## 3 System Model  
Let us consider a network comprised of three types of devices: a source, \(n\)-relays and a destination, all equipped with a single antenna. The relay nodes do not provide any additional traffic to the destination and the nodes are assumed to be half-duplex, that is, one node cannot transmit and receive simultaneously. The transmission of information occurs during two temporal phases of the same duration. The first phase is known as the \"broadcast phase\", where the source node broadcasts its information without redundancy during symbol periods. The relay nodes amplify (or detect and decode then re-encode) the symbols received and transmits it in the \"relaying phase\" where the relay nodes send the received symbols (refer Figure 1). It is assumed that the source-destination link is corrupt, so the destination does not take into account the signal originally transmitted in the broadcast phase. According to the given system model, the received signal at \(N^{\text{th}}\) relay in first phase can be given as:  
\[{}_{SR_{N}}=\sqrt{E_{sSR_{N}}}*\chi+\eta_{SRN} \tag{1}\]  
where \(H_{SR_{N}}\) is the channel coefficient present between source and \(N^{\text{th}}\) relay, \(\chi\) represent transmitted symbol from source, \(\eta_{SRN}\sim CN(0,N_{0})\) represents the additive white Gaussian noise (AWGN) noise with zero mean and \(N_{0}\) variance and \(\sqrt{E_{s}}\) presents the average energy per symbol. In the next step,the best relay to destination pair is selected using a Log Likelihood ratio to forward the received signal in its first phase. The best relay pair \(b_{i}\) is selected such that:  
\[b_{i}=arg\underset{i\in N}{\max}\{LLR_{N}\}\forall\ i\in(1,2) \tag{2}\]  
where \(LLR_{N}\) represents \(LLR\) magnitude of signal received at \(N^{\text{th}}\) relay in first phase given as:  
\[LLR_{N}=\frac{4\sqrt{E_{s}}}{N_{0}}Re\Big{\{}H_{SR_{N}}^{*}Z_{SR_{N}}\Big{\}}= \Big{|}H_{SR_{N}}\Big{|}^{2}\cdot\chi+Re\Big{\{}H_{SR_{N}}^{*},\eta_{SRN}\Big{\}} \tag{3}\]  
The magnitude of \(LLR\) in Equation (3) represents the reliability of hard decision whereas its sign represents hard decision value. Furthermore, the bit error probability at \(i\)th relay i.e., \(P_{bi}\) is related to \(LLR\) magnitude as [22]:  
\[P_{bi}=\frac{1}{1+e^{|LLR_{N}|}} \tag{4}\]  
According to the relationship provided in Equation (4), relay with maximum \(LLR\) magnitude will result in the lowest error possible. In the second phase, the selected relay pair will forward the signal received in the first phase with an amplification factor given as \(\beta_{i}=\frac{1}{\sqrt{\sqrt{E_{s}}|H_{\pi_{i}}|^{2}+N_{0}}}\ i\in(1,\ 2)\). Therefore, the received signal at the destination after the second phase of the transmission is given as:  
\[Z_{R_{i}D}=\sqrt{E_{s}}H_{R_{i}D}*\beta_{i}\Big{(}\sqrt{E_{s}}H_{SR_{N}}*\chi+ \eta_{SRN}\Big{)}+\eta_{R_{i}D}\ \forall\ i\in(1,\ 2) \tag{5}\]  
At the receiver end, the signal received from the two relays are combined using a maximal ratio combining (MRC), the combined signal will be:  
\[\widetilde{\chi}=\delta_{1}H_{R_{1}D}^{*}Z_{R_{1}D}+\delta_{2}H_{R_{2}D}^{*}Z_ {R_{2}D} \tag{6}\]  
where \(Z_{R_{1}D},Z_{R_{2}D}\) represents signal received at from relay-1 and relay-2 respectively and parameters \(\delta_{1}\) and \(\delta_{2}\) are known as link weights, the values of these link weights are determined such that signal strength of received signal at MRC output is maximum. These weights can be calculated as \(\delta_{1}=\frac{\sqrt{E_{s}}H_{R_{1}D}^{*}}{N_{0}}\) and \(\delta_{2}=\frac{\sqrt{E_{s}}H_{R_{2}D}^{*}}{N_{0}}\). Assuming the average energy of transmitted signal \(\chi\) to be unity, the value of instantaneous SNR \((\gamma_{t})\) at the destination node using the MRC receiver can be given as:  
Figure 1: General block diagram of cooperative diversity system.