# Extended Spatial-Index LED Modulation for Optical MIMO-OFDM Wireless Communication
## 2 System Model
### The Maximum a Posteriori Estimator  
In general, the transmitted vector \(\mathbf{u}\) is conveyed over optical wireless channel \(\mathbf{H}\in R_{+}^{n_{r}\times n_{I}}\) and is effected by an additive white Gaussian noise (AWGN) \(\boldsymbol{v}\in R^{n_{r}\times 1}\), where \(\boldsymbol{v}\sim\mathcal{N}(0,\sigma_{n}^{2})\), and \(n_{r}\) is the receiver PD number. The optical MIMO channel can be written as  
\[\mathbf{H}= \begin{bmatrix}h_{1,1}&\ldots&h_{1,n_{t}}\\ \vdots&\ddots&\vdots\\ h_{n_{r},1}&\ldots&h_{n_{r},n_{t}}\\ \end{bmatrix} , \tag{4}\]  
where \(h_{n_{r},n_{t}}\) represents the channel gain of the optical wireless link between the LEDs (\(n_{t}\)) and PDs (\(n_{r}\)). Thus, the expression of the received signal \(\mathbf{y}\in R^{n_{r}\times 1}\) in the proposed ESI-LED can be expressed as:  
\[\mathbf{y}=\mathbf{H}\mathbf{u}+\boldsymbol{v}, \tag{5}\]  
\[\mathbf{y}=\boldsymbol{h}_{i_{R}}\boldsymbol{c}_{n_{R}}+\boldsymbol{h}_{i_{I}} \boldsymbol{c}_{n_{I}}+\boldsymbol{v}, \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{3}{c}{**Data**} & \multicolumn{6}{c}{**Transmitted Data**} \\ \hline \(\mathbb{G}\)**/SB** & \(\mathbf{OFDM_{td}}\) & \(\mathbf{TX_{LED_{1}}}\) & \(\mathbf{TX_{LED_{2}}}\) & \(\mathbf{TX_{LED_{3}}}\) & \(\mathbf{TX_{LED_{4}}}\) & \(\mathbf{TX_{LED_{5}}}\) \\ \hline \multirow{4}{*}{\(13/1100\)} & \(c_{n_{R}}+jc_{n_{I}}\) & \(c_{n_{I}}\) & \(c_{n_{R}}/3  
where \(\mathbf{h}_{i_{R}}\) and \(\mathbf{h}_{i_{l}}\) are the aggregation of active optical wireless channel columns required for transmission the unipolar values of the real and imaginary components of the complex OFDM\({}_{td}\), respectively, and, for \(l^{th}\) combination \(\mathbb{C}_{l}\subseteq\{\mathbb{C}_{1},\mathbb{C}_{2},\ldots,\mathbb{C}_{l}, \ldots,\mathbb{C}_{2^{p+2}}\}\), they are given by:  
\[\mathbf{h}_{i_{R}}=\sum_{\forall j\in\mathbb{C}_{l}}^{N_{i_{R}}}\mathbf{h}_{j},\ \ \ \ \ \ \ \ \mathbf{h}_{i_{l}}=\mathbf{h}_{k},\ \ k\in\mathbb{C}_{l}\ \,k\
eq j, \tag{7}\]  
where \(N_{i_{R}}=1,2,\ldots,n_{t}-1\) is the number of active LEDs for the real part, and \(h_{i_{j}}\) is the active channel column for imaginary parts (only one active LED). In general, the electrical received signal to noise ratio \((SNR_{RX})\)[34] is expressed by:  
\[SNR_{RX}=\frac{P_{RX}^{elc}}{\sigma_{n}^{2}}=\frac{1}{\sigma_{n}^{2}}\zeta \left(\frac{1}{n_{r}}\sum_{m=1}^{n_{r}}\sum_{n=1}^{n_{t}}h_{m,n}\mathbf{I} \right)^{2}, \tag{8}\]  
where \(\zeta\) is the electrical-to-optical conversion coefficient, and it was assumed to be unity [31; 33], \(P_{RX}^{elc}\) is the average received electrical power, and Ica represents the mean emitted optical intensity. By normalizing the \(M\)-QAM constellation to have unity power, and at a high value of OFDM subcarriers \(N_{T}\), with the assumption that the power of the OFDM real part is equally divided between the active LEDs, then non-zero elements of the vector \(\mathbf{u}\) will follow clipped Gaussian distribution. Accordingly, the average electrical power emitted from the active LEDs [31; 33] for either \(c_{n_{R}}\) or \(c_{n_{I}}\) is given by:  
\[\mathbf{I}=E\{c_{n_{R}}^{\pm}(\mathbf{I})\}=\int_{0}^{\infty}v\mathcal{P}_{c_{ n_{R}}^{\pm}(\mathbf{v})}dv=\int_{0}^{\infty}v(\frac{1}{\sqrt{\pi}}e^{-v^{2}}u(v)+ \frac{1}{2}\delta(v))dv=\frac{1}{2\sqrt{\pi}}\, \tag{9}\]  
where \(u(v)\) and \(\delta(v)\) be the unit step and Dirac delta function, respectively. Full knowledge of the optical channel state information \((CSI)\) at the system receiver is presumed [31; 32; 33].