## 2 The 3D A2A MIMO Channel Model  
Figure 1 illustrates a typical A2A communication scenario, which includes an LoS ray, a specular reflection ray following Snell's Law, and multiple diffuse rays [2]. Without loss of generality, the projection of the Tx array center onto the \(xy\) plane is chosen as the origin of coordinates, denoted by \(O\). The horizontal separation between the centers of the Tx and Rx arrays is denoted as \(D\). The Tx (Rx) flies horizontally at an altitude \(h_{t}\) (\(h_{r}\)) with a speed \(v_{t}\) (\(v_{r}\)) and follows a specific moving direction denoted as \(\alpha_{t}\) (\(\alpha_{r}\)). Both the Tx and Rx employ uniform linear arrays comprising \(P\) and \(Q\) antennas, which are spaced at distances \(\delta_{t}\) and \(\delta_{r}\), respectively. Angles \(\xi_{t}\) and \(\xi_{r}\) define the Tx and Rx arrays' orientations in the horizontal plane, respectively.  
Owing to propeller rotation, UAVs often exhibit sinusoidal vibrations, which are particularly relevant in the study of small UAVs [14; 34]. These periodic motions of UAVs can be characterized by the displacement varying over time and are modeled as  
\[\mathbf{x}_{t}(t) =\mathbf{a}_{t}\cdot\sin(2\pi f_{t}\cdot t+\Theta_{t}) \tag{1}\] \[\mathbf{x}_{r}(t) =\mathbf{a}_{r}\cdot\sin(2\pi f_{r}\cdot t+\Theta_{r}). \tag{2}\]  
Here, \(f_{t}\) and \(f_{r}\) are the vibration frequencies of the Tx and Rx antennas, respectively; \(\Theta_{t}\) and \(\Theta_{r}\) are the initial phases, which follow uniform distributions within \([0,2\pi)\); and \(\mathbf{a}_{t}\) and \(\mathbf{a}_{r}\) are vibrational displacement amplitude vectors, which are expressed as  
\[\mathbf{a}_{t} =a_{t}[\cos\zeta_{t}^{E}\cos\zeta_{t}^{A},\cos\zeta_{t}^{E}\sin \zeta_{t}^{A},\sin\zeta_{t}^{E}]^{\mathrm{T}} \tag{3}\] \[\mathbf{a}_{r} =a_{r}[\cos\zeta_{r}^{E}\cos\zeta_{r}^{A},\cos\zeta_{r}^{E}\sin \zeta_{r}^{A},\sin\zeta_{r}^{E}]^{\mathrm{T}} \tag{4}\]  
where \((\cdot)^{\mathrm{T}}\) denotes the matrix transpose, \(a_{t}\) and \(a_{r}\) are vibrational displacement amplitudes following uniform distributions, and \(\zeta_{t}^{E}\) and \(\zeta_{t}^{A}\) are the vibration elevation and azimuth angles of the transmit UAV, respectively. Similarly, \(\zeta_{r}^{E}\) and \(\zeta_{r}^{A}\) are the vibration elevation and azimuth angles of the receiving UAV, respectively. Note that (1) and (2) describe a rapid oscillating or back-and-forth motion of the UAV rather than a rotation. This is different from existing models in which the UAVs undergo rotational motions [10; 11; 12; 13].  
The multiple-input multiple-output (MIMO) channel matrix \(\mathbf{h}(t,\tau)=[h_{qp}(t,\tau)]_{Q\times P}\) can be obtained as  
\[\mathbf{h}(t,\tau)=\mathbf{h}_{\text{LoS}}(t,\tau)+\mathbf{h}_{\text{NLoS}}(t,\tau) \tag{5}\]  
where \(\mathbf{h}_{\text{LoS}}(t,\tau)\) and \(\mathbf{h}_{\text{NLoS}}(t,\tau)\) are the channel matrices for the LoS and NLoS components of the channel, respectively. Additionally, \(h_{qp}(t,\tau)\) denotes the time-varying channel impulse response (CIR) between \(A_{p}^{t}\) and \(A_{q}^{r}\), where \(A_{p}^{t}\) denotes the \(p\)th Tx antenna and \(A_{q}^{r}\) accounts for the \(q\)th Rx antenna. For clarity, the essential parameters of the model are summarized in Table 1.  
Figure 1: The geometrical characteristics of the proposed A2A MIMO channel model.