# Two Low-Complexity Efficient Beamformers for an IRS- and UAV-Aided Directional Modulation Network
## 3 The Proposed Beamforming Methods: Max-SR-SLNR and MRT-NSP-PA
### The Proposed Max-SR-SLNR  
First, we optimized the AN beamforming vector, which is independent of \(\mathbf{\Theta}\) and \(\mathbf{v}_{CM}\). [PERSON] projects the \(\mathbf{v}_{AN  
Next, we designed the alternating iterative optimization problem with two variables, \(\mathbf{v}_{CM}\) and \(\mathbf{\Theta}\). The optimization problem with the criterion of maximizing the SR can be expressed as  
\[\max_{\mathbf{v}_{CM},\mathbf{\Theta}}R_{s}(\mathbf{v}_{CM}, \mathbf{\Theta}) \tag{19a}\] \[\text{such that }\mathbf{v}_{CM}^{H}\mathbf{v}_{CM}=1,\mathbf{ \theta}^{H}\mathbf{\theta}=N_{s}. \tag{19b}\]  
The SINR of Bob can be re-expressed as follows  
\[\gamma_{b}=\frac{\mathbf{v}_{CM}^{H}\mathbf{h}_{b1}\mathbf{h}_{ b1}^{H}\mathbf{v}_{CM}}{\mathbf{v}_{AN}^{H}\mathbf{h}_{b2}\mathbf{h}_{b2}^{H} \mathbf{v}_{AN}+\sigma_{b}^{2}}, \tag{20}\]  
where  
\[\mathbf{h}_{b1}^{H} =\Big{(}\sqrt{\beta_{1}P_{l}g_{ab}}\mathbf{h}_{ab}^{H}+\sqrt{\beta _{1}P_{l}g_{aib}}\mathbf{h}_{b0}^{H}\mathbf{\Theta}\mathbf{H}_{ai}\Big{)},\] \[\mathbf{h}_{b2}^{H} =\Big{(}\sqrt{\beta_{2}P_{l}g_{ab}}\mathbf{h}_{ab}^{H}+\sqrt{\beta _{2}P_{l}g_{aib}}\mathbf{h}_{b0}^{H}\mathbf{\[PERSON]}\mathbf{H}_{ai}\Big{)}. \tag{21}\]  
Accordingly, the rate of Bob can be rewritten as  
\[R_{b}=\log_{2}\!\left(1+\frac{\mathbf{v}_{CM}^{H}\mathbf{h}_{b1} \mathbf{h}_{b1}^{H}\mathbf{v}_{CM}}{\mathbf{v}_{AN}^{H}\mathbf{h}_{b2}\mathbf{ h}_{b2}^{H}\mathbf{v}_{AN}+\sigma_{b}^{2}}\right). \tag{22}\]  
Similarly, the SINR of [PERSON] can be rewritten as  
\[\gamma_{e}=\frac{\mathbf{v}_{CM}^{H}\mathbf{h}_{e1}\mathbf{h}_{ e1}^{H}\mathbf{v}_{CM}}{\mathbf{v}_{AN}^{H}\mathbf{h}_{e2}\mathbf{h}_{e2}^{H} \mathbf{v}_{AN}+\sigma_{e}^{2}}, \tag{23}\]  
where  
\[\mathbf{h}_{e1}^{H} =\Big{(}\sqrt{\beta_{1}P_{l}g_{ae}}\mathbf{h}_{ae}^{H}+\sqrt{ \beta_{1}P_{l}g_{aib}}\mathbf{h}_{ie}^{H}\mathbf{\Theta}\mathbf{H}_{ai}\Big{)},\] \[\mathbf{h}_{e2}^{H} =\Big{(}\sqrt{\beta_{2}P_{l}g_{ae}}\mathbf{h}_{ae}^{H}+\sqrt{ \beta_{2}P_{l}g_{aib}}\mathbf{h}_{ie}^{H}\mathbf{\Theta}\mathbf{H}_{ai}\Big{)}. \tag{24}\]  
Therefore, the rate of [PERSON] can be expressed as follows  
\[R_{e}=\log_{2}\!\left(1+\frac{\mathbf{v}_{CM}^{H}\mathbf{h}_{e1} \mathbf{h}_{e1}^{H}\mathbf{v}_{CM}}{\mathbf{v}_{AN}^{H}\mathbf{h}_{e2}\mathbf{ h}_{e2}^{H}\mathbf{v}_{AN}+\sigma_{e}^{2}}\right). \tag{25}\]  
According to (22) and (25), we designed the CM beamforming vectors using the criterion of maximizing the SR, and (19) can be transformed into  
\[\text{such that }\mathbf{v}_{CM}^{H}\mathbf{v}_{CM}=1,\mathbf{ \theta}^{H}\mathbf{\theta}=N_{s}. \tag{19b}\]0 \[\text{such that }\mathbf{v}_{CM}^{H}\mathbf{v}_{CM}=1,\mathbf{ \theta}^{H}\mathbf{\theta}=N_{s}. \tag{19b}\]1  
where \(a=\mathbf{v}_{AN}^{H}\mathbf{h}_{b2}\mathbf{h}_{b2}^{H}\mathbf{v}_{AN}\), and \(b=\mathbf{v}_{AN}^{H}\mathbf{h}_{e2}\mathbf{h}_{e2}^{H}\mathbf{v}_{AN}\). Accordingly, the Rayleigh-Ritz ratio theorem can be used, and \(\mathbf{v}_{CM}\) is the eigenvector corresponding to the largest eigenvalue of the following formula  
\[\text{such that }\mathbf{v}_{CM}^{H}\mathbf{v}_{CM}=1,\mathbf{ \theta}^{H}\mathbf{\theta}=N_{s}. \tag{19b}\]2The signal-to-leakage-noise ratio (SLNR) method was designed as the PSM of the IRS [41] as follows  
\[\text{such that }\mathbf{v}_{CM}^{H}\mathbf{v}_{CM}=1,\mathbf{ \theta}^{H}\mathbf{\theta}=N_{s}. \tag{19b}\]3 \[\text{such that }\mathbf{v}_{CM}^{H}\mathbf{v}_{CM}=1,\mathbf{ \theta}^{H}\mathbf{\theta}=N_{s}. \tag{19b}\]4  
where the objective function of (28) is  
\[\text{such that }\mathbf{v}_{CM}^{H}\mathbf{v}_{CM}=1,\mathbf{ \theta}^{H}\mathbf{\theta}=N_{s}. \tag{19b}\]5  
One obtains  
\[\text{such that }\mathbf{v}_{CM}^{H}\mathbf{v}_{CM}=1,\mathbf{ \theta}^{H}\mathbf{\theta}=N_{s}. \tag{19b}\]6  
where \(\mathbf{a}\in\mathbb{C}^{N_{s}\times 1}\), and \(\mathbf{b}\in\mathbb{C}^{N_{s}\times 1}\).