## 5 Computer Simulations
### Spectral Efficiency Performance in Rich Scattering Environment  
Figure 3 plots the spectral efficiency performance of the proposed hybrid beamforming scheme in a rich scattering environment, which is generated according to Rayleigh fading model, where the elements of the channel matrix are independent complex Gaussian numbers with zero mean and unit variance. Simulation results are obtained by changing the number of antennas at communicating nodes and the number of transmitted data streams as well. It is worth highlighting that the number of RF chains is equal to the number of transmitted data streams, while conducting computer simulations. The condition on the number of RF chains indicates the worst-case scenario, as the number of RF chains cannot be less than the number of data streams according to the condition (2) given in Section 2. Figure 3a shows the spectral efficiency achieved by the proposed method when \(N_{t}=N_{rel}=N_{r}=36\), and \(N_{s}=2,~{}4,~{}6\). Figure 2: (**a**) Spectral efficiency achieved by different relay selection techniques when \(N_{t}=N_{rel}=100\), \(N_{r}=36\), \(N_{s}=2,~{}3,~{}4\) with UPA at all communicating nodes. Mm-Wave channel matrix is generated by assuming \(N_{cl}=4,~{}N_{ny}=10\) and \(10^{\circ}\) spread angle. The number of RF-chains is equal to the number of RF-chains is equal to the number of data streams, and 10 relay nodes are present between the source and the destination for the selection of the best relay; (**c**) spectral efficiency achieved by different relay selection techniques when \(N_{t}=N_{rel}=256,~{}N_{r}=64,~{}N_{s}=5,~{}6,~{}7\) with UPA at all communicating nodes. Mm-Wave channel matrix is generated by assuming \(N_{cl}=4,~{}N_{ny}=10\), and a \(10^{\circ}\) spread angle. It is assumed that the number of RF-chains is equal to the number of data streams, and 10 relay nodes are present between the source and the destination for the selection of the best relay. Figure 3: (**a**) Spectral efficiency vs. SNR when \(N_{t}=N_{r}=N_{rel}=36\), \(N_{s}=2\), \(4\), \(6\) with Rayleigh fading channel and the number of RF-chains is equal to the number of data streams \(N_{t}^{RF}=N_{r}^{RF}=N_{rel}^{RF}=N_{s}\). There are 10 relay nodes between the source and the destination for the selection of the best relay; (**b**) spectral efficiency vs. SNR when \(N_{t}=N_{rel}=64,\ N_{r}=36,\ N_{s}=4,\ 6\), \(8\) with Rayleigh fading channel and the number of RF-chains is equal to the number of data streams \(N_{t}^{RF}=N_{r}^{RF}=N_{rel}^{RF}=N_{s}\). There are 10 relay nodes between the source and the destination for the selection of the best relay; (**c**) spectral efficiency vs. SNR when \(N_{t}=N_{rel}=100,\ N_{r}=25,\ N_{s}=6,\ 8,\ 10\) with Rayleigh fading channel and the number of RF-chains is equal to the number of data streams \(N_{t}^{RF}=N_{r}^{RF}=N_{rel}^{RF}=N_{s}\). There are 10 relay nodes between the source and the destination for the selection of the best relay. The proposed algorithm approaches the upper bound defined by full-complexity precoding, which is clear from the obtained results in Figure 3a. In the next use case, the number of antennas deployed at communicating nodes increases along with the transmitted data streams, in comparison to the previous one, for the evaluation of spectral efficiency under different system configuration parameters. Therefore, the number of antennas and data streams are set as \(N_{t}=N_{rel}=64\), \(N_{r}=36\), and \(N_{s}=4,\ 6,\ 8\), respectively, to generate simulation results as shown in Figure 3b. It is obvious from the obtained results that the proposed technique achieves a performance close to fully digital beamforming. Figure 3c illustrates the spectral efficiency performance by further increasing the number of antennas and data streams, in comparison to both the previous cases, such that \(N_{t}=N_{rel}=100,\ N_{r}=25\), and \(N_{s}=6,\ 8,\ 10\). Again, there is a minor performance gap between the proposed approach and its fully digital counterpart as depicted in Figure 3c.