## 3 Energy and Information Beamforming
### Strongest-Path Beamforming Scheme
#### 3.2.1. Average Useful Signal Power Maximization  
According to (13), the average power of useful signal can be expressed as:  
\[Q_{l}^{\text{U}}=\mathbb{E}\left[P_{l}\mathbf{w}_{D,l}^{H}\mathbf{h}_{l}\mathbf{h }_{l}^{H}\mathbf{w}_{D,l}\right]. \tag{25}\]  
Directly maximizing \(Q_{l}^{\text{U}}\) with respect to \(\mathbf{w}_{E,l}\) and \(\mathbf{w}_{D,l}\) is still difficult. By using the inequality \(\mathbb{E}\left[XY\right]-\mathbb{E}\left[X\right]\mathbb{E}\left[Y\right]= \text{cov}\left(X,Y\right)\geq 0\) for positive \(\left\{X,Y\right\}\), a lower bound on \(Q_{l}^{\text{U}}\) can be obtained as:  
\[Q_{l}^{\text{U}} \geq\mathbb{E}\left[P_{l}\right]\mathbb{E}\left[\mathbf{w}_{D,l}^ {H}\mathbf{h}_{l}\mathbf{h}_{l}^{H}\mathbf{w}_{D,l}\right] \tag{26}\] \[=\mathbf{w}_{E,l}^{H}\mathbf{K}_{l}\mathbf{w}_{E,l}\mathbf{w}_{D, l}^{H}\mathbf{C}_{l}\mathbf{w}_{D,l}\triangleq Q_{l}^{\text{U,LB}}.\]  
Since \(\mathbf{K}_{l}\) has the same eigenvectors as \(\mathbf{C}_{l}\), designing the energy and information beamformers to maximize the lower bound gives rise to the solution as follows:  
\[\mathbf{w}_{E,l}=\mathbf{w}_{D,l}=\mathbf{u}_{\max}\left\{\mathbf{C}_{l} \right\}. \tag{27}\]  
By exploiting the asymptotic property of the channel covariance matrix, we can further approximate the energy and information beamformers using the virtual AoD information of the sensor. This is stated in the following theorem. **Theorem 1**. : _When the number of AP antennas tends to infinity, i.e., \(\left\{N_{x},N_{y}\right\}\rightarrow\infty\), the energy and information beamformers converge to:_  
\[\mathbf{w}_{E,l}=\mathbf{w}_{D,l}=\frac{1}{\sqrt{N_{x}N_{y}}}\mathbf{b}\left( \rho_{x,l}\right)\otimes\mathbf{a}\left(\rho_{y,l}\right), \tag{28}\]  
_and the lower bound of average useful signal power can be expressed as:_  
\[Q_{l}^{\text{U,LB}} =\left\{\frac{\eta\tau}{1-\tau}p_{l}\left(\frac{\beta_{l}K_{l}}{ K_{l}+1}N_{x}N_{y}+\frac{\beta_{l}}{K_{l}+1}S_{l}\left(\rho_{x,l},\rho_{y,l} \right)\right)\right. \tag{29}\] \[\left.-\frac{E_{l}^{p}}{\left(1-\tau\right)T}\right\}\left(\frac{ \beta_{l}K_{l}}{K_{l}+1}N_{x}N_{y}+\frac{\beta_{l}}{K_{l}+1}S_{l}\left(\rho_{ x,l},\rho_{y,l}\right)\right).\]  
Proof. : This can be readily proven using the asymptotic expressions for the dominant eigenvector and eigenvalue of \(\mathbf{C}_{l}\), presented in (A6) of Appendix A. From Theorem 1, it is seen that the beamformers are designed to match the LoS path of the channel, i.e., to ensure that the largest beamforming gain is achieved at the LoS path. Moreover, for the commonly-used models, the PAS of NLoS channel \(S_{l}\left(\rho_{x},\rho_{y}\right)\) achieves the maximum at \(\left\{\rho_{x},\rho_{y}\right\}=\left\{\rho_{x,l},\rho_{y,l}\right\}\)[15; 36], i.e., the virtual AoDs corresponding to the sensors' physical AoDs in horizontal and vertical directions. As a result, the proposed beamformers also match the strongest NLoS path in the angular domain. Therefore, this scheme is named the strongest-path energy and information beamforming. On the other hand, it is seen that both the energy and information beamformers have a constant envelope, which can be realized using phase shifting networks in the analog-domain. The total number of required RF chains at AP can be reduced to \(L\), which is equal to the number of sensors [37]. Another important observation from Lemma 1 is that, if we neglect the processing energy \(E_{l}^{p}\), the average useful signal power decays with the squared path loss. Thus, a low altitude of AP is preferred to improve the efficiencies of WET and WIT of the single sensor. However, with the decrease of altitude, the number of covered sensors by AP is reduced as well, which may degrade the performance of the whole hybrid air-terrestrial network. Therefore, the planning of AP's altitude is an interestingand important research topic. If the prior of sensor density is available, a raw idea to solve this problem is presented as follows.