## 3 UEP Using Spatial Modulation
### ABER of SM Information Carrying Units
#### 3.2.1 ABER of the Bits Carried by the Antenna Index  
The occurred error probability on the estimated bits transmitted via the antenna index is derived as follows [13],  
\[BER_{AI}\ \leq\ E_{b}\left[\sum_{\hat{b}}N\left(b\rightarrow\overline{b} \right)PEP\left(\boldsymbol{x}_{b}\rightarrow\boldsymbol{x}_{\hat{b}}\right) \right]=\ \frac{1}{2^{R}}\ \sum_{v=1}^{2^{R}}\sum_{k=1}^{2^{R}}\frac{N_{e}\left(b \rightarrow\overline{b}\right)PEP\left(\boldsymbol{x}_{b}\rightarrow \boldsymbol{x}_{\overline{b}}\right)}{R} \tag{9}\]  
where \(E_{b}\) is the expected transmit antennas, data transmitted rate in case of \(b\) antenna index activated is \(R\) in Equation (9). The number of bit errors between antenna index \(b\) and estimated antenna index \(\overline{b}\) is indicates by \(N_{e}\left(b\rightarrow\overline{b}\right)\). \(PEP\left(\boldsymbol{x}_{b}\rightarrow\boldsymbol{x}_{\overline{b}}\right)\) is the pairwise error probability \(\left(PEP\right)\) which can be written as  
\[PEP\left(\boldsymbol{x}_{b}\rightarrow\boldsymbol{x}_{\overline{b}}\right) =\ E_{H}\left[PEP\left(\boldsymbol{x}_{b}\rightarrow\boldsymbol{x}_{ \overline{b}}/H\right)\right] \tag{10}\]  
where \(E_{H}\{.\}\) is the second-order statistic of pairwise error probability over the underwater acoustic channel, and \(PEP\left(\boldsymbol{x}_{b}\rightarrow\boldsymbol{x}_{\overline{b}}|\boldsymbol {H}\right)\) is the conditional \(PEP\), and can be written as  
\[PEP\left(\boldsymbol{x}_{b}\rightarrow\boldsymbol{x}_{\overline{b}}| \boldsymbol{H}\right)=PEP\left(\left\|\ \boldsymbol{r}-\ \sqrt{\rho}\hbar\boldsymbol{b}_{x}\ \|_{F}\!\!>\!\|\ \boldsymbol{r}-\ \sqrt{\rho}\hbar\boldsymbol{ \tau}_{\overline{b}}\boldsymbol{x}\ \|_{F}\right) \tag{11}\]  
Using \(\boldsymbol{r}\) as in (4), the conditional \(PEP\) can be written as  
\[PEP\left(\boldsymbol{x}_{b}\rightarrow\boldsymbol{x}_{\overline{b}}| \boldsymbol{H}\right)=\ PEP\left(\left\|\ \boldsymbol{v}\ \|_{F}\!\!>\!\|\ \boldsymbol{v}+\ \sqrt{\rho}\hbar\boldsymbol{v}\boldsymbol{x}-\ \sqrt{\rho}\hbar\boldsymbol{\tau}_{\overline{b}} \boldsymbol{x}\ \|_{F}\right)=\ Q \tag{12}\]  
where,  
\[Q(z)=\ \frac{1}{\pi}\int\limits_{0}^{\frac{\pi}{2}}e^{\frac{z^{2}}{sin^{2}\rho}}\ d\theta \tag{13}\]  
and (11) can be rewritten as  
\[PEP\left(\boldsymbol{x}_{b}\rightarrow\boldsymbol{x}_{\overline{b}}| \boldsymbol{H}\right)=Q\left(\sqrt{\frac{\rho}{2}\ \|\ \hbar\boldsymbol{v}\boldsymbol{x}-\ \hbar\boldsymbol{ \tau}_{\overline{b}}\boldsymbol{x}\ \|_{F}^{2}}\right)=Q\left(\sqrt{\bar{g}}\right)=\ \sum_{t=1}^{N_{r}}c_{n}^{2} \tag{14}\]where \(c_{n}\sim N\Big{(}0,\ \sigma_{c}^{2}\Big{)}\) with \(\sigma_{c}^{2}=\frac{\rho}{2}|\mathbf{x}|^{2}\) and \(g=\frac{\rho}{2}\parallel\mathbf{h}_{b}\mathbf{x}-\mathbf{h}_{\overline{b}}\mathbf{x} \parallel_{\overline{b}}^{2}\) defined in ([14]; Equation (15)) and it is assumed as a chi-squared random variable (RV) with \(2N_{r}\) degree of freedom (DOF), and its probability density function (PDF) can be given as in [13]:  
\[F_{g}(\psi)=\ \frac{\psi^{N_{r}-1}\ e^{\frac{-\psi}{2}}}{2^{N_{r}}\ \Gamma(N_{r})} \tag{15}\]  
\(\Gamma(. )\) is gamma function. Representing the pairwise error probability in terms of chi-squared RV, \(g\) can be written as [15]  
\[PEP\Big{(}\mathbf{x}_{b}\to\mathbf{x}_{\overline{b}}\Big{)}=\ \int\limits_{0}^{ \infty}Q\Big{(}\sqrt{\psi}\Big{)}\frac{\psi^{N_{r}-1}\ e^{\frac{-\psi}{2}}}{2^{N _{r}}\ \Gamma(N_{r})}\ d\psi=\frac{1}{\pi}\int\limits_{0}^{\infty}\int_{0}^{(\pi/2)}Q \Big{(}\sqrt{\psi}\Big{)}\frac{\psi^{N_{r}-1}\ e^{\frac{-\psi}{2}}}{2^{N_{r}} \ \Gamma(N_{r})}\ d\psi. \tag{16}\]  
Then the closed form of Equation (8) can be written as follows [16; 17],  
\[PEP\Big{(}\mathbf{x}_{b}\to\mathbf{x}_{\overline{b}}\Big{)}=\ \Bigg{[}\frac{1-\ \mu_{a}}{2}\Bigg{]}^{N_{r}}\sum_{w=0}^{N_{r}-1}\Bigg{(} \begin{array}{c}N_{r }-1+w\\ w\end{array} \Bigg{)}\Bigg{(}\frac{1-\ \mu_{a}}{2}\Bigg{)}^{w} \tag{17}\]  
and \(\mu_{a}=\Bigg{(}1-\ \sqrt{\frac{\sigma_{a}^{2}}{1+\sigma_{a}^{2}}}\Bigg{)}\).