## 2 System Model  
As indicated in Figure 1, a hybrid IRS and UAV-aided DM network was considered in which the IRS was installed on a UAV, assuming that the UAV operated at a sufficient altitude, and all channels were line-of-propagation. There were a base station (BS) with \(N\) antennas and a user Bob with single antenna. The hybrid IRS was equipped with \(M\) elements that consisted of \(M_{a}\) active and \(M_{p}\) passive IRS reflecting elements (\(1\leq M_{a}\leq M_{p}\), \(M=M_{a}+M_{p}\)). It was assumed that the active elements enabled adjustments in both the amplitude and phase, while the passive ones only tuned the phase of the incident signal. The signals that reflected more than or equal to twice on the hybrid IRS were negligible due to the severe PL [40]. We supposed that all the channel state information was completely accessible owing to the channel estimation [41]. Similar to a conventional fully passive IRS, it was assumed that each element of the hybrid IRS could independently reflect the incident signals. Let us denote the set of \(M_{a}\) active elements by \(\Omega\). \(\mathbf{\Theta}=\text{diag}\{\theta_{1},\ldots,\theta_{m},\ldots,\theta_{M}\} \in\mathbb{C}^{M\times M}\), \(\mathbf{\Psi}=\text{diag}\{\psi_{1},\ldots,\psi_{m},\ldots,\psi_{M}\}\in\mathbb{ C}^{M\times M}\), and \(\mathbf{\Phi}=\text{diag}\{\phi_{1},\ldots,\phi_{m},\ldots,\phi_{M}\}\in\mathbb{ C}^{M\times M}\) represent the reflection coefficients of total elements, active elements, and passive elements of hybrid IRS, respectively, where  
\[\theta_{m}=\left\{ \begin{array}{ll}|\beta_{m}|e^{j\mu_{m}},&\text{if }m\in\Omega,\\ e^{j\mu_{m}},&\text{otherwise},\end{array} \right. \tag{1}\]  
\(\mu_{m}\in[0,2\pi)\) is the phase, and \(|\beta_{m}|\) represents the amplifying coefficient that is subject to the power of the IRS active elements. Let us define  
\[\mathbf{\theta}=[\theta_{1},\ldots,\theta_{m},\ldots,\theta_{M}]^{H},\mathbf{\psi}=[\psi_{1},\ldots,\psi_{m},\ldots,\psi_{M}]^{H},\mathbf{\phi}=[\psi_{1},\ldots,\phi_{m},\ldots,\phi_{M}]^{H}, \tag{2}\] \[\mathbf{\Psi}=\mathbf{E}_{M_{a}}\mathbf{\Theta},\ \mathbf{\Phi}=\mathbf{E}_{M_{p}}\mathbf{\Theta}, \tag{3}\]where  
\[\mathbf{E}_{M_{a}}+\mathbf{E}_{M_{p}}=\mathbf{I}_{M_{r}},\ \mathbf{E}_{M_{a}} \mathbf{E}_{M_{p}}=\mathbf{0}_{M}, \tag{4}\]  
Non-zero elements of diagonal matrix \(\mathbf{E}_{M_{a}}\in\mathbb{C}^{M\times M}\) were a unity whose positions were determined via \(\Omega\). The transmission signal at BS is expressed as follows:  
\[\mathbf{s}=\sqrt{P}\mathbf{v}\mathbf{x}, \tag{5}\]  
where \(P\) represents the transmit power, \(\mathbf{v}\in\mathbb{C}^{N\times 1}\) and \(x\) are the beamforming vector and the information symbol satisfying \(\mathbf{v}^{H}\mathbf{v}=1\) and \(\mathbb{E}[\|x\|^{2}]=1\), respectively. In the presence of the PL, the received signal at Bob is given by  
\[y_{b} =(\sqrt{\rho_{srb}}\mathbf{h}_{rb}^{H}\mathbf{\Theta}\mathbf{H}_{ sr}+\sqrt{\rho_{sb}}\mathbf{h}_{sb}^{H})\mathbf{s}+\sqrt{\rho_{rb}}\mathbf{h }_{rb}^{H}\mathbf{\Psi}\mathbf{n}_{rb}+n_{b}\] \[=\sqrt{P}(\sqrt{\rho_{srb}}\mathbf{h}_{rb}^{H}\mathbf{\Psi} \mathbf{H}_{sr}+\sqrt{\rho_{srb}}\mathbf{h}_{rb}^{H}\mathbf{\Phi}\mathbf{H}_{ sr}+\sqrt{\rho_{sb}}\mathbf{h}_{sb}^{H})\mathbf{v}\mathbf{x}+\sqrt{\rho_{rb}} \mathbf{h}_{rb}^{H}\mathbf{\Psi}\mathbf{n}_{r}+n_{b}, \tag{6}\]  
where \(\rho_{srb}=\rho_{srb}\rho_{rb}\) represents the synthetic PL coefficient of BS-to-IRS channel and IRS-to-Bob channel, \(\rho_{sb}\) and \(\rho_{rb}\) stand for the PL coefficient of BS-to-Bob channel and IRS-to-Bob channel, respectively. \(\mathbf{n}_{r}\sim\mathcal{CN}(\mathbf{0},\sigma_{r}^{2}\mathbf{I}_{M_{a}})\) and \(n_{b}\sim\mathcal{CN}(\mathbf{0},\sigma_{b}^{2})\) denote the complex additive white Gaussian noise at the \(M_{a}\) active elements of the hybrid IRS and at Bob, respectively.