# Enhanced Fully Generalized Spatial Modulation for the Internet of Underwater Things
## 4 Performance Analysis of EFGSM for the Underwater Acoustic Channel
### ABER Performance Analysis
From (18), (20) and (21), the PEP can be expressed as follows:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k})=\ \int\limits_{0}^{\infty}Q\Bigg{(} \sqrt{\frac{\sigma_{h}{}^{2}\ f^{H}f}{4\ \sigma_{v}{}^{2}}}\Bigg{)}\frac{\psi^{N_{R}-1}\ e^{\frac{-\psi}{2}}}{2^{N_{R}} \ \Gamma(N_{R})}\ d\psi. \tag{22}\]  
The Q function can be alternatively expressed:  
\[Q(x)=\ \frac{1}{\pi}\int\limits_{0}^{\frac{\pi}{2}}e^{\frac{x^{2}}{sin\theta}}\ d\theta. \tag{23}\]As such, the PEP can be rewritten as follows:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k})=\ \frac{1}{2^{N_{R}}\ \pi\ \Gamma(N_{R})}\ \int\limits_{0}^{\pi}\int\limits_{0}^{\infty}e^{-( \frac{1}{2}+\frac{B}{sin^{2}\theta})\Psi}\ \psi^{N_{R}-1}d\theta d\psi, \tag{24}\]  
where \(B=\ \frac{\sigma_{h}^{2}\ell^{H}\ell}{4\ \sigma_{v}^{2}}\). Define \(I_{1}\) as:  
\[I_{1}=\ \int\limits_{0}^{\infty}e^{-(\frac{1}{2}+\frac{B}{sin^{2}\theta})\Psi} \ \psi^{N_{R}-1}d\psi. \tag{25}\]  
Then \(I_{1}\) in (25) can be computed according to [16; 24]:  
\[I_{1}=\ \frac{2^{N_{R}}\ \Gamma(N_{R})}{\Big{(}\frac{sin^{2}\theta+B}{sin^{2} \theta}\Big{)}^{N_{R}}}. \tag{26}\]  
Substituting (26) in (24), the PEP can be given as follows:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k})=\ \frac{1}{\pi}\int\limits_{0}^{\pi} \bigg{(}\frac{sin^{2}\theta}{sin^{2}\theta\ +B}\bigg{)}^{N_{R}}d\theta. \tag{27}\]  
The PEP in (28) can be expressed as follows:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k})=\ \frac{1}{2}\Bigg{[}1-U(B)\sum \limits_{k=0}^{N_{R}-1}\Bigg{(} \begin{array}{c}2k\\ k\end{array} \Bigg{)}\bigg{(}\frac{1-\ U^{2}(B)}{4}\bigg{)}^{k}\Bigg{]}, \tag{28}\]  
where,  
\[U(B)=\ \sqrt{\frac{B}{1+B}}. \tag{29}\]  
Furthermore, (29) can be rewritten as [24] as follows:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k})=\ \bigg{[}\frac{1-\ U(B)}{2}\bigg{]}^{N_{R}} \sum\limits_{k=0}^{N_{R}-1}\Bigg{(} \begin{array}{c}N_{R}-1+k\\ k\end{array} \Bigg{)}\bigg{(}\frac{1-\ U(B)}{2}\bigg{)}^{k}. \tag{30}\]  
Pair-wise error probability at high SNR is provided in (28) and (30), it can be expressed in an asymptotic form using a Taylor series (TS) as follows:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k})\ \cong\ \bigg{(}\frac{1}{2}\bigg{)} \bigg{(}\frac{\Gamma(N_{R}+0.5)}{\sqrt{\pi}\ N_{R}!}\bigg{)}\bigg{(}\frac{1}{B }\bigg{)}^{N_{R}} \tag{31}\]  
Each type of SMT has a different PEP which essentially depend on B, where B can be expressed as follows:  
\[B=\ \frac{\sigma_{h}^{2}}{4\ \sigma_{v}^{2}}\varepsilon, \tag{32}\]  
where \(\varepsilon=\ d^{H}d\) which can be specified based on the constellation symbols information utilized in the communication process of SMTs. For the proposed EFGSM, it is as defined in (33). In that expression \(N_{P_{M}}\) and \(N_{S_{M/2}}\) indicate the number of active antennas that are assigned to transmit the primary and secondary constellations of the data symbols, respectively. Moreover, \(\widetilde{N}_{\widetilde{P}_{M}}\) and \(\widetilde{N}_{\widetilde{S}_{M/2}}\) indicate the estimated number of antennas. \(\Upsilon\Big{(}P_{M},\ \widetilde{p}_{\widetilde{M}}\Big{)}\) and \(\Upsilon\Big{(}S_{M/2},\ \widetilde{S}_{\widetilde{M}/2}\Big{)}\) indicate the number of indices which are different between \(P_{M}\), \(\widetilde{p}_{\widetilde{M}}\) and \(S_{M/2}\), \(\widetilde{S}_{\widetilde{M}/2}\), respectively. Proof, in the proposed EFGSM, any random value can represent a different number between \(P_{M},~{}\widetilde{P}_{\widetilde{M}}\) or between \(S_{M/2},~{}\widetilde{S}_{\widetilde{M}/2}\). The chosen number of active antennas convey the data symbol of the primary constellation or the secondary constellation, while the equivalent of the ML decoder estimated antennas number of the EFGSM receiver side is uncertain.