# A Novel OFDM-Based Time Domain Quadrature GSM for Visible Light Communication System
## 3 Two Proposed OFDM-Based QG-MIMO Schemes
### Tb-Qgsmp  
The diagram of the OFDM-based TD-QGSMP scheme is shown in Figure 3. As shown in the figure, the input bits are first divided into two parts for mapping in two different dimensions. The first \(\lfloor log_{2}(\mathcal{C}(N_{t},N))\rfloor\) bits are encoded in the spatial dimension by using the gray code in each time slot. Different from the previous scheme, the proposed TD-QGSMP allocates \(2\times\mathrm{Nog}_{2}M\) bits for constellation mapping, and then a serial-to-parallel (S/P) conversion is performed to convert the serial \(2\times N\mathrm{log}_{2}M\) bits to \(N\) parallel bit streams for constellation mapping. After that, the \(N\) groups constellation symbols are split into real and imaginary parts, and then the constellation symbols vector \(\mathbf{c}_{\mathbf{z}}\) can be expressed as \(\mathbf{c}_{\mathbf{z}}=\mathbf{c}_{\mathbf{R},\mathbf{c}_{2}}+j\times \mathbf{c}_{\mathbf{m},z}\) with \(z\in\{1,2,\ldots N\}\) denoting the index of the modulated signal group. Subsequently, the two parts of the constellation symbols are separately performed regarding the OFDM modulation to generate the TD signal, which will be added by an appropriate DC bias to drive the LED. Finally, the \(z\)-th group in-phase-part-modulated signal is added to the \(z\)-th group quadrature-part-modulated signal to generate the signal vector \(\mathbf{u}_{\mathbf{z}}\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Spatial Bits** & **LED 1** & **LED 2** & **LED 3** & **LED 4** & **LED 5** & **LED 6** \\ \hline
000 & \(\beta_{l}\) & \(\beta_{l}\) & \(\beta_{l}\) & \(\beta_{l}\) & \(\beta_{l}\) & 0 & 0 \\
001 & \(\beta_{l}\) & \(\beta_{l}\) & \(\beta_{l}\) & 0 & \(\beta_{l}\) & 0 \\
010 & \(\beta_{l}\) & \(\beta_{l}\) & \(\beta_{l}\) & 0 & 0 & \(\beta_{l}\) \\
011 & \(\beta_{l}\) & \(\beta_{l}\) & 0 & \(\beta_{l}\) & \(\beta_{l}\) & 0 \\
100 & 0 & \(\beta_{l}\) & \(\beta_{l}\) & \(\beta_{l}\) & 0 & \(\beta_{l}\) \\
101 & 0 & \(\beta_{l}\) & \(\beta_{l}\) & 0 & \(\beta_{l}\) & \(\beta_{l}\) \\
110 & 0 & \(\beta_{l}\) & 0 & \(\beta_{l}\) & \(\beta_{l}\) & \(\beta_{l}\) \\
111 & 0 & 0 & \(\beta_{l}\) & \(\beta_{l}\) & \(\beta_{l}\) & \(\beta_{l}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: TD-QGSM mapping table in the case of \(N_{t}=6\) and \(N=4\). Figure 2: Schematic diagram of TD-QGSM. In the spatial domain, \(N\) out of total \(N_{t}\) LEDs will be chosen to transit different TD samples and the remaining LEDs are silent. After GSMP mapping, the elements of the resultant transmitted vector \(\mathbf{s}_{l}=\left[s_{l}^{1},s_{l}^{2},\ldots s_{l}^{N_{t}}\right]^{T}\) can be obtained by  
\[s_{l}^{k}= \begin{cases}\chi_{l}^{i},\text{ }if\text{ }k=v_{l}^{i}\\ 0,\text{ }other\end{cases} , \tag{6}\]  
where \(k\in\{1,2,\ldots N_{t}\}\) and \(i\in\{1,2,\ldots N\}\). Table 2 shows the mapping table that corresponds to \(N_{t}=6\) and \(N=4\). Take a set of spatial bits such as \"\(011\)\" from the mapping table as an example and the corresponding spatial index vector is \(\mathbf{v}_{l}=\left[1,2,4,5\right]^{T}\), which means that the 1 st, 2 nd, 4 th, 5 th LED transmitters will be selected to convey the sample \(\chi_{l}^{1},\)\(\chi_{l}^{2},\)\(\chi_{l}^{3},\)\(\chi_{l}^{4}\), respectively. Therefore, the transmitted vector can be obtained as \(\mathbf{s}_{l}=\left[\chi_{l}^{1},\chi_{l}^{2},0,\chi_{l}^{3},\chi_{l}^{4},0 \right]^{T}\). At the receiver side, the spatial index vector \(\mathbf{\hat{v}}_{l}\) and TD signal vector \(\mathbf{\hat{u}}_{z}\) are estimated by using a suitable detection algorithm, which will be demonstrated in Section 4. Split \(\mathbf{\hat{u}}_{z}\) into the in-phase and quadrature parts by using the separation operation of real and imaginary and then perform OFDM demodulation and constellation demapping on these two parts, respectively, to obtain the information bits. Furthermore, the spatial part bits are accessible by demapping the index vector \(\mathbf{\hat{v}}_{l}\) directly.