# Optical Wireless Communications Using Signal Space Diversity with Spatial Modulation
## 3 Results
### Analysis of the Theoretical and Numerical BER Results
However,  
such inaccuracy can be improved when \(\frac{F_{k}}{N_{0}}\) is increased. For most scenarios, the theoretical BER expressions agree well with the numerical simulations. \begin{table}
\begin{tabular}{c c c c c} \hline
**Items** & \(\mathbf{d}_{1}\) & \(\mathbf{d}_{2}\) & \(\mathbf{d}_{3}\) & \(\mathbf{d}_{1}\)\(<\)\(d_{3}\) \\ \hline h\({}_{1}\) =1, h\({}_{2}\) =0.63, 20\({}^{\circ}\) (Figure 4a, red) & 0.684\(\sqrt{E_{s}}\) & 0.964\(\sqrt{E_{s}}\) & 1.005\(\sqrt{E_{s}}\) & Yes (Equation (15)) \\ h\({}_{1}\) =1, h\({}_{2}\) =0.63, 45\({}^{\circ}\) (Figure 4a, blue) & 0.836\(\sqrt{E_{s}}\) & 0.836\(\sqrt{E_{s}}\) & 0.89\(\sqrt{E_{s}}\) & Yes (Equation (15)) \\ h\({}_{1}\) =1, h\({}_{2}\) =0.3, 20\({}^{\circ}\) (Figure 4b, black) & 0.443\(\sqrt{E_{s}}\) & 0.945\(\sqrt{E_{s}}\) & 0.711\(\sqrt{E_{s}}\) & Yes (Equation (15)) \\ h\({}_{1}\) =1, h\({}_{2}\) = 0.3, 30\({}^{\circ}\) (Figure 4b, red) & 0.563\(\sqrt{E_{s}}\) & 0.879\(\sqrt{E_{s}}\) & 0.549\(\sqrt{E_{s}}\) & No (Equation (16)) \\ h\({}_{1}\) =1, h\({}_{2}\) = 0.5, 30\({}^{\circ}\) (Figure 4b, blue) & 0.661\(\sqrt{E_{s}}\) & 0.901\(\sqrt{E_{s}}\) & 0.775\(\sqrt{E_{s}}\) & Yes (Equation (15)) \\ h\({}_{1}\) =1, h\({}_{2}\) = 0.5, 45\({}^{\circ}\) (Figure 4b, c, purple) & 0.791\(\sqrt{E_{s}}\) & 0.791\(\sqrt{E_{s}}\) & 0.707\(\sqrt{E_{s}}\) & No (Equation (16)) \\ h\({}_{1}\) =1, h\({}_{2}\) =0.3, 45\({}^{\circ}\) (Figure 4c, yellow) & 0.738\(\sqrt{E_{s}}\) & 0.738\(\sqrt{E_{s}}\) & 0.424\(\sqrt{E_{s}}\) & No (Equation (16)) \\ h\({}_{1}\) =1, h\({}_{2}\) = 0.4, 45\({}^{\circ}\) (Figure 4c, grey) & 0.762\(\sqrt{E_{s}}\) & 0.762\(\sqrt{E_{s}}\) & 0.566\(\sqrt{E_{s}}\) & No (Equation (16)) \\ h\({}_{1}\) =1, h\({}_{2}\) = 0.6, 45\({}^{\circ}\) (Figure 4c, green) & 0.825\(\sqrt{E_{s}}\) & 0.825\(\sqrt{E_{s}}\) & 0.849\(\sqrt{E_{s}}\) & Yes (Equation (15)) \\ h\({}_{1}\) =1, h\({}_{2}\) = 0.3, 29. 5\({}^{\circ}\) (Figure 4d, black) & 0.557\(\sqrt{E_{s}}\) & 0.883\(\sqrt{E_{s}}\) & 0.557\(\sqrt{E_{s}}\) & No (Equation (16)) \\ h\({}_{1}\) =1, h\({}_{2}\) = 0.4, 32. 4\({}^{\circ}\) (Figure 4d, red) & 0.633\(\sqrt{E_{s}}\) & 0.871\(\sqrt{E_{s}}\) & 0.632\(\sqrt{E_{s}}\) & No (Equation (16)) \\ h\({}_{1}\) =1, h\({}_{2}\) = 0.5, 37. 4\({}^{\circ}\) (Figure 4d, blue) & 0.726\(\sqrt{E_{s}}\) & 0.851\(\sqrt{E_{s}}\) & 0.725\(\sqrt{E_{s}}\) & No (Equation (16)) \\ h\({}_{1}\) =1, h\({}_{2}\) = 0.57, 44\({}^{\circ}\) (Figure 4d, orange) & 0.807 & 0.821 & 0.806 & No (Equation (16)) \\ \hline \end{tabular}
\end{table} 
Table 1: The values of \(d_{1}\), \(d_{2}\), and \(d_{3}\) with different channel gain distributions at different \(\varphi\). Figure 4: BER of the SSD scheme. (**a**) With different \(\varphi\). (**b**) With different channel gains and different \(\varphi\). (**c**) With \(d_{1}=d_{2}\) at \(\varphi=45^{\circ}\). (**d**) With \(d_{1}\cong d_{3}\). Note: num—numerical, theo—theoretical. Theoretical Investigation of Channel Gain Distributions at Different Signal Constellation Rotation Angles to Improve the BER Performance  
In addition to the study of theoretical BER expressions, Figure 4a,b also indicate the advantage of the SSD scheme for improving the BER performance. In this section, more theoretical investigations are conducted using different channel gain distribution settings together with different signal constellation rotation angles, which comprehensively show the capability of the SSD scheme to work with partial beam obstructions. For the first channel gain distribution setting, one of the channel gains remains constant as \(\mathrm{h}_{1}=1\), while the other channel gain \(\mathrm{h}_{2}\) changes from 0.2 to 0.8. This setting emulates the scenario when one channel is not obstructed while the other one is occasionally and partially obstructed. The signal constellation rotation angle also changes from \(0^{\circ}\) to \(45^{\circ}\) to comprehensively illustrate the BER trend.