# Differential Signalling in Free-Space Optical Communication Systems
## 7 Results and Discussion
For instance, at a BER of \(10^{-2}\), the SNR penalty is \(\sim\)14 dB for \(w_{eq}=3\) cm compared \(w_{eq}=7\) cm. Note that, in Figure 4c, the different SNR vales are obtained by varying the value of noise variance \(\sigma_{n}^{2}\) for \(\eta=1\). Figure 3: Single-input single-output (SISO) and differential signaling scheme (DSS) based free-space optical (FSO) links with moderate to strong 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 \(\sigma_{R}^{2}\). Figure 5a shows the variance of DTL for simultaneous PE and weak turbulence as a function of the channel correlation coefficient for range of 0 to 1. Again, it is seen that, for higher turbulence levels and PE, the DSS system offers improved performance compared to the SISO. The BER versus the SNR for a range of \(w_{eq}\) for the combined effects of PE and weak turbulence is plotted in Figure 5b, showing performance degradation at higher levels of turbulence and PE. As expected from Figure 4c, larger \(w_{eq}\) improves the BER performance of DSS. Compared to the cases where only turbulence or PE exist in the channel, we observe a significant degradation in the performance. For instance, in a weak turbulence case for \(\sigma_{R}^{2}=0.3\) and SNR of 10 dB, the BER is less than \(10^{-3}\). However, for the same turbulence situation, \(w_{eq}=1\), and SNR of 10 dB, the BER is less than \(10^{-2}\). Figure 4: Single-input single-output (SISO) and differential signaling scheme (DSS) based free-space optical (FSO) links with pointing errors (PE): (**a**) the normalised variance of detection threshold level (DTL) versus the correlation coefficient of the channels; (**b**) the jitter standard deviation of the equivalent DSS PE vs. correlation coefficient \(\rho_{\text{PE}}\), for the SISO system with DSS and for \(a=10\) cm, and \(w=100\) cm, and (**c**) bit-error-rate (BER) against signal-to-noise ratio (SNR) for \(\sigma_{j}=1,\,s=1,\,A_{0}=1\), and different values of \(w_{eq}\) with unit of cm. Finally, the combined effects of PE and moderate to strong turbulence on the variance of DTL as well as the BER performance are illustrated in Figure 6a,b, respectively. For these cases, we can observe that even DDS is not mitigating the camel fading efficiently. Therefore, a designer might consider additional methods such as using a larger aperture to improve the system performance by aperture averaging. The plots show further degradation in both the variance of DTL and the BER compared to Figure 5. The same trend of the larger \(w_{eq}\) and the better BER performance of DSS are also seen in Figure 6b. In a strong turbulence regime, PE are less dominant than turbulence. In all variance of DTL plots, one can see that once the channels are highly correlated, the normalised variance of DSS is reduced to the normalised variance of noise with the value of \(\sim\)0.06. Therefore, under a highly correlated turbulence channel with significant channel fading, it is highly advantageous to implement a DSS FSO scheme in comparison to a SISO one. Figure 5: Single-input single-output (SISO) and differential signaling scheme (DSS) based free-space optical (FSO) links with pointing errors (PE) and weak turbulence: (**a**) the normalised variance of detection threshold level (DTL) versus the correlation coefficient of the channels; and (**b**, **c**) bit-error-rate (BER) against signal-to-noise ratio (SNR) for different values of \(w_{eq}\) with unit of cm and \(\sigma_{R}^{2}\).