## 4 End-to-End SNR Correlation Analysis  
Based on the limited CSI feedback, \(l_{k}\)'s for all paths are able to be decided, and in path \(k\), an appropriate subset with size \(l_{k}\) in the entire set of hops \(\{1,\cdots,N_{k}\}\) can be determined for local CSI to provide the similar end-to-end SNR to global CSI (i.e., CSI for \(N_{k}\) hops), which can enable the PPS using local CSI to be done like the BPS using global CSI. In this section, we analyze the correlation coefficient between \(q_{k}\) and \(q_{k,l_{k}}\) to find the best set of \(l_{k}\) hops for local CSI. The correlation coefficient represents the quantity of the similarity between \(q_{k}\) and \(q_{k,l_{k}}\), and as it becomes larger, \(q_{k}\) and \(q_{k,l_{k}}\) become statistically more similar. For path \(k\), the correlation coefficient between \(q_{k}\) and \(q_{k,l_{k}}\) is expressed as  
\[\eta_{k,l_{k}}=\frac{E[q_{k}q_{k,l_{k}}]-E[q_{k}]E[q_{k,l_{k}}]}{\sqrt{\{E[q_{k }^{2}]-(E[q_{k}])^{2}\}\{E[q_{k,l_{k}}^{2}]-(E[q_{k,l_{k}}])^{2}\}}}, \tag{5}\]  
where \(-1\leq\eta_{k,l_{k}}\leq 1\), and \(q_{k}\) and \(q_{k,l_{k}}\) are highly correlated when \(\eta_{k,l_{k}}\) approaches 1. Using Equation (4), the \(t\)-th moment of \(f_{q_{k,l_{k}}}(x)\) is yielded as  
\[ \begin{array}{rcl}E[q_{k,l_{k}}^{t}]&=&\int_{0}^{\infty}x^{t}f_{q_{k,l_{k}}}( x)dx\\ &=&t\sum_{i_{1}=0}^{m_{k,1}-1}\cdots\sum_{i_{l_{k}}=0}^{m_{k,l_{k}}-1}\left\{ \prod_{j=1}^{L}\frac{1}{\Gamma(i_{j}+1)}\left(\frac{1}{\rho_{k,j}}\right)^{i_{ j}}\right\}\times\Gamma\left(t+\sum_{j=1}^{l_{k}}i_{j}\right)\left(\sum_{j=1 }^{L_{k}}\frac{1}{\rho_{k,j}}\right)^{-t-\sum_{j=1}^{L_{k}}i_{j}}.\end{array}  \tag{6}\]By replacing \(l_{k}\) with \(N_{k}\) in Equation (6), \(E[q_{k}^{t}]\) is also obtained. Then, \(E[q_{k}q_{k,l_{k}}]\) is derived as  
\[ \begin{split} E[q_{k}q_{k,l_{k}}]&=\,\int_{0}^{ \infty}\cdots\int_{0}^{\infty}\left[\min_{n=1,\cdots,N_{k}}\{x_{n}\}\right] \left[\min_{j=1,\cdots,l_{k}}\{x_{j}\}\right]f_{\gamma_{k,1}}(x_{1})\cdots f_ {\gamma_{k,N_{k}}}(x_{N_{k}})\\ &\quad\times dx_{1}\cdots dx_{N_{k}}\\ &=\,\int_{0}^{\infty}\cdots\int_{0}^{\infty}\left[\min_{n=l_{k}+1,\cdots,N_{k}}\{x_{n},y\}\right]yf_{\gamma_{k,l_{k}+1}}(x_{l_{k}+1})\cdots f_ {\gamma_{k,N_{k}}}(x_{N_{k}})f_{q_{k,l_{k}}}(y)\\ &\quad\times dx_{l_{k}+1}\cdots dx_{N_{k}}dy\\ &=\,\sum_{n=l_{k}+1}^{N_{k}}\int_{0}^{\infty}\int_{x_{n}}^{ \infty}\cdots\int_{x_{n}}^{\infty}x_{n}yf_{q_{k,l_{k}}}(y)f_{\gamma_{k,l_{k}+ 1}}(x_{l_{k}+1})\cdots f_{\gamma_{k,N_{k}}}(x_{N_{k}})\\ &\quad\times dydx_{l_{k}+1}\cdots dx_{n-1}dx_{n+1}\cdots dx_{N_{ k}}dx_{n}\\ &\quad+\int_{0}^{\infty}\int_{y}^{\infty}\cdots\int_{y}^{\infty}y^{2}f_{ \gamma_{k,l_{k}+1}}(x_{l_{k}+1})\cdots f_{\gamma_{k,N_{k}}}(x_{N_{k}})f_{q_{k, l_{k}}}(y)\\ &\quad\times dx_{l_{k}+1}\cdots dx_{N_{k}}dy.\end{split}  \tag{7}\]  
To derive the first part of Equation (7), using Equations (1) and (4), we first have  
\[ \begin{split} T_{1}&=\,\int_{x_{n}}^{\infty}yf_{q_{k,l_{k}}}(y)dy\\ &=\,\sum_{i_{1}=0}^{m_{k,1}-1}\cdots\sum_{i_{l_{k}}=0}^{m_{k,l_{ k}}-1}\left(\prod_{j=1}^{l_{k}}\frac{\rho_{k,j}^{-i_{j}}}{\Gamma(i_{j}+1)} \right)\\ &\quad\times\left\{e^{-\sum_{j=1}^{l_{k}}\frac{x_{n}}{\rho_{k,j} }}\sum_{u=0}^{1+\sum_{j=1}^{l_{k}}i_{j}}\frac{\Gamma\left(2+\sum_{j=1}^{l_{k}} i_{j}\right)}{\Gamma(u+1)}\left(\sum_{j=1}^{l_{k}}\frac{1}{\rho_{k,j}} \right)^{-1+u-\sum_{j=1}^{l_{k}}i_{j}}x_{n}^{u}\\ &\quad-\left(\sum_{j=1}^{l_{k}}i_{j}\right)e^{-\sum_{j=1}^{l_{k}} \frac{x_{n}}{\rho_{k,j}}}\sum_{u=0}^{\sum_{j=1}^{l_{k}}i_{j}}\frac{\Gamma \left(1+\sum_{j=1}^{l_{k}}i_{j}\right)}{\Gamma(u+1)}\times\left(\sum_{j=1}^{l _{k}}\frac{1}{\rho_{k,j}}\right)^{-1+u-\sum_{j=1}^{l_{k}}i_{j}}x_{n}^{u} \right\},\end{split}  \tag{8}\]  
\[