# Achievable Rate Maximization for Multi-Relay AF Cooperative SWIPT Systems with a Nonlinear EH Model
## 2 System Model  
We consider the multi-relay wireless cooperative system similar to [20; 21], where \(L\) relay nodes assist the source node \(S\) in transmitting the information data bits to the destination node \(D\), as shown in Figure 1. The \(L\) relay nodes, denoted by a set \(\Re=\{R_{1},R_{2},\ldots,R_{L}\}\), are assumed to be energy-limited and capable of harvesting energy from the received signal of \(S\). Each relay node uses the harvested energy to forward the source's information to \(D\) with AF protocol. It is assumed that there does not exist a direct link between the source and destination nodes due to limited transmit power or shadowing effects [20; 21]. Each node is assumed to be equipped with a single antenna and all radio links are assumed to be subjected to independent and frequency non-selective quasi-static fading that the channels keep invariant during the entire communication block with a time duration of \(T\)[20; 31]. The channel coefficients from \(S\) to the \(l\)th \(R_{l}\) and from \(R_{l}\) to \(D\) are denoted as \(h_{S,R_{l}}\) and \(h_{R_{l},D}\), respectively. All links are assumed to experience both small-scale Rician fading with Rician factor \(K_{R}\) and large-scale path loss effects with path loss exponent \(\beta\), i.e., the channel coefficients \(h_{i,j}\) can be denoted as \(h_{i,j}=g_{i,j}/\sqrt{d_{i,j}^{\beta}}\), where \((i,j)\in\{(S,R_{l}),(R_{l},D)\}\), \(g_{i,j}\) represents the small-scale Rician fading of the channel, and \(d_{i,j}\) is the distance between the nodes \(i\) and \(j\). We assume perfect channel state information (CSI) for all of the nodes. In fact, the CSI can be obtained by channel estimation and instantaneous channel feedback, and due to quasistatic fading, the time overhead for the channel estimation from the transmitters to the receivers and the channel feedback from the receivers to the transmitters is negligible compared to the total transmission time [32]. The data transmission of each communication block is divided into two time phases as in [11; 12]. A half-duplex mode is applied to the relay nodes for the elimination of the two time phases's mutual interference. In _phase_-I, the source node broadcasts the information bits while the \(L\) relay nodes receive the information and harvest the energy with the PS or TS receiver structure. In _phase_-II, \(S\) keeps silent and the \(L\) relay nodes amplify the received information signal and forward it to \(D\) using the energy harvested during _phase_-I. The time slot structure for PS and TS schemes is shown in Figure 2a,b, respectively, where \(\rho_{l}\) is the PS ratio for the PS receiver at \(R_{l}\) and \(\alpha\in[0,1]\) is the EH time ratio for the TS receiver at all relay nodes. Note that for TS scheme illustrated in Figure 2b, the same EH time ratio is considered for all relay nodes, i.e., \(\alpha_{l}\)=\(\alpha,\forall l\in\{1,2,\ldots,L\}\), where \(\alpha_{l}\) is the EH time ratio of \(R_{l}\). The reason for this is that the setting of the same EH time ratio can enhance the efficiency, since different EH time ratios may bring the problem that some relay has to wait until the other relays complete EH [20]. In _phase_-II, we assume that each relay uses a suitable space-time coding, e.g., distributed orthogonal space-time block coding (OSTBC) to achieve diversity gains for the data transmission from the relay nodes to the destination [21; 33; 34]. We first consider the PS scheme as illustrated in Figure 2a. For the transmission in _phase_-I, the received baseband signal at \(R_{l}\) with PS scheme is given as:  
\[y_{S,R_{l}}(n)=\sqrt{P_{s}}h_{S,R_{l}}x_{s}(n)+N_{R_{l},a}(n), \tag{1}\]  
Figure 1: System model. Figure 2: Time slot structure for TS and PS schemes: (**a**) PS scheme, (**b**) TS scheme. where \(P_{s}\) is the transmit power of the source node \(S\), \(x_{s}(n)\) is the information signal transmitted from \(S\), and \(N_{R_{l},a}(n)\sim\mathcal{CN}\left(0,\sigma_{R_{l},a}^{2}\right)\) is the receive antenna noise at \(R_{l}\).