## 2 System Model  
### Channel Model  
Let \(\mathbf{h}_{l}\in\mathbb{C}^{N_{x}N_{y}\times 1}\) denote the channel vector between AP and sensor \(l\). Using the air-terrestrial 3D MIMO channel model in [15; 31], \(\mathbf{h}_{l}\) can be expressed as:  
\[\mathbf{h}_{l}=\mathbf{h}_{l}^{\text{LoS}}+\mathbf{h}_{l}^{\text{NLoS}}. \tag{1}\]  
In (1), \(\mathbf{h}_{l}^{\text{LoS}}\in\mathbb{C}^{N_{x}N_{y}\times 1}\) denotes the deterministic LoS channel component and \(\mathbf{h}_{l}^{\text{NLoS}}\in\mathbb{C}^{N_{x}N_{y}\times 1}\) is the random vector characterizing the non-LoS channel component, which are given by:  
\[ \begin{split}&\mathbf{h}_{l}^{\text{LoS}}=\sqrt{\frac{\beta_{l}K_{ l}}{K_{l}+1}}\mathbf{b}\left(\varphi_{l},\theta_{l}\right)\otimes\mathbf{a}\left( \varphi_{l},\theta_{l}\right),\\ &\mathbf{h}_{l}^{\text{NLoS}}=\sqrt{\frac{\beta_{l}}{K_{l}+1}} \int\limits_{\varphi_{l}-\Delta\varphi_{l}}^{\varphi_{l}+\Delta\varphi_{l}} \int\limits_{\theta_{l}-\Delta\theta_{l}}^{\theta_{l}+\Delta\theta_{l}}r_{l} \left(\varphi,\theta\right)\mathbf{b}\left(\varphi,\theta\right)\otimes \mathbf{a}\left(\varphi,\theta\right)\mathrm{d}\varphi\mathrm{d}\theta,\end{split}  \tag{2}\]  
with:  
\[ \begin{split}&\mathbf{a}\left(\varphi,\theta\right)=\left[1,\exp \left(j\frac{2\pi d_{y}}{\lambda}\cos\theta\cos\varphi\right),\cdots,\exp \left(j\frac{2\pi d_{y}\left(N_{y}-1\right)}{\lambda}\cos\theta\cos\varphi \right)\right]^{T},\\ &\mathbf{b}\left(\varphi,\theta\right)=\left[1,\exp\left(j\frac{ 2\pi d_{x}}{\lambda}\cos\theta\sin\varphi\right),\cdots,\exp\left(j\frac{2\pi d _{x}\left(N_{x}-1\right)}{\lambda}\cos\theta\sin\varphi\right)\right]^{T}, \end{split}  \tag{3}\]  
where \(\beta_{l}\) and \(K_{l}\) denote the large-scale fading and Rician factor, respectively. \(\varphi_{l}\in[-\pi/2,\pi/2]\) and \(\theta_{l}\in(0,\pi/2]\) denote the AoDs of sensor \(l\) in the horizontal and vertical directions, and \(\Delta\varphi_{l}\) and \(\Delta\theta_{l}\) denote the corresponding angular spreads. Since AP is much higher than ground obstacles, there is little scatter around it. In this case, the physical signal propagation paths are mainly caused by the scattering process in the vicinity of sensors. This will result in very narrow angular spreads \(\Delta\varphi_{l}\) and \(\Delta\theta_{l}\)[11]. \(d_{x}\) and \(d_{y}\) denote the antenna spacings along the \(x\) axis and \(y\) axis, respectively. \(\lambda\) denotes the carrier wavelength, and \(r_{l}(\theta,\varphi)\) denotes the complex response gain associated with the direction \((\theta,\varphi)\). We assume that \(r_{l}(\theta,\varphi)\) has zero-mean, i.e., \(\mathbb{E}[r_{l}(\theta,\varphi)]=0\), and the response gains for different angles are uncorrelated, i.e.,  
\[\mathbb{E}\left[r_{l}^{*}\left(\varphi,\theta\right)r_{l}\left(\varphi^{\prime},\theta^{\prime}\right)\right]=S_{l}\left(\varphi,\theta\right)\delta\left( \varphi-\varphi^{\prime}\right)\delta\left(\theta-\theta^{\prime}\right), \tag{4}\]  
where \(S_{l}\left(\varphi,\theta\right)\) represents the channel power angle spectrum (PAS), which characterizes the channel power distribution in the angular domain. Define \(\rho_{x}=\frac{d_{x}}{\lambda}\cos\theta\sin\varphi\) and \(\rho_{y}=\frac{d_{y}}{\lambda}\cos\theta\cos\varphi\). Using a similar procedure as that in [16], the channel correlation matrix can be expressed as:  
\[