## 4 Numerical Simulations and Analysis
### Hybrid Precoding System
For the GSVD-Sparsity algorithm, in step 2, the fast selection algorithm was used to find the top \(N_{RF}^{t}\) largest elements (peak search) in the beamspace channel \(\mathbf{H}_{b}\), with a complexity of \(\mathcal{O}(N_{t}N_{r})\); in step 4, the least-squares solution required \(\mathcal{O}(N_{t}(N_{RF}^{t})^{2}+(N_{RF}^{t})^{3})\). Hence, the overall complexity of GSVD-Sparsity was \(\mathcal{O}(N_{t}N_{r}+N_{t}(N_{RF}^{t})^{2}+(N_{RF}^{t})^{3})\). In the exhaustive search algorithm, the analog precoder \(\mathbf{F}_{RF}\) was constructed by selecting \(N_{RF}^{t}\) columns from the dictionary matrix with \(N_{cl}N_{ray}\) candidates, resulting in a total of \(C_{N_{cl}N_{ray}}^{\text{N}_{RF}}\) possible combinations. The complexity of solving the least squares problem for each combination was \(\mathcal{O}(N_{t}(N_{RF}^{t})^{2}+(N_{RF}^{t})^{3})\), and thus, the overall complexity of the exhaustive search algorithm was \(C_{N_{cl}N_{ray}}^{\text{N}_{RF}}\mathcal{O}(N_{t}(N_{RF}^{t})^{2}+(N_{RF}^{t })^{3})\). Figure 10: Spectral efficiency of the legitimate user when using the GSVD-Sparsity algorithm in a hybrid precoding system. Clearly, the proposed GSVD-Sparsity algorithm demonstrates a significant advantage in time complexity. This is because, unlike the OMP algorithm which requires iterative updates to construct \(\mathbf{F}_{\text{RF}}\) and \(\mathbf{F}_{\text{BB}}\), GSVD-Sparsity uses a fast selection method to directly identify the top \(N_{RF}^{t}\) strongest beam paths in the beamspace channel, and uses the corresponding array response vectors to construct \(\mathbf{F}_{\text{RF}}\). This eliminates the need for complex iterative procedures, resulting in a considerable reduction in complexity. Moreover, selecting the strongest paths for transmission also aligns with the physical layer goal of improving transmission efficiency. Table 1 summarizes the computational complexity of the three algorithms. Figure 10(a),b show the constellation diagrams obtained by the legitimate user in a hybrid precoding system using the GSVD-Sparsity algorithm under \(\text{SNR}=0\) dB and \(\text{SNR}=15\) dB, respectively. Compared to Figures 4(c) and 4(c), it can be observed that the hybrid precoder obtained using the GSVD-Sparsity algorithm achieved a BER performance comparable to the optimal precoder \((\mathbf{F}_{opt},\mathbf{W}_{opt})\) in a fully digital system. To verify the robustness of the proposed GSVD-Sparsity algorithm under different channel conditions, Figure 12 presents the symbol error rate (SER) performance under both additive white Gaussian noise (AWGN) and multipath fading channels, with 16 QAM modulation employed at the legitimate user. The figure includes four curves, corresponding to GSVD-OMP with AWGN, GSVD-Sparsity with AWGN, GSVD-OMP with a fading channel, and GSVD-Sparsity with a fading channel. It can be observed that under AWGN channel conditions, the two curves on the left (i.e., GSVD-OMP and GSVD-Sparsity under AWGN) are almost overlapping, indicating that both methods achieved a similar SER performance when the channel was ideal. Under multipath fading conditions, the two curves on the right begin to diverge after 15 dB, showing that in worse channel conditions, GSVD-OMP \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Computational Complexity** \\ \hline GSVD-Sparsity & \(\mathcal{O}(N_{t}N_{r}+N_{t}{(N_{RF}^{t})}^{2}+{(N_{RF}^{t})}^{3})\) \\ \hline OMP & \(\mathcal{O}(N_{RF}^{t}N_{cl}N_{ray}N_{t}+N_{t}{(N_{RF}^{t})}^{3}+{(N_{RF}^{t})}^ {4})\) \\ \hline Exhaustive search & \(\mathrm{C}_{N_{cl}N_{ray}}^{N_{RF}^{t}}\mathcal{O}(N_{t}{(N_{RF}^{t})}^{2}+{(N_{ RF}^{t})}^{3})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Complexity comparison. Figure 11: The received constellation diagrams of the legitimate user employing the GSVD-Sparsity algorithm in a hybrid precoding system: (**a**) \(\text{SNR}=0\) dB; (**b**) \(\text{SNR}=15\) dB. achieved a slightly lower SER than GSVD-Sparsity. However, as the SNR increases, the SER for both methods drops rapidly to zero.