# Optical Wireless Communications Using Signal Space Diversity with Spatial Modulation
## 3 Results
### Analysis of the Theoretical and Numerical BER Results
The theoretical BER results using Equations (15) and (16) are shown in Figure 5a with a high \(\frac{E_{i}}{\mathrm{h}_{0}}=18\), where \(\frac{\mathrm{h}_{2}}{\mathrm{h}_{1}}\) denotes the channel-gain ratio where \(\mathrm{h}_{1}>\mathrm{h}_{2}\). The corresponding constellation distances of \(d_{1}\), \(d_{2}\), and \(d_{3}\) are shown in Figure 5b. From Figure 5a, when the rotation angle is fixed, the BER performance is improved with a higher \(\mathrm{h}_{2}\) (\(\frac{\mathrm{h}_{2}}{\mathrm{h}_{1}}\)) value, which is because a higher signal power is received with the increases in \(d_{1}\), \(d_{2}\), and \(d_{3}\). However, if SM is incorporated, the applicable range of \(\mathrm{h}_{2}\) must also comply with the condition of required channel gain difference to enable a relatively low channel correlation [8]. On the other hand, when the \(\mathrm{h}_{2}\) (\(\frac{\mathrm{h}_{2}}{\mathrm{h}_{1}}\)) value is fixed, there could be a turning point for the BER performance with an increasing rotation angle, depending on the values of \(d_{1}\) and \(d_{3}\). As shown in Figure 5b, an increasing rotation angle \(\varphi\) results in a larger \(d_{1}\) and a smaller \(d_{2}\) and \(d_{3}\). When \(\varphi\) is small, \(d_{1}\) is much smaller than \(d_{2}\) and \(d_{3}\), and thus, \(d_{1}\) contributes more to the BER. The turning point occurs when the condition \(d_{1}=d_{3}\) is met. When \(d_{1}\) is further increased with the increasing \(\varphi\) to achieve \(d_{1}>d_{3}\), the BER performance is degraded as it is now dominated by smaller \(d_{3}\)[21]. However, when \(d_{1}\) is always smaller than \(d_{3}\) at \(0\leq\varphi\leq 45^{\circ}\), such as when \(\mathrm{h}_{2}>0.57\) is met, there is no turning point. In brief, for the first channel gain setting, the SSD scheme works well when \(\mathrm{h}_{2}\) is reasonably high with the upper limit set by the channel gain imbalance required for SM. In addition, the BER performance of the SSD scheme can be improved at a larger rotation angle when \(d_{1}\leq d_{3}\) is met. The results show that the optimum BER improvement ranges from \(10^{-2.9}\) to \(10^{-4.9}\) with \(\frac{\mathrm{h}_{2}}{\mathrm{h}_{1}}=0.5\) and \(\varphi=45^{\circ}\). The average BER improvement ranges from \(10^{-x}\) to \(10^{-x-1.64}\) (\(1.18\leq x\leq 5.75\)), which is over one order of magnitude. For the second channel gain distribution setting, one of the channel gains \(\mathrm{h}_{1}\) is reduced, while the other channel gain \(\mathrm{h}_{2}\) is increased to achieve the constant (\(\mathrm{h}_{1}+\mathrm{h}_{2}\)) using two  
Figure 5: Results of the 1 st channel gain setting. (**a**) BER. (**b**) Constellation distance. consecutive time slots. This setting emulates the scenario when one channel is partially obstructed at one time slot while the other channel compensates for such optical loss at the adjacent time slot. To comprehensively illustrate the BER trend, the signal constellation rotation angle still changes between 0\({}^{\circ}\) and 45\({}^{\circ}\). The theoretical BER results are shown in Figure 6a with a high \(\frac{E_{\rm{e}}}{N_{0}}=18\) and \(\rm{h_{1}}+\rm{h_{2}}=1.5\). The corresponding constellation distances of \(d_{1}\), \(d_{2}\), and \(d_{3}\) are shown in Figure 6b with insets provided. Compared with the results in Figure 5a, the best achievable BER performance in Figure 6a is degraded due to the constraint of constant \(\rm{(h_{1}+h_{2})}\) that leads to a lower achievable received signal power, especially in the higher \(\frac{h_{2}}{\rm{h_{1}}}\) range. Besides, when the rotation angle is fixed, a turning point for the BER performance can occur with an increasing \(\frac{h_{2}}{\rm{h_{1}}}\), depending on the values of \(d_{1}\), \(d_{2}\), and \(d_{3}\). It is observed in Figure 6b that at a fixed rotation angle \(\varphi\), an increasing \(\frac{h_{2}}{\rm{h_{1}}}\) results in a monotonically decreasing \(d_{2}\) but nonmonotonic changes in \(d_{1}\) and \(d_{3}\), which depend on \(\varphi\), as shown in the inset of Figure 6b.