# Extended Spatial-Index LED Modulation for Optical MIMO-OFDM Wireless Communication
## 3 Simulation Results and Computational Complexity
### Computational Complexity
However, the computational complexity of FGIS is larger than the CC of the proposed ESI because the FGIS needs more LEDs to fulfill the same SE of the proposed ESI. The number of RS and RM of the conventional GLIM were \(RS=\beta(n_{r}+4)N_{T}+4 log_{2}(N_{T})\), and \(RM=\beta(2n_{r}+11)N_{T}+4N_{T}log_{2}(N_{T})\)[31], respectively, where \(\beta=4\) is the GLIM indices combinations and \(n_{r}=4\), which gives \(RS=32N_{T}+4 log_{2}(N_{T})\), and \(RM=76N_{T}+4N_{T}log_{2}(N_{T})\)[31]. Since the GLIM was adapted in [33] to support spatial modulation with its normal functionality, the computational complexity of the GLIM with \(p\) can be given by  
\[RS_{GLIM}=2^{p}\beta(n_{r_{GLIM}}+4)N_{T}+4 log_{2}(N_{T}), \tag{26}\]  
\[RM_{GLIM}=2^{p}\beta(2n_{r_{GLIM}}+11)N_{T}+4N_{T}log_{2}(N_{T}), \tag{27}\]  
where \(n_{r_{GLIM}}\) is the PD number at the GLIM receiver, and it is equal to the number of LEDs \(n_{t_{GLIM}}\) on the GLIM transmitter side. Note that, at the same \(p\), the number of LEDs \(n_{t_{GLIM}}\) at the GLIM transmitter side is much bigger than the proposed ESI transmitter side or \(n_{t_{GLIM}}>>n_{t_{ESI}}\). In accordance with the above CC analysis, the number of operations (NoO) for both the proposed ESI, the FGIS, and the GLIM are evaluated and compared against different numbers of \(p\) (i.e., different achievable data rate). The NoO of both the ESI, the FGIS, and the GLIM are piloted with OFDM subcarriers equal to \(N_{T}=256\) against different values of spatial bits i.e., \(0\), \(2\), \(4\), and \(6\) as shown in Figure 7a-d, respectively. At \(p=0\) spatial bits, the GLIM, FGIS, and ESI were achieved \(R=log_{2}(M)\) bpcu data rate, while they used different MIMO configurations i.e., the GLIM, FGIS, and ESI used \(4\times 4\), \(3\times 3\), and \(3\times 3\) MIMO LEDs configurations, respectively. Under the aforesaid parameters, the computational complexity of proposed ESI is the same as the CC of the FGIS since they used the same \(3\times 3\) MIMO LEDs. However, ESI and FGIS are \(40\%\) higher than the GLIM as shown in Figure 7a, while the computational complexity margin between the ESI and the GLIM is reduced to \(20\%\) as the spatial bit increased (i.e., \(p=2\)) as shown in Figure 7b. Moreover, the ESI achieves about \(12\%\) CC reduction compared to the FGIS as shown in Figure 7b. However, the proposed ESI CC improved compared to the FGIS and GLIM computational complexity as the number of spatial bits increases. For example, the NoO of ESI reduced by \(51\%\), and \(80\%\) compared to the GLIM as shown in Figure 7c,d, respectively, while it reduced by \(40\%\), and \(55\%\) compared to the FGIS as shown in Figure 7c,d, respectively. On average, the CC of the proposed ESI is reduced by \(30\%\) and \(77.5\%\) compared to the FGIS and GLIM, respectively, as Figure 7 shows. This is back-dated to the different nature of increment of the MIMO LEDs branches in ESI, FGIS, and GLIM, where the number of MIMO LEDs branches is increased exponentially with the increment of the spatial bits in the GLIM (see (22), while it increases linearly with the FGIS and semi-linearly in the proposed ESI. For instance, the GLIM needs theoretically \(16\times 16\), \(64\times 64\), and \(256\times 256\) MIMO-LED configurations to achieve spatial bits of \(p=2\), \(p=4\), and \(p=6\) bpcu, respectively. Furthermore, the FGIS needs a \(5\times 5\), \(7\times 7\), and \(9\times 9\) MIMO-LED configurations to achieve spatial bits of \(p=2\), \(p=4\), and \(p=6\) bpcu, respectively, while the proposed ESI needs \(4\times 4\), \(5\times 5\), and \(7\times 7\) MIMO-LEDs configurations to achieve the same aforementioned spatial bits, respectively. Since, according to (24)-(27) the number of LEDs-MIMO branches is the main player in CC analysis, CC of the GLIM increases exponentially compared to the CC of the proposed ESI as shown in Figure 7.