## 3 Energy and Information Beamforming
### Statistical Max-SINR Scheme
#### 3.1.1 Energy Beamforming Design  
Similar to the instantaneous SLNR [34; 35], the statistical SLNR is defined as follows:  
\[\text{SLNR}_{l}=\frac{\mathbb{E}\left[P_{l}\mathbf{w}_{D,l}^{H}\mathbf{h}_{l} \mathbf{h}_{l}^{H}\mathbf{w}_{D,l}\right]}{\mathbb{E}\left[P_{l}\sum\limits_{ k=1}^{L}\mathbf{w}_{D,k}^{H}\mathbf{h}_{l}\mathbf{h}_{l}^{H}\mathbf{w}_{D,k} \right]+\sigma^{2}}, \tag{15}\]where the first term on the denominator indicates the total power of interferences in the WIT phase caused by the transmission of sensor \(l\). For convenience, we define \(G_{l,k}=\mathbf{w}_{D,k}^{H}\mathbf{h}_{l}\mathbf{h}_{l}^{H}\mathbf{w}_{D,k}\) as the combining gain of information beamformer \(\mathbf{w}_{D,k}\) on the signal of sensor \(l\). Substituting the expression of \(P_{l}\) given by (10) into (15), the statistical SLNR can be rewritten as:  
\[ \begin{split}&\text{SLNR}_{l}\\ &=\frac{\mathbb{E}\left[\left(\frac{\eta\tau}{1-\tau}p_{l} \mathbf{w}_{E,l}^{H}\mathbf{h}_{l}\mathbf{h}_{l}^{H}\mathbf{w}_{E,l}-\frac{E_{ l}^{p}}{(1-\tau)\bar{T}}\right)G_{l,l}\right]}{\mathbb{E}\left[\left(\frac{\eta\tau}{1- \tau}p_{l}\mathbf{w}_{E,l}^{H}\mathbf{h}_{l}\mathbf{h}_{l}^{H}\mathbf{w}_{E,l} -\frac{E_{l}^{p}}{(1-\tau)\bar{T}}\right)\sum\limits_{k=1,k\
eq l}^{L}G_{l,k} \right]+\sigma^{2}}\\ &\geq\frac{\mathbb{E}\left[\frac{\eta\tau}{1-\tau}p_{l}\mathbf{ w}_{E,l}^{H}\mathbf{h}_{l}\mathbf{h}_{l}^{H}\mathbf{w}_{E,l}-\frac{E_{l}^{p}}{(1- \tau)\bar{T}}\right]\mathbb{E}\left[G_{l,l}\right]}{\mathbb{E}\left[\left(\frac {\eta\tau}{1-\tau}p_{l}\mathbf{w}_{E,l}^{H}\mathbf{h}_{l}^{H}\mathbf{w}_{E,l} -\frac{E_{l}^{p}}{(1-\tau)\bar{T}}\right)\sum\limits_{k=1,k\
eq l}^{L}G_{l,k} \right]+\sigma^{2}}\\ &=\frac{\mathbb{E}\left[G_{l,l}\right]\mathbf{w}_{E,l}^{H}\mathbf{ K}_{l}\mathbf{w}_{E,l}}{\mathbf{w}_{E,l}^{H}\left(\mathbf{K}_{l}\sum\limits_{k=1,k \
eq l}^{L}\mathbb{E}\left[G_{l,k}\right]+\sigma^{2}\mathbf{I}_{N_{x}N_{y}} \right)\mathbf{w}_{E,l}}\\ &\overset{\Delta}{=}\text{SLNR}_{l}^{\text{LB}},\end{split}  \tag{16}\]  
where \(\mathbf{K}_{l}=\frac{\eta\tau}{1-\tau}p_{l}\mathbf{C}_{l}-\frac{E_{l}^{p}}{(1 -\tau)\bar{T}}\mathbf{I}_{N_{x}N_{y}}\). The second line follows from the inequality \(\mathbb{E}\left[XY\right]-\mathbb{E}\left[X\right]\mathbb{E}\left[Y\right]= \text{cov}\left(X,Y\right)\geq 0\) when \(X\) and \(Y\) are positive. Based on (16), the energy beamformer is designed to maximize the lower bound of statistical SLNR, i.e.,  
\[\max_{\mathbf{w}_{E,l}}\text{SLNR}_{l}^{\text{LB}},\text{ s.t. }\left\|\mathbf{w}_{E,l}\right\|=1. \tag{17}\]  
It is easy to see that (17) is the generalized Rayleigh quotient problem, which can be solved as:  
\[\mathbf{w}_{E,l}=\mathbf{u}_{\max}\left\{\left(\mathbf{K}_{l}\sum\limits_{k=1,k\
eq l}^{L}\mathbb{E}\left[G_{l,k}\right]+\sigma^{2}\mathbf{I}_{N_{x}N_{y}} \right)^{-1}\mathbf{K}_{l}\right\}. \tag{18}\]  
Note that the matrices \(\mathbf{K}_{l}\) and \(\left(\mathbf{K}_{l}\sum\
olimits_{k=1,k\
eq l}^{L}\mathbb{E}\left[G_{l,k} \right]+\sigma^{2}\mathbf{I}_{N_{x}N_{y}}\right)^{-1}\) have the same eigenvectors with \(\mathbf{C}_{l}\). Let \(\lambda_{i}\) denote the \(i\)-th largest eigenvalue of \(\mathbf{C}_{l}\); the \(i\)-th largest eigenvalue of \(\left(\mathbf{K}_{l}\sum\
olimits_{k=1,k\
eq l}^{L}\mathbb{E}\left[G_{l,k} \right]+\sigma^{2}\mathbf{I}_{N_{x}N_{y}}\right)^{-1}\mathbf{K}_{l}\) can be expressed as:  
\[\lambda_{i}=\frac{\frac{\eta\tau}{1-\tau}p_{l}\lambda_{i}-\frac{E_{l}^{p}}{(1 -\tau)\bar{T}}}{\left(\frac{\eta\tau}{1-\tau}p_{l}\lambda_{i}-\frac{E_{l}^{p}}{ (1-\tau)\bar{T}}\right)\sum\limits_{k=1,k\
eq l}^{L}\mathbb{E}\left[G_{l,k} \right]+\sigma^{2}}. \tag{19}\]  
Therefore, (18) can be simplified as:  
\[\mathbf{w}_{E,l}=\mathbf{u}_{\max}\left\{\mathbf{C}_{l}\right\}. \tag{20}\]  
As discussed in [22], the MF energy beamforming aims to maximize the harvested energy at the intended sensor with instantaneous CSI.