# Enhanced Fully Generalized Spatial Modulation for the Internet of Underwater Things
## 4 Performance Analysis of EFGSM for the Underwater Acoustic Channel
### ABER Performance Analysis  
As in the conventional SMTs, the ABER of the proposed EFGSM can be evaluated by using a well-known union boundary technique as in [15; 16; 24]:  
\[ABER\leq\ \frac{1}{2^{R}}\ \sum_{v=1}^{2^{R}}\sum_{k=1}^{2^{R}}\frac{N(x_{v} \rightarrow\widetilde{x}_{k})p_{r}(x_{v}\rightarrow\widetilde{x}_{k})}{R}, \tag{10}\]  
where \(p_{r}(x_{v}\rightarrow\widetilde{x}_{k})\) represents the pairwise error probability (PEP) which defines the detection of erroneous symbol vector \(\widetilde{x}_{k}\) rather than the transferred symbol vector \(x_{v}\). \(N(x_{v}\rightarrow\widetilde{x}_{k})\) indicates how many bits in failure are within \(x_{v}\) and \(\widetilde{x}_{k}\), while \(R\) is the mean of the achievable bit rate. Accordingly, the conditional pairwise error probability (C-PEP) can be given as follows:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k}|\ H)=p_{r}\Big{(}||y-\ h_{l}S_{j}||^{ 2}>||y-\ \widetilde{h}_{l}\widetilde{S}_{j}||^{2}\Big{)}, \tag{11}\]  
where \(h_{l}S_{j}=(h_{l_{p}}S_{P_{M}}+\ h_{l_{S}}S_{S_{M/2}})\), is the received signal and includes both the primary and secondary constellations. Hence, using \(y\) as in (7), the reduced C-PEP can be written as:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k}|H)=\ p_{r}\Big{(}||n||^{2}>||n+\ h_{l}S_{j}-\ \widetilde{h}_{l}\widetilde{S}_{j}||^{2}\Big{)}. \tag{12}\]  
Letting \(\mathbf{z}=\ e_{l}S_{j}\) and \(\widetilde{\mathbf{z}}=\ e_{l}\widetilde{S}_{j}\), the right-hand side of (12) can be rewritten after expanding the norm operator as follows:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k}|\ H)=\ p_{r}\Big{(}||n||^{2}>||n+\ H(z-\ \widetilde{z})||^{2}\Big{)}, \tag{13}\]where \(\mathbf{z}\) and \(\widetilde{\mathbf{z}}\in\ \mathbb{R}^{N_{T}\times 1}\) defining the columns of identity matrix \(I_{N_{T}\times N_{R}}\). If we let \(f=(\mathbf{z}\ -\ \widetilde{\mathbf{z}})\), (13) can be rewritten as follows:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k}|\ H)=\ p_{r}\Big{(}-2\mathcal{R}\Big{(} n^{H}H\ f\Big{)}>f^{H}H^{H}Hf\Big{)}, \tag{14}\]  
where \(\mathcal{R}\big{(}n^{H}Hf\big{)}\) defines a Gaussian random variable (RV) distributed as \(\mathcal{R}\big{(}n^{H}Hf\big{)}\sim\Big{(}0,\ \frac{\sigma_{v}{}^{2}\ f^{H}H^{H}Hf}{2}\Big{)}\). Hence the expression of C-PEP in (14) can be rewritten as follows:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k}/H)=\ p_{r}\Bigg{(}-2\sqrt{\frac{ \sigma_{v}{}^{2}\ f^{H}H^{H}Hf}{2}}u>\ f^{H}H^{H}Hf\Bigg{)}, \tag{15}\]  
where \(u\) represents a regular Gaussian RV. The C-PEP in (15) can be rewritten after reducing it as:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k}|H)=\ p_{r}\Bigg{(}u>\ \sqrt{\frac{ \sigma_{v}{}^{2}\ f^{H}H^{H}Hf}{2\ \sigma_{v}{}^{2}}}\Bigg{)}. \tag{16}\]  
So,  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k}|H)=\ Q\Bigg{(}\ \sqrt{\frac{\sigma_{v}{}^{2}\ f^{H}H^{H}Hf}{2\ \sigma_{v}{}^{2}}}\Bigg{)}. \tag{17}\]  
Therefore, (17) can be represented as follows:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k}|H)=\ E_{H}\bigg{\{}Q\bigg{(}\ \frac{d^{H}d}{2\ \sigma_{v}{}^{2}}\bigg{)}\bigg{\}}, \tag{18}\]  
where \(E_{H}\{.\}\) indicates the expectation across the fading channel \(\mathbf{H}\), we can express \(d\) as follows:  
\[d=\mathbf{H}f=\sum\limits_{j=1}^{N_{R}}\mathbf{r}_{j}f, \tag{19}\]  
where \(\mathbf{r}_{j}\) is the \(j^{th}\) row of the channel matrix \(\mathbf{H}\). Moreover, the term \(d^{H}d\) is chi-squared RV with a \(2N_{R}\) degree of freedom (DOF). Hence, it can be written as a function of \(\psi\) as follows:  
\[p_{r}(x_{v}\rightarrow\widetilde{x}_{k}|\ H)=p_{r}\Big{(}||y-\ h_{l}S_{j}||^{ 2}>||y-\ \widetilde{h}_{l}\widetilde{S}_{j}||^{2}\Big{)}, \tag{11}\]0  
where \(\psi\) represents a standard chi-squared RV. Furthermore, the probability density function (PDF) of \(\psi\) can be calculated as:  
\[F(\psi)=\ \left\{ \begin{array}{ll}\frac{\psi^{N_{R}-1}\ e^{\frac{-\psi}{2}}}{2^ {N_{R}}\ \Gamma(N_{R})}&\ \ \ \psi>0\\ 0&\ \ \ otherwise\end{array} \right., \tag{21}\]  
where \(\Gamma(. )\) represents the public gamma function.