## 3 Enhancing Security Through Joint Precoding and Artificial Noise Design
### Secrecy Rate Optimization Through Joint Precoding and Artificial Noise Design
\begin{split}\frac{\partial F(\kappa)}{\partial\kappa}=& \frac{P_{t}\text{tr}\big{(}\mathbf{H}(\Theta_{b})\mathbf{P}\mathbf{P}^{H} \mathbf{H}^{H}(\Theta_{b})\big{)}\text{tr}\Big{(}\mathbf{H}(\Theta_{\epsilon })\mathbf{V}^{0}\big{(}\mathbf{V}^{0}\big{)}^{H}\mathbf{H}^{H}(\Theta_{ \epsilon})\Big{)}}{\sigma_{b}^{2}}\\ &\times\frac{-\text{tr}\big{(}\mathbf{H}(\Theta_{\epsilon}) \mathbf{P}\mathbf{P}^{H}\mathbf{H}^{H}(\Theta_{\epsilon})\big{)}\kappa^{2}-2 \sigma_{\epsilon}^{2}\kappa+\sigma_{\epsilon}^{2}}{(N_{t}-N_{r})(\sigma_{ \epsilon}^{2}+\zeta^{2}\text{tr}(\mathbf{H}(\Theta_{\epsilon})\mathbf{P} \mathbf{P}^{H}\mathbf{H}^{H}(\Theta_{\epsilon})))}\end{split}  \tag{18}\]  
Solving the equation \(\frac{\partial F(\kappa)}{\partial\kappa}=0\) under the condition \(0\leq\kappa\leq 1\), we obtain the closed-form solution for \(\kappa\), which is  
\[\kappa_{opt}=\frac{-\sigma_{\epsilon}^{2}+\sqrt{\sigma_{\epsilon}^{4}+\sigma _{\epsilon}^{2}\text{tr}(\mathbf{H}(\Theta_{\epsilon})\mathbf{P}\mathbf{P}^{H }\mathbf{H}^{H}(\Theta_{\epsilon}))}}{\text{tr}(\mathbf{H}(\Theta_{\epsilon}) \mathbf{P}\mathbf{P}^{H}\mathbf{H}^{H}(\Theta_{\epsilon}))} \tag{19}\]  
It can be noticed that when \(\kappa<\kappa_{opt}\), we have \(\frac{\partial F(\kappa)}{\partial\kappa}>0\), and when \(\kappa>\kappa_{opt}\), we have \(\frac{\partial F(\kappa)}{\partial\kappa}<0\). Thus, \(\kappa_{opt}\) is the largest value in \(0\leq\kappa\leq 1\), and the closed-form solution for \(\zeta\) is  
\[\zeta=\sqrt{\frac{-\sigma_{\epsilon}^{2}+\sqrt{\sigma_{\epsilon}^{4}+\sigma_{ \epsilon}^{2}\text{tr}(\mathbf{H}(\Theta_{\epsilon})\mathbf{P}\mathbf{P}^{H} \mathbf{H}^{H}(\Theta_{\epsilon}))}}{\text{tr}(\mathbf{H}(\Theta_{\epsilon}) \mathbf{P}\mathbf{P}^{H}\mathbf{H}^{H}(\Theta_{\epsilon}))}} \tag{20}\]  
Finally, we obtain the algorithm for joint optimization of \(\mathbf{P}\) and \(\zeta\), as shown in Algorithm 2. When the UAV is moving, there may be inaccuracies in the acquisition of [PERSON]'s and [PERSON]'s channel state information. During secrecy rate optimization, the actual channel matrices used for [PERSON] and [PERSON] are \(\mathbf{H}\big{(}\Theta_{b}^{\prime}\big{)}\) and \(\mathbf{H}(\Theta_{\epsilon}^{\prime})\), respectively, where \(\Theta_{b}^{\prime}\) and \(\Theta_{\epsilon}^{\prime}\) represent the estimated arrival angle information of [PERSON] and [PERSON] by the UAV. Note that in the previous section, we only presented the system model for a ULA configuration with a Line-of-Sight (LOS) channel. This model is applicable to scenarios with high flight altitudes and open ground areas, covering many practical applications. Although we have provided only this specific channel model, the AN-GSDM framework is actually applicable to any channel model, as long as the channel matrix for both the legitimate receiver and the eavesdropper can be accurately (or approximately) obtained. The system optimization design can then proceed based on this information. In the following simulations, we will introduce various types of channel models, including Rician [27], Kronecker [28], Saleh-Valenzuela (SV) [29], and CDL channels [30]. The Rician channel model represents rich scattering channels, considering multipath interference (both LOS and Non-Line-of-Sight (NLOS) paths), as well as the Kronecker channel correlation. The SV and CDL channels provide a detailed description of multi-path structures. The physical meaning of the precoding matrix varies across different channel models. For example, in rich scattering models, traditional beamforming becomes ineffective (due to the lack of directionality), and the precoding matrix controls the spatial power distribution rather than the physical beam directions. In the SV model, precoding may optimize several angle clusters, forming multiple beams that cover the main directions of these clusters,or it may focus on interference suppression. In the LOS model, [PERSON]'s location and angle are directly related, and precoding controls the beams to degrade the signal quality in [PERSON]'s direction.