# Extended Spatial-Index LED Modulation for Optical MIMO-OFDM Wireless Communication
## 2 System Model
### The Proposed ESI-LED System
In the example, 16 transmitted LEDs are required with the GLIM-OFDM [31] to achieve the same SE under the assumption that there are two spatial bits provided by the GLIM-OFDM [33], while five transmitted LEDs are needed with the FGIS (\(R_{FGIS}=log_{2}(M)+n_{t}-3\)) [33] to achieve the same SE. Back to our example as shown in Figure 1, the data are split into two groups; the OFDM frame data (\(N_{T}log_{2}(M)\)) and the extended \(p\left(N_{T}(\lceil log_{2}\left(\frac{n_{t}}{4}\right)\rceil+n_{t}-2)\right)\) and it is equal to \(2N_{T}\) in this example. For example, if the upcoming OFDM time domain signal is \(c_{n}=c_{n_{R}}-jc_{n_{I}}\) and the group section are \(p=\{1\;0\}_{n}\), then, as seen in Table 1, the \(p=\{1\;0\}_{n}\) will be used to map \(c_{n}\) to the 3 rd group. Moreover, based on the \(c_{n}\) signs information, the absolute value of the real part (\(c_{n_{R}}\)) will be transmitted over \(TX_{LED_{1}}\) and \(TX_{LED_{2}}\), while the positive unipolar imaginary part \(c_{n_{I}}\) will be conveyed over \(TX_{LED_{4}}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Data** & \multicolumn{4}{c}{**Transmitted Data**} \\ \hline \(\mathbb{G}\)**/SB** & \(\mathbf{OFDM_{td}}\) & \(\mathbf{TX_{LED_{1}}}\) & \(\mathbf{TX_{LED_{2}}}\) & \(\mathbf{TX_{LED_{3}}}\) & \(\mathbf{TX_{LED_{4}}}\) \\ \hline  & \(c_{n_{R}}+jc_{n_{I}}\) & \(c_{n_{R}}\) & \(c_{n_{I}}\) & - & - \\ \(1/00\) & \(c_{n_{R}}-jc_{n_{I}}\) & \(c_{n_{R}}\) & - & \(c_{n_{I}}\) & - \\  & \(-c_{n_{R}}+jc_{n_{I}}\) & \(c_{n_{R}}\) & - & - & \(c_{n_{I}}\) \\  & \(-c_{n_{R}}-jc_{n_{I}}\) & - & \(c_{n_{R}}\) & \(c_{n_{I}}\) & - \\ \hline  & \(c_{n_{R}}+jc_{n_{I}}\) & - & \(c_{n_{R}}\) & - & \(c_{n_{I}}\) \\ \(2/01\) & \(c_{n_{R}}-jc_{n_{I}}\) & - & - & \(c_{n_{R}}\) & \(c_{n_{I}}\) \\  & \(-c_{n_{R}}+jc_{n_{I}}\) & \(c_{n_{R}}/2\) & \(c_{n_{R}}/2\) & \(c_{n_{I}}\) & - \\  & \(-c_{n_{R}}-jc_{n_{I}}\) & \(c_{n_{R}}/2\) & \(c_{n_{I}}\) & \(c_{n_{R}}/2\) & - \\ \hline  & \(c_{n_{R}}+jc_{n_{I}}\) & \(c_{n_{I}}\) & \(c_{n_{R}}/2\) & \(c_{n_{R}}/2\) & - \\ \(3/10\) & \(c_{n_{R}}-jc_{n_{I}}\) & \(c_{n_{R}}/2\) & \(c_{n_{R}}/2\) & - & \(c_{n_{I}}\) \\  & \(-c_{n_{R}}+jc_{n_{I}}\) & \(c_{n_{R}}/2\) & \(c_{n_{I}}\) & - & \(c_{n_{R}}/2\) \\  & \(-c_{n_{R}}-jc_{n_{I}}\) & \(c_{n_{I}}\) & \(c_{n_{R}}/2\) & - & \(c_{n_{R}}/2\) \\ \hline  & \(c_{n_{R}}+jc_{n_{I}}\) & \(c_{n_{R}}/2\) & - & \(c_{n_{R}}/2\) & \(c_{n_{I}}\) \\ \(4/11\) & \(c_{n_{R}}-jc_{n_{I}}\) & \(c_{n_{R}}/2\) & - & \(c_{n_{I}}\) & \(c_{n_{R}}/2\) \\  & \(-c_{n_{R}}+jc_{n_{I}}\) & \(c_{n_{I}}\) & - & \(c_{n_{R}}/2\) & \(c_{n_{R}}/2\) \\  & \(-c_{n_{R}}-jc_{n_{I}}\) & - & \(c_{n_{R}}/2\) & \(c_{n_{R}}/2\) & \(c_{n_{I}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example for the ESI-LED transmission mechanism for \(n_{t}=4\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Data** & \multicolumn{5}{c}{**Transmitted Data**} \\ \hline \(\mathbb{G}/\)**SB** & \(\boldsymbol{OFDM_{td}}\) & \(\boldsymbol{TX_{LED_{1}}}\) & \(\boldsymbol{TX_{LED_{2}}}\) & \(\boldsymbol{TX_{LED_{3}}}\) & \(\boldsymbol{TX_{LED_{4}}}\) &  
Another example is shown in Table 2, where four addition bits (\(p=4\ bpcu\)) are gained by the proposed ESI by using only five transmitted LEDs. However, under the assumption of using the same modulation order to achieve the same SE, 7 and 64 transmitted LEDs are needed with FGIS [33] and the GLIM-OFDM [31], respectively. For example, if the upcoming OFDM complex time domain (OFDM\({}_{td}\)) signal is \(c_{n}=-c_{n_{R}}+jc_{n_{I}}\), and the spatial bits are \(p=\{1\ 1\ 0\ 1\}_{n}\), then, as seen in Table 2, the 14 th group will be selected. Moreover, the signs of real and imaginary of \(c_{n}\) will determine which of the LEDs will become active in the aforesaid selected group.