## 2 Transmission Model  
Consider a multiple relay cooperative communication where a source node \(S\) transmits information with the help of \(n_{R}\) EH relay nodes \(R_{1},\cdots,R_{n_{R}}\), as shown in Figure 1. The cooperation phase consists of two time-slots. In the first time-slot, the source node broadcasts a data message to the relay nodes. It is assumed that the direct links between source node and destination node are not available due to masking effects. In the second time-slot, the selected relay forwards the message to the destination node. Source and relay nodes are assumed to transmit through wireless orthogonal channels operating in a half-duplex mode. In the EH relay, the received source signal is divided with a power-splitter into two parts, where the first part is fed to the channel decoder and channel encoder, and the second part is used for supplying the power of message forwarding module. The power splitting ratio (PSR) of an EH relay \(R_{r}\) is denoted by \(\rho_{r},(0<\rho_{r}<1)\), which is defined as the ratio of the energy harvested for the information forwarding to the total received energy. The remaining fraction \(1-\rho_{r}\) of the received energy is allocated for the decoding and re-encoding of the message. Let \(\mathbf{u}=(u_{1},\ldots,u_{K})\) be the information sequence of the source node \(S\), where \(K\) denotes the information word length. In this work, the source node employs binary channel code of rate \(R_{S}=K/N_{S}\), where \(N_{S}\) denotes the source code-word length. Let \(\mathbf{c}=(c_{1},\ldots,c_{N_{S}})\) be the source code-word. The code bits of \(\mathbf{c}\) are mapped into a modulated sequence \(\mathbf{x}\). A binary phase-shift keying (BPSK) modulation is considered. During the first time-slot, the received sequence at the \(r\)th relay node is expressed as  
\[\mathbf{y}_{SR_{r}}=h_{SR_{r}}\sqrt{P_{SR_{r}}}\mathbf{x}+\mathbf{u}_{SR_{r}}, \tag{1}\]  
where \(h_{SR_{r}}\) is the channel fading coefficient at the \(S-R_{r}\) link, which is modeled as a Rayleigh distributed random variable with normalized second order moment, \(P_{SR_{r}}\) is the average received signal power at the EH relay node \(R_{r}\), and \(\mathbf{u}_{SR_{r}}\) is the additive noise, which is modeled as a zero-mean Gaussian random variable with variance \(\sigma_{n}^{2}\). In this paper, we consider quasi-static fading channels where the fading coefficients are constant within a block and change independently from one block to another. For the sake of fair comparison with the direct communication, let \(P_{SD}\) be the received signal power at the destination node of the direct link as if the direct link between the source node and destination node were available; \(P_{SD}\) is considered as the reference signal power, and, accordingly, the SNR of the  
Figure 1: Coded cooperative network with multiple energy-harvesting relays. received signal part at the EH relay node \(R_{r}\) used to process the data from the source is expressed as  
\[\Gamma_{SR_{r}} = |h_{SR_{r}}|^{2}(1-\rho_{r})G_{SR},\frac{P_{SD}}{\sigma_{n}^{2}}, \tag{2}\]  
where \(G_{SR_{r}}\) is referred to as the power gain of the \(S-R_{r}\) link and is given by \((d_{SD}/d_{SR_{r}})^{\alpha}\), whereby \(d_{SR_{r}}\) and \(d_{SD}\) are, respectively, the normalized distances of the \(S-R_{r}\) and \(S-D\) links, and \(\alpha\) is the path-loss exponent. It is worth noting that the normalized distance of a link \(T_{x}-R_{x}\) is given by \(\tilde{d}_{T_{x}R_{x}}/D_{T_{x}}\), where \(\tilde{d}_{T_{x}R_{x}}\) is the Euclidean distance separating the node \(T_{x}\) and \(R_{x}\), and \(D_{T_{x}}\) is a reference distance of the node \(T_{x}\)[28]. Let \(\tilde{\Gamma}_{SR_{r}}=(1-\rho_{r})G_{SR_{r}}P_{SD}/\sigma_{n}^{2}\) denote the average SNR of \(\Gamma_{SR_{r}}\).