# Resource Allocation for Relay-Based OFDMA Power Line Communication System
## 2 System Model and Problem Formulation  
### System Model  
Consider a downlink multi-user OFDMA PLC system with one BS, one non-regenerative relay and \(K\) users. Since we mainly focus on the resource allocation strategy for \(K\) users under one relay, this paper adopt a two-hop protocol for simplify. Assume that there are \(N\) subcarriers in each OFDM symbol and one subcarrier can only be used by one user. In the first hop, the source transmits OFDM symbols \(s_{n,k}\) with transmit power \(P_{s_{n,k}}\) through the backward channel \(h_{1}\) to the relay, where \(n\in\{1,\cdots,N\}\) is the subcarrier index and \(k\in\{1,\cdots,K\}\) is the user index. Then, the received signal of relay can be expressed as  
\[y_{n,k}=h_{1}\sqrt{P_{s_{n,k}}}s_{n,k}+n^{r}, \tag{1}\]  
where \(n^{r}\) denotes the noise at the relay node. In the PLC system, the multi-source impulsive noise is considered as the main factor. According to the model of [PERSON]'s Class A [21; 30], we can formulate the total color noise as the sum of Gaussian noise and the impulsive noise, and the probability density function of the color noise that with amplitude \(c\) can be given by  
\[p(c)=\sum_{c=0}^{\infty}\frac{e^{-A}A^{c}}{c!}\frac{1}{\sqrt{2\pi}\sigma_{c}} \exp(-\frac{z^{2}}{2\sigma_{c}^{2}}), \tag{2}\]  
where \(\sigma_{c}^{2}\) denotes the total color noise power, and  
\[\sigma_{c}^{2}=\sigma_{g}^{2}+\sigma_{I}^{2}\frac{\frac{c}{A}+\Gamma}{1+ \Gamma},\quad\Gamma=\frac{\sigma_{g}^{2}}{\sigma_{I}^{2}}, \tag{3}\]  
where \(\sigma_{g}^{2}\), \(\sigma_{I}^{2}\) denotes the Gaussian and impulsive noise power, respectively, \(A\) denotes the value of the detected impulsive noise. In the second hop, the relay amplifies the received signal with amplified factor \(\alpha_{n}\), and retransmits the amplified signal to the destination through the forward channel \(h_{n,k}\), the forward signal is given by  
\[g_{n,k}=\alpha_{n}h_{1}\sqrt{P_{s_{n,k}}}s_{n,k}+\alpha_{n}n^{r}. \tag{4}\]  
Let \(P_{r_{n,k}}\) denotes the transmit power on the relay node, and \(P_{r_{n,k}}=\mathbb{E}\left(|g_{n,k}|^{2}\right)\), the factor \(\alpha_{n}\) can be expressed as  
\[|\alpha_{n}|^{2}=\frac{P_{r_{n,k}}}{|h_{1}|^{2}P_{s_{n,k}}+\sigma^{2}}. \tag{5}\]  
After two-hop transmission, we can obtain the \(k\)-th user's received signal as  
\[d_{n,k}=h_{1}h_{n,k}\alpha_{n}\sqrt{P_{s_{n,k}}}s_{n,k}+h_{2}\alpha_{n}n^{r}+n _{n,k}^{d}, \tag{6}\]  
where \(n_{n,k}^{d}\) denotes the user node's noise, and \(n_{n,k}^{d}\sim\mathcal{CN}\left(0,\sigma^{2}\right)\). Based on the above description, we can obtain the \(k\)-th user's information rate on the \(n\)-th subcarrier  
\[R_{n,k}=\log_{2}\left(1+\frac{P_{s_{n,k}}P_{r_{n,k}}}{\frac{P_{s_{n,k}}\sigma ^{2}}{|h_{n,k}|^{2}}+\frac{P_{r_{n,k}}\sigma^{2}}{|h_{1}|^{2}}+\frac{\sigma^{ 4}}{|h_{1}|^{2}|h_{n,k}|^{2}}}\right). \tag{7}\]  
Consider the background noise and radio interferences in the PLC channel, the subcarrier signal might give rise to interference with other subcarriers. We can model the interference as out-of-band emissions, the interference from the \(m\)-th subcarrier to the \(n\)-th subcarrier can be expressed as  
\[I_{n,m}=P_{s_{n,k}}\eta_{n}, \tag{8}\]  
where \(\eta_{n}\) denotes the interference coefficient [31; 32], which is given by  
\[\eta_{n}=\int_{f_{n,m},-\frac{\triangle f}{2}}^{f_{n,m}+\frac{\triangle f}{2} }T_{s}\left(\frac{\sin\left(\pi fT_{s}\right)}{\pi fT_{s}}\right)^{2}df, \tag{9}\]  
where \(f_{n,m}=|f_{m}-f_{n}|\) denotes the spectral distance between the center frequency of subcarrier \(m\) and subcarrier \(n\). \(T_{s}\) denotes the symbol interval of OFDM and \(\triangle f\) denotes the channel bandwidth.