# Relay Selection Schemes for FSO Communications over Turbulent Channels
## 2 System Model  
The system consists of a source (S), relays (R), and a destination (D) working over the Log-normal channel and the Gamma-Gamma channel. We deem a relay-assisted FSO communication system, in which the transmitter transmits the signal from a point source, propagates that through an active relay from \(M\) parallel relays, before reaching the destination. An active relay will work in a time slot and the destination node will only receive the signal from the relay and will not receive anything from the source. Figures 1-3 illustrate the three relay selection cases S-R-D, S-R, and R-D. In the S-R-D scenario, there is a number of FSO \(M\) parallel relays in the communication system. We only choose one relay as the best selection to detect a transmitted data and forward the data to the destination, as depicted in Figure 1. The best relay selection \(R_{best}\) is selected by [12]  
\[R_{best}=\max_{i\in\{1:M\}}(\min(h_{SR_{M}},h_{R_{M}D})) \tag{1}\]  
where \(h_{SR_{M}}\) and \(h_{R_{M}D}\) are the channel fading coefficients between S and the \(M\) relay, \(R_{M}\), and between \(R_{M}\) and D, respectively [11]. Figure 1: Source-relay-destination scheme. Figure 2: Source-relay scheme. The nodes of relays are assumed to be located, where \(d_{SR}\) and \(d_{RD}\) denote the distances of S-\(R_{M}\) and \(R_{M}\)-D links, where \(i\) = 1, 2,..., \(M\)[11]. In our three schemes, we assume that all relays have full channel knowledge. In scheme (1), the user transmits data from the source directed towards the relays. Subsequently, one relay decodes-and-forward the signal and then retransmits it to the final destination. The best relay should have the highest end-to-end signal-to-noise ratio (SNR)  
In the Source-Relay scheme, there is a number of FSO parallel relays, \(M\), in the communication system. The best relay selection is determined in a left-hand side (transmission phase) when the data is transmitted from source to the best relay. Afterwards, this relay will complete the journey with retransmitting data to the final destination, as depicted in Figure 2. The best relay selection, \(R_{best}\) is selected by  
\[R_{best}=\max_{i\in\{1:M\}}(h_{SR_{M}}) \tag{2}\]  
In the relay-destination scheme, there is a number of FSO parallel relays, \(M\), in the communication system. The best relay selection is decided in a right-hand side (receiving phase). A pilot signal is transmitted from the source to all the relays at the same time. Subsequently, all of the relays beyond the destination choose which channel is the best to get the selected relay to pass the signal through this channel to the final destination, as depicted in Figure 3. The best relay that has the best channel coefficient is selected to complete the data to the receiver. The R-D is selected by  
\[R_{best}=\max_{i\in\{1:M\}}\left(h_{R_{M}D}\right) \tag{3}\]  
The normalized channel coefficient, \(h\), of the considered system can be formulated as [11]  
\[h=h_{a}h_{p}h_{\beta} \tag{4}\]  
where \(h_{a}\) and \(h_{P}\) are the channel fading coefficient due to atmospheric turbulence and the misalignment error, respectively. The path loss, \(h_{\beta}\) can be obtained by combining the weather attenuation with the geometric losses, as [11]  
\[h_{\beta}=10^{-ad/10}\times\frac{D_{R}^{2}}{\left(D_{T}+\theta_{T}d\right)^{2}} \tag{5}\]  
\[R_{best}=\max_{i\in\{1:M\}}(\min(h_{SR_{M}},h_{R_{M}D})) \tag{1}\]0 Relay-destination scheme. where \(d\) is the link distance for a hop, \(\alpha\) is the weather-dependent attenuation coefficient, \(D_{R}\) and \(D_{T}\) are the receiver and transmitter aperture diameters (in m), respectively, and \(\theta_{T}\) is the optical beam divergence angle (in mrad). The general model of misalignment fading is proposed in [13], as  
\[R_{best}=\max_{i\in\{1:M\}}(\min(h_{SR_{M}},h_{R_{M}D})) \tag{1}\]1  
where \(r\) is the radial displacement which is modelled by Rayleigh distribution.