# Optical Wireless Communications Using Signal Space Diversity with Spatial Modulation
## 3 Results
### Analysis of the Theoretical and Numerical BER Results  
The performance of the SSD scheme was investigated by comparing the theoretical and numerical BER results to verify the theoretical BER expressions. For the numerical simulation, upsampling = 4 and rolloff factor = 0.2 are set for the CAP modulation, with symbol rate = 2.5 GBaud/s. For the theoretical investigation, the results are from the BER Equations (15) and (16) in Section 2.3. The SSD scheme was first studied at different signal constellation rotation angles \(\varphi\) with fixed channel gains \(\mathrm{h}_{1}=1\) and \(\mathrm{h}_{2}=0.63\), where the 100% SSD interleaving ratio is applied. The BER results are shown in Figure 4a. Here, the theoretical BER expression is based on Equation (15) due to \(d_{1}\) being smaller than \(d_{3}\). As \(\varphi\) is increased from \(0^{\circ}\) to \(20^{\circ}\), there is a good match between the theoretical and numerical BER results. However, when \(\varphi\) is further increased to \(45^{\circ}\), a small gap is observed between the theoretical and numerical BER curves in the low \(\frac{E_{s}}{N_{0}}\) range. Table 1 shows the values of \(d_{1}\), \(d_{2}\), and \(d_{3}\) that are used in Section 3.1. From Table 1, \(d_{1}=d_{2}\) is obtained at \(\varphi=45^{\circ}\) when the small gap occurs. Moreover, the value of \(d_{1}\) (\(d_{2}\)) is also much closer to that of \(d_{3}\). From Equations (11) to (13), it is implied that the values of \(d_{1}\) and \(d_{3}\) are impacted by both the signal constellation rotation angle and the channel gain distribution. Therefore, the SSD scheme is also studied with different channel gain distributions at different signal constellation rotation angles \(\varphi\), where the 100% SSD interleaving ratio is still applied. The BER results are shown in Figure 4b. From Table 1, \(d_{1}<d_{3}\) is obtained for the black and blue curve pairs, and thus, the theoretical BER expressions are based on Equation (15). Otherwise, for the red and purple curve pairs, the theoretical BER expressions are based on Equation (16). It can be seen that the two theoretical BER expressions work well under different conditions of \(d_{1}\) and \(d_{3}\). Besides, for the \(\varphi=45^{\circ}\) scenario, a similar gap is observed between the theoretical and numerical BER curves due to the comparable \(d_{1}\), \(d_{2}\) and \(d_{3}\). To further explore the cause for the small gap, the SSD scheme was studied with the setting of \(d_{1}=d_{2}\) at \(\varphi=45^{\circ}\) using different channel gain distributions. The BER results are shown in Figure 4c. For the yellow and grey curve pairs, the theoretical and numerical BER results match well, where significant differences between the value of \(d_{1}\) (\(d_{2}\)) and that of \(d_{3}\) can be seen in Table 1. For the purple and green curve pairs, small gaps appear with the comparable \(d_{1}\), \(d_{2}\), and \(d_{3}\) observed in Table 1, which are similar to the \(\varphi=45^{\circ}\) scenarios in Figure 4a,b. In addition, the SSD scheme was also studied with the setting of \(d_{1}\cong d_{3}\). The BER results are shown in Figure 4d. Here, except for the orange curve pair, the other three curve pairs all exhibit good matches between the theoretical and numerical BER results. For these three curve pairs, obvious differences between the value of \(d_{1}\) (\(d_{3}\)) and that of \(d_{2}\) can be obtained from Table 1. As for the orange curve pair, a small gap occurs with the comparable \(d_{1}\), \(d_{2}\), and \(d_{3}\). Figure 4a-d clearly show that only in the scenario of comparable \(d_{1}\), \(d_{2}\), and \(d_{3}\) with low \(\frac{E_{s}}{N_{0}}\), the accuracy of theoretical BER expressions in Equations (15) and (16) is reduced. This is because when more noise is presented (low \(\frac{E_{s}}{N_{0}}\)) in the signal constellation with similar \(d_{1}\), \(d_{2}\), and \(d_{3}\), the constellation symbol can be incorrectly detected as the two neighboring constellation symbols instead of the correct constellation symbol.