## 2 System Model and Problem Formulation
### Wireless Power Transfer with Aerial Channel Model
Here, we have \(a_{2}<0\) and \(b_{2}>0\), with \(a_{2}\cong\alpha_{\frac{\pi}{2}}-\alpha_{0}\) and \(b_{2}\cong\alpha_{0}\) due to \(P_{\text{LoS}}(0)\to 0\) and \(P_{\text{LoS}}(\frac{\pi}{2})\to 1\). Moreover, in the A2G Rician fading channel, the Rician factor for the \(u\)-th EHD can be modeled as a function of the \(u\)-th EHD's elevation angle, \(\psi_{u}\)[8; 10; 18; 28]. By denoting \(a_{3}\) and \(b_{3}\) as the environment- and frequency-dependent constant parameters of the Rician factor, respectively, the Rician factor in dB units is expressed as  
\[\mathcal{K}_{u}^{\text{dB}}=a_{3}\exp\big{(}b_{3}\theta_{u}\big{)}. \tag{13}\]  
In (13), \(a_{3}\) and \(b_{3}\) are given as follows:  
\[a_{3}=\mathcal{K}_{0}^{\text{dB}},\,b_{3}=\frac{2}{\pi}\ln\Big{(}\frac{ \mathcal{K}_{\frac{\pi}{2}}^{\text{dB}}}{\mathcal{K}_{0}^{\text{dB}}}\Big{)}.\]  
where \(\mathcal{K}_{0}^{\text{dB}}\) and \(\mathcal{K}_{\frac{\pi}{2}}^{\text{dB}}\) denote the Rician factors in dB units when the elevation angles are 0 rad and \(\frac{\pi}{2}\) rad, respectively [28]. Acquiring the instantaneous channel state information of (8) for the UAV is not practical, as it requires additional resources and costs owing to the high computational complexity and energy consumption of channel estimation [2; 3; 18; 19]. To address this issue, in this study, we concentrated on waveform design for WPT by leveraging only the channel statistics of (8), not instantaneous CSI. Based on (7), the expected harvested power of the \(u\)-th EHD becomes  
\[\mathbf{h}_{u}=\sqrt{\beta_{0}d_{u}^{-\alpha_{u}}}\Bigg{(}\sqrt{\frac{\mathcal{ K}_{u}}{\mathcal{K}_{u}+1}}\mathbf{a}(\phi_{u})+\sqrt{\frac{1}{\mathcal{K}_{u}+1 }}\mathbf{h}_{u}^{\text{NLoS}}\Bigg{)}, \tag{8}\]0 \[\mathbf{h}_{u}=\sqrt{\beta_{0}d_{u}^{-\alpha_{u}}}\Bigg{(}\sqrt{\frac{\mathcal{ K}_{u}}{\mathcal{K}_{u}+1}}\mathbf{a}(\phi_{u})+\sqrt{\frac{1}{\mathcal{K}_{u}+1 }}\mathbf{h}_{u}^{\text{NLoS}}\Bigg{)}, \tag{8}\]1  
where we denote \(\mathbb{E}[\mathbf{h}_{u}\mathbf{h}_{u}^{H}]\) by \(\mathbf{G}_{u}(h)\), which is expressed as  
\[\mathbf{h}_{u}=\sqrt{\beta_{0}d_{u}^{-\alpha_{u}}}\Bigg{(}\sqrt{\frac{\mathcal{ K}_{u}}{\mathcal{K}_{u}+1}}\mathbf{a}(\phi_{u})+\sqrt{\frac{1}{\mathcal{K}_{u}+1 }}\mathbf{h}_{u}^{\text{NLoS}}\Bigg{)}, \tag{8}\]2  
In (15), we note that the channel statistics of \(d_{u}\), \(\alpha_{u}\), and \(\mathcal{K}_{u}\) depend on both the UAV altitude (\(h\)) and the \(u\)-th EHD's horizontal distance (\(r_{u}\)). Similar to the beam pattern matching design for MIMO radar sensing, the waveform design of the transmit beamforming of WPT signals, as well as UAV altitude, aims to optimize the received power of the EHDs by matching the desired received power. We denote the desired power at the \(u\)-th EHD by \(P_{u}^{\text{Des}}>0\); the WPT waveform is designed to minimize the normalized MSE between the desired and expected harvested power associated with all the EHDs, which is defined as follows:  
\[\mathbf{h}_{u}=\sqrt{\beta_{0}d_{u}^{-\alpha_{u}}}\Bigg{(}\sqrt{\frac{\mathcal{ K}_{u}}{\mathcal{K}_{u}+1}}\mathbf{a}(\phi_{u})+\sqrt{\frac{1}{\mathcal{K}_{u}+1 }}\mathbf{h}_{u}^{\text{NLoS}}\Bigg{)}, \tag{8}\]3 \[\mathbf{h}_{u}=\sqrt{\beta_{0}d_{u}^{-\alpha_{u}}}\Bigg{(}\sqrt{\frac{\mathcal{ K}_{u}}{\mathcal{K}_{u}+1}}\mathbf{a}(\phi_{u})+\sqrt{\frac{1}{\mathcal{K}_{u}+1 }}\mathbf{h}_{u}^{\text{NLoS}}\Bigg{)}, \tag{8}\]4