## 3 UAV AG Communication Channel Modeling and Characterization
### A UAV AG Multi-Mobility Multi-Trajectories Channel Model
\begin{split} h_{pq,MB}^{NLoS}(t,\tau)&=\sum_{k=1}^{K_{pq }(t)}\sum_{l=1}^{L_{k}}\Bigg{[}F_{q,V}(\phi_{E,l_{k}}^{R}(t),\phi_{A,l_{k}}^{ R}(t))\Bigg{]}^{T}\Bigg{[}\begin{matrix}e^{i\theta_{k}^{VV}}&\sqrt{ \kappa_{l_{k}}^{-1}}e^{i\theta_{k}^{VH}}\\ \sqrt{\kappa_{l_{k}}^{-1}}e^{i\theta_{k}^{HV}}&e^{i\theta_{k}^{ HH}}\\ \end{matrix}\Bigg{]}\\ &\times\Bigg{[}F_{p,V}(\phi_{E,l_{k}}^{T}(t),\phi_{A,l_{k}}^{T}(t)) \\ F_{p,H}(\phi_{E,l_{k}}^{T}(t),\phi_{A,l_{k}}^{T}(t))\Bigg{]}\sqrt{P_{pq,l_{k}} (t)}e^{2\pi f_{c}((D_{p,l_{k}}^{T,MB}(t)+D_{q,l_{k}}^{R,MB}(t))/c+\tau)}\\ &\cdot\delta(\tau-((D_{p,l_{k}}^{T,MB}(t)+D_{q,l_{k}}^{R,MB}(t))/c+ \tau))\end{split}  \tag{3}\]  
where \(\{\cdot\}^{T}\) denoted the transpose operation, \(f_{c}\) was the carrier frequency, and \(F_{p/q,V}\) and \(F_{p/q,H}\) denoted the antenna patterns of vertical and horizontal polarizations at the Tx or the Rx, respectively. Moreover, \(\kappa_{l_{k}}\) represented the cross-polarization power ratios (XPR) of the \(l\)th ray in cluster \(\mathbf{C}_{k}^{A/Z}\); and \(\theta_{l_{k}}^{T}\) (where \(Tr\) respected VV,VH,HV,HH) represented initial  
Figure 2: A 3D non-stationary GBSM with multi-trajectories. phases subject to uniform distribution in \((0,\,2\pi]\); while \(D_{p,l_{k}}^{T,MB}(t)\) and \(D_{q,l_{k}}^{R,MB}(t)\) were the distance of \(l\)th ray from the \(p\)th Tx antenna element to the scattering cluster \(\mathbf{C}_{k}^{A}(t)\), and the distance of the \(l\)th ray from the scattering cluster \(\mathbf{C}_{k}^{Z}(t)\) to the \(q\)th Rx antenna element, respectively; \(P_{pq,l_{k}}(t)\) were denoted as the powers of the \(l\)th ray in the \(k\)th cluster between the Tx and the Rx at time \(t\); \(c\) was the speed of light; and \(\bar{\tau}\) was the time delay of the virtual link. The LoS component was expressed as  
\[h_{pq}^{LoS}(t,\tau)= \begin{bmatrix}F_{q,V}(\phi_{E,LoS}^{R}(t),\phi_{A,LoS}^{R}(t))\\ F_{q,H}(\phi_{E,LoS}^{R}(t),\phi_{A,LoS}^{R}(t))\end{bmatrix} ^{T} \begin{bmatrix} e^{i\beta_{LoS}^{VV}}&0\\ 0&e^{i\beta_{LoS}^{HH}}\end{bmatrix}\begin{bmatrix}F_{p,V}(\phi_{E,LoS}^{T}(t ),\phi_{A,LoS}^{T}(t))\\ F_{p,H}(\phi_{E,LoS}^{T}(t),\phi_{A,LoS}^{T}(t))\end{bmatrix}  \tag{4}\] \[\times e^{l2\pi f_{r}D_{pq}^{LoS}(t)/c}\cdot\delta(\tau-D_{pq}^{LoS }(t)/c)\]  
where \(D_{pq}^{LoS}(t)\) was the distance between the \(p\)th Tx antenna element and the \(q\)th Rx antenna element. The calculations of \(D_{p,l_{k}}^{T,MB}(t)\), \(D_{q,l_{k}}^{R,MB}(t)\), and \(D_{pq}^{LoS}(t)\) were expressed as  
\[D_{p,l_{k}}^{T,MB}(t)=||\mathbf{C}_{l_{k}}^{A}(t)-\mathbf{A}_{p}^{T}(t)|| \tag{5}\]  
\[D_{q,l_{k}}^{R,MB}(t)=||\mathbf{A}_{q}^{R}(t)-\mathbf{C}_{l_{k}}^{Z}(t)|| \tag{6}\]  
\[D_{pq}^{LoS}(t)=||\mathbf{A}_{q}^{R}(t)-\mathbf{A}_{p}^{T}(t)|| \tag{7}\]  
where \(\mathbf{C}_{l_{k}}^{A}(t)\) and \(\mathbf{C}_{l_{k}}^{Z}(t)\) were coordinates of the scattering points of the \(l\)th ray in the cluster \(\mathbf{C}_{k}^{A}(t)\) and \(\mathbf{C}_{k}^{Z}(t)\), respectively. Furthermore, \(\mathbf{A}_{g}^{R}(t)\) and \(\mathbf{A}_{p}^{T}(t)\) were the coordinates of the \(q\)th Rx antenna element, and the coordinates of the \(p\)th Tx antenna element, respectively. The delay of the rays varied with time \(t\). The cluster power satisfied an exponential distribution with a single slope, and the total power was set to 1 by a normalization operation [27]. \(P_{pq,l_{k}}(t)\) changed with the moving trajectory. Here, we set different trajectories at the Tx and the Rx, respectively. The parameters involved in the channel modeling are listed in Table 1.