## 2 The 3D A2A MIMO Channel Model
### LoS Component
By substituting (7), (8), and (10) into (6), the CIR of the LoS component between \(A_{p}^{t}\) and \(A_{q}^{\tau}\) can be expressed as  
\[h_{qp}^{\mathrm{LoS}}(t,\tau) =\sqrt{P_{L}}\cdot e^{-j\frac{2\pi}{\lambda}d_{L}}\cdot e^{j \frac{2\pi}{\lambda}(p-1)\delta_{t}\sin\theta_{L}\cos(\phi_{L}-\xi_{t})}\cdot e ^{-j\frac{2\pi}{\lambda}(q-1)\delta_{r}\sin\theta_{L}\cos(\varphi_{L}-\xi_{r})}\] \[\times e^{j2\pi(v_{t}^{L}+v_{r}^{L})t}\cdot e^{j\frac{2\pi}{ \lambda}a_{t}\sin(2\pi f_{t}t)\cos(\gamma_{t}^{L})}\cdot e^{-j\frac{2\pi}{ \lambda}a_{r}\sin(2\pi f_{t}t)\cos(\gamma_{r}^{L})}\cdot\delta(\tau-\tau_{L}(t)) \tag{19}\]where \(v_{t}^{L}\) and \(v_{r}^{L}\) are the Doppler shifts at the Tx and Rx sides, respectively, which are given as  
\[v_{t}^{L} =\frac{v_{t}}{\lambda}\sin\theta_{L}\cos\alpha_{t} \tag{20}\] \[v_{r}^{L} =-\frac{v_{r}}{\lambda}\sin\theta_{L}\cos\alpha_{r}. \tag{21}\]  
Equation (19) yields insight into the phase variation in the A2A channel. Specifically, the phase of the channel is decomposed into the following components:  
* \(-\frac{2\pi}{\lambda}d_{L}\) accounts for the phase shift due to the initial propagation distance \(d_{L}\). * \(\frac{2\pi}{\lambda}(p-1)\delta_{t}\sin\theta_{L}\cos(\phi_{L}-\xi_{t})\) is the phase shift due to the relative locations of \(A_{p}^{t}\) in the Tx array. * \(-\frac{2\pi}{\lambda}(q-1)\delta_{r}\sin\theta_{L}\cos(\phi_{L}-\xi_{r})\) is the phase shift due to the relative locations of \(A_{q}^{r}\) in the Rx array. * \(2\pi(v_{t}^{L}+v_{r}^{L})t\) is the phase shift due to the large-scale movements of the UAVs in specific directions. * \(\frac{2\pi}{\lambda}a_{t}\sin(2\pi f_{t}t)\cos(\gamma_{t}^{L})\) is the phase shift stemming from the random vibration of the Tx antennas
* \(-\frac{2\pi}{\lambda}a_{r}\sin(2\pi f_{r}t)\cos(\gamma_{r}^{L})\) is the phase shift due to the random vibrations of the Rx antennas. We define the phase rotation stemming from the antenna vibrations as follows:  
\[\overline{d}_{L}(t)=|\mathbf{d}_{L}+\mathbf{m}_{r}t-\mathbf{m}_{t}t| \tag{15}\]0  
The antenna vibrations introduce an effective time-varying Doppler frequency, i.e.,  
\[\overline{d}_{L}(t)=|\mathbf{d}_{L}+\mathbf{m}_{r}t-\mathbf{m}_{t}t| \tag{15}\]1 \[\overline{d}_{L}(t)=|\mathbf{d}_{L}+\mathbf{m}_{r}t-\mathbf{m}_{t}t| \tag{15}\]2  
It is clear that the value of \(\
u_{\text{wob}}(t)\) depends on the vibration's amplitude, frequency, and direction, while the vibration frequency governs its temporal variation, which follows cosine functions. To further illustrate this, we employ the Jacobi-Anger expansion, i.e.,  
\[\overline{d}_{L}(t)=|\mathbf{d}_{L}+\mathbf{m}_{r}t-\mathbf{m}_{t}t| \tag{15}\]3  
where \(J_{n}(\cdot)\) is the \(n\)th-order Bessel function of the first kind [36], and (19) can be expanded as follows:  
\[\overline{d}_{L}(t)=|\mathbf{d}_{L}+\mathbf{m}_{r}t-\mathbf{m}_{t}t| \tag{15}\]4 \[\overline{d}_{L}(t)=|\mathbf{d}_{L}+\mathbf{m}_{r}t-\mathbf{m}_{t}t| \tag{15}\]5 \[\overline{d}_{L}(t)=|\mathbf{d}_{L}+\mathbf{m}_{r}t-\mathbf{m}_{t}t| \tag{15}\]6  
This expression indicates that the antenna vibration introduces a series of sidebands at frequency offsets \(k_{1}f_{t}+k_{2}f_{r}\). The PSDs of the LoS path with different carrier frequencies are shown in \[\overline{d}_{L}(t)=|\mathbf{d}_{L}+\mathbf{m}_{r}t-\mathbf{m}_{t}t| \tag{15}\]7 To facilitate the analysis, we set \(v_{t}=v_{r}=0\) m/s, \(f_{t}=f_{r}=20\) Hz, and the antennas vibrate with a fixed amplitude, i.e., \(a_{t}=a_{r}=0.01\) m. In accordance with the aforementioned analysis, the PSDs show that abundant harmonic components occur at integer multiples of 20 Hz. For a low carrier frequency, i.e., 2 GHz, the sidebands decrease significantly with the increase in the harmonics. However, under the same degree of vibration, increasing the carrier frequency, e.g., to 28 GHz, results in a large amount of power being transferred to the harmonic components due to the shorter wavelength.