# Extended Spatial-Index LED Modulation for Optical MIMO-OFDM Wireless Communication
## 3 Simulation Results and Computational Complexity
### Computational Complexity  
In this section, the computational complexity analyses (CC) of the proposed ESI receiver is evaluated and compared to the FGIS and conventional GLIM-OFDM receivers CC. The CC of ESI receiver is given by calculating the total number of the real multiplication (RM) and real summation (RS) that are required at its decoder. The main difference between the computational complexity of the proposed ESI or FGIS and GLIM is in the number of real operations required to calculate (\(h_{i_{R}}\)) in (7), and the CC of the MAP metric to determine the indices of the active LEDs. Based on [31], there is no need to compute all the variables in (16) as the optical indoor channel is a time-invariant channel and the indoor VLC channel state information is assumed to be known at the receiver side. Thus, the only parameters that need to be computed are just two parameters \(\mathcal{C}\) and \(\mathcal{D}\) in (16). To estimate the real channel path gains in (17), \(\mathcal{C}\), and \(\mathcal{D}\) are computed for every received OFDM subcarrier, while the other parameters i.e., \(\mathcal{A}\), \(\mathcal{B}\), and \(\mathcal{E}\) are pre-computed for all possible LED indices (\(\tilde{i}_{R}\), and \(\tilde{i}_{I}\)) and stored in the receiver. For fair comparison, the CC of the proposed ESI, the FGIS, and the GLIM [31] must be estimated at the same achievable data rate \(R\). Therefore, at the same modulation order \(M\), the proposed ESI, the FGIS, and the GLIM must have the same \(p\) to achieve the same \(R\). However, different MIMO LED configurations are employed with ESI, the FGIS, and the GLIM to achieve the same \(p\), where, at a modulation order of 2, i.e., \(M=4\), a \(4\times 4\) MIMO LED configuration is needed for the proposed ESI to achieve a rate of 4 bpcu (\(R=4\)), while \(5\times 5\) and \(16\times 16\) MIMO LEDs are needed with the FGIS and GLIM, respectively, to achieve the same rate \(R\) at the same \(M\) as shown in Figure 3.  
For the proposed ESI with \(p\), there will be \(2^{p+2}\) possible joint power allocation and LED combinations (\(\tilde{i}_{R}\), and \(\tilde{i}_{I}\)) with \(n_{t_{ESI}}\times n_{t_{ESI}}\) MIMO LED configuration. Thus, the number of real summation (RS) and number of real multiplication (RM) needed to decode any OFDM subcarrier (\(N_{i_{I}}\)) are dependent on RS and RM needed to find \(\mathcal{C}\), and \(\mathcal{D}\) in (16) plus the number of RS and RM needed to estimate the active LED indices i.e., \(\tilde{i}_{R}\), and \(\tilde{i}_{I}\). To find \(\mathcal{C}\) and \(\mathcal{D}\) in (16), the number of RS and RM needed are \(2^{p+2}(n_{t_{ESI}}n_{t_{ESI}})\) and \(2^{p+2}(2n_{t_{ESI}}+3)\), respectively. RS and RM are computed based on the values of \(\mathcal{C}\) and \(\mathcal{D}\) in (16), where the value of \(N_{i_{R}}\) is set to be equal to \(n_{t_{ESI}}\), which gives the upper limit for RS and RM. Moreover, to estimate the active LED indices i.e., \(\tilde{i}_{R}\), and \(\tilde{i}_{I}\) (17) under the same aforesaid assumptions, a \(2^{p+2}\times 4\) and a \(2^{p+2}\times 8\) RS and RM are needed, respectively. Therefore, the total number RS and RM needed to decode an OFDM symbol with \(N_{T}\) subcarrier are given as follows:  
Figure 6: Scenario 2 test 2 ABER of the proposed ESI compared to GLIM [31] and FGIS [33] at the same MIMO-LEDs configurations, but with different modulation orders to achieve the same SE for three physical channels A, B, and C.  
\[RS_{ESI} =2^{p+2}(n_{t_{ESI}}n_{r_{ESI}}+4)N_{T}+4 log_{2}(N_{T}), \tag{24}\] \[RM_{ESI} =2^{p+2}(2n_{r_{ESI}}+11)N_{T}+4N_{T}log_{2}(N_{T}), \tag{25}\]  
where \(4 log_{2}(N_{T})\) and \(4N_{T}log_{2}(N_{T})\) are the number of RS and RM needed for decoding the OFDM symbol, respectively. Likewise, the FGIS [33] CC (the total number RS and RM) can be estimated by the same aforementioned Equations (24) and 25), as the FGIS and the proposed ESI decoders have the same working mechanism.