# Enhanced Fully Generalized Spatial Modulation for the Internet of Underwater Things
## 4 Performance Analysis of EFGSM for the Underwater Acoustic Channel
### ABER Performance Analysis
\[\left\{\begin{array}{l}N_{N_{H}}|S_{N_{H}}-\widetilde{S}_{M}|^{2}+N_{H}|S_{S _{H}}-\widetilde{S}_{S}|^{2}\\ \\ \hskip 14.226378 ptif~{}h_{h_{h}}-\widetilde{S}_{M}|^{2}\\ \\ \hskip 14.226378 ptif~{}h_{h_{h}}-\widetilde{S}_{M}|^{2}+\left[\frac{N_{H}+ \widetilde{S}_{M}-\tau\left(y_{h}\cdot\widetilde{S}_{\widetilde{M}}\right)}{2} \right]|S_{N_{H}}-\widetilde{S}_{M}|^{2}+\left[\frac{\tau\left(y_{h}\cdot \widetilde{S}_{\widetilde{M}}\right)-N_{H}-\widetilde{S}_{M}}{2}\right]|S_{N_{ H}}|^{2}+\left[\frac{\tau\left(y_{h}\cdot\widetilde{S}_{\widetilde{M}}\right)-N_{H}+ \widetilde{S}_{M}}{2}\right]|\widetilde{S}_{N_{H}}|^{2}\\ \\ \hskip 14.226378 ptif~{}h_{h_{h}}-\widetilde{S}_{M}|^{2}+\left[\frac{N_{H}+ \widetilde{S}_{M}-\tau\left(y_{h}\cdot\widetilde{S}_{\widetilde{M}}\right)}{2} \right]|S_{N_{H}}-\widetilde{S}_{M}|^{2}+\left[\frac{\tau\left(y_{h}\cdot \widetilde{S}_{\widetilde{M}}\right)-N_{H}-\widetilde{S}_{M}}{2}\right]|S_{N_{ H}}|^{2}+\left[\frac{\tau\left(y_{h}\cdot\widetilde{S}_{\widetilde{M}}\right)-N_{H}+ \widetilde{S}_{M}}{2}\right]|\widetilde{S}_{N_{H}}|^{2}\\ \\ \hskip 14.226378 ptif~{}h_{h_{h}}+\widetilde{S}_{M}-\tau\left(y_{h}\cdot \widetilde{S}_{\widetilde{M}}\right)+\left[\frac{N_{H}+\widetilde{S}_{M}-\tau \left(y_{h}\cdot\widetilde{S}_{\widetilde{M}}\right)}{2}\right]|S_{N_{H}}|^{2} +\left[\frac{\tau\left(y_{h}\cdot\widetilde{S}_{\widetilde{M}}\right)-N_{H}+ \widetilde{S}_{M}}{2}\right]|\widetilde{S}_{N_{H}}|^{2}\\ \\ \hskip 14.226378 ptif~{}h_{h_{h}}+\widetilde{S}_{M}-\tau\left(y_{h}\cdot \widetilde{S}_{\widetilde{M}}\right)+\left[\frac{\tau\left(y_{h}\cdot \widetilde{S}_{\widetilde{M}}\right)-N_{H}-\widetilde{S}_{M}}{2}\right]|S_{N_{ H}}|^{2}+\left[\frac{\tau\left(y_{h}\cdot\widetilde{S}_{\widetilde{M}}\right)-N_{H}- \widetilde{S}_{M}}{2}\right]|\widetilde{S}_{N_{H}}|^{2}\\ \\ \hskip 14.226378 ptif~{}h_{h_{h}}+\widetilde{S}_{M}-\tau\left(y_{h}\cdot \widetilde{S}_{\widetilde{M}}\right)+\left[\frac{\tau\left(y_{h}\cdot \widetilde{S}_{\widetilde{M}}\right)-N_{H}-\widetilde{S}_{M}}{2}\right]|S_{N_{ H}}|^{2}+\left[\frac{\tau\left(y_{h}\cdot\widetilde{S}_{\widetilde{M}}\right)-N_{H}+ \widetilde{S}_{M}}{2}\right]|\widetilde{S}_{N_{H}}|^{2}\\ \\ \hskip 14.226378 ptif~{}h_{h_{h}}+\widetilde{S}_{M}-\tau\left(y_{h}\cdot \widetilde{S}_{\widetilde{M}}\right)+\left[\frac{\tau\left(y_{h}\cdot \widetilde{S}_{\widetilde{M}}\right)-N_{H}-\widetilde{S}_{M}}{2}\right]|S_{N_{ H}}|^{2}+\left[\frac{\tau\left(y_{h}\cdot\widetilde{S}_{\widetilde{M}}\right)-N_{H}- \widetilde{S}_{M}}{2}\right]|\widetilde{S}_{N_{H}}|^{2}\\ \\ \hskip 14.2263 decoder uses this definition for calculating the TNCO, and its receiver's complexity can be expressed as follows [16]:  
\[TNOO_{SM}=\ 8N_{R}(2)^{R_{SM}}. \tag{38}\]  
In the same way, the GSM receiver's complexity is calculated. However, the way of calculating the \(h_{li}\) in GSM is different than what is used in the conventional SM, where calculating \(h_{li}\) of the GSM ML decoder requires \(N_{T}-1\) complex summation with \(2(N_{u}-1)\) real summations [16; 21]. Accordingly, the TNCO of the GSM ML decoder is expressed as follows [16]:  
\[TNOO_{GSM}=\ 8N_{R}(2(N_{u}-1))(2)^{R_{GSM}}. \tag{39}\]  
The maximum required complex summation in the ML decoder of the FGSM is \(\frac{N_{t}}{2}-1\) all \(N_{t}\geq 3\). So, the TNCO of the ML decoder of the FGSM can be expressed as [16]:  
\[TNOO_{FGSM}=\ 8N_{R}\left(2\lceil\frac{N_{T}}{2}-1\rceil\right)(2)^{R_{FGSM}}, \tag{40}\]  
In the proposed EFGSM, to calculate the TNCO consider that there are two separate groups of antennas/combinations. First is the combination of the primary signal constellation and second is the combination of the secondary signal constellation. Both are similar to the FGSM, but each with different combinations. For these combinations, \(y-\ h_{i}S_{j}\) are needed to be computed for the ML decoder of the primary and the secondary signal constellations, wherein the proposed EFGSM two symbols with two signal constellations are transmitted from two or more active transmitter antennas.