## 4 Numerical Simulations and Analysis
### Full Digital System
Specifically, at \(\text{SNR}=0\) dB, the spectral efficiency loss for the legitimate user is approximately 5 bits/s/Hz, and within the SNR range of \([5,50]\), the loss increases to about 6.5 bits/s/Hz. However, in the GSVD-based scheme, the spectral efficiency of the eavesdropper is suppressed to zero, which demonstrates the security capability of the scheme. Figure 8 compares the secrecy rate performance of the GSVD-based secure beam precoding scheme with that of AN and zero-forcing (ZF) secure precoding in a fully digital system. Both AN and ZF precoding are widely used techniques in physical layer security. The principle of AN is to embed noise into the transmitted signal in such a way that it does not affect the legitimate user's ability to decode the information, but degrades the signal quality at the eavesdropper [23]. In contrast, the ZF projects the transmitted signal into the null space of the eavesdropper's channel, thereby ensuring that the eavesdropper receives no useful information [24]. In this simulation, the number of antennas at the eavesdropper is set to 60, since for the ZF secure precoding to be feasible, the system must satisfy certain conditions (e.g., \(N_{t}>\text{rank}(\mathbf{H_{c}})\)) to ensure that a non-trivial null space exists at the eavesdropper's side, enabling the design of ZF precoding. As shown in the figure, the secrecy rate achieved by the GSVD-based precoding is lower than that of the AN scheme. This is because the performance of GSVD is highly dependent on the structural differences between the legitimate user's channel and the eavesdropping channel, such as rank and singular value disparities. When the differ  
Figure 8: Secrecy rate comparison among GSVD-based secure beam precoding, AN, and ZF secure precoding. Figure 7: The spectral efficiency of both the legitimate user and the eavesdropper under the SVD-based scheme and the GSVD-based scheme in a fully digital system. ences between the two channels are not significant, GSVD is less effective in providing strong secrecy performance. In contrast, even when the legitimate user's channel and the eavesdropping channel are similar, the AN can still exploit spatial degrees of freedom to enhance secrecy, thus exhibiting greater robustness. It is worth noting that, unlike the AN scheme, the GSVD-based precoding does not allocate any transmit power to artificial noise; instead, the entire transmit power is dedicated to the information signal. Furthermore, to ensure a fair comparison, equal power allocation is applied to the initial data streams in the GSVD-based precoding. When power is properly allocated among the data streams, the GSVD-based precoding is capable of achieving the secrecy capacity [21]. The figure also shows that the GSVD-based precoding outperforms the ZF secure precoding. This is because ZF secure precoding requires the transmitted signal to be completely projected into the null space of the eavesdropping channel. When the eavesdropping channel has a large dimension or significantly overlaps with the legitimate user's channel, there may be insufficient degrees of freedom, making it impossible to construct a valid null space. In contrast, the GSVD-based approach jointly decomposes the legitimate user's channel and the eavesdropping channel and projects the signal onto subchannels that are strong for the legitimate user and weak for the eavesdropper. In this way, even when there is partial overlap between the legitimate user's channel and the eavesdropping channel, the GSVD-based method can still optimize the level of information leakage.