## 3 Dual-Branch SSC Receiver
### The \(q\)th Moments  
Based on the definition of the \(q\)th moments and using (17), \(\mu_{\gamma_{SCC}}(q)\) in the n.i.n.i.d. case can be expressed as:  
\[\mu_{\gamma_{sec}}(q)=\sum\limits_{i=1}^{2}{(P/F_{i})\mu_{\gamma_{sec}}^{( \gamma_{i})}(q)}, \tag{38}\]  
where \(\mu_{\gamma_{SSC}}^{(\gamma_{i})}(q)\) is defined as:  
\[\mu_{\gamma_{sec}}^{(\gamma_{i})}(q)=\int_{0}^{\infty}\gamma^{q}g_{\gamma_{i}} (\gamma)d\gamma+\int_{\gamma_{\tau}}^{\infty}\gamma^{q}f_{\gamma_{i}}(\gamma)d\gamma. \tag{39}\]  
By substituting \(\text{g}_{\gamma_{1}}(\gamma)\) in (19) (or \(\text{g}_{\gamma_{2}}(\gamma)\) in (20)) and \(f_{\gamma_{1}}(\gamma)\) (or \(f_{\gamma_{2}}(\gamma)\)) in (1) and using ([43], Equation (3.194.2/3)), after some mathematical manipulation, \(\mu_{\gamma_{SSC}}^{(\gamma_{1})}(q)\) and \(\mu_{\gamma_{SSC}}^{(\gamma_{2})}(q)\) are respectively yielded as:  
\[\begin{array}{rl}\mu_{\gamma_{sec}}^{(\gamma_{1})}(q)&=\sum\limits_{k=0}^{ \infty}\sum\limits_{l=0}^{\infty}\sum\limits_{i=0}^{\infty}\sum\limits_{j=0}^{ \infty}\sum\limits_{p=0}^{\infty}\frac{p_{N}^{k+i}p_{G}^{l+i}^{l+i}p_{G}^{l+i} \text{er}(\Lambda_{1})(m_{2}-m_{1})_{i}(n_{2}-n_{1})_{i}^{4}\pi_{2}^{m_{1}+k} \text{-}q_{\tau}\pi_{1}^{m_{1}+k}}{k\text{H}(m_{1})\Gamma(n_{1})B(m_{2}+k+i,n_ {2}+l+j)(m_{1}+k)}\\ &\times B(m_{2}+k+i+q,n_{2}+l+j-q)_{2}F_{1}[\Lambda_{1},m_{1}+k;1+m_{1}+k,- \alpha_{1}\gamma_{\tau}]\\ &+\frac{\gamma_{\tau}^{-n_{1}}\Lambda_{1}^{-n_{1}}}{B(m_{1}\pi_{1})(n_{1}-q)} F_{1}[m_{1}+n_{1},n_{1}-q;n_{1}-q+1;-(\Lambda_{1}\gamma_{\tau})^{-1}],\end{array} \tag{40}\]  
\[\begin{array}{rl}\mu_{\gamma_{sec}}^{(\gamma_{2})}(q)&=\sum\limits_{k=0}^{ \infty}\sum\limits_{l=0}^{\infty}\sum\limits_{j=0}^{\infty}\sum\limits_{j=0}^{ \infty}\frac{p_{N}^{k+i}p_{G}^{l+i}\text{er}(\Lambda_{1})(m_{2}-m_{1})_{i}(n_{ 2}-n_{1})_{i}^{4}\pi_{2}^{m_{2}+k+i}\gamma_{\tau}^{m_{2}+k+i}}{k\text{H}(m_{1 })\Gamma(n_{1})B(m_{2}+k+i,n_{2}+l+j)(m_{2}+k+i)}\\ &\times B(m_{1}+k+q,n_{1}+l-q)_{2}F_{1}[\lambda_{2}+i+j,m_{2}+k+i;m_{2}+k+i+1;- \alpha_{2}\gamma\\ &+\frac{\gamma_{\tau}^{-n_{2}}\Lambda_{2}^{-n_{2}}}{B(m_{2},n_{2})(n_{2}-q)} F_{1}[m_{2}+n_{2},n_{2}-q;n_{2}-q+1;-(\Lambda_{2}\gamma_{\tau})^{-1}].\end{array} \tag{41}\]  
In the case of n.i.i.d. Fisher-Snedecor composite fading channels, the \(q\)th order moment of \(\gamma_{SCC}\) can be expressed as:  
\[\mu_{\gamma_{sec}}(q)=\int_{0}^{\infty}\gamma^{q}g_{\gamma_{2}}(\gamma)d\gamma+ \int_{\gamma_{\tau}}^{\infty}\gamma^{q}f_{\gamma_{2}}(\gamma)d\gamma. \tag{42}\]  
By substituting (1) and (22) into (41) with the aid of ([43], Equation (3.194.3)), \(\mu_{\gamma_{SCC}}(q)\) can be obtained as:  
\[\begin{array}{rl}\mu_{\gamma_{sec}}(q)&=\sum\limits_{k=0}^{\infty}\sum\limits_{l=0 }^{\infty}\frac{\rho_{N}^{k}p_{G}^{l}\Gamma(\Lambda)B(m+k+q,n+l-q)}{k\text{H}(m )\Gamma(n)B(l+n,k+m)}\\ &\times\frac{\theta(\gamma_{\tau}\eta)^{2}}{\eta_{1}^{2}(m+k)}E_{1}[\lambda,m+k ;m+k+1;-\eta_{2}\gamma_{\tau}]\\ &+\frac{\gamma_{\tau}^{-n}\Lambda_{2}^{-n}}{B(m,n)(n-q)}E_{1}[m+n,n-q;n-q+1;-1/ \Lambda_{2}\gamma_{\tau}].\end{array} \tag{43}\]For the i.n.i.d. case, the \(q\)th order moment of \(\gamma_{SCC}\) in (38) can be simplified as:  
\[\mu_{\gamma_{sec}}(q)=P\sum_{i=1}^{2}\left(\int_{0}^{\infty}\gamma^{q}f_{\gamma_{ i}}(\gamma)d\gamma+F_{i}^{-1}\int_{\gamma_{\tau}}^{\infty}\gamma^{q}f_{\gamma_{ i}}(\gamma)d\gamma\right). \tag{44}\]  
By using (12) and ([43], Equation (3.194.2/3)) along with some mathematical manipulation, the closed-form expression of (44) can be calculated. Similarly, we also obtain the closed-form expression of the \(q\)th order moment of \(\gamma_{SCC}\) for the i.i.d. case.