## 4 Proposed Efficient Design for Transmit Waveform and UAV Altitude
To evaluate the proposed methods, we considered the following performance metrics, which are also detailed in the figures below:

* The transmit beam pattern in terms of the angular direction with the sample angle grids \([\theta_{i}]_{i=1}^{L}\). * The objective function value of problem \(\mathbf{P_{1}}\) in terms of the weighting factor \(\rho\). * The trade-off region between the beam pattern MSE, i.e., \(L_{\text{Rad}}(\mathbf{X},\gamma)\), and the harvested power MSE, i.e., \(L_{\text{WPT}}(\mathbf{X},h)\). * The execution time for computational complexity in terms of the weighting factor \(\rho\). The transmit beam pattern refers to the spatial power in the target directions achieved by each method. The objective function value indicates the quality of the solution found by that method. Hence, lower objective function values indicate better optimization performance. The trade-off region represents the range within which improving one metric results in a compromise in the other. It demonstrates the interaction and balance between radar sensing and WPT performance, highlighting the limitations in optimizing both simultaneously. The execution time of each method serves as a practical indicator of its computational complexity, with shorter execution times signifying lower complexity. Further, for performance comparisons, we considered the following four methods:

* The optimal waveform at UAV altitude \(H_{\text{min}}\), obtained by using problem \(\mathbf{P}_{2}^{\forall h}\) with \(h=H_{\text{min}}\). * The optimal waveform at UAV altitude \(H_{\text{max}}\), obtained by using problem \(\mathbf{P}_{2}^{\forall h}\) with \(h=H_{\text{max}}\). * The optimal design based on the proposed algorithm: the optimal waveform with the optimal UAV altitude obtained with Algorithm 1. * The proposed efficient design: the proposed waveform and UAV altitude obtained with Algorithm 2. Then, we evaluated the proposed methods with three sensing targets and two EHDs, i.e., Scenario 1 (\(Q=3\) and \(U=2\)). By referring to the benchmark scheme for the desired beam pattern [23; 24], the direction angles of the sensing targets were set to \([\theta_{1}^{\text{Tar}},\theta_{2}^{\text{Tar}},\theta_{3}^{\text{Tar}}]=[-40 ^{\circ},0^{\circ},40^{\circ}]\). Regarding the EHDs' locations, we set the horizontal distances from the UAV to \(r_{1}=50\) m and \(r_{2}=150\) m. In addition, to evaluate the proposed methods in various environments, we considered three different scenarios for the AoD pair of the EHDs, as follows:

* Scenario 1: \([\phi_{1},\phi_{2}]=[-20^{\circ},20^{\circ}]\). * Scenario 2: \([\phi_{1},\phi_{2}]=[-60^{\circ},60^{\circ}]\). * Scenario 3: \



As shown in Figure 2, with the proposed optimal and efficient designs, the peak values of the main beams in the dominant directions for radar sensing, i.e., \([-40^{\circ},0^{\circ},40^{\circ}]\), decreased compared to the radar-only design, while those of the beams in the direction of the EHDs, i.e., \([-20^{\circ},20^{\circ}]\), increased. The reason lies in the transmit power directed toward the EHDs. This result shows the effect of WPT on radar sensing. On the other hand, the peak values of the beams at \([20^{\circ}]\) were slightly larger than those at \([-20^{\circ}]\). This is because the UAV transmitted more power to the second EHD owing to the different horizontal distances of the two EHDs, i.e., \(r_{1}\
eq r_{2}\). Moreover, it was verified that the proposed efficient design demonstrated performance close to that of the optimal design. Figure 2(a)-c shows comparisons of the objective function values, the trade-off regions, and the execution times, respectively, in Scenario 1, where \([\phi_{1},\phi_{2}]=[-20^{\circ},20^{\circ}]\). Figure 2(a) shows that the optimal design based on the proposed method in Algorithm 1 yielded the smallest objective function value over the whole range of \(\rho\). The proposed efficient design outperformed the methods with UAV altitudes \(H_{\text{min}}\) and \(H_{\text{max}}\), achieving near-optimal performance.