## 2 The 3D A2A MIMO Channel Model
### Specular and Diffuse Components
#### 2.2.1 Lengths and Angles of the Rays  
The parameters of the specular ray are calculated deterministically. However, the diffuse rays are modeled in a stochastic manner. Channel measurements reveal that the properties of the specular and diffuse components are correlated in both delay and angular domains [38]. In general, the diffuse components are distributed around the specular component. Additionally, it has been shown that scatterers are not uniformly distributed. The scatterer density gradually decreases as the distance from the center of the scattering region increases [25; 39]. Based on the above-mentioned facts, the Gaussian scatter density model (GSDM) is adopted, which has been validated for modeling scatterer distribution in diverse communication scenarios [40; 41]. The GSDM is expressed as  
\[p_{XY}(x,y)=\frac{\exp\left(-\frac{\left(x-x_{0}\right)^{2}}{2\sigma_{x}^{2}} -\frac{y^{2}}{2\sigma_{y}^{2}}\right)}{2\pi\sigma_{x}\sigma_{y}} \tag{28}\]where \(x_{0}=-h_{t}\tan(\theta_{0})\) is the distance from the specular point to \(O\) and \(\theta_{0}\) is the ZoD of the specular ray, which is calculated as  
\[\theta_{0}=\frac{\pi}{2}+\arctan\biggl{(}\frac{h_{t}+h_{r}}{D}\biggr{)}. \tag{29}\]  
Additionally, \(\sigma_{x}\) and \(\sigma_{y}\) control the spread of scatterers along the directions of the \(x\) and \(y\) axes, respectively. The ZoA of the specular ray is \(\hat{\sigma}_{0}=\theta_{0}\) according to Snell's Law. Note that (28) cannot provide any angular information. Using the Jacobian of the transformation, the joint probability density function (PDF) of the ZoDs and AoDs can be expressed as  
\[p_{\Theta\Phi}(\theta,\phi)=|J(x,y)|\cdot p_{XY}(x,y) \tag{30}\]  
where  
\[J(x,y)= \begin{vmatrix}\frac{\partial x}{\partial\theta}&\frac{\partial x}{ \partial\phi}\\ \frac{\partial y}{\partial\theta}&\frac{\partial y}{\partial\phi}\\ \end{vmatrix} =\frac{h_{t}^{2}\tan\theta}{\cos^{2}(\theta)} \tag{31}\]  
\[x=-h_{t}\tan\theta\cos\phi \tag{32}\]  
\[y=-h_{t}\tan\theta\sin\phi. \tag{33}\]  
By substituting (28) and (31) into (30) and after several mathematical manipulations, the joint PDF of the ZoDs and AoDs is written as  
\[p_{\Theta\Phi}(\theta,\phi)=\frac{-h_{t}^{2}\tan(\theta)}{2\pi\cos^{2}(\theta )\sigma_{x}\sigma_{y}}\cdot\exp\left(-\frac{[h_{t}\tan(\theta)\cos(\phi)+x_{0} ]^{2}}{2\sigma_{x}^{2}}-\frac{[h_{t}\tan(\theta)\sin(\phi)]^{2}}{2\sigma_{y}^ {2}}\right) \tag{34}\]  
where \(\theta\in[\pi/2,\pi]\), \(\phi\in[-\pi,\pi]\). Note that this differs from previous works, such as [3, 42, 43], in which the scatterers are evenly distributed with a clear boundary. Figure 3 presents the marginal PDFs of the ZoD and AoD of the diffuse rays. Three cases were used for testing. In Case I, the following parameters were chosen: \(h_{t}=h_{r}=25\) m and \(D=50\) m. In Case II, the height of the transmit UAV was reduced to 20 m, while the other parameters remained unchanged. In Case III, the horizontal distance between the Tx and Rx was increased, with the parameters set as \(h_{t}=h_{r}=25\) m and \(D=70\) m. In all three cases, the scatterer dispersion parameters were set as \(\sigma_{x}=8\) and \(\sigma_{y}=6\). Compared to Case I, a lower altitude of the Tx in Case II yields smaller ZoDs. Additionally, as the altitude of the Tx decreases, the scattering region moves closer to the Tx side, causing a larger spread of the AoDs. In Case III, the horizontal distance between the Tx and Rx increases. The larger distances from the Tx to the scatterers result in more concentrated distributions of the ZoDs and AoDs. For the single-bounce propagation, the arrival and departure angles are interrelated as follows:  
\[\varphi_{n}=\left\{ \begin{array}{cc}-\pi-\arcsin\Bigl{(}\frac{\sin(\phi_{n}) \tilde{d}_{1,n}}{\tilde{d}_{2,n}}\Bigr{)},&0>\phi_{n}\geqslant-\pi\\ \pi-\arcsin\Bigl{(}\frac{\sin(\phi_{n})\tilde{d}_{1,n}}{\tilde{d}_{2,n}} \Bigr{)},&\pi>\phi_{n}\geqslant 0\end{array} \right. \tag{35}\]  
\[\vartheta_{n}=\frac{\pi}{2}+\arctan\Bigl{(}\frac{h_{r}}{\tilde{d}_{2,n}} \Bigr{)} \tag{36}\]  
where \(n=1,\dots,N\).