## 4 Energy Efficiency Optimization  
In this section, the transmit power, number of active antennas of AP and duration of the WET phase are jointly optimized to maximize the system EE. It is assumed that the low-complexity strongest-path beamforming scheme is employed at AP. Based on the theoretical analysis in Section 3.2, an approximate expression of average received SINR at AP can be obtained as:  
\[\mathbb{E}\left[\mathrm{SINR}_{l}\right]\approx\frac{Q_{l}^{\mathrm{U,LB}}}{Q_ {l}^{\mathrm{MUI}}+\sigma^{2}}, \tag{33}\]  
where the expressions of \(Q_{l}^{\mathrm{U,LB}}\) and \(Q_{l}^{\mathrm{MUI}}\) are given by (29) and (31), respectively. We assume that the MUI is effectively mitigated after the proposed MUI-aware sensor scheduling scheme, and as a result, the system is noise-limited (if the effect MUI is not negligible, by exploiting the analytical expression in (31), the EE optimization problem can be solved efficiently using the geometric programming based procedure developed in [40]). Thus, by substituting (29) into (33) and neglecting the low-order terms, the average received SINR can be further simplified as:  
\[\mathbb{E}\left[\mathrm{SINR}_{l}\right]\approx\frac{\tau N_{x}^{2}N_{y}^{2}}{ 1-\tau}\alpha_{l}p_{l}-\frac{N_{x}N_{y}}{1-\tau}\gamma_{l}, \tag{34}\]  
where \(\alpha_{l}=\frac{\eta}{\sigma^{2}}\left(\frac{\beta_{l}K_{l}}{K_{l}+1}\right)^ {2}\) and \(\gamma_{l}=\frac{E_{p}}{\sigma^{2}T}\frac{\beta_{l}K_{l}}{K_{l}+1}\). The EE is defined as the SE divided by the AP's total energy consumption, that is:  
\[\mathrm{EE}=\frac{\mathrm{SE}}{\tau T\sum_{l=1}^{L}p_{l}+TP_{c}}, \tag{35}\]  
where the SE is defined as the sum of all sensors' uplink achievable rates, that is:  
\[\mathrm{SE}=\left(1-\tau\right)\sum_{l=1}^{L}\log\left(1+\frac{\tau N_{x}^{2}N _{y}^{2}}{1-\tau}\alpha_{l}p_{l}-\frac{N_{x}N_{y}}{1-\tau}\gamma_{l}\right), \tag{36}\]and \(P_{c}\) denotes the circuit power consumption of AP. According to [21; 41], it is reasonable to assume that the circuit power consumption increases linearly with the number of the AP's active antennas. Thus, \(P_{c}\) can be expressed as:  
\[P_{c}=\bar{N}_{x}\bar{N}_{y}p_{\text{ant}}, \tag{37}\]  
where \(\bar{N}_{x}\) and \(\bar{N}_{y}\) denote the numbers of active antennas along the \(x\) axis and \(y\) axis, respectively, which satisfy \(\bar{N}_{x}\leq N_{x}\) and \(\bar{N}_{y}\leq N_{y}\). \(p_{\text{ant}}\) denotes the power consumption introduced by each active antenna. We consider the EE optimization problem with the per sensor achievable rate constraint, which is formulated as:  
\[\max_{\tau,\{p_{\}\}_{l=1}^{L},\bar{N}_{x},\bar{N}_{y}}\text{EE}, \tag{38}\] \[\text{s.t} \begin{cases}C_{1}:(1-\tau)\log\left(1+\frac{\tau N_{x }^{2}N_{y}^{2}}{1-\tau}\alpha_{l}p_{l}-\frac{N_{x}N_{y}}{1-\tau}\gamma_{l} \right)\geq r_{T},\\ l=1,\cdots,L,\\ C_{2}:\sum_{l=1}^{L}p_{l}\leq p_{\text{max}},p_{l}\geq 0,l=1,\cdots,L,\\ C_{3}:\bar{N}_{x}\leq N_{x},\bar{N}_{y}\leq N_{y},\text{ and }\bar{N}_{x},\bar{N}_{y }\in\mathbb{N}^{+},\\ C_{4}:\tau\in(0,1)\,.\end{cases} \]  
where \(r_{T}\) denotes the minimum data rate requirement of each sensor. \(p_{\text{max}}\) denotes the maximum transmit power constraint of AP. Note that the constraint \(C_{1}\) implicitly enforces the harvested energy to be greater than the processing energy. Thus, no additional constraint is needed. Due to the tremendous gain of a large-scale antenna array in a massive MIMO system, it is reasonable to assume that the uplink received SINR given by (34) is much greater than one. Thus, at a good operating point, the increase of \(N_{x}\), \(N_{y}\) and \(p_{l}\) will result in the decrease of EE since the achievable rate grows logarithmically with \(N_{x}\), \(N_{y}\) and \(p_{l}\). In this case, the constraint \(C_{1}\) will satisfy with equality at optimal \(N_{x}\), \(N_{y}\) and \(p_{l}\).