## 5 Performance Analysis and Optimization
### Average BEP/SEP
#### 5.3.2. ABEP for BPSK and BFSK  
For coherent modulation schemes, such as BPSK, BFSK, and BFSK with minimum correlation, the ABEP at the output of a dual-branch SSC system can be expressed by applying the MGF method as [1]:  
\[P_{e}(E)=\frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}MGF_{\gamma_{S\text{SC}}}\bigg{(} \frac{g}{\sin^{2}\theta}\bigg{)}d\theta. \tag{73}\]  
where \(g=1\) for BPSK, \(g=0.5\) for BFSK, and \(g=0.715\) for BFSK with minimum correlation.  
Here, only identically distributed fading is considered for feasible computation. By substituting (33) in (73) and applying the definition of the bivariate Meijer G-function, and after some mathematical manipulation, we can obtain the ABEP values of BPSK, BFSK, and BFSK with minimum correlation for SSC over correlated and identical Fisher-Snedecor composite fading channels as (the mathematical derivation is provided in Appendix B):  
\[\begin{array}{ll}P_{e}(E)&=\sum\limits_{k=0}^{\infty}\sum\limits_{l=0}^{ \infty}\frac{\rho_{k}^{l}\rho_{k}^{l}\rho_{\phi}^{l}(\gamma_{\tau}\eta_{2})^{ \pi+k_{2}}F_{1}[\lambda,m+k;1+m+k,-\eta_{2}\gamma_{\tau}]}{\hbar E2\sqrt{\pi} \Gamma(m)\Gamma(n)}G_{1,1:1,2:1,1}^{0,1:2:0,1:1}G_{2,3}^{3,1}[g/\eta_{1}]^{1-( m+k),1}_{0,n+l,0.5}\,]\\ &+\frac{\rho_{3}^{3,1}[g/\Lambda_{2}[\lambda,0.05]}{2\sqrt{\pi}\Gamma(m) \Gamma(n)}-\frac{\Lambda_{2}^{\pi}\gamma_{\tau}^{m}}{2\sqrt{\pi}\Gamma(m) \Gamma(n)}G_{1,1:2:1,1}^{0,1:2:0,1:1}[g\gamma_{\tau},\gamma_{\tau}\Lambda_{2} \big{|}^{1-m}_{-m}\big{|}_{0,0.5}\big{|}^{1-(m+n)}_{0,0.5}].\end{array} \tag{74}\]  
For the non-correlated case, Equation (74) can be simplified as:  
\[P_{e}(E)=\frac{(1+F_{\gamma}(\gamma_{\tau}))G_{2,3}^{3,1}[g/\Lambda\big{|}^{1 -m;1}_{0,n,0.5}]}{2\sqrt{\pi}\Gamma(m)\Gamma(n)}-\frac{\Lambda^{m}\gamma_{ \tau}^{m}}{2\sqrt{\pi}\Gamma(m)\Gamma(n)}G_{1,1:1,2:1,1}^{0,1:2:0,1:1}[g\gamma_ {\tau},\gamma_{\tau}\Lambda\big{|}^{1-m}_{-m}\big{|}_{0,0.5}\big{|}^{1-(m+n)}_{ 0}\big{|}. \tag{75}\]  
To obtain the optimum switching threshold minimizing the ABEP of the above coherent modulation schemes, by inserting (36) in (73) and differentiating with respect to \(\gamma_{\tau}\), then setting \(\partial P_{e}(E)/\partial\gamma_{\tau}\big{|}_{\gamma_{\tau}^{\tau}=\gamma_{ \tau}}=0\), we have:  
\[\gamma_{\tau}^{*}=\frac{1}{2g}\Bigg{[}Q^{-1}\left(\frac{G_{2,3}^{3,1}[g/ \Lambda\big{|}^{1-m,1}_{0,n,0.5}]}{2\sqrt{\pi}\Gamma(m)\Gamma(n)}\right) \Bigg{]}^{2}, \tag{76}\]  
where \(Q^{-1}(x)\) is the inverse function of the Gaussian \(Q\)-function defined in ([1], Equation (4.2)).  
Similar to (63), the ABEP of coherent modulation schemes for SEC can also be obtained by substituting (49) into (73) and following the approach in Appendix B, as follows:  
\[P_{e}(E)=\frac{P_{1}+P_{3}}{2\sqrt{\pi}\Gamma(m)\Gamma(n)}G_{2,3}^{3,1}[g/ \Lambda\big{|}^{1-(m+k),1}_{0,n+l,0.5}]-\frac{P_{3}\Lambda^{m}\gamma_{\tau}^{ m}}{2\sqrt{\pi}\Gamma(m)\Gamma(n)}G_{1,1:1,2:1,1}^{0,1:2:0,1:1}[g\gamma_{ \tau},\gamma_{\tau}\Lambda\big{|}^{1-m}_{-m}\big{|}_{0,0.5}\big{|}^{1-(m+n)}_{ 0}\big{|}. \tag{77}\]  
In order to find the ABEP values of coherent modulation schemes for SECps, we employ the multiple Mellin-Barnes-type contour integral of the multivariate Fox's H-function and the approach in Appendix B again. By plugging (58) in (73), after some algebraic manipulation, the exact and novel expression of the ABEP for SECps can be derived as:  
\[\begin{array}{ll}P_{e}(E)&=\frac{L(\Lambda/g)^{\mu}}{2\sqrt{\pi}\Gamma(m) \Gamma^{(\kappa)}}G_{2,1:2: