## Appendix A Parameters of \(\gamma_{D}\)
### Variance of \(\gamma_{D}\)
[19] as  
\[\mathbb{E}[c_{i}^{2}s_{k}^{2}]=\mathbb{E}[(\sum\limits_{l=1}^{N}\left|h_{il}^ {LD}\right|\left|\left|h_{l}^{SL}\right|\cos\theta_{il})^{2}(\sum\limits_{m=1 }^{N}\left|h_{lm}^{LD}\right|\left|\left|h_{m}^{SL}\right|\sin\theta_{km})^{2} ]\ldots\] (A50) \[+\mathbb{E}[(\sum\limits_{l=1}^{N}\left|h_{il}^{LD}\right|\left| \left|h_{l}^{SL}\right|\cos\theta_{il})^{2}]\frac{\sigma_{SD}^{2}}{2}+\mathbb{ E}[(\sum\limits_{m=1}^{N}\left|h_{km}^{LD}\right|\left|\left|h_{m}^{SL}\right|\sin \theta_{km})^{2}]\frac{\sigma_{SD}^{2}}{2}+\frac{\sigma_{SD}^{4}}{4}\]  
The in-phase term is  
\[\mathbb{E}[(\sum\limits_{l=1}^{N}\left|h_{il}^{LD}\right|\left|h_{l}^{SL} \right|\cos\theta_{il})^{2}]=\mathbb{E}[\sum\limits_{l=1}^{N}\sum\limits_{l=1 }^{N}\left|h_{il}^{LD}\right|\left|\left|h_{l}^{SL}\right|\left|h_{m}^{SL} \right|\cos\theta_{il}\cos\theta_{il}]=\frac{N}{2}\chi_{LD}\chi_{SL}(1+\alpha _{2})+N(N-1)\mu_{LD}^{2}\mu_{SL}^{2}\alpha_{1}^{2}\] (A51)  
and the quadrature term is  
\[\mathbb{E}[(\sum\limits_{m=1}^{N}\left|h_{km}^{LD}\right|\left| \left|h_{m}^{SL}\right|\sin\theta_{km})^{2}]=\mathbb{E}[\sum\limits_{d=1}^{N} \sum\limits_{m=1}^{N}\left|h_{kd}^{LD}\right|\left|\left|h_{km}^{LD}\right| \left|\left|h_{d}^{SL}\right|\left|\left|h_{m}^{SL}\right|\sin\theta_{kd}\sin \theta_{km}\right|\right.=N(\chi_{LD}\chi_{SL}\frac{1}{2}(1-\alpha_{2}))\] (A52)  
where  
\[\chi_{LD}=\mathbb{E}[\left|h_{il}^{LD}\right|^{2}]=\Omega_{LD}\] (A53)  
\[\chi_{SL}=\mathbb{E}[\left|h_{d}^{SL}\right|^{2}]=\Omega_{SL}\] (A54)  
\[\xi_{LD}=\mathbb{E}[\left|h_{il}^{LD}\right|^{3}]=\frac{\Gamma(1.5+m_{LD})}{( \frac{m_{LD}}{\Omega_{LD}})\Gamma(m_{LD})}\] (A55)  
\[\mathbb{E}[(\sum\limits_{l=1}^{N}\left|h_{il}^{LD}\right|\left| \left|h_{l}^{SL}\right|\cos\theta_{il})^{2}(\sum\limits_{m=1}^{N}\left|h_{km} ^{LD}\right|\left|\left|h_{m}^{SL}\right|\sin\theta_{km})^{2}\right|=\]0 (A56)  
\[\mathbb{E}[(\sum\limits_{l=1}^{N}\left|h_{il}^{LD}\right|\left| \left|h_{l}^{SL}\right|\cos\theta_{il})^{2}(\sum\limits_{m=1}^{N}\left|h_{km} ^{LD}\right|\left|\left|h_{m}^{SL}\right|\sin\theta_{km})^{2}\right|=\]1 (A57)  
\[\mathbb{E}[(\sum\limits_{l=1}^{N}\left|h_{il}^{LD}\right|\left| \left|h_{l}^{SL}\right|\cos\theta_{il})^{2}(\sum\limits_{m=1}^{N}\left|h_{km} ^{LD}\right|\left|\left|h_{m}^{SL}\right|\sin\theta_{km})^{2}\right|=\]2 (A58)  
The term \(\mathbb{E}[s_{i}^{2}s_{k}^{2}]\) can be easily obtained because \(s_{i}\) and \(s_{k}\) are uncorrelated and zero mean, then \[\mathbb{E}[s_{i}^{2}s_{k}^{2}]= \begin{cases}(\mathbb{E}[s_{i}^{2}])^{2}&i\
eq k\\ \mathbb{E}[s_{k}^{4}]=3(\text{var}(s_{k}))^{2}&i=k\end{cases}  \tag{100}\]  
and, by substituting the variances; consequently, (101). \[\mathbb{E}[s_{i}^{2}s_{k}^{2}]= \begin{cases}\frac{1}{4}(\sigma_{SD}^{2}+N(1-a_ {2})(\sigma_{LD}^{2}\sigma_{SL}^{2}+\sigma_{LD}^{2}\mu_{SL}^{2}+\sigma_{SL}^{2} \mu_{LD}^{2}+\mu_{SL}^{2}\mu_{LD}^{2}))^{2}&i\
eq k\\ \\ \frac{3}{4}(\sigma_{SD}^{2}+N(1-a_{2})(\sigma_{LD}^{2}\sigma_{SL}^{2}+\sigma_{LD }^{2}\mu_{SL}^{2}+\sigma_{SL}^{2}\mu_{LD}^{2}+\mu_{SL}^{2}\mu_{LD}^{2}))^{2}&i=k \end{cases}  \tag{102}\]  
Finally, the term \(\mathbb{E}[Z_{i}Z_{k}]\) can be calculated by using the formulas of \(\mathbb{E}[c_{i}^{2}c_{k}^{2}]\), \(\mathbb{E}[c_{i}^{2}s_{k}^{2}]\) and \(\mathbb{E}[s_{i}^{2}s_{k}^{2}]\), resulting in (102). \[\mathbb{E}[Z_{i}Z_{k}]=