## 2 System Model and Problem Formulation
### Uplink Wireless Information Transfer  
After harvesting energy from the H-AP, the \(k\)th user sends an information signal utilizing the energy harvested in the downlink phase. Every \(k\)th user transmits an independent complex baseband signal, \(x_{k}\). We assume the Gaussian input as \(x_{k}\) is a circularly symmetric complex Gaussian (CSCG) random variable with mean 0 and variance, \(\sqrt{P_{k}}\). We assume that \(U_{k}\) utilizes all the energy harvested on downlink to send the information signal to the H-AP. Thus, \(P_{k}\) denotes the average power at the \(k\)th user's transmitter, which is given by  
\[P_{k}\ =\ \frac{E_{k}}{\tau_{k}},\ k=1,2,3,\ldots\ K \tag{4}\]  
The received signal at the H-AP sent by the \(k\)th user in the \(k\)th uplink slot is given by  
\[y_{Ak}=h_{k}x_{k}+\ z_{A,k} \tag{5}\]  
where \(z_{A,k}\) denotes the noise at the H-AP, which is a complex Gaussian-distributed random variable with zero mean and variance, \(\sigma^{2}\). The achievable uplink throughput for the \(k\)th user in the \(k\)th slot can be defined as  
\[\begin{split} R_{k}\ \left(\tau\right)&=\ \tau_{k}log_{2}\left[1+\frac{\left|h_{k}\right|^{2}P_{k}}{\sigma^{2}} \right]\\ &=\ \tau_{k}log_{2}\left[1+\alpha_{k}P_{A}\frac{\tau_{k}}{\tau_{0}} \right]\end{split} \tag{6}\]  
where \(\alpha_{k}=\ \frac{\zeta_{k}\left|h_{k}\right|^{4}}{\sigma^{2}}\). We can see from the above expression that \(R_{k}\ \left(\tau\right)\) would increase with an increase in transmit power for the given allocated time slots on uplink and downlink. However, as the BS has limited resources and cannot transmit much power to users, there is a need to control the transmit power while maintaining QoS. There is a need to optimize power while maintaining uplink throughput and downlink harvested energy above a specified threshold. Here, we can formulate the problem for transmit power optimization as Equation (7):\[\min_{\tau,P_{A}} P_{A}\] (7a) s.t. \[\tau_{k}log_{2}\left[1+\alpha_{k}P_{A}\frac{\tau_{k}}{\tau_{0}} \right]\geq R_{thre} \tag{7b}\] \[\varsigma_{k}\left|h_{k}\right|^{2}P_{A}\tau_{0}\geq\ E_{thre}\] (7c) \[\sum_{j=0}^{k}\tau_{j}\leq 1\] (7d) \[0\leq P_{A}\leq P_{max} \tag{7e}\]  
In Equation (7), we are minimizing the transmit power while optimizing two variables: the transmit power and downlink-uplink time slot allocation. By optimizing these two parameters, we optimize the transmit power. In Equation (7), (7b, and 7c) confirm the QoS while managing uplink throughput and harvested energy to maintain them above threshold levels \(R_{thre}\) and \(E_{thre}\), respectively. Equation (7d) satisfies the normalization of allocation time based on the frame length. \(P_{max}\) is the maximum power resources available at the BS.  
However, Equation (7) is a non-convex problem, and it is analytically very difficult to find an optimal solution for \((P^{*},\tau^{*})\). Traditionally, non-convex problems can be solved using an exhaustive search algorithm, which requires high computational complexity. For this problem, we propose a deep-learning-based scheme that can give an accurate (and less complex) solution for Equation (7). The analytical results discussed in Section 5 prove this approach to be a promising technique for solving Equation (7) with less computational power compared to traditional approaches.