# Secure Beamforming Design for UAV-Empowered Integrated Sensing and Communication
## 2 System Model and Problem Formulation
### Problem Formulation  
From Equations (7), (9) and (10), it can be observed that the UAV communication and sensing beamforming (\(\omega_{k}^{c}\), \(\forall k\) and \(\omega_{n}^{r},\forall n\)) and its position (\(q\)) collectively influence the system's communication and sensing functionalities. Furthermore, the presence of an eavesdropper means these variables also significantly impact the system's security.  
Therefore, we propose a joint optimization of the UAV communication and sensing beamforming (\(\omega_{k}^{c},\forall k\) and \(\omega_{n}^{r},\forall n\)) and the UAV position (\(q\)), aiming to minimize the maximum eavesdropper SINR while adhering to the QoS requirements of legitimate users, meeting the sensing metric constraints for the eavesdropper, and complying with the UAV transmit power limitations. Consequently, the optimization problem is formulated as follows:  
\[\begin{split}\text{P1}:\min_{\omega_{k}^{c},\omega_{n}^{c},q} \max_{j,k}(\text{SINR}_{j,k}^{c}=\frac{\left|g_{j}^{\text{H}}(\mathbf{q},\mathbf{e}_{j} )\omega_{k}^{c}\right|^{2}}{\left|\sum\limits_{i=1,j\
eq k}^{K}g_{j}^{\text{ H}}(\mathbf{q},\mathbf{e}_{j})\omega_{i}^{c}\right|^{2}+\left|\sum\limits_{n=1}^{N} \delta_{j}^{\text{H}}(\mathbf{q},\mathbf{e}_{j})\omega_{n}^{r}\right|^{2}+\sigma_{e}^ {2}})\\ \text{s.t.}\ \ \text{C1}:\ \text{SINR}_{k}=\frac{\left|\mathbf{h}_{k}^{ \text{H}}(\mathbf{q},\mathbf{u}_{k})\omega_{k}^{c}\right|^{2}}{\left|\sum\limits_{i=1,j\
eq k}^{K}h_{k}^{\text{H}}(\mathbf{q},\mathbf{u}_{k})\omega_{i}^{c}\right|^{2}+ \left|\sum\limits_{n=1}^{N}h_{k}^{\text{H}}(\mathbf{q},\mathbf{u}_{k})\omega_{n}^{r} \right|^{2}+\sigma_{k}^{2}}\geq\Gamma_{k}^{\text{com}},\forall k\\ \text{C2}:\text{c}\big{(}\theta(\mathbf{q},\mathbf{e}_{j})\big{)}=\mathbf{a}^ {\text{H}}(\mathbf{q},\mathbf{e}_{j})\bigg{(}\sum\limits_{k=1}^{K}\omega_{k}^{c} \omega_{k}^{c}\,{}^{\text{H}}+\sum\limits_{n=1}^{N}\omega_{n}^{r}\omega_{n}^{ r}\,{}^{\text{H}}\bigg{)}\mathbf{a}\big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\geq d^{2} \big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\Gamma_{e}^{\text{sen}},\forall j\\ \text{C3}:\sum\
olimits_{k=1}^{K}\left\|\omega_{k}^{c}\right\|^{2 }+\sum\
olimits_{n=1}^{N}\left\|\omega_{n}^{r}\right\|^{2}\leq P_{\text{max}} \end{split} \tag{11}\]  
where \(\Gamma_{k}^{\text{com}}\) represents the minimum SINR threshold required for the legitimate user \(k\) to satisfy the communication quality of service, and \(\Gamma_{e}^{\text{sen}}\) represents the sensing threshold. The constraint C1 stipulates the QoS requirement of UAV and legitimate user \(k\) communication, constraint C2 indicates that the beam pattern gain of the UAV toward the eavesdropper \(j\) must be greater than the product of the sensing threshold \(\Gamma_{e}^{\text{sen}}\) and the square of the distance between the UAV and the eavesdropper \(j\), and constraint C3 ensures that the power of the signal transmitted by the UAV must not exceed the maximum transmit power \(P_{\text{max}}\).  
The objective function and constraint C1 in problem P1 are quadratic fractional form, and the steering vector \(\mathbf{a}\big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\) in constraint C2 is highly coupled with the variable \(\mathbf{q}\). Additionally, \(\omega_{k}^{c},\forall k\) and \(\omega_{n}^{r},\forall n\) are also highly coupled, rendering problem P1 a highly non-convex problem which is challenging to solve directly. To solve P1, we decompose it into two subproblems: beamforming optimization and UAV positioning optimization. These subproblems are solved iteratively to obtain a suboptimal solution.