# Secure Beamforming Design for UAV-Empowered Integrated Sensing and Communication
## 2 System Model and Problem Formulation
### Communication Model
\(\cos\theta(\mathbf{q},\mathbf{u}_{k})\) denotes the cosine of the Angle of Departure (AoD) for the user \(k\), given by:  
\[\cos\theta(\mathbf{q},\mathbf{u}_{k})=\frac{H}{\sqrt{H^{2}+\left\|\mathbf{q}-\mathbf{u}_{k} \right\|^{2}}} \tag{5}\]  
The signal received by the legitimate user \(k\) can be expressed as:  
\[y_{k} =\mathbf{h}_{k}^{\mathrm{H}}(\mathbf{q},\mathbf{u}_{k})\mathbf{x}+n_{k} \tag{6}\] \[=\underbrace{\mathbf{h}_{k}^{\mathrm{H}}(\mathbf{q},\mathbf{u}_{k})\mathbf{\omega }_{k}^{c}s_{k}^{c}}_{\text{desired $k$-th user signal}}+\underbrace{\sum\limits_{i=1,i \
eq k}^{K}\mathbf{h}_{k}^{\mathrm{H}}(\mathbf{q},\mathbf{u}_{k})\mathbf{\omega}_{i}^{c}s_{i} ^{c}}_{\text{interference signals from other users}}\] \[+\underbrace{\sum\limits_{n=1}^{N}\mathbf{h}_{k}^{\mathrm{H}}(\mathbf{q},\mathbf{u}_{k})\mathbf{\omega}_{n}^{r}s_{n}^{r}}_{\text{sensing interference signal}}+\underbrace{n_{k}}_{\text{AWGN}}\]  
where \(n_{k}\sim\mathcal{CN}\big{(}0,\sigma_{k}^{2}\big{)}\) represents the additive white Gaussian noise (AWGN) received by the user \(k\). From (6), it can be observed that the user \(k\) receives not only its desired communication signal but also the communication signals from other users, the sensing signals, and the additive noise. Therefore, the SINR for the user \(k\) can be expressed as:  
\[\text{SINR}_{k}=\frac{\left|\mathbf{h}_{k}^{\mathrm{H}}(\mathbf{q},\mathbf{u}_{k})\mathbf{ \omega}_{k}^{c}\right|^{2}}{\left|\sum\limits_{i=1,i\
eq k}^{K}\mathbf{h}_{k}^{ \mathrm{H}}(\mathbf{q},\mathbf{u}_{k})\mathbf{\omega}_{i}^{c}\right|^{2}+\left|\sum \limits_{n=1}^{N}\mathbf{h}_{k}^{\mathrm{H}}(\mathbf{q},\mathbf{u}_{k})\mathbf{\omega}_{n}^{r }\right|^{2}+\sigma_{k}^{2}} \tag{7}\]  
Analogous to the signal received by legitimate users, the expression for the signal intercepted by the \(j\)-th eavesdroppers is as follows:  
\[y_{e,j} =\sum\limits_{i=1}^{K}\mathbf{g}_{j}^{\mathrm{H}}(\mathbf{q},\mathbf{e}_{j}) \mathbf{\omega}_{i}^{c}s_{i}^{c}+\sum\limits_{n=1}^{N}\mathbf{g}_{j}^{\mathrm{H}}(\bm {q},\mathbf{e}_{j})\mathbf{\omega}_{n}^{r}s_{n}^{r}+z_{e} \tag{8}\] \[=\underbrace{\mathbf{g}_{j}^{\mathrm{H}}(\mathbf{q},\mathbf{e}_{j})\mathbf{\omega }_{k}^{c}s_{i}^{c}}_{\text{eavesdropped signal}}+\underbrace{\sum\limits_{i=1,i \
eq k}^{K}\mathbf{g}_{j}^{\mathrm{H}}(\mathbf{q},\mathbf{e}_{j})\mathbf{\omega}_{i}^{c}s_{ i}^{c}}_{\text{interference signals from other users}}\] \[+\underbrace{\sum\limits_{n=1}^{N}\mathbf{g}_{j}^{\mathrm{H}}(\mathbf{q},\mathbf{e}_{j})\mathbf{\omega}_{n}^{r}s_{n}^{r}}_{\text{sensing interference signal}}+\underbrace{z_{e}}_{\text{AWGN}}\]  
where \(\mathbf{g}_{j}(\mathbf{q},\mathbf{e}_{j})=\sqrt{\beta d^{-2}\big{(}\mathbf{q},\mathbf{e}_{j}\big{)} }\mathbf{a}(\mathbf{q},\mathbf{e}_{j})\in\mathbb{C}^{N\times 1}\) represents the eavesdropping channel, \(d\big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\) represents the distance between the UAV and the eavesdropper \(j\), and \(\mathbf{a}\big{(}\mathbf{q},\mathbf{e}_{j}\big{)}\) represents the beamforming steering vector of the UAV toward the eavesdropper \(j\), with an expression similar to (4). \(z_{e}\sim\mathcal{CN}(0,\sigma_{e}^{2})\) represents the AWGN received by the eavesdropper. Therefore, SINR for the eavesdropper \(j\) listening to the user \(k\) can be expressed as:  
\[\mathbf{a}(\mathbf{q},\mathbf{u}_{k})=\left[1,\ \exp\left(j\frac{2\pi d^{arr}}{\lambda}\cos \theta(\mathbf{q},\mathbf{u}_{k})\right),\ \ldots,\ \exp\left(j\frac{2\pi d^{arr}}{\lambda}(N-1)\cos \theta(\mathbf{q},\mathbf{u}_{k})\right)\right]^{\mathrm{T}} \tag{4}\]0  
To safeguard the communication security of the ISAC system, it is essential to minimize the maximum eavesdropping SINR while maintaining effective communication between the UAV and legitimate users.