# 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 MRT-NSP-PA  
In the above subsection, the iterative optimization process between variables \(\mathbf{v}_{CM}\) and \(\mathbf{\theta}\) led to a high computational complexity. In order to reduce the complexity of the Max-SR-SLNR, a low-complexity MRT-NSP-PA method is presented in which the three variables \(\mathbf{v}_{CM}\), \(\mathbf{v}_{AN}\), and \(\mathbf{\theta}\) were designed independently as follows.  
Let us define  
\[\mathbf{h}_{ai}=\mathbf{h}^{H}\big{(}\theta_{AI}^{t}\big{)}. \tag{35}\]  
First, the MRT method was used to design \(\mathbf{v}_{CM}\). Taking the transmit power limit into account, the final CM beamforming vector can be directly given by  
\[\mathbf{v}_{CM}=\frac{\mathbf{h}_{ai}}{\|\mathbf{h}_{ai}\|}. \tag{36}\]  
In the same manner, the AN beamforming method based on the MRT and NSP is  
\[\mathbf{v}_{AN}=\frac{\mathbf{T}_{-ae}\mathbf{h}_{ae}}{\|\mathbf{T}_{-ae} \mathbf{h}_{ae}\|}. \tag{37}\]  
Next, we designed the IRS phase matrix \(\mathbf{\Theta}\), which was completely different to the Max-SR-SLNR method. The received CM power via the cascaded path at Bob is equal to  
\[P_{b}=\beta_{1}P_{Sabi}\mathbf{v}_{CM}^{H}\mathbf{h}_{ai}^{H}\mathbf{\Theta}^{ H}\mathbf{h}_{ib}\mathbf{h}_{ib}^{H}\mathbf{\Theta}\mathbf{H}_{ai}\mathbf{v}_{ CM}. \tag{38}\]  
Equation (38) can be rewritten as  
\[P_{b}= \beta_{1}P_{Sabi}\mathbf{\theta}^{H}\operatorname{diag}(\mathbf{ H}_{ai}\mathbf{v}_{CM})\mathbf{h}_{ib}. \tag{39}\] \[\mathbf{h}_{ib}^{H}\operatorname{diag}(\mathbf{H}_{ai}\mathbf{v} _{CM})\mathbf{\theta}. \tag{40}\]  
To maximize the received CM power along the cascaded path from Alice to Bob via the IRS at Bob, the PA method directly gives the value of \(\mathbf{\theta}\) as follows:  
\[\mathbf{\theta}=e^{-j\arg\left(\mathbf{h}_{ib}^{H}\operatorname{diag}( \mathbf{H}_{ai}\mathbf{v}_{CM})\right)^{H}}. \tag{41}\]  
The complexity of this MRT-NSP-PA method is  
\[\mathcal{O}\left(2N_{s}^{2}+2N_{a}N_{s}-2N_{s}+4N_{a}+2N_{a}^{2}-2\right) \tag{42}\]  
FLOPs.  
From (34) and (42), it can be noticed that the computational complexity of the Max-SR-SLNR method was higher than that of the MRT-NSP-PA method. Because the beamforming vectors of the MRT-NSP-PA method were designed as closed form solutions, they have the advantage of low computational complexity. However, there was some performance loss of the SR in the MRT-NSP-PA method. Of the two methods, the MRT-NSP-PA method was more feasible. This is because in the MRT-NSP-PA algorithm, the CM, the AN beamforming vector, and the PSM of the IRS were designed as closed form solutions.  
In the above, the CM beamforming was only phase-aligned to the Alice-to-IRS channel, ignoring the direct path to the desired user Bob, etc. Therefore, designing the CM beamforming in this case led to performance loss. In order to explore the CM beamforming direction on the SR performance, we explored the relationship between the number of IRS reflecting elements and the direction of the CM beamforming. Thus, the CM beamforming was allowed to rotate in the angle range \([0,\pi]\). In this case, the direction of the CM beamforming \(\theta_{CM}\) is written as  
\[\theta_{CM}\in[0,\pi]. \tag{43}\]  
In what follows, we adopted three methods to design \(\mathbf{v}_{CM}\):  
\[\mathbf{v}_{CM}=\frac{\mathbf{h}_{ab}}{\|\mathbf{h}_{ab}\|}, \tag{44}\]  
\[\mathbf{v}_{CM}=\frac{(\mathbf{h}_{ai}+\mathbf{h}_{ab})}{\|(\mathbf{h}_{ai}+ \mathbf{h}_{ab})\|}, \tag{45}\]  
and  
\[\mathbf{v}_{CM}=\frac{\mathbf{h}_{ai}}{\|\mathbf{h}_{ai}\|}. \tag{46}\]