# Joint Transmit Power Allocation and Splitting for SWIPT Aided OFDM-IDMA in Wireless Sensor Networks
## 2 System Model and Problem Formulation
### Problem Formulation  
In this subsection, we formulate the optimization problem of the proposed SWIPT aided OFDM-IDMA system. Recall that our purpose is to minimize the transmit power of SN, in the condition that both EH and ID requirements of RNs are satisfied. We will first derive these two constraints before formulating the power minimization problem. * The Energy Harvesting (EH) Constraint In the proposed system, the harvest energy is utilized to recharge the circuits of MUD, so that the iterative decoding can work under sufficient number of iterations. To achieve that, the harvested DC energy should be no less than the one needed for iterative MUD, which can be expressed as: \[\Phi_{k}\geq E_{c}Q_{k}\] (12) where \(E_{c}\) is the minimum required energy for MUD to work per iteration and \(Q_{k}\) is the iteration number of the \(k\)-th RN. So far the EH constraint can be expressed as (12). Nevertheless, note that the objective is to minimize the transmit power of RF signals from SN. Hence when formulating the EH requirement, the direct and explicit relationship between \(E_{c}Q_{k}\) and \(P_{SN}\) is preferred, rather than that between \(E_{c}Q_{k}\) and \(\Phi_{k}\). Hence we further take (3)-(5) into (12) to substitute \(\Phi_{k}\) with \(P_{SN}\), and the resulted inequation can be expressed as: \[(1-\rho_{k})|H_{k}|^{2}P_{SN}\geq b_{k}-\frac{1}{a_{k}}\ln(\frac{M_{k}(1+e^{a_ {k}b_{k}})}{E_{c}Q_{k}(1+e^{a_{k}b_{k}}-M_{k})+M_{k}^{2}}-1)\,\ \ \forall k\in 1,\cdots,K\] (13)
* The Information Decoding (ID) Constraint ID constraint guarantees the BER performance of the \(k\)-th RN, where the BER of RN \(k\) is expected to be lower than a given target. Recall that in Section 2.2, the \(g(\cdot)\) function is introduced to depict the relationship between the BER performance and achievable SINR. Assuming the BER target of the \(k\)-th RN is \(BER_{k^{\prime}}^{l}\), the corresponding SINR threshold to reach that can be obtained as \(\Gamma_{k}\ =\ g^{-1}(BER_{k}^{t})\). According to [2,16], \(g(\cdot)\) is the monotonically decreasing function. As a consequence, the final SINR value of RN \(k\) after iterative decoding should be greater than \(\Gamma_{k}\), so that to guarantee the desired BER performance. Based on (11), the ID constraint can be represented as:  
\[\Lambda_{k}^{Q_{k}}=\frac{\rho_{k}|H_{k}|^{2}P_{k}}{\rho_{k}|H_{k}|^{2}\sum_{k^{ \prime}\
eq k}^{K}P_{k^{\prime}}f(\Lambda_{k^{\prime}}^{Q_{k^{\prime}}-1})+N \sigma^{2}}\geq\Gamma_{k}\,\ \ \ \ \forall k\in 1,\cdots,K \tag{14}\]  
With the preliminaries provided above, the optimization problem can then be formulated as follows:  
**Original Problem Formulation (\(\mathcal{OP}\))**: _Given the minimum harvested energy constraints along with the Quality of Service (QoS) requirements of each RN, our objective is to minimize the total transmit power of SN, by jointly determining the **power allocation, power splitting and numbers of iterations** among all RNs. Such optimization problem can be summarized as follows:_  
\[\mathcal{OP}: \min_{\{\rho_{k}\},\{P_{k}\},\{Q_{k}\}}\ P_{SN}=\sum_{k=1}^{K}P_ {k} \tag{15}\] \[s.t. \eqref{eq:power_allocation_power_splitting}\, \tag{14}\]  
where the first constraint guarantees the EH requirements and the second constraint denotes the ID requirements of RNs. It can be observed that the iterative decoding process is taken into account in (14), where the final SINR value of each RN is sensitive to not only the initial power assignment and PS ratio, but also to their number of iterations \(\{Q_{k}\}\), as well as the achievable SINR in previous iteration. In addition, \(\{Q_{k}\}\) also affects the EH requirements in (13). As a result, to get a promising solution to \(\mathcal{OP}\), \(\{Q_{k}\}\) should be optimized along with \(\{P_{k}\}\) and \(\{\rho_{k}\}\). **3. The proposed Power Allocation and Splitting Algorithm**  
We in this section, propose the sub-optimal algorithm to solve the optimization problem concluded in \(\mathcal{OP}\).