## 5 Computer Simulations  
In this section, computer simulations are conducted to evaluate the performance of the proposed algorithm for hybrid beamforming and relay selection mechanism in a single-user multi-relay MIMO system. In addition to the fully digital precoding, the performance of the proposed scheme is compared with conventional relay selection techniques (e.g., best-harmonic-mean (BHM) and SVD-based approaches). Sparse mm-Wave channel and rich scattering environment are considered by changing system parameters to generate simulation results. It is assumed that relay nodes are spatially distributed, and hence, there is no interference among the channels in the first hop. To capture the mathematical structure of mm-Wave propagation environment, a narrowband clustered channel is adopted based on the extended Saleh-Valenzuela model given in [23]. The mathematical model of mm-Wave channel with uniform planer (UPA) can be expressed as  
\[\mathbf{H}_{mm-Wave}=\sqrt{\frac{N_{t}N_{r}}{N_{cl}N_{ray}}}\sum_{i=0}^{N_{cl}-1} \sum_{i=1}^{N_{ray}}\alpha_{il}\wedge_{t}\big{(}\varphi_{il}^{t},\theta_{il}^{t }\big{)}\wedge_{r}(\varphi_{il}^{r},\theta_{il}^{r})\mathbf{a}_{r}(\varphi_{il}^{ r},\theta_{il}^{r})\ \ \mathbf{a}_{t}\big{(}\varphi_{il}^{t},\theta_{il}^{t}\big{)}^{H}, \tag{55}\]  
where \(\alpha_{il}\) represents complex gain of the \(l^{th}\) propagation path in the \(i\)th scattering cluster with \(\alpha_{il}\sim\mathcal{CN}\big{(}0,\sigma_{a,i}^{2}\big{)}\), and \(\big{(}\varphi_{il}^{t},\varphi_{il}^{t}\big{)}\) and \(\big{(}\theta_{il}^{r},\theta_{il}^{r}\big{)}\) are its angles of departure \(AoDs\) and angles of arrival \(AoAs\) in azimuth and elevation, respectively. The vectors \(\mathbf{a}_{t}\big{(}\varphi_{il}^{t},\theta_{il}^{t}\big{)}\) and \(\mathbf{a}_{r}\big{(}\varphi_{il}^{r},\theta_{il}^{r}\big{)}\) are the normalized transmit and receive array response vectors, and the functions \(\wedge_{t}\big{(}\varphi_{il}^{t},\theta_{il}^{t}\big{)}\) and \(\wedge_{r}\big{(}\varphi_{il}^{r},\theta_{il}^{r}\big{)}\) specify transmit and receive antenna gains, respectively. Moreover, \(N_{t},\ N_{r},\ N_{cl}\), and \(N_{ray}\) denote the number of transmit antennas, receive antennas, number of clusters, and the number of rays per cluster, respectively. From the perspective of conventional relay selection strategies in MIMO communication networks, the optimal relay can be selected using the BHM method. The mathematical formulation of this process is given as  
\[h_{BHM}=\max_{k\in\{1,\ldots,K\}}\Bigg{\{}\frac{1}{\Big{(}\left|\mathbf{H}_{k} \right|^{-2}+\left|\mathbf{G}_{k}\right|^{-2}\Big{)}}\Bigg{\}}, \tag{56}\]  
where \(\left|\mathbf{H}_{k}\right|\) denotes the channel gain from source to the \(k\)th relay node, and \(\left|\mathbf{G}_{k}\right|\) represents the channel gain from the \(k\)th relay node to the destination. Another method that selects the best relay, for spectral efficiency maximization, is based on the sum of singular values of the channel matrix between the source and relay node. Mathematical formulation of this scheme can be expressed as  
\[\mathbf{H}_{k}=\mathbf{U}_{1k}\mathcal{S}_{1k}\mathbf{V}_{1k}^{H}\] \[SR=\max_{k\in\{1,\ldots,K\}}\{tr(\mathcal{S}_{1k})\}\, \tag{57}\]  
where \(SR\) stands for the best selected relay that attains a maximum sum of singular values over all the channels from the source to relay nodes. It is worthwhile to mention here that relay selection, in multi-relay MIMO networks, is a non-convex problem regardless of beamforming strategy (fully digital/hybrid). From this perspective, there is no guarantee for the global optimal solution even with fully digital precoding. This feature can be exploited to obtain the full-complexity solution from the corresponding hybrid transceiver design. Following this approach, a hybrid beamforming is designed by maximizing end-to-end SNR with the associated essential constraints. There exists a high probability of obtaining near-optimal solution for a carefully derived hybrid precoding scheme under given constraints.