# Differential Signalling in Free-Space Optical Communication Systems
## 7 Results and Discussion  
In this section, we will present and further discuss the equations derived in the previous section. For each studied case, the variance of DTL as well as the BER of the link is obtained. The wavelengths adopted are 850 and 830 nm and we have assumed that both links have identical parameters. Furthermore, the normalised noise variance is set to 0.06. For the lossy channel and with the ambient light, where channels are inherently highly correlated, the variance of DTL and BER expressions remains the same as under the clear channel condition; therefore, here we focus on the other channel conditions of PE and turbulence. Regarding the series expressions with infinite upper limit in Table 1, it worth mentioning that we did not study the convergence of them. However, we considered between 30-50 terms to calculate the results. Figure 2 illustrates the results for weak turbulence. The predicted data shown are obtained using the equations given in Section 4.1. Using Label (17b), the predicted normalised variance of DTL against the turbulence for a range of correlation coefficients for SISO and DSS based schemes is shown in Figure 2a. Regardless of turbulence the variance of DTL approaches the minimum value in the cases of highly correlated channels. Note that, on the contrary, the variance of a SISO remains constant, whereas the DSS based scheme offers improved performance at higher turbulence levels. Figure 2b illustrates the BER as a function of SNR for the DSS system for a range turbulence levels. Also displayed are the simulated results, which show a close match with the predicted data. As can be observed, the DSS link, which is assumed to have a high correlation coefficient, display negligible deterioration at higher turbulence levels. Figure 2: Single-input single-output (SISO) and differential signaling scheme (DSS) based free-space optical (FSO) links with weak turbulence: (**a**) the normalised variance of detection threshold level (DTL) versus the correlation coefficient of the channels; and (**b**) bit-error-rate (BER) against signal-to-noise ratio (SNR) for a range of \(\sigma_{R}^{2}\). As discussed previously, in the case of moderate to strong turbulence, other models should be adopted to assess the link performance. Here, we have used a GG turbulence model as outlined in Section 6.2 to determine the variance of DTL and the BER of a DSS link as shown in Figure 3. As in the weak turbulence case, for higher levels of turbulence in highly correlated channels, DSS offers improved performance compared with SISO. Figure 4a depicts the variance of DTL versus the \(\rho_{\text{PE}}\) for both SISO and DSS links for a range of equivalent beamwidth \(w_{eq}\). The plots display the same profile as in Figure 2a with DSS offering improved performance at more significant PE. The benefit of highly correlated DSS compared to the SISO link with the same \(w_{eq}\) is more outstanding when \(w_{eq}\) reduces. Smaller \(w_{eq}\) can be interpreted as the reduced effect of PE on the optical beam. Using the predicted \(\gamma_{\text{DSS}}\) from Label (29), the variance of equivalent DSS PE (\(\sigma_{j,\text{DSS}}\)) can be determined for a range of \(\rho_{\text{PE}}\). Figure 4b depicts the jitter standard deviation against \(\rho_{\text{PE}}\) for a SISO link and an equivalent DSS link with \(a=10\) cm, \(w=100\) cm, and \(\sigma_{j,1}=\sigma_{j,2}=10\) cm. It is observed that the PE induced fading effect reduces with increasing value of \(\rho\). For instance, for an uncorrelated channel with \(w_{eq}=0.3\), the normalised variance is \(\sim\)0.16, whereas, in highly correlated channels, the normalised variance is reduced to \(\sim\)0.06. Also from both Label (29) and Figure 4b and for \(\rho_{\text{PE}}\geq 0.5\), we have \(\gamma_{\text{DSS}}\leq\gamma_{1}\). The predicted BER results against the SNR for a range of \(w_{eq}\) are displayed in Figure 4c showing a good match with the simulated data. Note that, for lower values of \(w_{eq}\), the SNR penalty is rather high for a given BER.