## 4 Proposed Efficient Design for Transmit Waveform and UAV Altitude
First, by substituting \(\mathbf{X}_{\text{Prop}}\) into (5) for MIMO radar sensing operation, we have  
\[p^{\text{Rad}}(\theta_{l};\mathbf{X}_{\text{Prop}}) =\mathbf{a}^{H}(\theta_{l})\mathbf{X}_{\text{Prop}}\mathbf{a}( \theta_{l})\] \[=\mathbf{a}^{H}(\theta_{l})\mathbf{X}_{\text{Rad}}\mathbf{a}( \theta_{l})+\sum_{k=1}^{U}w_{k}\mathbf{\hat{a}}^{H}(\theta_{l})\big{\{} \mathbf{\hat{a}}(\phi_{k})\mathbf{\hat{a}}^{H}(\phi_{k})-\mathbf{X}_{\text{Rad }}\big{\}}\mathbf{a}(\theta_{l})\] \[=b_{\text{Rad},l}+\epsilon_{\text{Rad},l}^{T}\mathbf{w},\,\forall l \in[1,2,\ldots,L]\triangleq\mathcal{L}, \tag{41}\]  
where \(\mathbf{w}=[w_{1},\ldots w_{U}]^{T}\), and \(b_{\text{Rad},l}\in\mathbb{R}\) and \(\epsilon_{\text{Rad},l}\in\mathbb{R}^{U\times 1}\) (\(\forall l\in\mathcal{L}\)) are expressed as  
\[b_{\text{Rad},l} =\mathbf{a}^{H}(\theta_{l})\mathbf{X}_{\text{Rad}}\mathbf{a}( \theta_{l}), \tag{42}\] \[\mathbf{P}_{5}:\min_{\mathbf{X}_{\text{Rad}},\gamma} \rho L_{\text{Rad}}(\mathbf{X}_{\text{Rad}},\gamma)\]0  
Similarly, by substituting \(\mathbf{X}_{\text{Prop}}\) into (14) for WPT operation, we have  
\[\mathbf{P}_{5}:\min_{\mathbf{X}_{\text{Rad}},\gamma} \rho L_{\text{Rad}}(\mathbf{X}_{\text{Rad}},\gamma)\]1 \[\mathbf{P}_{5}:\min_{\mathbf{X}_{\text{Rad}},\gamma} \rho L_{\text{Rad}}(\mathbf{X}_{\text{Rad}},\gamma)\]2  
where \(b_{\text{WPT},u}(h)\in\mathbb{R}\) and \(\epsilon_{\text{WPT},u}^{T}(h)\in\mathbb{R}^{U\times 1}\) (\(\forall u\in\mathcal{U}\)) are functions with respect to the UAV altitude (\(h\)), expressed as\[\mathbf{P}_{5}:\min_{\mathbf{X}_{\text{Rad}},\gamma} \rho L_{\text{Rad}}(\mathbf{X}_{\text{Rad}},\gamma)\]3 \[\mathbf{P}_{5}:\min_{\mathbf{X}_{\text{Rad}},\gamma} \rho L_{\text{Rad}}(\mathbf{X}_{\text{Rad}},\gamma)\]4  
Accordingly, from (41) and (44), the objective function of problem \(P_{1}\) can be rewritten as a function of \(\mathbf{w}\) as follows:  
\[\mathbf{P}_{5}:\min_{\mathbf{X}_{\text{Rad}},\gamma} \rho L_{\text{Rad}}(\mathbf{X}_{\text{Rad}},\gamma)\]5 \[\mathbf{P}_{5}:\min_{\mathbf{X}_{\text{Rad}},\gamma} \rho L_{\text{Rad}}(\mathbf{X}_{\text{Rad}},\gamma)\]6 \[\mathbf{P}_{5}:\min_{\mathbf{X}_{\text{Rad}},\gamma} \rho L_{\text{Rad}}(\mathbf{X}_{\text{Rad}},\gamma)\]7 \[\mathbf{P}_{5}:\min_{\mathbf{X}_{\text{Rad}},\gamma} \rho L_{\text{Rad}}(\mathbf{X}_{\text{Rad}},\gamma)\]8 \[\mathbf{P}_{5}:\min_{\mathbf{X}_{\text{Rad}},\gamma} \rho L_{\text{Rad}}(\mathbf{X}_{\text{Rad}},\gamma)\]9 \[[\mathbf{X}_{\text{Rad}}]_{n,n}=\frac{P_{T}}{N},\,\forall n\in \mathcal{N}, \tag{38}\]0  
As a result, the weighting factors \([w_{i}]_{i=1}^{U}\) for the proposed waveform, scaling factor (\(\gamma\)), and UAV altitude (\(h\)) can be jointly optimized by solving the following problem:  
\[ \begin{split}\mathbf{P}_{6}:\min_{\mathbf{w},\gamma,h}& t(\mathbf{w},\gamma,h)\\ \text{s.t.}& 0\leq w_{k}\leq 1,\;\forall k\in\mathcal{ U},\\ &\sum_{k=1}^{U}w_{k}\leq 1,\\ & H_{\min}\leq h\leq H_{\max}.\end{split}  \tag{48}\]  
Similar to the original problem, \(\mathbf{P}_{1}\), \(t(\mathbf{w},\gamma,h)\) in problem \(\mathbf{P}_{6}\) is also a non-convex, very complex, non-linear function with respect to \(h\); therefore, it is very hard to obtain an optimal solution by directly solving it. However, for a given \(h\in[H_{\min},H_{\max}]\), the objective function of problem \(\mathbf{P}_{6}\) is convex because \(t(\mathbf{w},\gamma,h)\) is a quadratic function with respect to \(w\); thus, we have  
\[\frac{\rho}{L}\sum_{l=1}^{L}\epsilon_{\text{Rad},l}\epsilon_{\text{Rad},l}^{T} +\frac{1-\rho}{U}\sum_{u=1}^{U}\frac{\epsilon_{\text{WPT},\mu}(h)\epsilon_{ \text{WPT},\mu}^{T}(h)}{(P_{u}^{\text{Des}})^{2}}\succeq\mathbf{0}, \tag{50}\]  
because \(\rho\in[0,1]\), \(\epsilon_{\text{Rad},l}\epsilon_{\text{Rad},l}^{T}\succeq\mathbf{0}\), and \(\epsilon_{\text{WPT},\mu}(h)\epsilon_{\text{WPT},\mu}^{T}(h)\succeq\mathbf{0}\). Hence, with a given \(h\), we can obtain the optimal \(\mathbf{w}\) and \(\gamma\) by solving the following problem:  
\[