# Differential Signalling in Free-Space Optical Communication Systems
## Appendix A
### BER Expression in PE  
Let us consider the transmission of the OOK signal, where the received signal can be written corresponding to both states of \(s\) as  
\[y_{1}=\eta h_{1}+n_{1}-n_{2}\;\text{if}\;S=\;1, \tag{15a}\] \[y_{2}=-\eta l_{2}+n_{1}-n_{2}\;\text{if}\;S=0. \tag{15b}\]  
The decision for the value of \(S\) is taken by comparing \(Y=y_{1}-y_{2}\) with  
\[P(e|h_{1},h_{2})= 0.5[P\{n_{1}-n_{2}<-\eta h_{1}\}+P\{n_{1}-n_{2}>\eta h_{2}\}], \tag{10b}\] \[P(e|h_{1},h_{2})== 0.5[1-P\{n_{1}-n_{2}>-\eta h_{1}\}+P\{n_{1}-n_{2}>\eta h_{2}\}]. \tag{10c}\]  
By substituting Label (11a) and Label (11b) in Label (10a), we obtain Label (10b); Label (10c) is a consequence of the property that \(P\{x<v\}=1-P\{x>v\}\), where \(P\{x\}\) stands for the probability of \(x\). Since \(n_{i}\sim N\big{(}0,\sigma_{n}^{2}\big{)}\), \(i=1,\ 2\), it can be easily shown that \(n_{1}-n_{2}\sim N\big{(}0,2\sigma_{n}^{2}\big{)}\), and the PDF of \(n_{1}-n_{2}\) is given by  
\[f_{n_{1}-n_{2}}(x)=\frac{1}{2\sqrt{\pi}\sigma_{n}}\mathrm{e}^{-x^{2}/(4\sigma_ {n}^{2})}. \tag{11}\]  
From Label (10), we have  
\[P\{n_{1}-n_{2}>-\eta h_{1}\}=\frac{1}{2\sqrt{\pi}\sigma_{n}}\int_{-\eta h_{1}}^ {\infty}\mathrm{e}^{-x^{2}/(4\sigma_{n}^{2})}dx. \tag{12}\]  
Substituting \(\frac{x}{\sqrt{2}\sigma_{n}}=y\) in Label (12) and following some algebra, we have  
\[P\{n_{1}-n_{2}>-\eta h_{1}\}=\frac{1}{\sqrt{2\pi}}\int_{\frac{-\eta h_{1}}{ \sqrt{2}\sigma_{n}}}^{\infty}e^{y^{2}/2}dy=Q\bigg{(}\frac{-\eta h_{1}}{\sqrt{ 2}\sigma_{n}}\bigg{)}, \tag{13}\]  
\[P\{n_{1}-n_{2}>\eta h_{2}\}=Q\bigg{(}\frac{\eta h_{2}}{\sqrt{2}\sigma_{n}} \bigg{)}. \tag{14}\]  
Hence, from Label (10c), Label (13), and Label (14):  
\[P(e|h_{1},h_{2})=\frac{1}{2}\bigg{[}1-Q\bigg{(}\frac{-\eta h_{1}}{\sqrt{2} \sigma_{n}}\bigg{)}+Q\bigg{(}\frac{\eta h_{2}}{\sqrt{2}\sigma_{n}}\bigg{)} \bigg{]}. \tag{15}\]  
Since \(1-Q(-x)=Q(x)\), therefore, Label (15) can be rewritten as  
\[P(e|h_{1},h_{2})=\frac{1}{2}\bigg{[}Q\bigg{(}\frac{\eta h_{1}}{\sqrt{2}\sigma _{n}}\bigg{)}+Q\bigg{(}\frac{\eta h_{2}}{\sqrt{2}\sigma_{n}}\bigg{)}\bigg{]}. \tag{16}\]  
The PDF of \(h_{i}\) (unconditional on the instantaneous value of \(r\)) under the boresight, PE (i.e., Rician distribution of \(r\)) is given by ([8] Equation (5)),  
\[y_{2}=-\eta l_{2}+n_{1}-n_{2}\;\text{if}\;S=0. \tag{15b}\]0  
where \(A=\frac{\gamma^{2}}{A_{0}^{\gamma^{2}-1}}e^{-s^{2}/2\sigma_{j}^{2}}\), \(B=\frac{s}{\sigma_{j}^{2}}\sqrt{\frac{-\eta\sigma_{n}^{2}}{2}}\). For fully correlated \(h_{1}\) and \(h_{2}\)\(\rho=1\) and \(h_{1}=h_{2}\), thus the average BER is given by  
\[y_{2}=-\eta l_{2}+n_{1}-n_{2}\;\text{if}\;S=0. \tag{15b}\]1  
Using Label (17) and the relation \(Q\bigg{(}\frac{\eta h_{1}}{\sqrt{2}\sigma_{n}}\bigg{)}=\frac{1}{2}erfc\bigg{(} \frac{\eta h_{1}}{2\sigma_{n}}\bigg{)}\), where \(erfc(\cdot)\) is the complimentary error function, in Label (18), we have  
\[y_{2}=-\eta l_{2}+n_{1}-n_{2}\;\text{if}\;S=0. \tag{15b}\]2  
where \(\ln(\cdot)\) is a natural logarithm function. Employing a substitution \(x=B\bigg{(}\ln\bigg{(}\frac{h_{i}}{A_{0}}\bigg{)}\bigg{)}^{1/2}\) and \(=-\frac{1}{B^{2}}=\frac{\sigma_{1}^{4}}{s^{2}}\frac{2}{w_{\alpha q}^{2}}\), and following simplifications in Label (A16), we have  
\[y_{2}=-\eta l_{2}+n_{1}-n_{2}\;\text{if}\;S=0. \tag{15b}\]3  
where \(D=AA\gamma^{2}C=\frac{2\gamma^{2}}{s^{2}}\frac{\sigma_{1}^{4}}{w_{\alpha q}^{2 }}e^{-s^{2}/2\gamma_{j}^{2}}\). Let us now use a power series expansion of the complimentary error function  
\[y_{2}=-\eta l_{2}+n_{1}-n_{2}\;\text{if}\;S=0. \tag{15b}\]4  
in Label (A17), which results in  
\[y_{2}=-\eta l_{2}+n_{1}-n_{2}\;\text{if}\;S=0. \tag{15b}\]5  
Utilizing an identity  
\[y_{2}=-\eta l_{2}+n_{1}-n_{2}\;\text{if}\;S=0. \tag{15b}\]6  
in Label (A19) results in the power series based BER expression of the considered scheme.