# Joint Transmit Power Allocation and Splitting for SWIPT Aided OFDM-IDMA in Wireless Sensor Networks
## 2 System Model and Problem Formulation  
### System Model  
We consider a wireless access network with one SN transmitting to \(K\) RNs. The SN who has constant power supplies, can act as both the information transmitter and power beacon. While the RNs are battery-powered who expect to receive the information from SN and also be wirelessly recharged by SN. The channel between SN and each RN is assumed to experience both large scale fading and frequency-selective multi-path Rayleigh-fading. The number of fading paths equals to \(L\). To combat such frequency selective fading, Orthogonal Frequency Division Multiplexing (OFDM) is employed. At the SN, the data for each RN is first encoded by the same encoder but permuted by different chip-level interleavers. The resulted sequences are then linearly superimposed together and modulated onto the subcarriers readily for transmission. At the \(k\)-th RN, the received signal on the \(n\)-th subcarrier is:  
\[\begin{split} Y_{k}(n)&=H_{k}(n)\sum_{i=1}^{K}\sqrt{P_ {i}}X_{i}(n)+\omega\\ &=H_{k}(n)\sqrt{P_{k}}X_{k}(n)+\zeta_{k}(n)\,\ \ n=1,\cdots,N\end{split} \tag{1}\]  
where \(H_{k}(n)\) contains not only the frequency channel response of the \(k\)-th RN on subcarrier \(n\), but also the path-loss from SN to RN \(k\). \(X_{k}(n)\) is the transmitted symbol for the \(k\)-th RN on subcarrier \(n\). \(P_{k}\) is the transmitt power per-subcarrier assigned to the \(k\)-th RN. \(N\) is the number of subcarriers in one OFDM block. \(\omega\) denotes the Additive White Gaussian Noise (AWGN) follows \(\mathcal{N}\sim(0,\sigma^{2})\). \(\zeta_{k}(n)\) represents the interference plus noise part with respect to \(X_{k}(n)\), which can be further expressed as:  
\[\zeta_{k}(n)=H_{k}(n)\sum_{k^{\prime}\
eq k}^{K}\sqrt{P_{k^{\prime}}}X_{k^{ \prime}}(n)+\omega\,\ \ n=1,\cdots,N. \tag{2}\]  
It is assumed that the proposed SWIPT aided system works under the PS mode, as shown in Figure 1. Then the received signal at each RN is partitioned into two parts through a power splitter. To be specific, an \(\rho_{k}\) portion of the signal power is fed to the iterative MUD for ID while the remaining signal power is gathered by the Energy Receiver (ER), where \(0\leq\rho_{k}\leq 1,\forall k\). The average received RF power at the ER of each SN during one OFDM block transmission is expressed as:  
\[P_{k}^{EH}=(1-\rho_{k})|H_{k}|^{2}\sum_{k=1}^{K}P_{k} \tag{3}\]  
where \(|H_{k}|^{2}=\sum_{n=1}^{N}|H_{k}(n)|^{2}\) is the accumulated channel gain of RN \(k\) across the entire OFDM block [17].  
Note that the harvested RF power has to be converted to the direct current (DC) electricity, in order to be stored and utilized by RNs. It has been proved by field experiments that such energy conversion is a non-liner process, where the conversion efficiency would increase at first and then decrease as the input RF power rises [18; 19]. To describe such behavior, we adopt the non-linear EH model proposed in [20; 21]. Based on that, the actually received DC energy of \(k\)-th RN can be expressed as:  
\[\Phi_{k}=\frac{(1+e^{a_{k}b_{k}})\Psi_{k}-M_{k}^{2}}{1+e^{a_{k}b_{k}}-M_{k}} \tag{4}\]  
where  
\[\Psi_{k}=\frac{M_{k}}{1+e^{-a_{k}(P_{k}^{EH}-b_{k})}} \tag{5}\]  
is the logistic function with respect to the received RF power \(P_{k}^{EH}\). \(M_{k}\), \(a_{k}\) and \(b_{k}\) are the fixed parameters determined by the detailed EH circuit specifications.  
On the other hand for information decoding, the initial received SINR for the \(k\)-th RN at subcarrier \(n\) can be expressed as:  
\[\lambda_{k}^{0}(n)=\frac{\rho_{k}|H_{k}(n)|^{2}P_{k}}{\rho_{k}|H_{k}(n)|^{2} \sum_{k^{\prime}\
eq k}^{K}P_{k^{\prime}}+\sigma^{2}} \tag{6}\]  
where the bandwidth of subcarrier is assumed to be unity for the ease of analysis.