# A Novel OFDM-Based Time Domain Quadrature GSM for Visible Light Communication System
## 3 Two Proposed OFDM-Based QG-MIMO Schemes
### SE Analysis  
As for the OFDM-based TD-GSM scheme, Hermitian symmetry is not a constraint on the spatial symbol since the GSM procedure is performed in the time domain; the SE contributed by the spatial symbols is \(\lfloor\log_{2}(\mathcal{C}(N_{t},N))\rfloor\). However, Hermitian symmetry still affects the symbols in the constellation mapping part as only \(\frac{L}{2}-1\) subcarriers out of total subcarriers can transmit valid data, meaning that the SE contributed by each symbol is \(\frac{L}{L}\), and then the SE contributed by the symbols used for constellation mapping is \(\frac{L}{L}\text{log}_{2}(M)\). Therefore, the total SE of TD-GSM can be expressed by  
\[\begin{split} R_{TD-\text{GSM}}&=\frac{\frac{L}{2} -1}{L}\text{log}_{2}(M)+\lfloor\log_{2}(\mathcal{C}(N_{t},N))\rfloor\\ &\approx\frac{1}{2}\text{log}_{2}(M)+\lfloor\log_{2}(\mathcal{C}( N_{t},N))\rfloor.\end{split} \tag{7}\]  
As for the OFDM-based TD-QGSM scheme, a total of \(2\times\log_{2}M\) bits will be assigned in each time slot to perform the constellation mapping, and two groups of OFDM modulators will be deployed to modulate the constellation symbols to obtain the information symbols. Thus, the SE contributed by constellation part for TD-QGSM will be double that of the TD-GSM, which can be presented as \(\frac{2L}{L}\text{log}_{2}(M)\). Due to the constant number of bits used to select the activation LEDs, the SE contributed by the spatial part remains the same as TD-GSM. The total SE of the proposed TD-QGSM can be expressed by  
\[\begin{split} R_{TD-QGSM}&=\frac{2L}{L}\text{ log}_{2}(M)+\lfloor\log_{2}(\mathcal{C}(N_{t},N))\rfloor\\ &\approx\log_{2}(M)+\lfloor\log_{2}(\mathcal{C}(N_{t},N))\rfloor.\end{split} \tag{8}\]  
As for the OFDM-based TD-QGSMP scheme, the constellation symbols will be divided into \(N\) different segments for transmission; the SE contributed by the spatial part remains the same as TD-QGSM. However, the SE contributed by constellation part becomes \(N\) times that of TD-QGSM, which can be expressed as \(\frac{2L}{L}N\text{log}_{2}(M)\); hence, the total SE of the proposed TD-QGSMP can be provided as  
\[\begin{split} R_{TD-QGSMP}&=\frac{\frac{2L}{2}-2 }{L}N\log_{2}(M)+\lfloor\log_{2}(\mathcal{C}(N_{t},N))\rfloor\\ &\approx N\log_{2}(M)+\lfloor\log_{2}(\mathcal{C}(N_{t},N))\rfloor.\end{split} \tag{9}\]  
In summary, both the two proposed transmission schemes enhance the overall SE by improving the SE of the constellation part, which results in a significant improvement effect, and the specific comparison will be presented in Section 5.1. Finally, Table 3 demonstrates the theoretical SE of the proposed schemes. In addition, the SEs of some existing OFDM-based generalized MIMO transmission schemes proposed in [15,19] are also presented here for easy comparison.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Transmission Scheme** & **Achievable Rate** & **Feature** \\ \hline FD-GSM & \(\frac{1}{2}\text{log}_{2}(M)+\frac{1}{2}\lfloor\log_{2}(\mathcal{C}(N_{t},N ))\rfloor\) & GSM performed in frequency domain. \\ \hline FD-GSMP & \(\frac{N}{2}\text{log}_{2}(M)+\frac{1}{2}\lfloor\log_{2}(\mathcal{C}(N_{t},N))\rfloor\) & 1. Constellation mapping symbols divided into \(N\) groups; 2. GSM performed in frequency domain. \\ \hline FD-QGSM & \(\frac{1}{2}\text{log}_{2}(M)+\lfloor\log_{2}(\mathcal{C}(N_{t},N))\rfloor\) & 1. I/Q separation performed before OFDM modulation; 2. GSM performed in frequency domain. \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of SE for different OFDM-based generalized MIMO transmission schemes.