# Enhanced Fully Generalized Spatial Modulation for the Internet of Underwater Things
## 2 Related Work  
In recent years, spatial modulation techniques (SMTs) such as SM [18; 19], GSM [21; 22], quadrature spatial modulation (QSM) [23], FGSM [15; 16], and fully quadrature spatial modulation (FGSM) [16] have been considered as promising MIMO techniques that achieve better performances in terms of spectral and energy efficiency as well as in reducing system complexity in comparison with the conventional MIMO counterparts. In SM [18; 19] one single RF chain, out of the \(N_{T}\) antennas, is activated to convey the data constellation symbol, and the index of this active RF chain carries additional data bits. The available data rate on the traditional SM technique \(R_{SM}\) is relative to signal constellation \(M\)-ary and the number of transmitted antennas \(N_{T}\). It allows poor communication since just one transmitter antenna is working at any given time. Consequently the SM achievable data rate, \(R_{SM}\), can be expressed as follows [18; 19]:  
\[R_{SM}=\ log_{2}(M)+\ log_{2}(N_{T}), \tag{1}\]  
where \(M\) represents the modulation order of the conveyed symbol. The GSM [21; 22] was proposed to improve the SM achievable data rate by activating multiple antennas. In GSM, the same data is transmitted through two or more antennas using the same data signal constellation. However, the data rate improvement is marginal, moreover, activating more than one antenna without efficient use will rapidly consume the energy source. The GSM achievable data rate, \(R_{GSM}\), is considered as proportional to the logarithm of the binomial coefficients, and can be written as follows [21; 22]:  
\[R_{GSM}=\ log_{2}(M)+\ \big{\lfloor}log_{2}\Bigg{(} \begin{array}{c}N_{T}\\ N_{u}\end{array} \Bigg{)}\big{\rfloor}, \tag{2}\]  
where \(\big{\lfloor}\cdot\big{\rfloor}\), \(\Bigg{(} \begin{array}{c}\cdot\\ \cdot\end{array} \Bigg{)}\)and \(N_{u}\) indicate the floor operator, the binomial coefficient, and the number of transmitting active antennas, respectively. The spatial constellation diagram was extended into two orthogonal dimensions in QSM where real and imaginary parts are transmitted through one or more transmitter antenna [23]. The real part of the data constellation symbol is transmitted via one dimension, and the imaginary part is transmitted via another dimension. By this data bit transmission method, the achievable data rate is enhanced compared to SM and GSM, but regrettably, the enhanced data rate is logarithmic with the square of the number of \(N_{T}\). Accordingly, the QSM achievable data rate, \(R_{QSM}\) can be expressed as follows [23]:  
\[R_{QSM}=\ log_{2}(M)+\Big{(}log_{2}N_{T}^{2}\Big{)}, \tag{3}\]  
In enhanced SM (ESM) [13; 17], the information bits are not only transmitted by the active transmitter antennas index(es) but also by using the signal constellation type. Hence, an integer of the power of two transmitter antenna \(N_{T}\) out of the transmitting antennas can be selected to be used in information transmission. Half of the activated transmitter antennas are forced to transmit information based on the primary signal constellation, and the others transmit information using the secondary signal constellation which is obtained by a single-step geometric insertion among the points of the primary signal constellation. The points of the secondary constellation are located at the midpoints of the intersections created by neighbor points of the \(M\)-QAM which are used as a primary constellation [17]. For example, if there are four transmitter antennas used, the maximum number of potential working antennas is two to transmit antenna conveyed information, one activated by the primary signal constellation, and the other is activated by secondary signal constellation. The ESM marginally increases the achievable data rate with significant improvement in the average energy per transmitted symbol \((E_{s})\).