# Joint Transmit Power Allocation and Splitting for SWIPT Aided OFDM-IDMA in Wireless Sensor Networks
## 2 System Model and Problem Formulation
### Optimization of \(\{P_{k}\}\) with Fixed \(\{\rho_{k}\}\) and \(\{Q_{k}\}\)  
We first optimize \(\{P_{k}\}\) with pre-determined \(\{\rho_{k}\}\) and \(\{Q_{k}\}\). As stated in \(\mathcal{OP}\), the optimal solution to \(\mathcal{OP}\) should satisfy both the EH constraint (13) and ID one (14). However, with the assumption that \(\{\rho_{k}\}\) and \(\{Q_{k}\}\) are already known, we can decouple these two constraints with each other and denote \(P_{\text{EH}}^{*}\) and \(P_{\text{ID}}^{*}\) are the minimum transmit power by solely considering the EH and ID requirements, respectively. Then the feasible solution to \(\mathcal{OP}\) can be approximated as:  
\[P_{\text{SN}}^{*}\approx\max\left\{P_{\text{EH}}^{*}\,P_{\text{ID}}^{*} \right\}. \tag{16}\]  
Note that (16) can hold only if \(\{\rho_{k}\}\) and \(\{Q_{k}\}\) are already fixed. Apparently, the minimum transmit power of SN is bounded by either EH or ID function at RN who requires more energy to achieve the target performance. We will leave the discussion on the relationship between \(P_{\text{EH}}^{*}\) and \(P_{\text{ID}}^{*}\) as well as their impact on the final solution at the end of this subsection, and focus on the approach to achieve them two here. According to (13), \(P_{\rm EH}^{*}\) is bounded by the RN who owns worst channel condition and largest iteration numbers, and can be easily expressed as:  
\[P_{\rm EH}^{*}=\max\limits_{\{k=1,\cdots k\}}\left\{\frac{\Pi_{k}}{(1-\rho_{k})| H_{k}|^{2}N}\right\} \tag{17}\]  
where  
\[\Pi_{k}=b_{k}-\frac{1}{a_{k}}ln(\frac{M_{k}(1+e^{a_{k}b_{k}})}{E_{c}Q_{k}(1+e^{ a_{k}b_{k}}-M_{k})+M_{k}^{2}}-1). \tag{18}\]  
We then focus on optimizing \(P_{\rm ID}^{*}\) in the following. Note that the initial \(\{Q_{k}\}\) should be set large enough to make the iterative MUD converge (To guarantee the convergence of MUD, \(\{Q_{k}\}\) might be necessarily set over-large at initial stage. However, they will be optimized in the following steps. ), where additional iterations would not bring extra benefits on cancelling the interference, i.e.,  
\[f(\Lambda_{k}^{Q_{k}})\approx f(\Lambda_{k}^{Q_{k}-1})\, \tag{19}\]  
In such a case, we can approximate \(\Lambda_{k}^{Q_{k}}\) with \(\Lambda_{k}\), where  
\[\Lambda_{k}=\lim\limits_{Q_{k}\rightarrow\infty}\Lambda_{k}^{Q_{k}}\,\forall k \tag{20}\]  
is the theoretical upper bound of the achieved SINR when iteration number goes to infinity. Then, we can rewrite the first constraint in \(\mathcal{OP}\) with (20) and get the sub-problem that optimizes \(\{P_{k}\}\) solely with ID constraint as:  
\[\min\limits_{\{P_{k}\}}\ P_{\rm ID}=\sum\limits_{k=1}^{K}P_{k} \tag{21}\] \[s.t.\qquad\Lambda_{k}=\frac{\rho_{k}|H_{k}|^{2}P_{k}}{\rho_{k}|H _{k}|^{2}\sum\limits_{k^{\prime}\
eq k}^{K}P_{k^{\prime}}f(\Lambda_{k^{\prime} })+N\sigma^{2}}\geq\Gamma_{k} \tag{22}\]  
It can be found that the constraint (22) is relaxed without involving \(\{Q_{k}\}\) and iterative decoding, which greatly facilitates our analysis. Intuitively speaking, to minimize the transmit power, the best option is to let the SINR of each RN to be just equal to its threshold as \(\Lambda_{k}=\Gamma_{k}\). Unfortunately such favorable solution is almost inevitable to achieve. The reason falls in the fact that the achievable SINR of each RN is related not only to the initial power assignment \(\{P_{k}\}\), but also to the SINR of other RNs, in the form of \(P_{k^{\prime}}f(\Lambda_{k^{\prime}})\). In other words, the SINR of RNs are associated with each other, and it is difficult to get the solution to the problem, by solely optimize \(\{P_{k}\}\) with the only constraint (22).