# Secure Beamforming Design for UAV-Empowered Integrated Sensing and Communication
## 2 System Model and Problem Formulation
### Sensing Model  
The entire sensing process of the UAV is divided into two stages. In the first stage, the UAV emits an omnidirectional scanning signal that can detect multiple users, including both legitimate and illegitimate ones. The UAV distinguishes between legitimate users and eavesdroppers through three scenarios:  
1. Several legitimate users will actively transmit signals to the UAV, providing their identity and location information. The UAV authenticates this information, and if it passes, the users are confirmed as legitimate. Otherwise, they are considered eavesdroppers.
2. The UAV then sends an authentication request to the remaining users. Some users will respond to the authentication request, and the UAV will authenticate this information; if it passes, they are identified as legitimate users. Otherwise, they are considered eavesdroppers.
3. If a user neither actively transmits identity information nor responds to the UAV authentication request, the UAV considers them an eavesdropper.  
In the second stage, to ensure secure communication and further sense the eavesdroppers, the UAV employs beamforming technology. We formulate an optimization problem aimed at minimizing the maximum eavesdropping SINR while satisfying the communication threshold for legitimate users, the sensing threshold for the target, and the power budget. This approach is designed to maximize the system's security performance.  
However, with the method described above, we cannot distinguish between actual eavesdroppers and non-responding users, such as regular people who are neither system users nor eavesdroppers, as they do not interact with the system. This limitation can result in false alarms.  
In the second stage, the composite signals transmitted by the UAV are reflected by the sensing target. Analyzing these reflected signals enables the estimation of relevant target information. The power of the reflected signals can be determined by calculating the beam pattern gain from the UAV directed toward the target, combined with the path loss. Given the requirement for the UAV to sense the presence of the eavesdropper, we employ the beam pattern gain in the eavesdropper's direction as the metric for sensing performance [22]. In this paper, the communication and sensing signals transmitted by the UAV are jointly utilized for the sensing function. Therefore, for any sensing angle \(\theta\big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\), the beam pattern gain \(\varsigma\big{(}\theta\big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\big{)}\) can be expressed as:  
\[\begin{split}&\varsigma\big{(}\theta\big{(}\mathbf{q},\mathbf{e}_{j} \big{)}\big{)}=\mathbb{E}\Big{(}\left|\mathbf{a}^{\text{H}}\big{(}\mathbf{q},\mathbf{e}_{j} \big{)}\mathbf{x}\right|^{2}\Big{)}\\ &=\mathbf{a}^{\text{H}}\big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\left(\sum \limits_{k=1}^{K}\mathbf{\omega}_{k}^{c}\omega_{k}^{c}\,{}^{\text{H}}+\sum\limits_{ n=1}^{N}\mathbf{\omega}_{n}^{r}\mathbf{\omega}_{n}^{r}\,{}^{\text{H}}\right)\mathbf{a}\big{(}\mathbf{q}, \mathbf{e}_{j}\big{)}\end{split} \tag{10}\]  
The power of the signals reflected from the sensing target to the UAV, incorporating path loss, is described by (10) and the distance between the two. Detailed computational constraints are provided in constraint C2.