## 2 System Model and Problem Formulation
This target can be achieved by converting the composite channel \(\mathbf{H}_{comp2}=\mathbf{W}_{BB,k}^{H}\mathbf{W}_{RF}^{H}\mathbf{G}_{k} \ \mathbf{F}_{2k}\mathbf{G}_{BB2,k}\in\mathbb{C}^{N_{b}\times N_{s}}\) into an equivalent parallel SISO sub-channels, and it facilitates in achieving SNR maximization at the destination. Therefore, the final received signal at the destination corresponding to the \(n\)th data stream \(s_{n,k}\) in \(\mathbf{s}_{k}\) (1) can be expressed as  
\[N_{s}\leq min\Big{(}N_{t}^{RF},\ N_{rel}^{RF},\ N_{r}^{RF}\Big{)}\ll min(N_{t}, \ N_{rel},\ N_{r}). \tag{2}\]1where \(\rho_{n,k}\triangleq\|\mathbf{W}_{BB,k}^{H}(:,n)\mathbf{W}_{RF}^{H}\mathbf{G}_{k}\;\mathbf{F}_{2k} \mathbf{G}_{BB2,k}(:,n)\|\) and \(v_{2,k}\sim\mathcal{CN}\big{(}0,\,\sigma_{d}^{2}\big{)}\). Using (8), (11) can also be written as  
\[N_{s}\leq min\Big{(}N_{t}^{RF},\ N_{rel}^{RF},\ N_{r}^{RF}\Big{)}\ll min(N_{t}, \ N_{rel},\ N_{r}). \tag{2}\]2  
Therefore, the end-to-end SNR corresponding to \(N_{s}\) data streams through the \(k\)th relay node can be expressed as  
\[N_{s}\leq min\Big{(}N_{t}^{RF},\ N_{rel}^{RF},\ N_{r}^{RF}\Big{)}\ll min(N_{t}, \ N_{rel},\ N_{r}). \tag{2}\]3  
where \(\beta_{1k}=\frac{\|G_{BB1,k}^{H}\mathbf{H}_{k}\;\mathbf{V}_{RF}\mathbf{V}_{BB,k}\|_{F}^{2} }{\sigma_{r}^{2}}\) and \(\beta_{2k}=\frac{\|W_{BB,k}^{H}\mathbf{W}_{RF}^{H}\mathbf{G}_{k}\;\mathbf{F}_{2k}\mathbf{G}_{ BB2,k}\|_{F}^{2}}{\sigma_{r}^{2}}\) are the SNRs at the \(k\)th relay node and destination, respectively. As \(\beta_{1k},\,\beta_{2k}>0\), the overall SNR from the source to the destination achieves maximum level when both \(\beta_{1k}\) and \(\beta_{2k}\) attain their maximum values. This target can be achieved by decomposing end-to-end SNR (13) into two independent sub-problems. One sub-problem tries to maximize \(\beta_{1k}\) and the other attempts to maximize \(\beta_{2k}\). Let \(\mathbf{V}_{k}=V_{RF}\mathbf{V}_{BB,k}\in\mathbb{C}^{N_{t}\times N_{s}}\) and \(\mathbf{W}_{k}=\mathbf{W}_{RF}\mathbf{W}_{BB,k}\in\mathbb{C}^{N_{r}\times N_{t}}\) be the hybrid beamforming matrices at the source and destination, respectively. Therefore, the compact representation of the received signal at the destination can be written as  
\[N_{s}\leq min\Big{(}N_{t}^{RF},\ N_{rel}^{RF},\ N_{r}^{RF}\Big{)}\ll min(N_{t}, \ N_{rel},\ N_{r}). \tag{2}\]4  
The power constraint at the \(k\)th relay node is given as  
\[N_{s}\leq min\Big{(}N_{t}^{RF},\ N_{rel}^{RF},\ N_{r}^{RF}\Big{)}\ll min(N_{t}, \ N_{rel},\ N_{r}). \tag{2}\]5  
where \(P_{r}\) is the transmit power at the \(k\)th relay station. Using (14), the achievable rate, when transmitted data streams pass through the selected \(k\)th relay node, can be obtained by the following relation  
\[N_{s}\leq min\Big{(}N_{t}^{RF},\ N_{rel}^{RF},\ N_{r}^{RF}\Big{)}\ll min(N_{t}, \ N_{rel},\ N_{r}). \tag{2}\]6  
where \(\mathbf{R}_{n}=\sigma_{r}^{2}\big{(}\mathbf{W}_{k}{}^{H}\mathbf{G}_{k}\mathbf{F}_{k}\big{)} \big{(}\mathbf{W}_{k}{}^{H}\mathbf{G}_{k}\mathbf{F}_{k}\big{)}^{H}+\sigma_{d}^{2}\mathbf{W}_{k }{}^{H}\mathbf{W}_{k}\) is the equivalent noise covariance matrix. The optimization problem to maximize the achievable rate through the \(k\)th relay node can be formulated as  
\[N_{s}\leq min\Big{(}N_{t}^{RF},\ N_{rel}^{RF},\ N_{r}^{RF}\Big{)}\ll min(N_{t}, \ N_{rel},\ N_{r}). \tag{2}\]7 \[N_{s}\leq min\Big{(}N_{t}^{RF},\ N_{rel}^{RF},\ N_{r}^{RF}\Big{)}\ll min(N_{t}, \ N_{rel},\ N_{r}). \tag{2}\]8 \[N_{s}\leq min\Big{(}N_{t}^{RF},\ N_{rel}^{RF},\ N_{r}^{RF}\Big{)}\ll min(N_{t}, \ N_{rel},\ N_{r}). \tag{2}\]9  
where \(P_{s}\) denotes the transmit power at the source. Generally, the interest lies in joint optimization of the problem in (17) for deriving the required analog RF and digital baseband precoders/combiners. In the presence of several element-wise constant amplitude constraints, joint optimization of (17) is quite challenging for finding the global optimal solution.