# 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 \(\{Q_{k}\}\)
To overcome this, a new set of variables \(\{\gamma_{k}\}\) is introduced to further decompose (22) into the following two constraints [25]:  
\[\frac{\rho_{k}|H_{k}|^{2}P_{k}}{\rho_{k}|H_{k}|^{2}\sum\limits_{k ^{\prime}\
eq k}^{K}P_{k^{\prime}}f(\Lambda_{k^{\prime}})+N\sigma^{2}}\geq \gamma_{k} \tag{23}\] \[\Gamma_{k}\leq\gamma_{k}\leq\infty\,\quad\forall k\in 1,\cdots,K \tag{24}\]  
where the strict constraint on \(\{\Gamma_{k}\}\) is now relaxed with the new introduced variables \(\{\gamma_{k}\}\). The promising solution to problem (21) could be obtained by minimizing both \(\{\Gamma_{k}\}\) and \(\{P_{k}\}\) jointly. Consequently, the sub-problem is re-formulated as follows [16]:  
**Sub-Problem Formulation (\(\mathcal{SP}\))**: _Given the fixed iteration number, PS ratio, as well as the minimum SINR requirements \(\Gamma_{k}\) with respect to certain BER performance of each RN, the objective of minimizing the total transmit power of SN can be achieved by jointly considering the **actually achievable SINR**\(\{\gamma_{k}\}\) and **power allocation**\(\{P_{k}\}\) among all RNs. The minimum transmit power, obtained through power allocation, should guarantee the actually achievable SINR of each RN to be also minimized but greater than \(\Gamma_{k}\). Such a problem can be formulated as a cascaded one as:  
\[\mathcal{SP}:\min_{\{\gamma_{k}\}}\left\{ \begin{array}{l}\min_{\{P_{k}\}} \sum_{k=1}^{K}P_{k}\\ s.t.\ \frac{\rho_{k}|H_{k}|^{2}P_{k}}{\rho_{k}|H_{k}|^{2}\sum\limits_{k^{ \prime}\
eq k}^{K}P_{k^{\prime}}f(\Lambda_{k^{\prime}})+N\sigma^{2}}\geq\gamma_ {k}\,\\ s.t.\ \Gamma_{k}\leq\gamma_{k}\leq\infty\,\ \ \ \ \forall k\in 1,\cdots,K \end{array} \right. \tag{25}\]  
where \(\{\gamma_{k}\}\) is also involved into the optimization process together with \(\{P_{k}\}\). So far \(\mathcal{SP}\) is still a non-convex problem. But fortunately a similar problem for IDMA in uplink transmission has been successfully tackled with in [25], according to which \(\mathcal{SP}\) can also be converted to a convex one by proper mathematical derivation. To do so we first give the expression of optimal power allocation according to (23) as  
\[P_{k}^{*}=\frac{\sigma^{2}\gamma_{k}^{*}}{(1-\sum_{k=1}^{K}\alpha_{k})(1+ \tilde{f}(\gamma_{k}^{*}))|H|_{k}^{2}\rho_{k}} \tag{26}\]  
where \(\{\gamma_{k}^{*}\}\) is one feasible SINR solution to the outer problem illustrated in (25) and  
\[\tilde{f}(\gamma_{k}^{*})=\Lambda_{k}^{*}f(\gamma_{k}^{*}) \tag{27}\]  
\[\alpha_{k}=\frac{\tilde{f}(\gamma_{k}^{*})}{1+\tilde{f}(\gamma_{k}^{*})}. \tag{28}\]  
Based on (27) and (28), two new functions can be defined as  
\[F(\alpha_{k})=(1-\alpha_{k})\tilde{f}^{-1}(\frac{\alpha_{k}}{1-\alpha_{k}}) \tag{29}\]  
\[G(\mathbf{\alpha})=1-\sum_{k=1}^{K}\alpha_{k} \tag{30}\]  
where \(\mathbf{\alpha}=[\alpha_{1},\cdots,\alpha_{K}]\) is the vector containing feasible set \(\{\alpha_{k}\}\). It has been proved in [25] that \(F(\cdot)\) is a convex function and \(G(\cdot)\) is linear with respect to \(\mathbf{\alpha}\). Hence \(\mathcal{SP}\) is equivalent to the following quasi-convex problem:  
\[\min_{\mathbf{\alpha}}\ \frac{\sigma^{2}}{G(\mathbf{\alpha})}\sum_{k=1}^{K} \frac{F(\alpha_{k})}{|H|_{k}^{2}\rho_{k}} \tag{31}\] \[s.t.\ \ \ 0<\alpha_{k}<\hat{\alpha}_{k}=\frac{\tilde{f}(\gamma_{k}) }{1+\tilde{f}(\gamma_{k})}\]  
where the upper bound \(\hat{\alpha}_{k}\) is given according to constraint (22) in \(\mathcal{OP}\). Through utilizing standard convex optimization tool [26], the optimal solution to such quasi-convex problem can be obtained. The detailed solution procedure for such a convex problem can also be found in [25], which we would like to omit here for simplicity. Then with \(\mathbf{\alpha}^{*}=[\alpha_{1}^{*},\cdots,\alpha_{K}^{*}]\) on hand, the optimal power assignment \(\{P_{k}^{*}\}\) can also be achieved by plugging \(\mathbf{\alpha^{*}}\) into (28)-(26) inversely and consequently. Please confirm if it is (26)-(28) We then go back to (16) to check whether the EH and ID constraints are both satisfied with \(\{P_{k}^{*}\}\).