# Differential Signalling in Free-Space Optical Communication Systems
## 5 The Case of Pointing Errors  
In a terrestrial FSO link, the joint geometrical loss and PE induced fading within the channel is given by [23]  
\[h_{i}(r;L)\approx A_{0}exp\left(-\frac{2r^{2}}{w_{eq}^{2}}\right), \tag{25}\]  
where \(r\), \(A_{0}\) and \(w_{eq}\) correspond to beam displacement, the geometrical loss and equivalent beamwidth, respectively. Note that \(A_{0}=[\text{erf}(v)]^{2}\) and \(w_{\text{eq}}^{2}=w^{2}\frac{\sqrt{\pi}\text{erf}(v)}{2v\exp(-v^{2})}\), where \(v=\sqrt{\pi/2}\frac{a}{w}\) and \(\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt\)[23]. Note that, if the geometrical loss is taken into account by using Label (9), \(A_{0}=1\) in Label (25). The PE induced displacement has two major components: (i) the boresight \(r_{b}\)--displacement between the beam centre and centre of the detector; and (ii) jitter \(r_{j}\)--offset of the beam centre at detector plane [24]. The mean offset of PE \(r_{b}\) represents a deviation originated from thermal expansion of buildings [8], whereas \(r_{j}\) is a RV originated from building sway and vibrations [8]. From the statistical point of view, the jitter corresponds to the random variation of the optical beam footprint around the boresight direction with the jitter variance of \(\sigma_{j}^{2}\)[43]. It is shown in [8] that \(r\) in Label (25) has a Rician PDF as below  
\[f_{r}(r)=\frac{r}{\sigma_{j}^{2}}\left(\frac{-\left(r^{2}+s^{2}\right)}{2 \sigma_{j}^{2}}\right)I_{0}\left(\frac{r_{S}}{\sigma_{j}^{2}}\right) \tag{26}\]  
with the mean and 2 nd moment given by  
\[Mean(h_{i})=E[h_{i}]=\frac{A_{0}\gamma^{2}}{1+\gamma^{2}}exp\left(-\frac{r_{ b}^{2}}{2(1+\gamma^{2})\sigma_{j}^{2}}\right), \tag{27a}\] \[E\left[{h_{i}}^{2}\right]=\frac{{A_{0}}^{2}\gamma^{2}}{2+\gamma^{2}}exp\left(- \frac{2r_{b}^{2}}{2(2+\gamma^{2})\sigma_{j}^{2}}\right), \tag{27b}\]  
where \(s\) is the boresight displacement and \(I_{0}(\cdot)\) is the modified Bessel function of the 1 st kind with order zero and \(\gamma=w_{\text{eq}}/2\sigma_{j}\). Therefore, the variance of \(h_{i}\) is  
\[Var(h_{i})=\frac{A_{0}^{2}\gamma^{2}}{2+\gamma^{2}}exp\left(-\frac{r_{b}^{2}}{ (2+\gamma^{2})\sigma_{j}^{2}}\right)-\frac{A_{0}^{2}\gamma^{4}}{\left(1+\gamma ^{2}\right)^{2}}exp\left(-\frac{r_{b}^{2}}{(1+\gamma^{2})\sigma_{j}^{2}} \right). \tag{28}\]  
Substituting Label (27) and Label (28) in Label (6) yields in the DTL average and variance expressions as given in Table 1. Note that we have used \(\rho_{\text{PE}}\) to refer to the correlation between PE fading effects within the channels. By using the tail probability of the error events, we can derive the BER for this special scenario where only the PE is considered. The simplified BER expression is outlined in Table 1, whereas detailed derivation for the BER is given in Appendix A.  
It is also possible to estimate the equivalent parameters of the DSS based FSO link. For this, we have adopted a single input single output (SISO) link with Rayleigh PE PDF and derived the parameters, which have the same PE variance as DSS. For the simplified case, where \(r_{b,1}=r_{b,2}=0\), assuming that \(\gamma_{1}=\gamma_{2}\) and \(A_{1}=A_{2}=A_{\text{DSS}}\), and simplifying \(\text{Var}(h_{\text{DSS}})=\text{Var}(h_{1}-h_{2})\), the following equation is derived from which \(\gamma_{\text{DSS}}\) of the equivalent PE PDF as  
\[\left(2+{\gamma_{1}}^{2}\right)\left(1+{\gamma_{1}}^{2}\right)^{2}/\left(2+{ \gamma_{DSS}}^{2}\right)\left(1+{\gamma_{DSS}}^{2}\right)^{2}=2(1-\rho){ \gamma_{1}}^{2}/{\gamma_{DSS}}^{2}. \tag{29}\]For \(r_{b,1}=r_{b,2}=0\), Rician distribution becomes Rayleigh and we assume that the PDF of equivalent PE is also Rayleigh for comparison.