## Appendix A Parameters of \(\gamma_{D}\)
### Variance of \(\gamma_{D}\)  
Let \(Z_{i}=|h_{i}|^{2}\); then,  
\[\text{var}(\gamma_{D})=\text{var}(\|\mathbf{h}\|^{2})=\text{var}(\sum_{i=1}^ {M}Z_{i}). \tag{105}\]  
Therefore, the variance of the sum of the terms \(Z_{i}\) is given by  
\[\text{var}(\sum_{i=1}^{M}Z_{i})=\sum_{i=1}^{M}\text{var}(Z_{i})+\;2\sum_{1 \leq i<k\leq M}\text{cov}(Z_{i},Z_{k}), \tag{106}\]  
whose magnitudes are equally distributed. Therefore,  
\[\text{var}(\sum_{i=1}^{M}Z_{i})=M\;\text{var}(Z_{i})+M(M-1)\;\text{cov}(Z_{i },Z_{k}). \tag{107}\]  
The covariance can be obtained by  
\[\text{cov}(Z_{i},Z_{k})=\mathbb{E}[Z_{i}Z_{k}]-\mathbb{E}[Z_{i}]\mathbb{E}[Z_ {k}], \tag{108}\]  
where  
\[\mathbb{E}[Z_{i}Z_{k}]=\mathbb{E}[c_{i}^{2}c_{k}^{2}]+2\mathbb{E}[c_{i}^{2}s_ {k}^{2}]+\mathbb{E}[s_{i}^{2}s_{k}^{2}]. \tag{109}\]The expected value of the product of two different in-phase coefficients can be written as  
\[\mathbb{E}[c_{i}c_{k}]=\mathbb{E}[(\sum_{l=1}^{N}\left|h_{il}^{LD}\right|\left|h_{ l}^{SL}\right|\cos\theta_{il}+Re\{h_{i}^{SD}\})\times(\sum_{m=1}^{N}\left|h_{km}^{ LD}\right|\left|h_{m}^{SL}\right|\cos\theta_{km}+Re\{h_{k}^{SD}\})].\] (A35)  
By expanding the product,  
\[\mathbb{E}[c_{i}c_{k}]=\mathbb{E}[Re\{h_{i}^{SD}\}Re\{h_{k}^{SD}\}]+\sum_{l=1} ^{N}\sum_{m=1}^{N}\mathbb{E}[\left|h_{il}^{LD}\right|\left|h_{km}^{LD}\right| \left|h_{l}^{SL}\right|\left|h_{m}^{SL}\right|\cos\theta_{il}\cos\theta_{km}],\] (A36)  
where the independent terms can be separated as  
\[\mathbb{E}[\left|h_{il}^{LD}\right|\left|h_{km}^{LD}\right|\left|h_{l}^{SL} \right|\left|h_{m}^{SL}\right|\cos\theta_{il}\cos\theta_{km}]=\mathbb{E}[ \left|h_{km}^{LD}\right|^{2}]\mathbb{E}[\left|h_{m}^{SL}\right|^{2}]\mathbb{E }[\cos\theta_{km}]^{2}=\mu_{LD}^{2}\mu_{SL}^{2}\alpha_{1}^{2}\forall l\
eq m,\] (A37)  
and  
\[\mathbb{E}[\left|h_{il}^{LD}\right|\left|h_{km}^{LD}\right|\left|h_{l}^{SL} \right|\left|h_{m}^{SL}\right|\cos\theta_{il}\cos\theta_{km}]=\mathbb{E}[ \left|h_{km}^{LD}\right|^{2}]\mathbb{E}[\left|h_{m}^{SL}\right|^{2}]\mathbb{E }[\cos\theta_{km}]=\mu_{LD}^{2}(\sigma_{SL}^{2}+\mu_{SL}^{2})\alpha_{1}^{2} \forall i\
eq k,l=m,\] (A38)  
where the term \(\mathbb{E}[\left|h_{m}^{SL}\right|^{2}]=\sigma_{SL}^{2}+\mu_{SL}^{2}\) and the variance of the Nakagami-\(m\) distributed term is  
\[\sigma_{SL}^{2}=\Omega_{SL}(1-\frac{1}{m_{SL}}(\frac{\Gamma(m_{SL}+0.5)}{ \Gamma(m_{SL})})^{2}).\] (A39)  
For \(i=k\),  
\[\text{var}(\sum_{i=1}^{M}Z_{i})=\sum_{i=1}^{M}\text{var}(Z_{i})+\;2\sum_{1 \leq i<k\leq M}\text{cov}(Z_{i},Z_{k}), \tag{106}\]0 (A40)  
where  
\[\text{var}(\sum_{i=1}^{M}Z_{i})=\sum_{i=1}^{M}\text{var}(Z_{i})+\;2\sum_{1 \leq i<k\leq M}\text{cov}(Z_{i},Z_{k}), \tag{106}\]1 (A41)  
To obtain the variance according to (A34), the term \(\mathbb{E}[\left.c_{i}^{2}c_{k}^{2}\right|\) needs to be computed. The terms are approximately correlated Gaussian random variables by the central limit theorem (CLT), for large values of \(N\), therefore  
\[\text{var}(\sum_{i=1}^{M}Z_{i})=\sum_{i=1}^{M}\text{var}(Z_{i})+\;2\sum_{1 \leq i<k\leq M}\text{cov}(Z_{i},Z_{k}), \tag{106}\]2 (A42)  
where \(f_{c_{i},c_{k}}(x,y)\) is the joint distribution of the two correlated Gaussian variables \(c_{i}\) and \(c_{k}\). The result of the integral is  
\[\text{var}(\sum_{i=1}^{M}Z_{i})=\sum_{i=1}^{M}\text{var}(Z_{i})+\;2\sum_{1 \leq i<k\leq M}\text{cov}(Z_{i},Z_{k}), \tag{106}\]3 (A43)  
where \(\mu_{c_{k}}=\mathbb{E}[c_{k}]\), \(\sigma_{c_{k}}^{2}=\text{var}(c_{k})\), and since \(\text{var}(c_{i})=\text{var}(c_{k})\) and \(\mathbb{E}[c_{i}]=\mathbb{E}[c_{k}]\), the correlation coefficient \(\rho_{c_{i},c_{k}}\) can be obtained as  
\[\text{var}(\sum_{i=1}^{M}Z_{i})=\sum_{i=1}^{M}\text{var}(Z_{i})+\;2\sum_{1 \leq i<k\leq M}\text{cov}(Z_{i},Z_{k}), \tag{106}\]4 (A44)  
where \(\mathbb{E}[c_{i}c_{k}]\) can be calculated by (A45) in the next page. \[\mathbb{E}[c_{i}c_{k}]=