# Two Low-Complexity Efficient Beamformers for an IRS- and UAV-Aided Directional Modulation Network
## 2 System Model  
As illustrated in Figure 1, an IRS- and UAV-aided DM network is sketched, where the IRS was mounted on a UAV, and the UAV hovered at a suitable height such that the UAV could see [PERSON], [PERSON], or both. Alice was a transmitter equipped with \(N_{a}\) antennas, Bob and Eve were the legal and illegal receivers equipped with a single antenna, respectively, and the IRS was equipped with \(N_{s}\) passive reflecting elements. The baseband signal sent from Alice is given by  
\[\mathbf{x}=\sqrt{\beta_{1}P_{t}}\mathbf{v}_{CM}s+\sqrt{\beta_{2}P_{t}}\mathbf{ v}_{AN}z, \tag{1}\]  
where \(\beta_{1}\) denotes the power allocation (PA) factor of the CM, \(\beta_{2}\) represents the PA factor of the AN, with \(\beta_{1}+\beta_{2}=1\), \(P_{t}\) stands for the transmit power, \(\mathbf{v}_{CM}\) represents the precoding vector of the CM with \(\mathbf{v}_{CM}\in\mathbb{C}^{N_{a}\times 1}\), \(\mathbf{v}_{AN}\) represents the precoding vector of the AN with \(\mathbf{v}_{AN}\in\mathbb{C}^{N_{a}\times 1}\), \(s\) represents the CM, with a constraint that \(\mathbb{E}\big{[}|s|^{2}\big{]}=1\), and \(z\) is the AN with \(\mathbb{E}\big{[}|z|^{2}\big{]}=1\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Conjugate Transpose** & **Inverse** & **Pseudo-Inverse** & **Two-Norm** & **Diagonal Operator** \\ \hline \((\cdot)^{H}\) & \((\cdot)^{-1}\) & \((\cdot)^{\ddagger}\) & \(\|\cdot\|\) & \(\mathrm{diag}(\cdot)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Explanation of mathematical symbols. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Previous Work** & **Aim of the Method** & **Strong Point** & **Limitation** \\ \hline Existing method in [36] & Boost SR & Low complexity & Poor SR \\ \hline SDR in [37] & Maximize SR & High SR & High complexity \\ \hline \hline \end{tabular}
\end{table} 
Table 2: A comparison table with previous works. Figure 1: System model drawing of an IRS- and UAV-aided DM network. The signal received at Bob is represented as  
\[y_{b}=\left(\sqrt{g_{ab}}\mathbf{h}_{ab}^{H}+\sqrt{g_{aib}}\mathbf{h }_{ib}^{H}\mathbf{\Theta}\mathbf{H}_{ai}\right)\mathbf{x}+n_{b}\] \[=\sqrt{g_{ai}\beta_{1}P_{1}}\mathbf{h}_{ab}^{H}\mathbf{v}_{CM} \mathbf{s}+\sqrt{g_{aib}\beta_{1}P_{1}}\mathbf{h}_{ib}^{H}\mathbf{\Theta} \mathbf{H}_{ai}\mathbf{v}_{CM}\mathbf{s}+\] \[\sqrt{g_{ab}\beta_{2}P_{1}}\mathbf{h}_{ab}^{H}\mathbf{v}_{AN}+ \sqrt{g_{aib}\beta_{2}P_{1}}\mathbf{h}_{ib}^{H}\mathbf{\Theta}\mathbf{H}_{ai} \mathbf{v}_{AN}z+n_{b}, \tag{2}\]  
where \(\mathbf{h}_{ab}\in\mathbb{C}^{N_{x}\times 1}\), \(\mathbf{H}_{ai}=\ln\left(\theta_{A1}^{\prime},\varphi_{A1}^{\prime}\right) \mathbf{h}^{H}\left(\theta_{A1}^{\prime},\varphi_{A1}^{\prime}\right)\in \mathbb{C}^{N_{x}\times N_{x}}\), and \(\mathbf{h}_{ib}\in\mathbb{C}^{N_{x}\times 1}\) represent the Alice-to-Bob, Alice-to-IRS, and IRS-to-Bob channels, respectively. \(g_{ab}\) and \(g_{aib}=g_{ai}g_{ib}\) are the path loss coefficients of the Alice-to-Bob direct channel and the cascade channel, respectively. \(n_{b}\sim\mathcal{CN}\left(0,\sigma_{b}^{2}\right)\) is the additive white Gaussian noise (AWGN) at Bob. The phase shifting matrix (PSM) of the IRS is given by  
\[\mathbf{\Theta}=\text{diag}\left(e^{i\phi_{1}},\cdots,e^{i\phi_{m}},\cdots,e^ {i\phi_{N_{x}}}\right), \tag{3}\]  
where \(\phi_{m}\) is the reflected phase of the m-th cell on the IRS, and \(\mathbf{\theta}^{H}=\left(e^{i\phi_{1}},\cdots,e^{i\phi_{N_{x}}}\right)\). The normalized channel vector can be written as  
\[\mathbf{h}(\theta,\varphi)=\frac{1}{\sqrt{N}}\left[e^{i2\pi\Phi_{\theta, \varphi}(1)},\cdots,e^{i2\pi\Phi_{\theta,\varphi}(2)},\cdots,e^{i2\pi\Phi_{ \theta,\varphi}(N)}\right]^{H}, \tag{4}\]  
where the phase shift \(\Phi_{\theta,\varphi}(n)\) is given by  
\[\Phi_{\theta,\varphi}(n)=-\frac{d}{\lambda}\left(n-\frac{N+1}{2}\right)\cos \theta\cos\varphi,\quad n=1,\cdots,N, \tag{5}\]  
where \(d\) is the array element spacing, \(\lambda\) is the carrier wavelength, \(\theta\) is the angle of departure, and \(\varphi\) is the pitch angle.