## 4 Proposed Efficient Design for Transmit Waveform and UAV Altitude
20:if\(p_{m}^{(j)}<H_{\text{min}}\)then
21:\(p_{m}^{(j)}-H_{\text{min}}\)
22:elseif\(p_{m}^{(j)}>H_{\text{max}}\)then
23:\(p_{m}^{(j)}\gets H_{\text{max}}\)
24:endif
25: Compute \(b_{\text{WPT},u}(p_{m}^{(j)})\) and \(\mathbf{c}_{\text{WPT},u}(p_{m}^{(j)})\) (\(\forall u\in\mathcal{U}\)) from (45) and (46), respectively
26: Compute \(\mathbf{w}^{\star}(p_{m}^{(j)})\) from (55)
27:if\(\mathbf{w}^{\star}(p_{m}^{(j)})\) satisfies (57) and (58) then
28: Compute \(\gamma^{\star}(p_{m}^{(j)})\) from (56)
29:else
30: Solve problem \(\mathbf{P}_{7}^{y_{h}}\) with a given \(h=p_{m}^{(j)}\) and obtain \(\mathbf{w}^{\star}(p_{m}^{(j)})\) and \(\gamma^{\star}(p_{m}^{(j)})\)
31:endif
32: Compute the fitness function value \(f_{\text{Fit}}^{\text{Prop}}(p_{m}^{(j)})\) from (59)
33:if\(f_{\text{Fit}}^{\text{Prop}}(p_{m}^{(j)})<f_{\text{Fit}}^{\text{Prop}}(p_{m}^{ best})\)then
34:\(p_{m}^{best}\gets p_{m}^{(j)}\)
```  
**Algorithm 2** Algorithm for proposed efficient design method ```
1:end if
2:end for
3: Update \(g^{best}\leftarrow\arg\min_{[p_{m}^{best\top}]_{m=1}^{M}}f_{\text{Fit}}^{\text{Prop}}(p_{m}^ {best})\)
4:end for
5:Obtain an optimal altitude \(h_{\text{Prop}}^{*}=g^{best}\)
6:Repeat Line 25 - 31 with \(p_{m}^{(j)}=h_{\text{Prop}}^{*}\), and then obtain \(\mathbf{w}^{*}(h_{\text{Prop}}^{*})\) and \(\gamma^{*}(h_{\text{Prop}}^{*})\)
7:Obtain the proposed waveform \(\mathbf{X}_{\text{Prop}}^{*}\) from (37)
8:Output: Proposed waveform \(\mathbf{X}_{\text{Prop}}^{*}\), scaling factor \(\gamma_{\text{Prop}}^{*}\), and altitude \(h_{\text{Prop}}^{*}\)
```

**Algorithm 2**_Cont. ## 5 Numerical Results

In this section, we present the numerical evaluation of the proposed joint design methods. To obtain the numerical results reported in this section, we used the simulation settings listed in Table 1, similar to [5; 6; 7; 8; 9; 10; 11; 32; 33]. In addition, the carrier wavelength and the space between adjacent antennas were set to \(\frac{d_{\alpha}}{\lambda_{c}}=0.5\) to construct an array steering vector in (4). For the desired MIMO radar beam pattern, the sample angle grids \([\theta_{i}]_{i=1}^{L}\) were set to uniform samples in the range of \([-90^{\circ},90^{\circ}]\) with a resolution of \(0.1^{\circ}\), and the desired beam pattern was formed by utilizing the dominant peak directions with its beam width [23; 24]. By denoting \([\theta_{q}^{\text{Tar}}]_{q=1}^{Q}\) and \(\Delta\) as the dominant peak directions and beam width of the desired beam pattern (\(\chi(\theta_{i})\)), respectively, the latter can be formed as follows:

\[\chi(\theta_{i})= \begin{cases}1,&\theta_{q}^{\text{Tar}}-\frac{\Delta}{2}\leq \theta_{l}\leq\theta_{q}^{\text{Tar}}+\frac{\Delta}{2},\;q=1,2,\ldots,Q,\\ 0,&\text{otherwise},\end{cases}  \tag{60}\]

where we set the beam width to \(\Delta=10^{\circ}\)[23; 24]. Moreover, we set the desired power of the \(u\)-th EHD (i.e., \(P_{u}^{\text{Des}},\;\forall u\in\mathcal{U}\)) to 25 % of the maximum desired power, which can be obtained by solving problem \(P_{4}^{\prime\prime}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Description** & **Symbol** & **Value** \\ \hline Transmit sum power & \(P_{T}\) & 1 W \\ \hline Path loss at reference distance & \(\beta_{0}\) & 1 \\ \hline Path loss exponents of ground links & \(\alpha_{0}\) & 3.5 \\ \hline Path loss exponents of aerial links & \(\alpha_{\frac{\alpha}{2}}\) & 2 \\ \hline LoS probability-related parameters & \(a_{1}\) & 9.61 \\ when considering the urban environment & \(b_{1}\) & 0.16 \\ \hline Rician factor-related parameters & \(\mathcal{K}_{0}^{\text{dB}}\) & 5 dB \\  & \(\mathcal{K}_{\frac{\alpha}{2}}^{\text{dB}}\) & 15 dB \\ \hline Minimum altitude of UAV & \(H_{\text{min}}\) & 50 m \\ \hline Maximum altitude of UAV & \(H_{\text{max}}\) & 250 m \\ \hline Number of antennas of UAV & \(N\) & 8 \\ \hline Coefficients in PSO algorithm & \(\sigma\) & 0.7 \\  & \(z_{1}\) & 2 \\  & \(z_{2}\) & 2 \\ (Number of particles) & \(M\) & 8 \\ (Number of iterations) & \(J\) & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation settings.