## 4 Proposed Efficient Design for Transmit Waveform and UAV Altitude
\begin{split}\mathbf{P}_{\gamma}^{\forall h}:\min_{\mathbf{w},T}& t(\mathbf{w},\gamma,h)\\ \text{s.t.}& 0\leq w_{k}\leq 1,\;\forall k\in\mathcal{ U},\\ &\sum_{k=1}^{U}w_{k}\leq 1.\end{split}  \tag{51}\]  
We can numerically obtain the optimal solutions to problem \(\mathbf{P}_{\gamma}^{\forall h}\) with a well-known convex optimization problem solver, such as CVX, and denote the solutions by \(\mathbf{w}^{\star}(h)\) and \(\gamma^{\star}(h)\). Moreover, by relaxing constraints (51) and (52), the Karush-Kuhn-Tucker (KKT) optimality conditions of problem \(P_{7}^{y_{th}}\) are obtained as follows:  
\[\frac{\partial t(\mathbf{w},\gamma,h)}{\partial\mathbf{w}} =2\Bigg{[}\frac{\rho}{L}\sum_{l=1}^{L}\mathbf{c}_{\text{Rad},l} \mathbf{c}_{\text{Rad},l}^{T}+\frac{1-\rho}{U}\sum_{u=1}^{Ut}\frac{\mathbf{c}_ {\text{WPT},u}(h)\mathbf{c}_{\text{WPT},u}^{T}(h)}{(P_{u}^{\text{Des}})^{2}} \Bigg{]}\mathbf{w}\] \[\quad-\big{(}2\frac{\rho}{L}\sum_{l=1}^{L}\chi(\theta_{l}) \mathbf{c}_{\text{Rad},l}^{T}\big{)}\gamma+2\frac{\rho}{L}\sum_{l=1}^{L}b_{ \text{Rad},l}\mathbf{c}_{\text{Rad},l}^{T}\big{)}\] \[\quad-2\frac{1-\rho}{U}\sum_{u=1}^{U}\Big{(}1-\frac{b_{\text{WPT},u}(h)}{P_{u}^{\text{Des}}}\Big{)}\frac{\mathbf{c}_{\text{WPT},u}^{T}(h)}{P _{u}^{\text{Des}}}\] \[=0, \tag{53}\] \[\frac{\partial t(\mathbf{w},\gamma,h)}{\partial\gamma} =-\Big{(}2\frac{\rho}{L}\sum_{l=1}^{L}\chi(\theta_{l})\mathbf{c} _{\text{Rad},l}^{T}\Big{)}\mathbf{w}+2\frac{\rho}{L}\sum_{l=1}^{L}(\gamma \chi(\theta_{l})-b_{\text{Rad},l})\chi(\theta_{l})\] \[=0. \tag{54}\]  
From (53) and (54), we obtain the optimal \(\mathbf{w}^{\star}(h)\) and \(\gamma^{\star}(h)\) with a closed-form solution, which are expressed as  
\[\mathbf{w}^{\star}(h) =\Bigg{[}\frac{\rho}{L}\sum_{l=1}^{L}\mathbf{c}_{\text{Rad},l} \mathbf{c}_{\text{Rad},l}^{T}-\frac{\rho}{L}\frac{\big{(}\sum_{l=1}^{L}\chi( \theta_{l})\mathbf{c}_{\text{Rad},l}^{T}\big{)}\big{(}\sum_{l=1}^{L}\chi( \theta_{l})\mathbf{c}_{\text{Rad},l}^{T}\big{)}}{\sum_{l=1}^{L}\chi^{2}( \theta_{l})}\] \[\quad\quad+\frac{1-\rho}{U}\sum_{u=1}^{Ut}\frac{\mathbf{c}_{\text {WPT},u}(h)\mathbf{c}_{\text{WPT},u}^{T}(h)}{(P_{u}^{\text{Des}})^{2}}\Bigg{]} ^{-1}\] \[\quad\quad\times\Bigg{[}\frac{\rho}{L}\frac{\big{(}\sum_{l=1}^{L} b_{\text{Rad},l}\chi(\theta_{l})\big{)}\big{(}\sum_{l=1}^{L}\chi(\theta_{l}) \mathbf{c}_{\text{Rad},l}\big{)}}{\sum_{l=1}^{L}\chi^{2}(\theta_{l})}-\frac{ \rho}{L}\sum_{l=1}^{L}b_{\text{Rad},l}\mathbf{c}_{\text{Rad},l}\] \[\quad\quad+\frac{1-\rho}{U}\sum_{u=1}^{U}\Big{(}1-\frac{b_{\text{ WPT},u}(h)}{P_{u}^{\text{Des}}}\Big{)}\frac{\mathbf{c}_{\text{WPT},u}^{T}(h)}{P _{u}^{\text{Des}}}\Bigg{]}, \tag{55}\] \[\quad-\big{(}2\frac{\rho}{L}\sum_{l=1}^{L}\chi(\theta_{l}) \mathbf{c}_{\text{Rad},l}^{T}\big{)}\gamma+2\frac{\rho}{L}\sum_{l=1}^{L}b_{ \text{Rad},l}\mathbf{c}_{\text{Rad},l}^{T}\big{)}\]0  
Here, if \(\mathbf{w}^{\star}(h)\) in (55) satisfies constraints (51) and (52) such that  
\[\quad-\big{(}2\frac{\rho}{L}\sum_{l=1}^{L}\chi(\theta_{l}) \mathbf{c}_{\text{Rad},l}^{T}\big{)}\gamma+2\frac{\rho}{L}\sum_{l=1}^{L}b_{ \text{Rad},l}\mathbf{c}_{\text{Rad},l}^{T}\big{)}\]1 \[\quad-\big{(}2\frac{\rho}{L}\sum_{l=1}^{L}\chi(\theta_{l}) \mathbf{c}_{\text{Rad},l}^{T}\big{)}\gamma+2\frac{\rho}{L}\sum_{l=1}^{L}b_{ \text{Rad},l}\mathbf{c}_{\text{Rad},l}^{T}\big{)}\]2  
then \(\mathbf{w}^{\star}(h)\) and \(\gamma^{\star}(h)\) are the optimal solutions to problem \(P_{7}^{y_{th}}\). Otherwise, we numerically obtain \(\mathbf{w}^{\star}(h)\) and \(\gamma^{\star}(h)\) by solving problem \(P_{7}^{y_{th}}\) when \(\mathbf{w}^{\star}(h)\) in (55) does not meet (57) and (58). Note that (55) and (56) may significantly reduce the computational complexity due to the derivation of a closed-form solution.