# Extended Spatial-Index LED Modulation for Optical MIMO-OFDM Wireless Communication
## 2 System Model
### The Proposed ESI-LED System  
An illustration of the proposed ESI-LED system structure is shown in Figure 1, where the incoming data bits are split into two separated groups. The first group is formed from \(N_{T}log_{2}(M)\) bits (data bits), where \(N_{T}\) is the OFDM sub-carrier number and \(M\) refers to the modulation order. As shown in Figure 1, the OFDM time domain (OFDM\({}_{td}\)) vector \(\mathbf{c_{T}}\) is generated by applying IFFT on the resultant modulated vector \(\mathbf{c_{F}}\). The next group of bits consists of \(N_{T}\big{(}\lceil log_{2}(\frac{n_{t}}{4})\rceil+n_{t}-2\big{)}\), which are defined as spatial bits (\(p\)), where \(n_{t}\) is the number of transmit LEDs and \(\lceil\.\ \rceil\) is the ceiling operator. The OFDM time domain vector \(\mathbf{c_{K}}\) is conveyed by LED indices that are selected based on both \(p\) and the OFDM signal signs information. The LED indices in the proposed ESI-LED are varied from the situation where just two LEDs are activated to the situation where many or all LEDs are activated. The proposed system represents a combination of the traditional fully generalized SM [26; 27; 28] and the optical SM systems which are depending on a fixed number of active LEDs (only two active LEDs). The reliability of the proposed ESI communication channel will be increased by varying the number of transmit LEDs [26; 27; 28]. In the ESI, all prospective LED indices combinations \(\mathbb{C}\) are exploited to improve the utilization of the resultant SD. Moreover, to increase the effectiveness of the resultant SD in improving the proposed system SE, the power allocations of the OFDM real part component (\(c_{n_{R}}\)) are also varied \(q_{i}\) times over the same active LED indices \(\mathbb{C}_{i}\), where \(q_{i}\) is equal to the number of the active LEDs in the LEDs combination \(\mathbb{C}_{i}\) (i.e., \(q_{i}=n_{t_{i}}\) when \(n_{t_{i}}\geq 3\) and \(q_{i}=1\) when \(n_{t_{i}}=2\)). Hence, there will be \(\sum_{i}q_{i}\mathbb{C}_{i}\) available joint LED indices and power allocations combinations. The resultant combinations are split into various \(\mathbb{G}\) groups, and every group involves four unparalleled combinations to include all four potential likelihoods for OFDM\({}_{td}\) signs information (i.e., \(\pm c_{n_{R}}\ \pm jc_{n_{I}}\)). Thus, the resultant \(p\) from the proposed ESI can be given by \(\lfloor log_{2}(\sum q_{i}\mathbb{C}_{i}/4)\rfloor\) (group selection bits), where \(\lfloor.\rfloor\) denotes the floor process. The extended SM over the joint LED power allocations combinations grouping is proposed in the ESI model, which differs from the traditional SM schemes that utilize the spatial modulation domain over the LED indices only. This would reinforce the SE of the ESI modulation technique. In this context, the maximum achievable data rate of the ESI (\(R_{ESI}\)) could be given as:  
Figure 1: The extended spatial-index light emitting diode (ESI-LED) modulation system structure with \(n_{t}\times n_{r}\) configuration. \[R_{ESI}=log_{2}(M)+\left|log_{2}\left(\frac{\sum_{i}q_{i}\mathbb{C}_{i}}{4} \right)\right|, \tag{1}\]  
\[R_{ESI}=log_{2}(M)+\left|log_{2}\left(\left[\binom{n_{t}}{2}+\sum_{q=3}^{n_{t}}q \binom{n_{t}}{q}\right]/4\right)\right|, \tag{2}\]  
\[R_{ESI}= \begin{cases}\underbrace{log_{2}\left(M\right)}_{\text{data bits}}&n_{t}=3,\\ \underbrace{log_{2}\left(M\right)}_{\text{data bits}}+\underbrace{\left[ log_{2}\left(\frac{n_{t}}{4}\right)\right]+n_{t}-2}_{\text{spatial bits}}&n_{t}>3,\end{cases}  \tag{3}\]  
where \(\binom{\cdot}{\cdot}\) is the binomial operator. Table 1 shows an example for the ESI-LED transmission functionality, where an extra one and two bits per channel use (bpcu) are gained by the ESI-LED system over the FGIS [33] and the GLIM-OFDM [31], respectively, when \(n_{t}=4\). Essentially, the proposed ESI-LED scheme achieves SE equal to \(R_{ESI}=log_{2}(M)+2\) bpcu using only four transmit LEDs.