## 2 The 3D A2A MIMO Channel Model
### LoS Component  
The channel matrix for the LoS component is written as  
\[\mathbf{h}_{\text{LoS}}(t,\tau)= \mathbf{v}_{r}(\theta_{L},\varphi_{L})\mathbf{v}_{t}^{\text{H}}( \theta_{L},\phi_{L})\cdot\sqrt{P_{L}}\cdot e^{-j\frac{2\pi}{\lambda}d_{L}(t)} \cdot\delta(\tau-\tau_{L}(t)) \tag{6}\]  
where \((\cdot)^{\text{H}}\) denotes the conjugate transpose, \(\lambda=c/f_{c}\) denotes the wavelength, \(c\) accounts for the speed of light, and \(f_{c}\) denotes the carrier frequency. In (6), the transmit and receive array steering vectors are given as  
\[\mathbf{v}_{t}(\theta_{L},\phi_{L})= [1,e^{-j\frac{2\pi}{\lambda}\delta\sin\theta_{L}\cos(\phi_{L}- \xi_{t})},...,e^{-j\frac{2\pi}{\lambda}(P-1)\delta\sin\theta_{L}\cos(\phi_{L}- \xi_{t})}]^{\text{T}} \tag{7}\] \[\mathbf{v}_{r}(\theta_{L},\varphi_{L})= [1,e^{-j\frac{2\pi}{\lambda}\delta\sin\theta_{L}\cos(\varphi_{L}- \xi_{r})},...,e^{-j\frac{2\pi}{\lambda}(Q-1)\delta\tau\sin\theta_{L}\cos( \varphi_{L}-\xi_{r})}]^{\text{T}} \tag{8}\]  
where \(\theta_{L}\) is the zenith angle of departure (ZoD--the departure angle relative to the \(+z\) axis), i.e.,  
\[\theta_{L}=\frac{\pi}{2}+\arctan\left(\frac{h_{t}-h_{r}}{D}\right). \tag{9}\]  
The parameter \(\theta_{L}\) is the zenith angle of arrival (ZoA) and is calculated as \(\theta_{L}=\pi-\theta_{L}\). The azimuth angle of departure (AoD) and azimuth angle of arrival (AoA) are \(\phi_{L}=0\) and \(\varphi_{L}=\pi\), respectively. Due to the large Tx-Rx distance, it is considered that the angles remain unchanged over the small observation time interval. The delay of the LoS path is \(\tau_{L}(t)=d_{L}(t)/c\). _where \(d_{L}\) is the length of the LoS path at the initial time and is calculated as_  
\[d_{L}=\sqrt{(h_{t}-h_{r})^{2}+D^{2}}. \tag{11}\]  
_Additionally, \(\gamma_{t}^{L}\) is the angle between \(\mathbf{a}_{t}\) and the LoS ray and is determined as_  
\[\cos(\gamma_{t}^{L})=\frac{\mathbf{a}_{t}^{T}\mathbf{\Phi}_{L}}{|\mathbf{a}_{t }|\cdot|\mathbf{\Phi}_{L}|} \tag{12}\]  
_where \(\mathbf{\Phi}_{L}\) is the departure angle unit vector of the LoS ray, i.e.,_  
\[\mathbf{\Phi}_{L}= \begin{bmatrix}\sin\theta_{L}\cos\phi_{L}\\ \sin\theta_{L}\sin\phi_{L}\\ \cos\theta_{L}\end{bmatrix} . \tag{13}\]  
_Similarly, \(\gamma_{r}^{L}\) is the angle between \(\mathbf{a}_{r}\) and the LoS ray and is determined as_  
\[\cos(\gamma_{r}^{L})=\frac{\mathbf{a}_{r}^{T}\mathbf{\Phi}_{L}}{|\mathbf{a}_{ r}|\cdot|\mathbf{\Phi}_{L}|}. \tag{14}\]  
**Proof of Proposition 1. ** Owing to the UAV's movement, the mean distance from the Tx to Rx at time \(t\) can be expressed as  
\[\overline{d}_{L}(t)=|\mathbf{d}_{L}+\mathbf{m}_{r}t-\mathbf{m}_{t}t| \tag{15}\]  
where \(\mathbf{d}_{L}=d_{L}\mathbf{\Phi}_{L}\) is the vector from the Tx to the Rx at the initial time instant, and \(\mathbf{m}_{t}=v_{t}[\cos\alpha_{t},\sin\alpha_{t},0]^{\mathrm{T}}\) and \(\mathbf{m}_{r}=v_{r}[\cos\alpha_{r},\sin\alpha_{r},0]^{\mathrm{T}}\) are the velocity vectors of the Tx and Rx, respectively. Considering the short observation time interval, \(\overline{d}_{L}(t)\) can be calculated as [35]  
\[\overline{d}_{L}(t)\approx d_{L}+v_{r}t\sin\vartheta_{L}\cos\alpha_{r}-v_{t}t \sin\theta_{L}\cos\alpha_{t}. \tag{16}\]  
The time-varying distance between the Tx and the Rx, considering the transmit UAV's vibration, is obtained as  
\[d_{L}^{\prime}(t) =|\overline{\mathbf{d}}_{\mathbf{L}}(\mathbf{t})-\mathbf{x}_{t} (t)|\] \[=\sqrt{\overline{d}_{L}(t)^{2}+|\mathbf{x}_{t}(t)|^{2}-2\overline {d}_{L}(t)|\mathbf{x}_{t}(t)|\cos(\gamma_{t}^{L})} \tag{17}\]  
where \(\overline{\mathbf{d}}_{L}(t)=\overline{d}_{L}(t)\mathbf{\Phi}_{L}\). Note that \(a_{t}\ll\overline{d}_{L}(t)\). Using the approximation \(\sqrt{1-x}\approx 1-\frac{x}{2}\) for small x, and substituting (16) into (17), we have  
\[d_{L}^{\prime}(t)\approx d_{L}+v_{r}t\sin\theta_{L}\cos\alpha_{r}-v_{t}t\sin \theta_{L}\cos\alpha_{t}-a_{t}\sin(2\pi f_{t}t)\cos(\gamma_{t}^{L}). \tag{18}\]  
Equation (10) can be derived by applying a similar procedure that takes both the transmit and receiving UAVs' vibrations into account.