## 3 System Model and Problem Formulation  
As shown in Figure 1, we consider a dual-hop SWIPT enabled multi-relay OFDM based IoT system, where relays operate in time-division half-duplex mode based DF protocol. This system consists of one source \(S\), one destination \(D\), and \(K\) relays. Let \(\mathcal{K}=\{1,2,\ldots,K\}\) denote the set of \(K\) number of relays. The set of \(N\) subcarriers is denoted by \(\mathcal{N}=\{1,2,\ldots,N\}\). Due to the limited coverage and shadowing attenuation, the direct link between \(S\) and \(D\) is infeasible. In addition, perfect channel state information (CSI) is available for acquisition.  
Figure 1: System model. Source \(S\), destination \(D\), and \(K\) relays.  
The transmission between \(S\) and \(D\) are performed through the following two timeslots. In the first timeslot, \(S\) broadcasts the signal over all the subcarriers. \(n\in S_{1,k}^{I}\) denotes the subcarriers allocated by relay \(k\) (\(k\in\mathcal{K}\)) for decoding the received information, and \(n\in S_{1,k}^{I}\) denotes the remaining subcarriers which are utilized to harvest energy. In the second timeslot, relay \(k\) forwards the re-encoded information over the subcarriers \(n\in S_{2,k}\) to \(D\). In order to prevent mutual interference, a subcarrier utilized for information transmission will be only assigned to one relay in each timeslot.  
Let the channel gain and the allocated power on subcarrier \(n\) associated with relay \(k\) in hop \(t\) be denoted as \(h_{n,k,t}\) and \(p_{n,k,t}\) respectively, for \(n\in\mathcal{N}\), \(k\in\mathcal{K}\) and \(t=\{1,2\}\). Thus, the transmission rate \(r_{n,k,t}\) can be expressed as  
\[r_{n,k,t}=\ln\left(1+\frac{h_{n,k,t}p_{n,k,t}}{\sigma^{2}}\right) \tag{1}\]  
where \(\sigma^{2}\) represents the variance of additive white Gaussian noise and \(p_{n,k,1}\) is equally allocated.  
The harvested energy at relay \(k\) is given by  
\[Q_{k}=\sum_{n\in S_{1,k}^{E}}\xi\left(h_{n,k,1}p_{n,k,1}+\sigma^{2}\right) \tag{2}\]  
where \(\xi\) is the energy conversion efficiency of relays.  
The transmission rate from source to destination via relay \(k\) is the minimum of the rate achieved over two hops, which can be written as  
\[r_{k}=\frac{1}{2}\min\left(\sum_{n\in S_{1,k}^{I}}r_{n,k,1},\sum_{n\in S_{2,k} }r_{n,k,2}\right). \tag{3}\]  
Our goal is to maxmize the system transmission rate through optimizing the joint resource allocation, including power allocation \(\mathbf{p}=\left\{p_{n,k,2}\right\}\) and subcarrier allocation \(\mathbf{\Omega}=\left\{S_{1,k}^{I},S_{1,k}^{E},S_{2,k}\right\}\). Thus, the optimization problem can be mathematically formulated as  
\[\textbf{(P1)}: \max_{\{p,\mathbf{\Omega}\}}\ \sum_{k=1}^{K}r_{k}\] \[\text{s.t.} C1:Q_{k}\geq\sum_{n\subseteq S_{2,k}}p_{n,k,2},\forall k \tag{4}\] \[C2:S_{1,k}^{I}\cap S_{1,k^{\prime}}^{I}=\varnothing,\forall k \
eq k^{\prime}\] (5) \[C3:S_{2,k}\cap S_{2,k}=\varnothing,\forall k\
eq k^{\prime}\] (6) \[C4:S_{1,k}^{I}\cap S_{1,k}^{P}=\varnothing,\forall k\] (7) \[C5:S_{1,k}^{I}+S_{1,k}^{P}=\mathcal{N},\forall k\] (8) \[C6:p_{n,k,2}\geq 0,\forall n,k \tag{9}\]  
where \(C1\) represents the energy used by each relay to forward \(S\) signal in the second timeslot should be not larger than the energy it harvests in the first timeslot, and \(C2\sim C5\) indicate the constraints of the subcarrier set.