# Multi-User AF Relay Networks with Power Allocation and Transfer: A Joint Approach
## 2 Preliminaries  
### System Model  
Consider an EH dual-hop AF-assisted relay network consisting of \(N\) EH source nodes \((S_{k},\ \mathrm{for}\ k=1,\ldots,N)\), one relay node \((R)\) and \(N\) destination nodes \((D_{k},\ \mathrm{for}\ k=1,\ldots,N)\), as depicted in Figure 1. A single antenna is deployed at each node in the network. As a result of long distance and deep fading, there is no direct link between \(S_{k}\) and \(D_{k}\), \(\forall k\). The nodes \(S_{k}\), \(\forall k\), and \(R\) can harvest energy from ambient energy sources and store them in batteries. We adopt a time-slotted model with a unit time duration for each time slot. The total time slot length is \(N_{T}\). Furthermore, the battery capacity i.e., energy queues, at each \(S_{k}\), \(\forall k\), and \(R\) are \(B_{k,max}^{s}\) and \(B_{r,max}\) units of energy, respectively. At the end of each slot, the status of batteries is updated. We further assume frequency flat fading channels between all links. In each time slot, the relay \(R\) works in half-duplex (HD) fashion. Suppose \(i\), for \(i=1,2,\ldots,N_{T}\), indicates the time slot index. In the first phase, the signal received by the relay node in the \(i^{th}\) time slot is written as  
\[r_{i}=\sum_{k=1}^{N}\sqrt{P_{k,i}}h_{k,i}s_{k,i}+n_{r,i}, \tag{1}\]  
where \(P_{k,i}\) represents the transmission power of the node \(S_{k}\), \(s_{k,i}\) is the transmitted message with \(E\left\{|s_{k,i}|^{2}\right\}=1\), \(h_{k,i}\) is the channel coefficient from \(S_{k}\) node to \(R\), \(n_{r,i}\in\mathbb{C}\left(0,\sigma_{r}^{2}\right)\) is AWGN at node \(R\). The signal transmitted by node \(R\) after amplification is given by \(x_{r,i}=\alpha_{i}r_{i}\), where \(\alpha_{i}^{2}=\frac{W_{r,i}}{\sum_{k=1}^{N}P_{k,i}|h_{k,i}|^{2}+\sigma_{r}^{2}}\), and \(W_{r,i}\) is the transmission power of node \(R\). In the second phase, the received signal at the \(i^{th}\) destination node in the \(i^{th}\) time slot is expressed as  
\[y_{j,i}=\alpha_{i}\Big{(}\sqrt{P_{j,i}}g_{j,i}h_{j,i}s_{j,i}+\sum_{k=1,k\
eq j }^{N}\sqrt{P_{k,i}}g_{j,i}h_{k,i}s_{k,i}+g_{j,i}n_{r,i}\Big{)}+n_{d,i}\,, \tag{2}\]  
where \(g_{j,i}\), \(n_{d,i}\) and \(\sigma_{d_{j}}^{2}\) are defined similar to \(h_{k,i}\), \(n_{r,i}\) and \(\sigma_{r}^{2}\) for the link from \(R\) to the \(D_{j}\).  
Using (2), we can compute the sum rate over \(N_{T}\) consecutive time slots as  
\[f_{R}(\mathbf{P_{s}},\mathbf{W_{r}})=\frac{1}{2}\sum_{j=1}^{N}\sum_{i=1}^{N_{T}}\log_ {2}\left(1+\mathrm{Y}_{j,i}\right)\,, \tag{3}\]  
Figure 1: Energy harvesting multi-user amplify-and-forward (AF) relay network.  
where \(\mathbf{P_{s}}\in\mathbb{C}^{N\times N_{T}}\), \(\mathbf{W_{r}}\in\mathbb{C}^{1\times N_{T}}\), and signal-to-interference-plus-noise ratio (SINR) at node \(D_{j}\) is obtained as  
\[\mathrm{Y}_{j,i}=\frac{P_{j,i}W_{r,i}|g_{j,i}h_{j,i}|^{2}}{\mathrm{Y}_{1}+ \mathrm{Y}_{2}}\,,\qquad i=1,\ldots,N_{T}\,, \tag{4}\]  
with \(\mathrm{Y}_{1}=W_{r,i}\sum_{k=1,k\
eq j}P_{k,i}|g_{j,i}h_{k,i}|^{2}+W_{r,i}|g _{j,i}|^{2}\sigma_{r}^{2}\) and \(\mathrm{Y}_{2}=\sigma_{d_{j}}^{2}\left(\sum_{k=1}^{N}P_{k,i}|h_{k,i}|^{2}+ \sigma_{r}^{2}\right)\).