## 3 UEP Using Spatial Modulation
### Spatial Modulation  
The SM has been introduced based on the constellation set of data symbols sent via the active antenna out of sending antennas [12]. The index of the antenna in a transmission is used to send extra data bits. Therefore, high data rate can be attained in SM compared to space shift keying modulation (SSK) or the conventional multiple-input-multiple-output (MIMO). In SM, the upcoming bitstream sent at a time immediate is divided within two different groups of bits. One of these groups is applied to modulate a signal constellation symbol from a signal constellation diagram of an optional \(M\) -ary quadrature amplitude modulation (\(M\) -QAM), or any further signal constellation design. These modulated-based bits embed \(log_{2}(M)\) bits, which are called data bits. The other set of bits embeds, \(log_{2}(Nt)\) bits, are characterized as spatial bits, and \(Nt\) is the number of broadcasting antennas used at the transmitter side. The spatial bits are applied to choose the antenna subset, which will be used to transmit the constellation symbol. The possible data rate achieved using SM, \(R_{SM}\), can be represented as [12]  
\[R_{SM}=log_{2}(M)+log_{2}(Nt) \tag{3}\]  
The SM has a wireless link with \(Nt\) antenna transmission and \(Nr\) antenna receiver and SM map constellation vector are \(\mathbf{x}=\left[x_{1}\ x_{2}\ldots\ldots\ x_{Nt}\right]^{T}\) assuming unity constraint power (i.e., \(E_{X}\!\left[X^{H}X\right]=1\)). SM activate only one antenna, in this case just one of \(x_{a}\) of constellation vector \(\mathbf{x}\) is nonzero to evade power transmission loss. The received signal of SM can be expressed as  
\[\mathbf{r}=\sqrt{\rho}HX+v \tag{4}\]  
where \(\rho\) is the average signal to noise ratio (SNR), \(H\) is the uncorrelated underwater acoustic channel impulse response, and \(v\) is the additive white Gaussian noise with independent and identically distributed (iid) entries according to \(\mathcal{CN}(0,\ 1)\). During SM time slot, the active constellation vector can be express as  
\[x_{b,q}\triangleq\left[0\ 0\ \ldots x_{b,q}\ldots\ 0\ 0\right]^{T} \tag{5}\]  
where \(b\) is the active antenna index and \(x_{b,q}\) is the \(q\) -th constellation symbol transmitted over the \(b\) antenna. The received signal \(\mathbf{r}\) in the case of \(x_{q}\) is transmitted from the \(b\)-th antenna and can be written as  
\[\mathbf{r}=\sqrt{\rho}h_{b}x_{q}+v \tag{6}\]  
where \(\mathbf{h}_{b}\) is the \(b\)-th column of the underwater acoustic channel \(\mathbf{H}\). With constant modulation signaling assumption, the received signal detection can be obtained as \[\hat{b}=\arg_{b}\max\left|\frac{\mathbf{h}_{b}^{H}\mathbf{r}}{\left|\mathbf{h}_{b }\right|_{F}}\right| \tag{7}\]  
\[\hat{q}=\arg_{q}\max\ Re\left\{\left|\left(\left(\mathbf{h}_{b}x_{q}\right)^{H} \right|^{H}\right)\right. \tag{8}\]  
where \(\hat{b}\) and \(\hat{q}\) is the estimated antenna and constellation symbol, respectively, and \(\left|h_{b}\right|_{F}\) represents the Frobenius norm. The error in the estimated antenna is based on the underwater acoustic channel and the number of the antennas transmission, whereas the error of data symbols is based on the underwater acoustic channel and the constellation modulation order.