## 4 Proposed Efficient Design for Transmit Waveform and UAV Altitude
In addition, Figure 2(b) illustrates that the proposed efficient design resulted in a considerable performance improvement compared with the methods with \(H_{\text{min}}\) and \(H_{\text{max}}\), achieving near-optimal performance in terms of the trade-off region. It is evident that the beam pattern MSE increased as the harvested power MSE decreased, and vice versa. Further, Figure 2(c) shows a comparison of the execution time of each method for evaluating the computational complexity. As expected, the proposed efficient design considerably reduced the execution time compared with the optimal design, which yielded an execution time of about 1400 s, while the proposed design had an execution time within \([20,230]\) s. It was shown that the execution time of the proposed design was comparable to those of the methods with \(H_{\text{min}}\) and \(H_{\text{max}}\), except when \(\rho=0.9\). Next, we conducted performance evaluations in Scenario 2, where \([\phi_{1},\phi_{2}]=[-60^{\circ},60^{\circ}]\). Figure 4 shows the transmit beam patterns in this scenario. Figure 3: Performance comparison of various methods in Scenario 1, where \([\phi_{1},\phi_{2}]=[-20^{\circ},20^{\circ}]\). As expected, with the optimal and proposed designs, the peak values of the main beams for radar sensing were lower than those obtained with the radar-only design. The main beams at \(-40^{\circ}\) and \(40^{\circ}\) were broader compared with the radar-only design. This is because the UAV transmitted signals not only for radar sensing, i.e., in the \(-40^{\circ}\) and \(40^{\circ}\) directions, but also in the directions of the EHDs for WPT, i.e., \(-60^{\circ}\) and \(60^{\circ}\). Notably, the main beam width at \(40^{\circ}\) was much greater than that at \(-40^{\circ}\) in order to transfer more power to the second EHD at \(60^{\circ}\) compared with the first EHD at \(-60^{\circ}\). On the other hand, the beam pattern obtained with the proposed efficient design was relatively different from that obtained with the optimal design, especially in the ranges of \([-90^{\circ},-35^{\circ}]\) and \([35^{\circ},90^{\circ}]\). As such, Figure 5a-c illustrates the performance evaluations in Scenario 2, where \([\phi_{1},\phi_{2}]=[-60^{\circ},60^{\circ}]\). As shown in Figure 5a,b, the proposed efficient design outperformed the methods with \(H_{\text{min}}\) and \(H_{\text{max}}\) in terms of the objective function value and the trade-off region, respectively. Figure 4: Beam patterns of various methods with \(\rho=0.6\) in Scenario 2, where \([\phi_{1},\phi_{2}]=[-60^{\circ},60^{\circ}]\). However, it displayed performance degradation compared with the optimal design. The reason is that the main beams at \(-40^{\circ}\) and \(40^{\circ}\) obtained with the proposed efficient design were notably sharper than those obtained with the optimal design, as shown in Figure 4. However, the performance gap decreased as the weighting factor (\(\rho\)) increased. Figure 5c shows that the execution time of the optimal design was about \(1400\) s, whereas that of the proposed efficient design was in the range of \([58,760]\), where the time increased as \(\rho\) increased. As expected, it was verified that the proposed efficient design considerably outperformed the optimal design in terms of execution time. Finally, we conducted performance evaluations in Scenario 3, where \([\phi_{1},\phi_{2}]=[-80^{\circ},80^{\circ}]\). Figure 6 illustrates the transmit beam patterns in this scenario. As shown in Figure 6, with both the proposed optimal and efficient designs, the peak values of the main beams in the dominant directions for radar sensing, i.e., \([-40^{\circ},0^{\circ},40^{\circ}]\), were comparable to those obtained with the radar-only design. On the other hand, the beam pattern in the side-lobe regions within \([-90^{\circ},-59^{\circ}]\) and \([57^{\circ},90^{\circ}]\) was enhanced to transmit more power. The reason is that the UAV transmitted signals in the directions of the EHDs for WPT, i.e., \(-80^{\circ}\) and \(80^{\circ}\).