## 4 End-to-End SNR Correlation Analysis
\begin{split} E[q_{k}q_{k,l_{k}}]&=\ \left[\sum_{n=l_{k}+1}^{N_{k}}\sum_{i_{1}=0}^{m_{k,1}-1}\cdots\sum_{i_{n-1}-1}^ {m_{k,n-1}-1}\sum_{i_{k+1}=0}^{m_{k,n+1}-1}\cdots\sum_{i_{N_{k}}=0}^{m_{k,N_{k} }-1}\frac{\rho_{k,n}^{-m_{k,n}}}{\Gamma(m_{k,n})}\left(\prod_{j=1,j\
eq n}^{N_{ k}}\frac{\rho_{k,j}^{-i_{j}}}{\Gamma(i_{j}+1)}\right)\left\{\Gamma\left(\omega_{k,n} \right)\right.\\ &\quad\times\left.\left(\sum_{j=1}^{N_{k}}\frac{1}{\rho_{k,j}} \right)^{-\omega_{k,n}}+\sum_{i=0}^{\sum_{j=1}^{l_{k}}j}\frac{\Gamma\left(1+ \sum_{j=1}^{N_{k}}i_{j}\right)}{\Gamma\left(i_{k}+1\right)}\left(\sum_{j=1}^{ l_{k}}\frac{1}{\rho_{k,j}}\right)^{-1+n-\sum_{j=1}^{l_{k}}i_{j}}\Gamma \left(\lambda_{k,n}\right)\left.\left(\sum_{j=1}^{N_{k}}\frac{1}{\rho_{k,j}} \right)^{-\lambda_{k,n}}\right\}\right]\\ &+\ \left[\sum_{i_{1}=0}^{m_{k,1}-1}\cdots\sum_{i_{N_{k}}=0}^{m_{k,N_{k} }-1}\left(\prod_{j=1}^{N_{k}}\frac{\rho_{k,j}^{-i_{j}}}{\Gamma(i_{j}+1)} \right)\left\{\left(2+\sum_{j=1}^{N_{k}}i_{j}\right)\left(\sum_{j=1}^{l_{k}} \frac{1}{\rho_{k,j}}\right)\left(\sum_{j=1}^{N_{k}}\frac{1}{\rho_{k,j}}\right) ^{-1}-\sum_{j=1}^{l_{k}}i_{j}\right\}\right.\\ &\quad\times\left.\Gamma\left(2+\sum_{j=1}^{N_{k}}i_{j}\right) \left(\sum_{j=1}^{N_{k}}\frac{1}{\rho_{k,j}}\right)^{-2-\sum_{j=1}^{l_{k}}i_{ j}}\right].\end{split}  \tag{12}\]  
Therefore, using \(E[q_{k}q_{k,l_{k}}]\) in Equation (12) and \(E[q_{k}]\), \(E[q_{k,l_{k}}]\), \(E[q_{k}^{2}]\) and \(E[q_{k,l_{k}}^{2}]\) given from Equation (6), a closed-form expression for the correlation coefficient \(\eta_{k,l_{k}}\) in Equation (5) is obtained. As mentioned in Section 3, it is assumed that the subset of hops for CSI feedback in path \(k\) is set to \(\{1,\cdots,l_{k}\}\) for the simplification of the mathematical expressions, but it can be easily extended to any subset of hops with size \(l_{k}\), denoted by \(\{\delta_{1},\cdots,\delta_{l_{k}}\}\), where \(\delta\) denotes the hop index. Then, the possible subsets \(\{\delta_{1},\cdots,\delta_{l_{k}}\}\) are respectively used to calculate \(\eta_{k,l_{k}}\) using its closed-form expression, and the best subset of hops with size \(l_{k}\) in path \(k\) is found to maximize \(\eta_{k,l_{k}}\), where it implies that \(q_{k,l_{k}}\) for the best subset provides the best similarity to \(q_{k}\). Thus, the PPS using only instantaneous CSI for the best subset of hops in each of paths can work most analogously to the BPS using global instantaneous CSI for every path. It is noted that although the closed-form expression for \(\eta_{k,l_{k}}\) may be complicated, only the Nakagami parameters \(m_{k,n}\)'s and the average SNRs \(\rho_{k,n}\)'s are required to calculate \(\eta_{k,l_{k}}\), where all the values of the parameters may slowly change over time. Furthermore, it is noted that the PPS based on the SNR correlation can be also employed in AF relaying systems since the end-to-end SNR of AF relaying is tightly upper-bounded by that of DF relaying [31]. For Rayleigh fading channels (i.e., \(m_{k,n}=1\) for all \(k\) and \(n\)), using Equations (6) and (12), the closed-form expression for \(\eta_{k,l_{k}}\) in Equation (5) can be simplified as follows:  
\[ \begin{split}\eta_{k,l_{k}}&=\ \left[\left(\sum_{j=l_{k}+1}^{N_{k}}\frac{1}{\rho_{k,j}} \right)\left\{2\left(\sum_{j=1}^{N_{k}}\frac{1}{\rho_{k,j}}\right)^{-3}+\left( \sum_{j=1}^{l_{k}}\frac{1}{\rho_{k,j}}\right)^{-1}\right.\left(\sum_{j=1}^{N_{ k}}\frac{1}{\rho_{k,j}}\right)^{-2}\right\}+2\left(\sum_{j=1}^{l_{k}}\frac{1}{\rho_{k,j}} \right)\left(\sum_{j=1}^{N_{k}}\frac{1}{\rho_{k,j}}\right)^{-3}\\ &\quad-\left(\sum_{j=1}^{l_{k}}\frac{1}{\rho_{k,j}}\right)^{-1} \left(\sum_{j=1}^{N_{k}}\frac{1}{\rho_{k,j}}\right)^{-1}\right]\left(\sum_{j=1}^{ l_{k}}\frac{1}{\rho_{k,j}}\right)\left(\sum_{j=1}^{N_{k}}\frac{1}{\rho_{k,j}} \right)\left(\sum_{j=1}^{l_{k}}\frac{1}{\rho_{k,j}}\right)\left(\sum_{j=1}^{N_{ k}}\frac{1}{\rho_{k,j}}\right)^{-1},\end{split}  \tag{13}\]  
where it is noted that the correlation coefficient \(\eta_{k,l_{k}}\) becomes larger as \(\rho_{k,j}\)'s for \(j=1,\cdots,l_{k}\) decrease.