# Enhanced Fully Generalized Spatial Modulation for the Internet of Underwater Things
## 3 Proposed EFGSM
For a better explanation of the proposed EFGSM transmission procedure, assume, without loss of generality, \(N_{T}=N_{R}=4\) as shown in Figure 1, where \(N_{R}\) represents the receiver antennas. Using 4-QAM modulator as a primary signal constellation type, the secondary signal constellation is equivalent to the throughput of half of the primary signal constellation, such as BPSK. In this case, the antenna indexing of the proposed EFGSM can be represented as shown in Table 1, for 7 bpcu. For illustration, in the example mentioned, consider the following generated bits, \(\left[ \begin{array}{cc}010&1010\\ \end{array} \right]\), are needed to be transmitted. The first three bits [010] represents the bits carried via the primary and secondary signal constellations. [01] is equivalent to \(log_{2}(P_{M})\), and conveyed using the primary signal constellation, and the other part [0] (equal to the half of primary signal constellation bits), equivalent to \(log_{2}(S_{M/2})\), transmitted via the secondary signal constellation. The remaining four bits, [1010], are transmitted via the antenna index \(T_{x1}T_{x4}\). The secondary signal constellation bits will be transmitted over the active antenna, \(T_{x1}\), and primary signal constellation will be transmitted over the active transmitting antenna \(T_{x4}\). Therefore, the \(N_{T}\times 1\) transmitted vector of the EFGSM is \(x=\left[ \begin{array}{cc}S_{M/2}&0&0&P_{M}\\ \end{array} \right]\). That vector of the EFGSM is sent via \(N_{T}x\)\(N_{R}\) over an uncorrelated underwater acoustic channel, H contaminated by an additive white Gaussian noise with \(n\sim CN(0,N_{o})\). The received signal \(y\in\mathbb{C}^{N_{R}x1}\) at receiver side is given by:  
\[y=\ h_{I_{D}}S_{P_{M}}+\ h_{I_{S}}S_{S_{M/2}}+n, \tag{7}\]  
and,  
\[h_{I_{D}}=\ \sum_{i=1}^{N_{I_{D}}}h_{i}\,h_{I_{S}}=\ \sum_{k=1}^{N_{S}}h_{k}, \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Data Bits**} & **Antennas** & **Signal** & \multirow{2}{*}{**Data Bits**} & **Antennas** & **Signal** \\  & **Combination** & **Constellation** & & **Combination** & **Constellation** \\ \hline
0000 & \(T_{x1}T_{x2}\) & \(P_{M}:S_{M/2}\) & 1000 & \(T_{x1}T_{x2}\) & \(S_{M/2}:\ P_{M}\) \\ \hline
0001 & \(T_{x1}T_{x3}\) & \(P_{M}:S_{M/2}\) & 1001 & \(T_{x1}T_{x3}\) & \(S_{M/2}:\ P_{M  
where, \(h_{l_{p}}\) and \(h_{l_{S}}\) indicate the summation of columns of activated antennas channel required to transmit the primary signal constellation and secondary constellation, respectively. \(N_{l_{p}}\), \(N_{l_{S}}=1\), \(2\ldots\ \frac{N_{l}}{2}\) with. denotes the ceiling operator and \(h_{i}\) denotes the \(i^{th}\) columns of channel H.  
The knowledge of the channel at the receiver side is assumed to be perfect. Therefore, the maximum-likelihood (ML) decoder can be used in the EFGSM system and is expressed as:  
\[\left[ \begin{array}{cc}\widetilde{h}_{l_{p}}&\widetilde{h}_{l_{S}}& \widetilde{S}_{P_{M}}&\widetilde{S}_{S_{M/2}}\\ \end{array} \right]=\arg\underbrace{min}_{h_{l_{p}},h_{l_{S}}},\underbrace{ min}_{S_{P_{M}}},\ \underbrace{||y-\ h_{l_{p}}S_{P_{M}}-\ h_{l_{S}}S_{S_{M/2}}||^{2}}. \tag{9}\]  
In the ML decoder, the indexed information bits are recovered using the estimated antenna indices combination and the primary and secondary estimated data symbol constellations.