## 2 System and Channel Model  
### Signal and System Assumptions  
The proposed mixed DF relay-assisted FSO system transmits the same information-bearing optical signal through \(N\) different parallel paths. Additionally, each path employs \(H\)-1 serially connected DF relays, with \(H\geq 1\), and thus each of the \(N\) paths consists of \(H\) intermediate links, commonly known as hops, as can be seen in Figure 1. Thus, the specific case of \(N=1\) describes the single input single output (SISO) link configuration, which is either a single-hop or a multi-hop one, for \(H=1\) and \(H>1\), respectively. Similarly, for the generalized case of \(N>1\) along with \(H>1\), the system is considered as a mixed single input-multiple output (SIMO) and multiple input single output (MISO) one, respectively. Indeed, from the transmitter to each first DF relay node of each parallel path, the system consists of one source (i.e., the transmitter) and \(N\) different destinations (i.e., the \(N\) parallel first relay nodes),and thus, it can be considered as a SIMO one. Then, each DF relay decodes its received signal's copy and retransmits it to the next, serially connected DF relay of the same path. This procedure continues until the final retransmissions, i.e., from the final DF relay nodes to the final receiver node of the total system. Thus, from the last DF node of each parallel path to the final receiver node, the system consists of multiple transmitter DF nodes and one receiver node, i.e., this part of the system is a multiple input single output (MISO) one. It is notable that we consider OOK signaling with an intensity modulation and direct detection (IM/DD) format in order to overcome any interference of the arriving information signal copies, from different paths, at the receiver node of the total system. Under these assumptions, the received signal, \(y_{\{i,j\}}\),at the \(j\)th hop in the \(i\)th path is expressed as [10; 15]:  
\[y_{\{i,j\}}=\eta I_{\{i,j\}}x_{\{i,j\}}+n_{\{i,j\}} \tag{1}\]  
where \(i\) indicates a specific parallel path, i.e., \(i\) = {1,2...\(N\)}, \(j\) represents a specific hop in the parallel path, i.e., \(j\) = {1,2...\(H\)}, \(\eta\) stands for the effective photo-current conversion ratio, \(I_{\{i,j\}}\), is the normalized irradiance in the \(j\)th hop in the \(i\)th path, while \(x_{\{i,j\}}\) and \(n_{\{i,j\}}\) denote the corresponding modulated signal and the additive white Gaussian noise (AWGN) with zero mean and variance \(N_{0}/2\), respectively [3; 4].  
Owing to atmospheric turbulence, pointing errors and attenuation effects, the normalized irradiance, \(I_{\{i,j\}}\), is written as [3; 4; 10]:  
\[I_{\{i,j\}}=I_{a,\{i,j\}}I_{p,\{i,j\}}I_{l,\{i,j\}} \tag{2}\]  
where \(I_{a,\{i,j\}}\) and \(I_{p,\{i,j\}}\) represent the normalized received irradiance in the \(j\)th hop in the \(i\)th path due to turbulence-induced and misalignment-induced fading, respectively, while \(I_{l,\{i,j\}}\) stands for the corresponding deterministic weather-dependantpath loss parameter, [23]. However, the investigation of the path losses is not the scope of this work and thus, without loss of generality, it is assumed that is normalized to unity (\(I_{l,m}\) = 1) [4; 24].