# Optical Wireless Communications Using Signal Space Diversity with Spatial Modulation
## 1 Introduction
### Theoretical BER Expressions  
To further illustrate the performance of the proposed SSD scheme, the theoretical BER expressions of the SSD scheme are derived in this section. Assuming that two spatial bits differ in a group, the corresponding two complex-valued QAM signals can be written as  
\[\mathrm{y_{1-QAM}}=\mathrm{a_{1}}+\mathrm{b_{1}}\times j \tag{1}\]  
\[\mathrm{y_{2-QAM}}=\mathrm{a_{2}}+\mathrm{b_{2}}\times j \tag{2}\]  
where \(\mathrm{a_{1}}\) and \(\mathrm{a_{2}}\) are the values in the horizontal real axis (in-phase component), and \(\mathrm{b_{1}}\) and \(\mathrm{b_{2}}\) are the values in the vertical imaginary axis (quadrature component). To implement SSD, the two QAM signals above are first rotated by angle \(\varphi\), written as  
\[\mathrm{y_{1}}=\mathrm{e}^{i\varphi}\times\mathrm{y_{1-QAM}}=(\mathrm{a_{1}} \cos\varphi-\mathrm{b_{1}}\sin\varphi)+(\mathrm{b_{1}}\cos\varphi+\mathrm{a_{1 }}\sin\varphi)\times j \tag{3}\]  
\[\mathrm{y_{2}}=\mathrm{e}^{i\varphi}\times\mathrm{y_{2-QAM}}=(\mathrm{a_{2}} \cos\varphi-\mathrm{b_{2}}\sin\varphi)+(\mathrm{b_{2}}\cos\varphi+\mathrm{a_{2 }}\sin\varphi)\times j \tag{4}\]  
Then, diversity interleaving is conducted, given as  
\[\mathrm{y^{\prime}_{1}}=(\mathrm{a_{1}}\cos\varphi-\mathrm{b_{1}}\sin\varphi) +(\mathrm{a_{2}}\cos\varphi-\mathrm{b_{2}}\sin\varphi)\times j \tag{5}\]  
\[\mathrm{y^{\prime}_{2}}=(\mathrm{b_{1}}\cos\varphi+\mathrm{a_{1}}\sin\varphi) +(\mathrm{b_{2}}\cos\varphi+\mathrm{a_{2}}\sin\varphi)\times j \tag{6}\]  
After signal constellation rotation and diversity interleaving, the SSD signals are obtained. Such SSD signals can be transmitted in the OWC channels via CAP modulation and converted back to the complex signals at the receiver after going through the CAP-matched filters (MFs), as described in Section 2.2. Assume that the total channel gains of two spatial channels, including the channel gains passing through the electrical-optical-electrical conversion, are \(\mathrm{h_{1}}\) and \(\mathrm{h_{2}}\)[7]. The CAP MF outputs are written as  
\[\mathrm{y_{Rx1}}=\frac{1}{2}\mathrm{h_{1}}\times(\mathrm{a_{1}}\cos\varphi- \mathrm{b_{1}}\sin\varphi)+\frac{1}{2}\mathrm{h_{1}}\times(\mathrm{a_{2}}\cos \varphi-\mathrm{b_{2}}\sin\varphi)\times j \tag{7}\]  
\[\mathrm{+n_{1}}cos\varphi+\mathrm{n_{1}}sin\varphi\times j\]  
\[\mathrm{y_{Rx2}}=\frac{1}{2}\mathrm{h_{2}}\times(\mathrm{b_{1}}\cos\varphi+ \mathrm{a_{1}}\sin\varphi)+\frac{1}{2}\mathrm{h_{2}}\times(\mathrm{b_{2}}\cos \varphi+\mathrm{a_{2}}\sin\varphi)\times j+\mathrm{n_{2}}\cos\varphi+\mathrm{n _{2}}\sin\varphi\times j \tag{8}\]  
where \(\mathrm{n_{1}}\) and \(\mathrm{n_{2}}\) are the additive white Gaussian noise. The subsequent SSD diversity deinterleaving outputs can be written as  
\[\mathrm{y^{\prime}_{Rx1}}= \left[\frac{1}{2}\mathrm{h_{1}}\times(\mathrm{a_{1}}\cos\varphi- \mathrm{b_{1}}\sin\varphi)+\mathrm{n_{1}}\cos\varphi\right]+ \tag{9}\] \[\mathrm{y_{2-QAM}}=\mathrm{a_{2}}+\mathrm{b_{2}}\times j \tag{2}\]0 (10) \[\mathrm{y_{2-QAM}}=\mathrm{a_{2}}+\mathrm{b_{2}}\times j \tag{2}\]1  
Here, without loss of generality, we assume that 4-QAM is used at the transmitters with \(|\mathrm{a_{1}}|\)=\(|\mathrm{a_{2}}|\)=\(|\mathrm{b_{1}}|\)=\(|\mathrm{b_{2}}|\) = 1, and \(\mathrm{h_{1}}>\mathrm{h_{2}}\). Thus, the noise-free signal constellation after the SSD diversity deinterleaving is illustrated in Figure 3, where \(E_{s}\) is the mean transmitted electrical energy of the 4-QAM signal [8]. It can be seen that the viable range of the \(\varphi\) angle is \(0\leq\varphi\leq 45\). The BER performance of the SSD scheme is determined by the distance of the neighboring constellation symbols [21]. The distances between two edge symbols, denoted as \(d_{1}\) and \(d_{2}\), respectively, can be expressed as  
\[\mathrm{y_{2-QAM}}=\mathrm{a_{2}}+\mathrm{b_{2}}\times j \tag{2}\]2  
\[\mathrm{y_{2-QAM}}=\mathrm{a_{2}}+\mathrm{b_{2}}\times j \tag{2}\]3  
It can be deduced that \(d_{1}<d_{2}\) is obtained when \(0\leq\varphi\leq 45^{\circ}\).