# Joint Transmit Power Allocation and Splitting for SWIPT Aided OFDM-IDMA in Wireless Sensor Networks
## 2 System Model and Problem Formulation
### Iterative Multi-User Detection and SNR Evolution  
Recall that an iterative MUD is employed at each RN of the proposed system, which is comprised of an Gaussian Approximation Detector (GAD) and K signal-user _a posteriori probability_ decoders (APP DECs) working in an iterative manner. To decode the desired signal, say \(X_{k}(n)\) for RN \(k\), the GAD first treats \(\zeta_{k}(n)\) in (2) as the Gaussian random variable based on Central Limit Theorem. The extrinsic information about \(X_{k}(n)\); defined as \(L_{e}(X_{k}(n))\), can be generated following an soft-input soft-output (SISO) estimation procedure given the observation of \(Y_{k}(n)\) and the _a priori_ information provided by APP DECs [22]. Then \(L_{e}(X_{k}(n))\)s are de-interleaved and fed to the APP DECs as the _a priori_ information. According to the standard APP decoding rule [23], the DEC outputs the corresponding extrinsic information and sends back its interleaved version \(L_{c}(X_{k}(n))\) to the GAD as the _a priori_ information to update the estimate of \(\zeta_{k}(n)\). The MAI will be mitigated gradually with such extrinsic information exchanging in an iterative manner. After the \(q\)-th iteration, the residual interference with respect to the \(k\)-th RN on subcarrier \(n\) can be denoted as [24]:  
\[V^{q}\{\zeta_{k}(n)\}=\rho_{k}|H_{k}(n)|^{2}\sum_{k^{\prime}\
eq k}^{K}P_{k^{ \prime}}V^{q}\{X_{k^{\prime}}(n)\}+\sigma^{2} \tag{7}\]  
where \(V\{\cdot\}\) denotes the variance of the corresponding variables. Note that \(V^{q}(X_{k^{\prime}}(n))\) represents the residual interference contributed by RN \(k^{\prime}\) in current iteration, which can be calculated based on \(L_{c}^{(q-1)}(X_{k^{\prime}}(n))\) in the previous iteration as:  
\[V^{q}\{X_{k^{\prime}}(n)\}=1-\tanh^{2}(\frac{L_{c}^{(q-1)}(X_{k^{\prime}}(n))} {2}). \tag{8}\]  
As a consequence, the final SINR with respect to \(X_{k}(n)\) after the \(Q\)-th iteration can be obtained as:  
\[\lambda_{k}^{Q}(n)=\frac{\rho_{k}|H_{k}(n)|^{2}P_{k}}{\rho_{k}|H_{k}(n)|^{2} \sum_{k^{\prime}\
eq k}^{K}P_{k^{\prime}}V^{Q}\{X_{k^{\prime}}(n)\}+\sigma^{2} }. \tag{9}\]  
We can find that to get the final SINR of one specific RN \(k\), it should take into account the variance value of other RNs on each subcarrier per-iteration. Apparently, one can utilize brute-force simulations to get (9). However, such an approach is time consuming and impractical to be employed for performance optimization. In fact, the performance of OFDM-IDMA is related to the _averaged SINR_ per-subcarrier, thanks to the low-rate coding as well as the chip-level random permutation over the entire OFDM block. Based on that idea, an SNR evolution method was proposed in [2], where the  
Figure 1: The receiver structure of the \(k\)-th RN. SINR as well as the variance value of RN \(k\) per each subcarrier are approximated by their expectations over the entire OFDM block as:  
\[\Lambda_{k}^{q}=\mathbb{E}[\Lambda_{k}^{q}(n)]\,\ \ \ \ \ \ \bar{V}_{k}^{q}= \mathbb{E}[V_{k}^{q}(n)]. \tag{10}\]  
The recursive relationship between them during each iteration can be characterized by the defined \(f(\cdot)\) function as \(\bar{V}_{k}^{q}=f(\Lambda_{k}^{q-1})\), where \(f(\cdot)\) is obtained by simulating the single RN system under the same channel conditions [2]. By substituting (9) and \(f(\cdot)\) function into (10), we can get the averaged SINR updating rule as:  
\[\Lambda_{k}^{q}=\frac{\rho_{k}|H_{k}|^{2}P_{k}}{\rho_{k}|H_{k}|^{2}\sum_{k^{ \prime}\
eq k}^{K}P_{k^{\prime}}f(\Lambda_{k^{\prime}}^{q-1})+N\sigma^{2}}. \tag{11}\]  
It can be observed in (11) that the SINR updating of RN \(k\) is now related to the SINR rather than the variance values of its counterparts. It helps to facilitate the performance analysis of iterative decoding in OFDM-IDMA, with the easily obtained \(f(\cdot)\) function.