## Appendix A Statistical Parameters of \(Z_{k}\)
### Variance of \(Z_{k}\)  
To compute the variance of \(Z_{k}\), we consider that  
\[\sigma_{Z_{k}}^{2}=var(Z_{k})=var\Bigg{(}\frac{P}{K}\sum_{l=1}^{M}|r_{lk}|^{2} \Bigg{)},\] (A25)  
in terms of the real and imaginary parts, the equation can be rewritten as  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}var\Bigg{(}\sum_{l=1}^{M}c_ {lk}^{2}+\sum_{l=1}^{M}s_{lk}^{2}\Bigg{)},\] (A26)  
since the variance of the sum of random variables can be computed in terms their covariances, then  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}[var(C_{k})+var(S_{k})+2 cov (C_{k},S_{k})],\] (A27)  
where  
\[var(C_{k})=var\Bigg{(}\sum_{l=1}^{M}c_{lk}^{2}\Bigg{)}=M(M-1)cov(c_{lk},c_{lk })+Mvar(c_{lk}),\] (A28)  
\[var(S_{k})=var\Bigg{(}\sum_{l=1}^{M}s_{lk}^{2}\Bigg{)}=M(M-1)cov(s_{lk},s_{lk })+Mvar(s_{lk}),\] (A29)  
and  
\[cov(C_{k},S_{k})=E[C_{k}S_{k}]-E[C_{k}]E[S_{k}].\] (A30)  
To compute the previous variances, the covariances \(cov(c_{lk},c_{ik})\) and \(cov(s_{lk},s_{ik})\) are needed; we consider that  
\[cov(c_{lk},c_{ik})=E[c_{lk}c_{ik}]-E[c_{lk}]E[c_{ik}]\to cov(c_{lk},c_{ik})=E [c_{lk}c_{ik}]-(E[c_{lk}])^{2}\] (A31)  
and  
\[cov(s_{lk},s_{ik})=E[s_{lk}s_{ik}]-E[s_{lk}]E[s_{ik}]\to cov(s_{lk},s_{ik})=E [s_{lk}s_{ik}]-(E[s_{lk}])^{2},\] (A32)  
since the expected value \(E[C_{k}S_{k}]\) can be obtained as  
\[E[C_{k}S_{k}]=E\Bigg{[}\left(\sum_{l=1}^{M}\Bigg{(}\sum_{p=1}^{ N}\sum_{q=1}^{N}\Big{|}h_{lp}\Big{|}\big{|}\big{|}\delta_{gk}\big{|}\cos\delta_{pq} +\Re\{h_{(d),d}\}\Bigg{)}^{2}\right)\times\ldots\\ \left(\sum_{l=1}^{M}\Bigg{(}\sum_{p=1}^{N}\sum_{q=1}^{N}\Big{|}h _{lp}\Big{|}\big{|}\big{|}\delta_{gk}\big{|}\sin\delta_{pq}+\Im\{h_{(d),d}\} \Bigg{)}^{2}\right]\Bigg{)},\] (A33)thus  
\[E[c_{lk}c_{ik}]=E\Bigg{[}\left(\sum_{p=1}^{N}\sum_{q=1}^{N}\Big{|}h_ {ip_{1}}\Big{|}\Big{|}g_{qk}\Big{|}\cos\delta_{pq}+\Re\{h_{(d),kl}\}\right)\times\ldots\\ \left(\sum_{p=1}^{N}\sum_{q=1}^{N}\Big{|}h_{ip_{1}}\Big{|}\Big{|}g_ {qk}\Big{|}\cos\delta_{pq}+\Re\{h_{(d),kl}\}\right)\Bigg{]}. \tag{100}\]  
To compute the expected value \(E[c_{lk}c_{ik}]\), it is needed to change the summation indexes as  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}var\Bigg{(}\sum_{l=1}^{M}c_ {lk}^{2}+\sum_{l=1}^{M}s_{lk}^{2}\Bigg{)},\]0  
After applying the product and considering the independent indexes, it follows that  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}var\Bigg{(}\sum_{l=1}^{M}c_ {lk}^{2}+\sum_{l=1}^{M}s_{lk}^{2}\Bigg{)},\]1  
by expanding the previous equations, then  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}var\Bigg{(}\sum_{l=1}^{M}c_ {lk}^{2}+\sum_{l=1}^{M}s_{lk}^{2}\Bigg{)},\]2  
Assuming that \(l\
eq i\)  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}var\Bigg{(}\sum_{l=1}^{M}c_ {lk}^{2}+\sum_{l=1}^{M}s_{lk}^{2}\Bigg{)},\]3which can be simplified to  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}var\Bigg{(}\sum_{l=1}^{M}c_ {lk}^{2}+\sum_{l=1}^{M}s_{lk}^{2}\Bigg{)},\]4  
The term \(E[s_{lk}s_{ik}]\), assuming that \(l\
eq i\), can be computed as follows:  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}var\Bigg{(}\sum_{l=1}^{M}c_ {lk}^{2}+\sum_{l=1}^{M}s_{lk}^{2}\Bigg{)},\]5  
by expanding the previous equation, then  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}var\Bigg{(}\sum_{l=1}^{M}c_ {lk}^{2}+\sum_{l=1}^{M}s_{lk}^{2}\Bigg{)},\]6  
and can be rewritten as  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}var\Bigg{(}\sum_{l=1}^{M}c_ {lk}^{2}+\sum_{l=1}^{M}s_{lk}^{2}\Bigg{)},\]7  
by substituting the statistical moments of the random variables, it follows that  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}var\Bigg{(}\sum_{l=1}^{M}c_ {lk}^{2}+\sum_{l=1}^{M}s_{lk}^{2}\Bigg{)},\]8  
that can be simplified to \[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}var\Bigg{(}\sum_{l=1}^{M}c_ {lk}^{2}+\sum_{l=1}^{M}s_{lk}^{2}\Bigg{)},\]9  
since \(\beta_{1}=0\), therefore  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}[var(C_{k})+var(S_{k})+2 cov (C_{k},S_{k})],\]0  
The covariance \(cov(c_{lk},c_{ik})\) can be computed as  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}[var(C_{k})+var(S_{k})+2 cov (C_{k},S_{k})],\]1  
then  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}[var(C_{k})+var(S_{k})+2 cov (C_{k},S_{k})],\]2  
and covariance \(cov(s_{lk},s_{ik})\) can be calculated as  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}[var(C_{k})+var(S_{k})+2 cov (C_{k},S_{k})],\]3  
The variances \(var(c_{lk})\) and \(var(s_{lk})\) can be computed as  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}[var(C_{k})+var(S_{k})+2 cov (C_{k},S_{k})],\]4  
that can be rewritten as  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}[var(C_{k})+var(S_{k})+2 cov (C_{k},S_{k})],\]5  
since  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}[var(C_{k})+var(S_{k})+2 cov (C_{k},S_{k})],\]6  
then  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}[var(C_{k})+var(S_{k})+2 cov (C_{k},S_{k})],\]7  
The variance \(var(C_{k})\) can be calculated by  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}[var(C_{k})+var(S_{k})+2 cov (C_{k},S_{k})],\]8The variance \(var(S_{k})\) can be defined as  
\[\sigma_{Z_{k}}^{2}=\bigg{(}\frac{P}{K}\bigg{)}^{2}[var(C_{k})+var(S_{k})+2 cov (C_{k},S_{k})],\]9  
and the correlation \(E[C_{k}S_{k}]\) is  
\[var(C_{k})=var\Bigg{(}\sum_{l=1}^{M}c_{lk}^{2}\Bigg{)}=M(M-1)cov(c_{lk},c_{lk })+Mvar(c_{lk}),\]0  
The term \(E[C_{k}S_{k}]\) can be calculated as  
\[var(C_{k})=var\Bigg{(}\sum_{l=1}^{M}c_{lk}^{2}\Bigg{)}=M(M-1)cov(c_{lk},c_{lk })+Mvar(c_{lk}),\]1  
the term \(E\Big{[}s_{l_{1}k}^{2}c_{l_{2}k}^{2}\Big{]}\) can be computed as  
\[var(C_{k})=var\Bigg{(}\sum_{l=1}^{M}c_{lk}^{2}\Bigg{)}=M(M-1)cov(c_{lk},c_{lk })+Mvar(c_{lk}),\]2  
then, \(\mu_{C_{k}S_{k}}\) will be given by (100).