# Secure Beamforming Design for UAV-Empowered Integrated Sensing and Communication
## 2 System Model and Problem Formulation
### Communication Model  
We consider a scenario in which the UAV transmits information signals to legitimate users while simultaneously sending sensing signals to detect eavesdroppers. We employ two different signals for communication and sensing. Let \(s_{k}^{r}\) denote the desired communication signal sent by the UAV to the legitimate user \(k\), \(\mathbf{\omega}_{k}^{r}\in\mathbb{C}^{N\times 1}\) denote the corresponding communication beamforming vector, \(s_{n}^{r}\) denote the dedicated radar sensing signal sent by the UAV, \(\mathbf{\omega}_{n}^{r}\in\mathbb{C}^{N\times 1}\) and denote the corresponding sensing beamforming vector. The transmitted signal of the UAV is then given by [12; 19]:  
\[\mathbf{x}=\sum_{k=1}^{K}\mathbf{\omega}_{k}^{r}s_{k}^{c}+\sum_{n=1}^{N}\mathbf{\omega}_{n} ^{r}s_{n}^{r} \tag{1}\]  
Assuming, without loss of generality, that both the communication signal \(s_{k}^{c}\), \(\forall k\) and the sensing signal \(s_{n}^{r},\forall n\) are independent and identically distributed complex Gaussian random variables with zero mean and unit variance, i.e., \(s_{k}^{c}\sim\mathcal{CN}(0,1)\)[18]. Therefore, the average transmit power of the UAV is \(\mathbb{E}\left(\left\|\mathbf{x}\right\|^{2}\right)=\sum_{k=1}^{K}\left\|\mathbf{ \omega}_{k}^{c}\right\|^{2}+\sum_{n=1}^{N}\left\|\mathbf{\omega}_{n}^{r}\right\|^ {2}\). Assuming the maximum transmit power of the UAV is \(P_{\text{max}}\), the UAV transmit power constraint can be expressed as:  
\[\sum_{k=1}^{K}\left\|\mathbf{\omega}_{k}^{c}\right\|^{2}+\sum_{n=1}^{N}\left\|\mathbf{ \omega}_{n}^{r}\right\|^{2}\leq P_{\text{max}} \tag{2}\]  
We consider a three-dimensional (3D) Cartesian coordinate system, assuming the UAV flight altitude is \(H\) meters. The position of the UAV is denoted as \((x,y,H)\), \(\mathbf{q}=(x,y)\) representing its horizontal position. The position of the legitimate user \(k\) is fixed at \((x_{k},y_{k},0)\), \(\mathbf{u}_{k}=(x_{k},y_{k})\) representing its horizontal position, and the position of the \(j\)-th eavesdropper is fixed at \(\left(x_{e,j},y_{e,j},0\right)\), \(\mathbf{e}_{j}=\left(x_{e,j},y_{e,j}\right)\) representing its horizontal position. Due to the relatively high deployment altitude of the UAV, there exists a strong Line of Sight (LOS) link between the UAV and each user. Therefore, we model the channel as an LOS channel. The LOS communication channel from the UAV to the user \(k\) is given by:  
\[ \begin{array}{ll}\mathbf{h}_{k}(\mathbf{q},\mathbf{u}_{k})&=\sqrt{\beta d^{-2}(\mathbf{q}, \mathbf{u}_{k})}\mathbf{a}(\mathbf{q},\mathbf{u}_{k})\\ &=\sqrt{\frac{\beta}{H^{2}+\left\|\mathbf{q}-\mathbf{u}_{k}\right\|^{2}}}\mathbf{a}(\mathbf{q},\mathbf{u}_{k})\end{array}  \tag{3}\]  
where \(\mathbf{h}_{k}(\mathbf{q},\mathbf{u}_{k})\in\mathbb{C}^{N\times 1}\), \(\beta\) represents the signal power attenuation gain at the reference distance \(d_{0}=1\) m, \(d(\mathbf{q},\mathbf{u}_{k})=\sqrt{H^{2}+\left\|\mathbf{q}-\mathbf{u}_{k}\right\|^{2}}\) represents the distance between the UAV  
Figure 1: System model. and the user \(k\), and \(\mathbf{a}(\mathbf{q},\mathbf{u}_{k})\in\mathbb{C}^{N\times 1}\) represents the beamforming steering vector of the UAV toward the user \(k\), which can be expressed as:  
\[\mathbf{a}(\mathbf{q},\mathbf{u}_{k})=\left[1,\ \exp\left(j\frac{2\pi d^{arr}}{\lambda}\cos \theta(\mathbf{q},\mathbf{u}_{k})\right),\ \ldots,\ \exp\left(j\frac{2\pi d^{arr}}{\lambda}(N-1)\cos \theta(\mathbf{q},\mathbf{u}_{k})\right)\right]^{\mathrm{T}} \tag{4}\]  
where \(d^{arr}\) represents the distance between two adjacent antennas in the linear array, \(\lambda\) represents the wavelength of the carrier signal, typically \(d^{arr}=\lambda/2\).