## 3 Optimal Design of Transmit Waveform and UAV Altitude
### Analysis of Maximum Desired Power of an EHD  
In this subsection, we focus on the maximum desired power for the \(u\)-th EHD, i.e., the maximum value of \(P_{u}^{\mathrm{Des}}\). When considering only WPT operation with a single EHD, i.e., \(\rho=0\) and \(U=1\), the expected harvested power of the \(U\)-th EHD, i.e., (14), depends on the transmit waveform (\(\mathbf{X}\)) and the UAV altitude (\(h\)). Therefore, the maximum harvested power of the \(U\)-th EHD can be obtained by solving the following problem:  
\[P_{4}:\max_{\mathbf{X},h} \mathrm{Tr}\left(\mathbf{G}_{U}(h)\mathbf{X}\right)\] s.t. \[[\mathbf{X}]_{n,n}=\frac{P_{T}}{N},\;\forall n\in\mathcal{N}, \tag{30}\] \[\mathbf{X}\succeq\mathbf{0},\] (31) \[H_{\min}\leq h\leq H_{\max}. \tag{32}\]To solve problem \(P_{4}\) using mathematical analysis, we first relax the per-antenna power constraints (30) and then consider the sum power constraint, i.e., \(\mathrm{Tr}(\mathbf{X})=P\). Here, (30) is a special case of the sum power constraint. Accordingly, problem \(P_{4}\) is relaxed as follows:  
\[P_{4}^{\prime}: \max_{h,\mathbf{X}} \mathrm{Tr}\left(\mathbf{G}_{U}(h)\mathbf{X}\right)\] s.t. \[\mathrm{Tr}(\mathbf{X})=P,\;\mathbf{X}\succeq\mathbf{0},\] \[H_{\min}\leq h\leq H_{\max}.\]  
As shown in (23), the objective function of problem \(P_{4}^{\prime}\) is a non-convex, very complex, non-linear function of \(h\), so finding optimal solutions for \(\mathbf{X}\) and \(h\) by directly solving the problem is very difficult. However, the optimal waveform (\(\mathbf{X}\)) of problem \(P_{4}^{\prime}\) is obtained from Lemma 1 in [18] as follows:  
\[\mathbf{X}^{\star}=P_{T}\bar{\mathbf{a}}(\phi_{U})\bar{\mathbf{a}}^{H}(\phi_{ U}), \tag{33}\]  
where \(\bar{\mathbf{a}}(\phi_{U})=\frac{\mathbf{a}(\phi_{U})}{\|\mathbf{a}(\phi_{U} )\|}\). In other words, it is verified that optimal beamforming for a single EHD is the well-known maximum-ratio transmission (MRT) beamforming. Intuitively, this result is reasonable because MRT beamforming is known to maximize the received power of a single device in MIMO communication systems.  
In this case, by (33), we have  
\[[\mathbf{X}^{\star}]_{n,n} =\big{[}P_{T}\bar{\mathbf{a}}(\phi_{U})\bar{\mathbf{a}}^{H}(\phi _{U})\big{]}_{n,n}\] \[=\frac{P_{T}}{\|\mathbf{a}(\phi_{U})\|^{2}}\big{[}\mathbf{a}(\phi _{U})\mathbf{a}^{H}(\phi_{U})\big{]}_{n,n}\] \[=\frac{P_{T}}{N}\big{[}\mathbf{a}(\phi_{U})\mathbf{a}^{H}(\phi_{ U})\big{]}_{n,n} \tag{34}\] \[=\frac{P_{T}}{N}, \tag{35}\]  
where (34) and (35) are due to the fact that \(\|\mathbf{a}(\phi_{U})\|^{2}=N\) and \(\big{[}\mathbf{a}(\phi_{U})\mathbf{a}^{H}(\phi_{U})\big{]}_{n,n}=1\), respectively, which can be readily obtained from (4).  
As a result, the optimal waveform of (33) satisfies the per-antenna power constraints (30); therefore, (33) is the optimal waveform of problem \(P_{4}\). By substituting (33) into (23), the objective function of problem \(P_{4}\) can be rewritten as follows:  
\[\mathrm{Tr}\left(\mathbf{G}_{U}(h)\mathbf{X}^{\star}\right) =\beta_{0}\big{(}r_{U}^{2}+h^{2}\big{)}-\frac{1}{2}\Big{(}\frac{a _{2}}{\Psi_{r_{U}}(h)+1}+b_{2}\Big{)}\,\frac{P_{T}\big{(}N\Omega_{r_{U}}(h)+1 \big{)}}{\Omega_{r_{U}}(h)+1}\] \[\triangleq g_{U}(h). \tag{36}\]  
Thus, the optimal altitude (\(h\)) can be obtained by solving the following problem:  
\[P_{4}^{\prime\prime}: \max_{h}\,g_{U}(h)\] s.t. \[H_{\min}\leq h\leq H_{\max}.\]  
Problem \(P_{4}^{\prime\prime}\) has only one variable, \(h\), with \(h\in[H_{\min},H_{\max}]\), so we can easily adopt a 1D exhaustive line search method to obtain the optimal altitude \(h^{\star}\). Finally, the maximum desired power of an EHD is expressed as \(g_{U}(h^{\star})\); therefore, we need to have \(P_{u}^{\mathrm{Des}}\) less than \(g_{u}(h^{\star})\) for all \(u\in U\) in WPT operation.