## 3 Proposed Max-SNR-FP Scheme
### Optimize \(\mathbf{v}\) Given \(\mathbf{\Phi}\) and \(\mathbf{\Psi}\)
## 6 Simulation Results

Simulation results are shown regarding the performance of three proposed schemes in this section. Simulation parameters were as follows: \(N=8\), \(M=128\), \(M_{a}=32\), \(d=\lambda/2\), \(\sigma_{b}^{2}=-70\) dBm, \(\sigma_{r}^{2}=2\sigma_{b}^{2}\), \(P=25\) dBm, \(P_{r}^{\max}=30\) dBm. The base station, IRS (or UAV), and Bob were located at (0, 0, 0 m), (100\(\sqrt{2}\) m, 0 m, 100\(\sqrt{2}\) m), and (110\(\sqrt{3}\) m, 0 m, 110 m), respectively. The PL at distance \(d_{ab}\) is given by \(\rho(d_{ab})=\mathrm{PL}_{0}-10\gamma\mathrm{log}_{10}\frac{d_{ab}}{d_{0}}\), where \(\gamma\) represents the PL exponent, and \(\mathrm{PL}_{0}=-30\) dB represents PL reference distance \(d_{0}=1\) m. The PL exponents of all channels were 2. We selected the positions of the hybrid IRS active elements to be \(\Omega=\{1,\ldots,M_{a}\}\). To measure the performance of the proposed schemes, the following benchmark schemes were taken into account. 1. **No-IRS:** Without the IRS, the channel matrix and vector of IRS-related links become a zero matrix and zero vector, i.e., \(\mathbf{H}_{sr}=\mathbf{0}\) and \(\mathbf{h}_{rb}=\mathbf{0}\), and only beamforming is optimized. 2. **Random phase:** the hybrid IRS PSM is set to \(\mathbf{\Theta}=[\phi^{j_{1}},\ldots,\phi^{j_{m}},\ldots,\phi^{j_{M}}]\), where \(\mu_{m}\) is randomly selected from \([0,2\pi)\), and only beamforming is optimized. 3. **Passive IRS:** the number \(M_{a}\) of active IRS elements is equal to 0, and only beamforming and passive IRS PSM are optimized. 4. **Active IRS:** the number \(M_{p}\) of passive IRS elements is equal to 0, and only beamforming and active IRS PSM are optimized. First, the convergence behavior of the three proposed algorithms, namely, Max-SNR-FP, Max-SNR-EAR, and Max-SNR-MM, is investigated. Figure 2 presents the comparison of the convergence of the proposed algorithms at different base station powers: \(P=20\) and \(25\) dBm. Figure 2 shows that all proposed methods converged needing only a finite number of iterations. Regardless of \(P=20\) or \(25\) dBm, the proposed Max-SNR-EAR and Max-SNR-MM methods had faster convergence rates compared to that of the Max-SNR-FP method. Figure 3 demonstrates the curves of the achievable rate versus the base station transmit power \(P\), where \(M=32\) and \(M_{a}=10\). This figure reveals that the achievable rates of the three proposed methods and benchmark schemes increased as \(P\) gradually increased, and the achievable rates of the three proposed methods and the existing method in [34] were superior to those of the other benchmark methods of passive IRS, random-phase, and no IRS. Their achievable rates were about \(15\%\) better than that of the passive IRS and about \(25\%\) better than those of the no-IRS and random-phase schemes when \(P=20\). In addition, the achievable rate of the proposed Max-SNR-FP method exceeded those of the proposed Max-SNR-MM method, the existing method in [34], and the proposed Max-SNR-EAR method regardless of the value of \(P\). Figure 4 shows the curves of the achievable rate versus the IRS power budget \(P_{r}^{\max}\) ranging from 20 to 30 dBm, where \(M=32\) and \(M_{a}=10\). The achievable rates of the three proposed algorithms and the existing algorithm in [34] increased with the increase in maximal transmission power \(I_{r}^{\max}\). This was due to the fact that the hybrid IRS with active elements achieved a greater performance gain with increasing \(P_{r}^{\max}\). As \(P_{r}^{\max}\) increased, the difference among the achievable rates of the proposed Max-SNR-FP, Max-SNR-EAR, and Max-SNR-MM schemes, and the existing scheme in [34] gradually decreased. The decreasing order of their achievable rates was: Max-SNR-FP, Max-SNR-MM, the existing scheme in [34], and Max-SNR-EAR. Figure 5 depicts the curves of the achievable rate versus the number \(M\) of hybrid IRS phase-shift elements, where \(M=2M_{a}\).