## 3 Proposed Max-SNR-FP Scheme
### Optimize \(\mathbf{v}\) Given \(\mathbf{\Phi}\) and \(\mathbf{\Psi}\)
We compare the achievable rates of three proposed methods with those of the benchmark schemes of active IRS, passive IRS, no IRS, random-phase IRS, and the existing method in [34]. The achievable rates of the proposed Max-SNR-FP, Max-SNR-EAR, and Max-SNR-MM schemes gradually increased with the increase in \(M\), and the first proposed scheme was better than the rest and the existing method in [34]. The achievable rates of the three proposed schemes outperform those of the passive IRS, random-phase IRS, and that without an IRS. In addition, when \(M\) tended towards a large scale, the difference in achievable rates between the three proposed schemes and active IRS gradually decreased. This was due to the fact that, when a limited power budget is shared by a large number of active elements, the amplitudes of these elements shrink. \[=\frac{|\widehat{\mathbf{\theta}}^{H}\mathbf{h}_{b}|^{2}}{\sigma_{r}^ {2}|\widehat{\mathbf{\theta}}^{H}\mathbf{H}_{b}|^{2}+\sigma_{b}^{2}}. \tag{47}\]6 Achievable rate versus the numbers of IRS phase shift elements. \[=\frac{|\widehat{\mathbf{\theta}}^{H}\mathbf{h}_{b}|^{2}}{\sigma_{r}^ {2}|\widehat{\mathbf{\theta}}^{H}\mathbf{H}_{b}|^{2}+\sigma_{b}^{2}}. \tag{47}\]7 Convergence of the proposed methods at different base station power levels. Figure 6 presents the achievable rate versus the number \(M_{a}\) of active elements on the hybrid IRS of three proposed schemes and benchmark schemes. When \(M=2\), the achievable rate of the proposed Max-SNR-EAR method was similar to that of the passive IRS. The difference of the achievable rates between the proposed Max-SNR-MM method and the existing method in [34] method was trivial regardless of the value of \(M_{a}\). With the increase in \(M_{a}\), the hybrid IRS gradually transformed into an active IRS, and the achievable rates of the three proposed methods and the existing method in [34] gradually increased, while those of the passive IRS, random-phase IRS, and no IRS remained constant. This reveals the advantages of active elements in hybrid IRS for enhancing the performance of the system. Figure 7 plots the curves of the computational complexity versus the number \(M\) of hybrid IRS elements. This figure reveals that the difference between the computational complexity of the proposed Max-SNR-EAR and Max-SNR-MM methods gradually increased with the increase in \(M\). Moreover, when \(M\) tended towards a large scale, the computational complexities of the existing method in [34] and the proposed Max-SNR-FP method were far higher than those of the proposed Max-SNR-MM and proposed Max-SNR-EAR methods. \[=\frac{|\widehat{\mathbf{\theta}}^{H}\mathbf{h}_{b}|^{2}}{\sigma_{r}^ {2}|\widehat{\mathbf{\theta}}^{H}\mathbf{H}_{b}|^{2}+\sigma_{b}^{2}}. \tag{47}\]8 Achievable rate versus the numbers of IRS phase shift elements. \[=\frac{|\widehat{\mathbf{\theta}}^{H}\mathbf{h}_{b}|^{2}}{\sigma_{r}^ {2}|\widehat{\mathbf{\theta}}^{H}\mathbf{H}_{b}|^{2}+\sigma_{b}^{2}}. \tag{47}\]9 Achievable rate versus the numbers of IRS phase shift elements. ## 7 Conclusions

In this paper, we investigated a hybrid IRS UAV-assisted DM network. To fully explore the advantages of the hybrid IRS and maximize the achievable rate, three alternating iterative schemes, namely, Max-SNR-FP, Max-SNR-EAR, and Max-SNR-MM, were proposed to jointly design the beamforming vector, passive IRS PSM, and active IRS PSM by alternately optimizing one and giving the rest. The active and passive IRS phase-shift matrices of the first two proposed methods are optimized separately, while the third method optimizes them jointly. The simulation results reveal that the achievable rates of the three proposed methods increased with the number of hybrid IRS elements and were superior to those of without IRS, random-phase IRS, and passive IRS. When the number of IRS phase shift elements tended towards a large scale, the difference in achievable rates among the three proposed methods was trivial.