## 3 Joint Optimization of Altitude and Energy Beamforming for a Single EHD
### Optimal Design with a Closed-Form Energy Beamforming Solution  
For a single-EHD scenario, i.e., \(U=1\), we have \(t=\tau_{U}=\frac{Q_{U}^{\mathrm{Req}}}{\mathrm{Tr}\left(\mathbf{G}_{U}(h) \mathbf{X}\right)}\) in the problem \(\mathbf{P}_{2}\), so it can be reformulated into the problem \(\mathbf{P}_{3}\):  
\[\mathbf{P}_{3}:\min_{h,\mathbf{X}} \frac{Q_{U}^{\mathrm{Req}}}{\mathrm{Tr}\left(\mathbf{G}_{U}(h) \mathbf{X}\right)}\] s.t. \[\mathrm{Tr}(\mathbf{X})\leq P,\ \mathbf{X}\succeq\mathbf{0},\] \[H_{\min}\leq h\leq H_{\max}.\]  
Since \(\mathrm{Tr}\left(\mathbf{G}_{U}(h)\mathbf{X}\right)>0\) must be satisfied at an optimal value, the objective function of the problem \(\mathbf{P}_{3}\) can be recast as  
\[\min_{h,\mathbf{X}}\frac{Q_{U}^{\mathrm{Req}}}{\mathrm{Tr}\left(\mathbf{G}_{U}(h) \mathbf{X}\right)}=\max_{h,\mathbf{X}}\ \mathrm{Tr}\left(\mathbf{G}_{U}(h)\mathbf{X}\right). \tag{24}\]  
From (24), the problem \(\mathbf{P}_{3}\) is equivalent to the problem \(\mathbf{P}_{3^{\prime}}\):  
\[\mathbf{P}_{3^{\prime}}:\max_{h,\mathbf{X}} \mathrm{Tr}\left(\mathbf{G}_{U}(h)\mathbf{X}\right)\] s.t. \[\mathrm{Tr}(\mathbf{X})\leq P,\ \mathbf{X}\succeq\mathbf{0},\] \[H_{\min}\leq h\leq H_{\max}.\]  
It is also difficult to obtain an optimal solution for \(h\) and \(\mathbf{X}\) by directly solving the problem \(\mathbf{P}_{3^{\prime}}\), because \(\mathbf{G}_{U}(h)\) is a non-convex and complex non-linear function of \(h\). However,an optimal covariance matrix for \(\mathbf{X}\) in relation to the problem \(\mathbf{P}_{\mathcal{Y}}\) is obtained in closed-form solution via the following lemma. **Lemma 1**. : _Optimal energy beamforming in the case of a single EHD is_  
\[\mathbf{X}^{\star}=P\tilde{\mathbf{a}}_{U}\tilde{\mathbf{a}}_{U}^{ H} \tag{25}\]  
_where \(\tilde{\mathbf{a}}_{U}=\dfrac{\mathbf{a}_{U}}{\|\mathbf{a}_{U}\|}\). In addition, with a fixed \(h\in[H_{\min},H_{\max}]\), the optimal value of the problem \(\mathbf{P}_{\mathcal{Y}}\) is given by_  
\[\mathrm{Tr}\left(\mathbf{G}_{U}(h)\mathbf{X}^{\star}\right)=P \beta\omega d_{U}^{-\alpha_{U}}\dfrac{\mathcal{K}_{U}\|\mathbf{a}_{U}\|^{2}+1} {\mathcal{K}_{U}+1} \tag{26}\]  
_where \(d_{U}\), \(\alpha_{U}\), and \(\mathcal{K}_{U}\) depend on \(h\). In (26), \(\frac{\mathcal{K}_{U}\|\mathbf{a}_{U}\|^{2}}{\mathcal{K}_{U}+1}\) and \(\frac{1}{\mathcal{K}_{U}+1}\) indicate the impact of the LoS and NLoS components, respectively._  
Proof of Lemma 1.: Refer to Appendix A. From Lemma 1, it is verified that optimal energy beamforming for a single-EHD case, i.e., \(\mathbf{X}^{\star}\), depends only on the LoS component, i.e., \(\mathbf{a}_{U}\), of the EHD and is not affected by the UAV altitude \(h\). Consequently, by inserting (25) and (26) in Lemma 1 into the problem \(\mathbf{P}_{\mathcal{Y}}\), the optimal UAV altitude for a single EHD can be obtained by solving the following problem:  
\[\mathbf{P}_{\mathcal{Y}}^{h}: \max_{h}P\beta_{0}d_{U}^{-\alpha_{U}}\dfrac{\mathcal{K}_{U}\| \mathbf{a}_{U}\|^{2}+1}{\mathcal{K}_{U}+1}\] \[\mathrm{Tr}(\mathbf{X})\leq P,\ \mathbf{X}\succeq\mathbf{0},\]0  
To tackle the problem \(\mathbf{P}_{\mathcal{Y}}^{h}\), we omit \(P\beta_{0}\), which is a constant term with respect to \(h\), and we define the objective function in the problem \(\mathbf{P}_{\mathcal{Y}}^{h}\) as  
\[\mathrm{Tr}(\mathbf{X})\leq P,\ \mathbf{X}\succeq\mathbf{0},\]1 \[\mathrm{Tr}(\mathbf{X})\leq P,\ \mathbf{X}\succeq\mathbf{0},\]2  
where (27) is obtained by substituting (5) and (8) into \(d_{U}\) and \(\alpha_{U}\), respectively. In this case, \(\theta_{U}\) is the function of the variable \(h\) as well as the EHD's distance \(r_{U}\), while \(P_{\text{LoS}}(\theta_{U})\) and \(\mathcal{K}_{U}\) are complex functions with respect to \(\theta_{U}\).