg{(}\frac{m^{m-1}}{B(m,n)(n-1)^{m}}{ }_{2}F_{1}(m+n,m;m+1;-\frac{m}{n-1})+\frac{(n-1)^{n-1}}{B(m,n)m^{n}}{}_{2}F_{1} (m+n,n-1;n;-\frac{n-1}{m})\bigg{)}. \tag{64}\]  
Based on (48), the average output SNR at the SEC can also readily obtained by setting \(q=1\). To obtain the optimum switching threshold \(\gamma_{\tau}^{*}\), the derivative of (47) with respect to \(\gamma_{\tau}\) when \(q=1\) can be expressed as:  
\[\begin{array}{ll}\frac{\partial\mu_{\gamma_{SCC}}(1)}{\partial\tau_{\tau}}& =(L-1)F(\gamma_{\tau})^{L-2}f_{\gamma}(\gamma_{\tau})\int_{0}^{\infty}\gamma f _{\gamma}(\gamma)d\gamma\\ &+\sum\limits_{i=0}^{L-2}\Big{(}iF(\gamma_{\tau})^{i-1}f_{\gamma}(\gamma_{ \tau})\int_{\gamma_{\tau}}^{\infty}\gamma f_{\gamma}(\gamma)d\gamma+\gamma_{ \tau}F(\gamma_{\tau})^{i}f_{\gamma}(\gamma_{\tau})\Big{)}.\end{array} \tag{65}\]  
In order to simplify the analysis process, we apply the method in [50] and let \(\gamma_{\tau}^{*}=\alpha\overline{\gamma}\), where \(\alpha\) is a function of \(L\) and \(\overline{\tau}\). By inserting \(\gamma_{\tau}^{*}=\alpha\overline{\gamma}\) into (65), the average output SNR at the SEC can be given over Fisher-Snedecor \(\mathcal{F}\) fading channels as:  
\[\overline{\tau}_{SEC}=\overline{\tau}\bigg{(}P_{1}+\frac{P_{3}((n-1)/\alpha)^ {n-1}}{B(m,n)m^{n}}{}_{2}F_{1}(m+n,n-1;n;-(n-1)/\alpha m)\bigg{)}. \tag{66}\]  
In Table 3, it can be observed that the value of \(\alpha\) increases as \(L\) grows, regardless of the fading conditions. However, the value of \(\alpha\) gets smaller with the increase of the values of the shadowing parameters (\(n\)), and it approaches that of the case of Nakagami-m (\(n\rightarrow\infty\)) in [50].