# A Novel OFDM-Based Time Domain Quadrature GSM for Visible Light Communication System
## 3 Two Proposed OFDM-Based QG-MIMO Schemes
### Td-Qgsm  
The schematic diagram of the OFDM-based TD-QGSM system is shown in Figure 2, where \(M\)-ary quadrature amplitude modulation (\(M\)-QAM) is adopted in OFDM modulation. As shown in Figure 2, first, the input bits are divided into two parts for mapping in two different dimensions. In each time slot, we choose \(N\) LEDs for activation. Therefore, the first \(\lfloor log_{2}(\mathcal{C}(N_{t},N))\rfloor\) bits are encoded in the spatial dimension based on gray coding while the remaining \(2\times\log_{2}M\) bits are used to map the constellation symbols. We split all the constellation symbols into real and imaginary parts; thus, the total constellation symbols vector \(\mathbf{c}\) can be described as \(\mathbf{c}=\mathbf{c}_{Re}+j\times\mathbf{c}_{Im}\), where \(\mathbf{c}_{Re}\) and \(\mathbf{c}_{Im}\) denote the real part and the imaginary part of \(\mathbf{c}\), respectively. Then, two groups of OFDM modulators are employed to perform the OFDM modulation according to \(\mathbf{c}_{Re}\) and \(\mathbf{c}_{Im}\) so as to obtain the in-phase and quadrature modulated signals. Since the intensity modulation is usually deployed in VLC system to generate the real-valued intensity waveform to drive the LED transmitters, a DC bias is required to be added to the final modulated signal. The modulated signal can be expressed as  
\[s_{l}=\begin{cases}x_{l}+b,\ x_{l}\geqslant-b\\ 0,\ x_{l}<-b\end{cases}, \tag{4}\]  
where \(l\in\{0,1,\ldots,L-1\}\) denotes the index of the time slot. \(s_{l}\) denotes the TD sample in the \(l\)-th time slot, \(x_{l}\) denotes the real-valued bipolar TD signal generated at the output of IFFT, and \(b\) signifies the added DC bias. Finally, the in-phase-part-modulated signal is added to the quadrature-part-modulated signal to generate the signal vector \(\mathbf{u}\).  
In the spatial domain, the spatial information will be encoded by selecting the activated LED transmitters. Assume that \(N\) out of \(N_{t}\) total LEDs are chosen to convey the same TD sample in each time slot while the rest LEDs are set to be zeros. Let \(\mathbf{v}_{l}=\left[v_{l}^{1},v_{l}^{2},\ldots,v_{l}^{N}\right]^{T}\) denote spatial index vector at the \(l\)-th time slot, \(\beta_{l}\) denote the corresponding TD samples for a given time slot, and the elements of the vector \(\mathbf{s}_{l}=\left[s_{l}^{1},s_{l}^{2},\ldots s_{l}^{N_{t}}\right]^{T}\) that jointly mapped by \(\mathbf{v}_{l}\) and \(\mathbf{u}\) can be provided by  
\[s_{l}^{k}=\begin{cases}\beta_{l},\ if\ k=v_{l}^{i}\\ 0,\ other\end{cases}, \tag{5}\]  
where \(k\in\{1,2,\ldots N_{t}\}\) and \(i\in\{1,2,\ldots N\}\) denote the index of the elements of the signal vector and the index vector, respectively. Table 1 demonstrates the mapping table in the case of \(N_{t}=6\) and \(N=4\). Take a set of spatial bits such as \"110\" from the mapping table as an example; the spatial index vector can be expressed by \(\mathbf{v}_{l}=\left[2,4,5,6\right]^{T}\), which means that the joint vector is transmitted by the 2 nd, 4 th, 5 th, and 6 th LED transmitters simultaneously,while the input bits of the 1 st and 3 rd LED transmitters will be set to zeros. Therefore, the transmitted vector can be obtained as \(\mathbf{s}_{l}=[0,\beta_{l},0,\beta_{l},\beta_{l},\beta_{l}]^{T}\). Finally, the vector \(\mathbf{s}_{l}\) is used to modulate those \(N\) active LEDs.  
At the receiver side, the TD signal vector \(\mathbf{\hat{u}}\) and the spatial index vector \(\mathbf{\hat{v}}_{l}\) are first estimated using a suitable detection algorithm, where the details are presented in Section 4. We split \(\mathbf{\hat{u}}\) into the in-phase and quadrature parts by using the separation operation of real and imaginary and then perform the OFDM demodulation on these two parts, respectively. After constellation demapping and parallel-to-serial (P/S) conversion, the information bits can be recovered accordingly. As for the spatial part bits, they can be produced by demapping the index vector \(\mathbf{\hat{v}}_{l}\) directly.