# Extended Spatial-Index LED Modulation for Optical MIMO-OFDM Wireless Communication
## 2 System Model
### The Maximum a Posteriori Estimator
Thus, for a certain joint power allocation and LED indices (i.e., the LED indices for both the real part \(i_{R}\), and for the imaginary part \(i_{l}\) ), the conditional maximum a posteriori \((MAP)\) estimation for the values \(\hat{c}_{n_{R}}\) and \(\hat{c}_{n_{I}}\)[31; 33] can be written as follows:  
\[\left(\hat{c}_{n_{R}}^{(i_{R},i_{l})},\hat{c}_{n_{l}}^{(i_{R},i_{l})}\right)= arg\underset{\hat{c}_{n_{R}}\hat{c}_{n_{I}}}{\max}\mathcal{P}\left(\tilde{c}_{n_{R}},\hat{c}_{n_{l}}/\mathbf{y}\right), \tag{10}\]  
where \(\mathcal{P}\left(\tilde{c}_{n_{R}},\tilde{c}_{n_{l}}/\mathbf{y}\right)\) is the conditional probability density function( _p.d.f._) of \(\tilde{c}_{n_{R}}\) and \(\tilde{c}_{n_{l}}\) when \(\mathbf{y}\) is known. By considering the independence between \(\tilde{c}_{n_{R}}\), and \(\tilde{c}_{n_{I}}\), (10) and by utilizing the Bayes' rule, Equation (10) can be rewritten as follows:  
\[\left(\hat{c}_{n_{R}}^{(i_{R},i_{l})},\hat{c}_{n_{l}}^{(i_{R},i_{l})}\right)= arg\underset{\hat{c}_{n_{R}}\hat{c}_{n_{I}}}{\max}\mathcal{P}\left(\mathbf{y}/ \tilde{c}_{n_{R}},\tilde{c}_{n_{I}}\right)\mathcal{P}\left(\tilde{c}_{n_{R}} \right)\mathcal{P}\left(\tilde{c}_{n_{I}}\right). \tag{11}\]  
Since \(\mathcal{P}\left(\mathbf{y}/\tilde{c}_{n_{R}},\tilde{c}_{n_{I}}\right)\sim \ \mathcal{N}\big{(}\mathbf{h}_{i_{R}}\tilde{c}_{n_{R}}+\mathbf{h}_{i_{l}}\tilde{c}_{n_{l}},\mathbf{v}\big{)}\) (note that the constants in (11) are ignored since they are not effected on the estimation process),  
\[\left(\hat{c}_{n_{R}}^{(i_{R},i_{l})},\hat{c}_{n_{l}}^{(i_{R},i_{l})}\right)= arg\underset{\hat{c}_{n_{R}}\hat{c}_{n_{I}}}{\max}e^{\left(-\left[\left(\hat{c}_{n_{R} }^{2}+\hat{c}_{n_{I}}^{2}\right)\right]\right)}\times e^{\left(-\left\|\mathbf{y }-\mathbf{h}_{i_{R}}\tilde{c}_{n_{R}}-\mathbf{h}_{i_{l}}\tilde{c}_{n_{l}}\right\|^{2} /2\sigma_{n}^{2}\right)}. \tag{12}\]  
After taking the logarithm of Equation (12),  
\[\left(\hat{c}_{n_{R}}^{(i_{R},j_{l})},\hat{c}_{n_{l}}^{(i_{R},j_{l})}\right)= arg\underset{\hat{c}_{n_{R}}\hat{c}_{n_{I}}}{\min}\mathcal{M}_{MAP}\left(i_{R},i_{l}, \tilde{c}_{n_{R}},\tilde{c}_{n_{I}}\right), \tag{13}\]where \(\mathcal{M}_{MAP}\left(.\right)\) is the MAP estimation metric and it is given by:  
\[\mathcal{M}_{MAP}=\left\|\mathbf{y}-\boldsymbol{h}_{i_{R}}\tilde{c}_{n_{R}}- \boldsymbol{h}_{i_{I}}\tilde{c}_{n_{I}}\right\|^{2}+2\sigma_{n}^{2}\left(\tilde{ c}_{n_{R}}^{2}+\tilde{c}_{n_{I}}^{2}\right). \tag{14}\]  
By using the same analysis as in [31; 33], Equation (14) can be written as  
\[\mathbf{y}=\boldsymbol{h}_{i_{R}}\boldsymbol{c}_{n_{R}}+\boldsymbol{h}_{i_{I}} \boldsymbol{c}_{n_{I}}+\boldsymbol{v}, \tag{6}\]0  
where  
\[\mathbf{y}=\boldsymbol{h}_{i_{R}}\boldsymbol{c}_{n_{R}}+\boldsymbol{h}_{i_{I}} \boldsymbol{c}_{n_{I}}+\boldsymbol{v}, \tag{6}\]1 \[\mathbf{y}=\boldsymbol{h}_{i_{R}}\boldsymbol{c}_{n_{R}}+\boldsymbol{h}_{i_{I}} \boldsymbol{c}_{n_{I}}+\boldsymbol{v}, \tag{6}\]2 \[\mathbf{y}=\boldsymbol{h}_{i_{R}}\boldsymbol{c}_{n_{R}}+\boldsymbol{h}_{i_{I}} \boldsymbol{c}_{n_{I}}+\boldsymbol{v}, \tag{6}\]3  
To solve (13), the first order derivative of (15) with respect to \(\tilde{c}_{n_{R}}\), and \(\tilde{c}_{n_{I}}\) is equating to zero, then the solution can be written as  
\[\mathbf{y}=\boldsymbol{h}_{i_{R}}\boldsymbol{c}_{n_{R}}+\boldsymbol{h}_{i_{I}} \boldsymbol{c}_{n_{I}}+\boldsymbol{v}, \tag{6}\]4  
where \(\left[a\right]^{+}=\max(0,\ a)\). It is worth mentioning here that, due to the transmitted signals of the proposed ESI, the FGIS [33], and the GLIM-OFDM [31] have the same distributions (i.e., clipped Gaussian probability density function), therefore the above \(MAP\) derived analysis is almost the same as the \(MAP\) derived analysis given in [31; 33]. However, there is a major difference between the proposed ESI-LED and both FGIS [33] and the GLIM-OFDM [31] in the cardinality size of the spatial domain (group selection bits).