# Differential Signalling in Free-Space Optical Communication Systems
## 2 Differential Signalling Scheme  
The proposed system block diagram is depicted in Figure 1. The NRZ-OOK signal \(S\in\{0,1\}\) and its inverted version \(\overline{S}\) are used for intensity-modulation of two optical sources at wavelengths of \(\lambda_{1}\) and \(\lambda_{2}\), respectively. It is worth mentioning that by comparing \(S\) to the optimal DTL \(S_{thresh}=E[S]\) where \(E[\cdot]\) denotes the expected value, the original data bits (i.e., bit is \"0\" for \(S<S_{thresh}\) and \"1\" elsewhere) can be retrieved. Note that the optimal DTL for \(\overline{S}\) is also \(S_{thresh}\). The output intensities \(I_{i}\) (\(i=1,\ 2\)) of optical sources (OSs) are given by  
\[\left[ \begin{array}{c}I_{1}\\ I_{2}\end{array} \right]=\left[ \begin{array}{cc}\Gamma_{1}&0\\ 0&\Gamma_{2}\end{array} \right]\left[ \begin{array}{c}S\\ \overline{S}\end{array} \right], \tag{1}\]  
where \(\Gamma_{i}\) denotes the electrical-to-optical conversion coefficient of \(i\)-th OS. Outputs of OSs are then passed through a beam combiner (BC) to ensure that both beams are transmitted over the \(L\)-length FSO channel. The optical signals \(\mathbf{y}\) at the Rx are given by  
\[\left[ \begin{array}{c}y_{1}\\ y_{2}\end{array} \right]=\left[ \begin{array}{cc}h_{1}&0\\ 0&h_{2}\end{array} \right]\left[ \begin{array}{cc}\Gamma_{1}&0\\ 0&\Gamma_{2}\end{array} \right]\left[ \begin{array}{c}S\\ \overline{S}\end{array} \right], \tag{2}\]  
where \(0\leq h_{i}\leq 1\) denotes the channel coefficient (response) including the effects of geometrical and atmospheric losses, PE, and atmospheric turbulence. \(\mathbf{y}\) is passed through a 50/50 beam splitter (BS) and optical filters (OF) with the centre wavelengths of \(\lambda_{1}\) and \(\lambda_{2}\) prior to being collected by an optical Rx. The generated photocurrents are amplified by transimpedance amplifiers (TIA) with outputs given by  
\[\left[ \begin{array}{c}v_{1}\\ v_{2}\end{array} \right]=\frac{1}{2}\left[ \begin{array}{cc}\Re_{1}G_{1}h_{1} \Gamma_{1}&0\\ 0&\Re_{2}G_{2}h_{2}\Gamma_{2}\end{array} \right]\left[ \begin{array}{c}S\\ \overline{S}\end{array} \right]+\left[ \begin{array}{c}n_{1}\\ n_{2}\end{array} \right], \tag{3}\]  
where \(\Re_{i}\) is the photodetector responsivity, \(G_{i}\) is gain of TIA, \(n_{i}\) is the additive white Gaussian noise (AWGN) with zero mean and variance \(\sigma_{n,i}^{2}=N_{0}/2\), and \(N_{0}\) denotes the noise power spectral density. The combined output \(v_{t}=v_{1}-v_{2}\) is given by  
\[v_{t}=0.5\big{[}\Gamma_{1}h_{1}\mathcal{R}_{1}G_{1}S-\Gamma_{2}h_{2}\mathcal{R }_{2}G_{2}\overline{S}\big{]}+\Delta n, \tag{4}\]  
where \(\Delta n=n_{1}-n_{2}\). A sampler, sampling at the centre of bit duration, and a threshold detector are used to regenerate the transmit data. From Label (4), the optimal DTL for \(v_{t}\) is given by  
\[V_{thresh}=0.5S_{thresh}(\Gamma_{1}h_{1}\mathcal{R}_{1}G_{1}-\Gamma_{2}h_{2} \mathcal{R}_{2}G_{2})+\Delta n. \tag{5}\]For simplicity, we make the following assumptions: (i) links are identical; therefore, we let \(0.5\Gamma_{i}\mathcal{R}_{i}G_{i}=\eta\); (ii) channels (i.e., transmission path) are very close to each other with \(h_{1}\) and \(h_{2}\) having the same statistical properties, thus \(\mathrm{E}[\cdot]\) being the expected value, \(\mathrm{E}[h_{1}{}^{n}]=\mathrm{E}[h_{2}{}^{n}]\), where \(n\in\mathbb{N}\)); and (iii) the chosen wavelengths are close enough to ensure that the channel effects on the propagating optical beams are almost the same. Next, we have [27]  
\[\mathrm{Mean}(V_{thresh})=\eta S_{thresh}[\mathrm{Mean}(h_{1})-\mathrm{Mean}( h_{2})], \tag{6a}\] \[\mathrm{Var}(V_{thresh})=\eta^{2}S_{thresh}{}^{2}\Big{[}\mathrm{Var}(h_{1})+ \mathrm{Var}(h_{2})-2\rho\sqrt{\mathrm{Var}(h_{1})\mathrm{Var}(h_{2})}\Big{]} +\sigma_{n,1}^{2}+\sigma_{n,2}^{2}, \tag{6b}\]  
where \(\mathrm{Mean}(\cdot)\) and \(\mathrm{Var}(\cdot)\) are the mean and variance, respectively, and \(\rho\) is the correlation coefficient between \(h_{1}\) and \(h_{2}\).