## 2 System Model
### Energy Transfer and Information Transmission  
We adopt frame-based transmission where the length of one frame is denoted by \(T\). Each frame consists of a WET phase and a WIT phase. In the WET phase, the AP charges sensors via downlink energy beamforming. The duration of WET phase is \(\tau T\) with \(0<\tau<1\). According to the law of energy conservation, the harvested energy at sensor \(l\) can be expressed as [21; 22; 23] (As in [21; 22; 23], we have neglected the non-linearity in the energy harvesting process. This is a good assumption for long-range WET, as shown in [32]. ):  
\[ \begin{split} E_{l}&=\eta\tau T\sum_{j=1}^{L}p_{j} \Big{|}\mathbf{h}_{l}^{H}\mathbf{w}_{E,j}\Big{|}^{2}\\ &=\eta\tau Tp_{l}\Big{|}\mathbf{h}_{l}^{H}\mathbf{w}_{E,j}\Big{|} ^{2}+\eta\tau T\sum_{j=1,j\
eq l}^{L}p_{j}\Big{|}\mathbf{h}_{l}^{H}\mathbf{w} _{E,j}\Big{|}^{2},\end{split}  \tag{8}\]  
where \(\mathbf{w}_{E,j}\) denotes the energy beamformer for sensor \(j\). \(\eta\) (\(0<\eta<1\)) denotes the energy conversion efficiency, i.e., the ratio between the harvested energy and received energy. \(p_{j}\) denotes the power allocated to the beamformer of sensor \(j\), which satisfies \(\sum_{j=1}^{L}p_{j}=p_{\text{sum}}\). The first term of the right-hand side indicates the expected energy that can be harvested at sensor \(l\), and the second term indicates the energy leakage from the signals sent to other sensors. Note that the schemes in [19; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30] assume that all harvested energy is used for the power amplifier of the sensor during the WIT phase. Different from the previous works, in this paper, we employ a more practical power consumption model for the sensor. In particular, after the WET phase, a part of the harvested energy is used for necessary processing at the sensor to achieve its main task, and the remaining energy is used for uplink transmission during the WIT phase. A similar model has also been applied in [20]. Let \(E_{l}^{p}\) denote the processing energy, in the WIT phase of time period \((1-\tau)T\), the transmit power of sensor \(l\) can be expressed as:  
\[ \begin{split} P_{l}&=\frac{E_{l}-E_{l}^{p}}{(1- \tau)\;T}\\ &=\frac{\eta\tau}{1-\tau}\sum_{j=1}^{L}p_{j}\Big{|}\mathbf{h}_{l} ^{H}\mathbf{w}_{E,j}\Big{|}^{2}-\frac{E_{l}^{p}}{(1-\tau)\;T}.\end{split}  \tag{9}\]  
Equation (9) has assumed that the harvested energy \(E_{l}\) is greater than the processing energy \(E_{l}^{p}\). Otherwise, \(P_{l}\) should be set to zero, and an outage occurs. To simplify the problem, we assume that the energy leakage, i.e., the second term of (8), contributes little to the total harvested energy. As will be seen later, this assumption is well satisfied under the proposed sensor scheduling scheme. Therefore, the transmit power of sensor \(l\) in (9) can be approximated as:  
\[P_{l}\approx\frac{\eta\tau}{1-\tau}p_{l}\Big{|}\mathbf{h}_{l}^{H}\mathbf{w}_{E,l}\Big{|}^{2}-\frac{E_{l}^{p}}{(1-\tau)\;T}. \tag{10}\]  
The received signal at AP during the WIT phase is:  
\[\mathbf{y}=\mathbf{H}\boldsymbol{\Lambda}^{1/2}\mathbf{x}+\mathbf{n}, \tag{11}\]  
where \(\mathbf{H}=[\mathbf{h}_{1},\mathbf{h}_{2},\cdots,\mathbf{h}_{L}]\) and \(\boldsymbol{\Lambda}=\text{diag}\left\{P_{1},P_{2},\cdots,P_{L}\right\}\). \(\mathbf{x}=[x_{1},x_{2},\cdots,x_{L}]^{T}\), and \(x_{l}\) denotes the unit-power transmit signal of sensor \(l\). \(\mathbf{n}\) denotes the additive white Gaussian noise (AWGN) vector with distribution \(\mathcal{CN}\left(0,\sigma^{2}\mathbf{I}_{N_{x}N_{y}}\right)\). To decode the signal of sensor \(l\), AP combines the received signal by multiplying the information beamformer \(\mathbf{w}_{D,l}\), i.e.,  
\[y_{l}=\sqrt{P_{l}}\mathbf{w}_{D,l}^{H}\mathbf{h}_{l}x_{l}+\mathbf{w}_{D,l}^{H }\sum_{j=1,j\
eq l}^{L}\sqrt{P_{j}}\mathbf{h}_{j}x_{j}+\mathbf{w}_{D,l}^{H} \mathbf{n}. \tag{12}\]  
For convenience, we normalize \(\mathbf{w}_{E,l}\) and \(\mathbf{w}_{D,l}\) as \(\big{\|}\mathbf{w}_{E,l}\big{\|}=\big{\|}\mathbf{w}_{D,l}\big{\|}=1\).