## 4 Numerical Simulations and Analysis
### Full Digital System  
First, we compare the performance of the GSVD-based scheme and the SVD-based scheme in a fully digital system. Figure 5a,b show the constellation diagrams of the legitimate user and the eavesdropper, respectively, formed by \((\vec{\mathbf{F}},\vec{\mathbf{W}})\) when \(\text{SNR}=0\) dB, where \((\vec{\mathbf{F}},\vec{\mathbf{W}})\) is obtained through SVD (Equation (18)). Figure 5c,d depict the constellation diagrams formed by \((\mathbf{F}_{opt},\mathbf{W}_{opt})\), which is obtained through GSVD (Equation (14)). The signal modulation scheme is 16 QAM. Figure 6 illustrates the scenario where \(\text{SNR}=15\) dB. When the GSVD-based scheme is used to design the precoder, a comparison between Figures 5c and 6c shows that the constellation points for the legitimate user cluster together when the SNR is low, making proper demodulation impossible. However, when the SNR is high, proper demodulation becomes achievable. From Figures 5d and 6d, it can be observed that all received signals fall at the \((0,0)\) point, rendering the eavesdropper completely unable to obtain a valid constellation diagram. Therefore, the GSVD-based scheme effectively prevents the eavesdropper from receiving useful signals. This is achieved at the cost of reducing the received SNR for legitimate users, ensuring that the eavesdropper is completely unable to demodulate correctly. Figure 5: Received constellation diagrams in a fully digital system when \(\text{SNR}=0\) dB: (**a**) constellation diagram of the legitimate user using the SVD-based scheme; (**b**) constellation diagram of the eavesdropper using the SVD-based scheme; (**c**) constellation diagram of the legitimate user using the GSVD-based scheme; (**d**) constellation diagram of the eavesdropper using the GSVD-based scheme. When using an SVD-based scheme for precoder design, a comparison of Figure 6a,b shows that, although the eavesdropper's received SNR is lower than that of the legitimate user, it does not affect the eavesdropper's ability to correctly demodulate. Comparing Figures 5b and 6b, it is evident that even at a low SNR level, the eavesdropper can still demodulate the constellation diagram, with only a slight increase in the bit error rate (BER). Therefore, the SVD-based scheme cannot prevent the eavesdropper from receiving useful signals and fails to meet the requirements for secure transmission. Figures 5a and 6a illustrate the constellation diagrams received by the legitimate user, where normal demodulation is achievable. Figure 7 illustrates the spectral efficiency variations for both the legitimate user and the eavesdropper under the SVD-based scheme and GSVD-based scheme as the SNR changes. The optimal curve in the figure represents the channel capacity of the system, serving as a benchmark. The solid line with \(*\) indicates the spectral efficiency of the eavesdropper under the GSVD-based scheme, with most values being 0 or close to 0. This is consistent with the constellation diagram results shown in Figures 5d and 6d, where the eavesdropper is unable to correctly demodulate the received signals. Compared to the SVD-based scheme, although the GSVD-based scheme sacrifices part of the transmission efficiency, it ensures transmission security. On the other hand, the SVD-based scheme, while maintaining a higher user rate, completely fails to prevent eavesdropping. Figure 6: Received constellation diagrams in a fully digital system when \(\mathrm{SNR}=15\) dB: (**a**) constellation diagram of the legitimate user using the SVD-based scheme; (**b**) constellation diagram of the eavesdropper using the SVD-based scheme; (**c**) constellation diagram of the legitimate user using the GSVD-based scheme; (**d**) constellation diagram of the eavesdropper using the GSVD-based scheme. Compared to the SVD-based scheme, the GSVD-based scheme incurs a certain spectral efficiency loss.