## 2 System Model
The elements of the overall channel matrix for each user are  
\[\mathbf{y}=\left(\mathbf{G}^{H}\mathbf{\Phi}^{H}\mathbf{H}^{H}+\mathbf{H}_{d}^{H}\right)\mathbf{\
abla}+ \mathbf{\eta}, \tag{1}\]1  
where  
\[\mathbf{y}=\left(\mathbf{G}^{H}\mathbf{\Phi}^{H}\mathbf{H}^{H}+\mathbf{H}_{d}^{H}\right)\mathbf{\
abla}+ \mathbf{\eta}, \tag{1}\]2  
and \(\mathbf{h}_{(d),k}=\big{[}h_{(d),k1},h_{(d),k2},\dots h_{(d),kM}\big{]}\) are the lines of the matrix \(\mathbf{H}_{d}\). The norm of the overall channel for each user can be written as  
\[\mathbf{y}=\left(\mathbf{G}^{H}\mathbf{\Phi}^{H}\mathbf{H}^{H}+\mathbf{H}_{d}^{H}\right)\mathbf{\
abla}+ \mathbf{\eta}, \tag{1}\]3  
and the SINR at the user antenna \(k\) is  
\[\mathbf{y}=\left(\mathbf{G}^{H}\mathbf{\Phi}^{H}\mathbf{H}^{H}+\mathbf{H}_{d}^{H}\right)\mathbf{\
abla}+ \mathbf{\eta}, \tag{1}\]4  
where the terms  
\[\mathbf{y}=\left(\mathbf{G}^{H}\mathbf{\Phi}^{H}\mathbf{H}^{H}+\mathbf{H}_{d}^{H}\right)\mathbf{\
abla}+ \mathbf{\eta}, \tag{1}\]5  
Therefore,  
\[\mathbf{y}=\left(\mathbf{G}^{H}\mathbf{\Phi}^{H}\mathbf{H}^{H}+\mathbf{H}_{d}^{H}\right)\mathbf{\
abla}+ \mathbf{\eta}, \tag{1}\]6  
The SINR can be rewritten as  
\[\mathbf{y}=\left(\mathbf{G}^{H}\mathbf{\Phi}^{H}\mathbf{H}^{H}+\mathbf{H}_{d}^{H}\right)\mathbf{\
abla}+ \mathbf{\eta}, \tag{1}\]7where \(Z_{i}={\mathbf{v_{i}}}^{2}\). Since the summation terms of Equation (10) are complex variables, then  
\[\mathbf{y}=\left(\mathbf{G}^{H}\mathbf{\Phi}^{H}\mathbf{H}^{H}+\mathbf{H}_{d}^{H}\right)\mathbf{\
abla}+ \mathbf{\eta}, \tag{1}\]8  
where  
\[\mathbf{y}=\left(\mathbf{G}^{H}\mathbf{\Phi}^{H}\mathbf{H}^{H}+\mathbf{H}_{d}^{H}\right)\mathbf{\
abla}+ \mathbf{\eta}, \tag{1}\]9  
Considering that the complex number \(r_{lk}=c_{lk}+js_{lk}\), thus  
\[\mathbf{Y}=\mathbf{H}\mathbf{\Phi}\mathbf{G}+\mathbf{H}_{d}, \tag{2}\]0  
in which  
\[\mathbf{Y}=\mathbf{H}\mathbf{\Phi}\mathbf{G}+\mathbf{H}_{d}, \tag{2}\]1  
and  
\[\mathbf{Y}=\mathbf{H}\mathbf{\Phi}\mathbf{G}+\mathbf{H}_{d}, \tag{2}\]2  
By substituting the imaginary and real parts of the complex scalar \(r_{lk}\), it follows that  
\[\mathbf{Y}=\mathbf{H}\mathbf{\Phi}\mathbf{G}+\mathbf{H}_{d}, \tag{2}\]3  
since the real and imaginary parts of the direct path \(h_{(d),kl}\) are Gaussian and uncorrelated. Let be \(C_{k}=\sum\limits_{l=1}^{M}c_{lk}^{2}\) and \(S_{k}=\sum\limits_{l=1}^{M}s_{lk}^{2}\); therefore, we can rewrite \({\mathbf{v_{k}}}^{2}\) as  
\[\mathbf{Y}=\mathbf{H}\mathbf{\Phi}\mathbf{G}+\mathbf{H}_{d}, \tag{2}\]4  
then  
\[\mathbf{Y}=\mathbf{H}\mathbf{\Phi}\mathbf{G}+\mathbf{H}_{d}, \tag{2}\]5  
we consider that  
\[\mathbf{Y}=\mathbf{H}\mathbf{\Phi}\mathbf{G}+\mathbf{H}_{d}, \tag{2}\]6  
Therefore, the SINR can be written in terms of these two coefficients as  
\[\mathbf{Y}=\mathbf{H}\mathbf{\Phi}\mathbf{G}+\mathbf{H}_{d}, \tag{2}\]7  
Since \(Z_{k}\) is the sum of squared Gaussian random variables, it is reasonable to assume that \(Z_{k}\) is Gamma distributed with parameters \(\alpha_{Z}\) and \(\beta_{Z}\). The term \(F_{k}\) is the sum of Gamma random variables, and it is supposed to be also Gamma distributed with parameters \(\alpha_{F}\) and \(\beta_{F}\). The SINR \(\gamma_{k}\) is the ratio between the variables \(Z_{k}\) and \(F_{k}\) and also is supposed to be Gamma distributed. This study proposes an approximation for the SINR distribution to obtain the error and secrecy outage probability. Although not an exact solution, the approximation is very accurate, even for small values of \(M\), \(N\), and \(K\). The parameters of the SINR distribution can be obtained by calculating the moments of \(\gamma_{k}\). Since \(\gamma_{k}\) is the ratio of \(Z_{k}\) and \(F_{k}\), therefore, the statistics of the numerator \(Z_{k}\) and the denominator \(F_{k}\) must be evaluated. The variables \(Z_{k}\) and \(F_{k}\) are correlated, and because of this, the covariance between them must be taken into account. According to [PERSON] et al.