# Pilot-Based Adaptive Channel Estimation for Underwater Spatial Modulation Technologies
## 2 Spatial Modulation Technologies
### Generalized Spatial Modulation  
GSSK [35; 36] is related to SSK, but GSSK does not require the generation of the habitual data constellations symbols. However, the transmit antennas are used to embed information in GSSK. In SSK a single transmitting antenna is operated to transfer information. Therefore, RF chains are no more required in the transmission side for GSSK. Moreover, to route the transmitted data, the single RF splitter is entailed to the corresponding transmit antennas using RF switches. Mathematically, the GSSK data rate is stated as follows [35; 36]:  
\[m_{GSSK}=\left\lfloor\log_{2}\left(\begin{array}{c}N_{t}\\ N_{a}\end{array}\right)\right\rfloor, \tag{6}\]  
where \(N_{a}\) is the number of active antennas for data transmission, \(\left(\begin{array}{c}\cdot\\ \cdot\end{array}\right)\)represents the binomial coefficient and \(\lfloor\cdot\rfloor\) represents the floor operator.  
The GSM [21; 22; 37; 38] is the combination of GSSK and SM that attains spectral efficiency higher than SM by activating a greater number of antennas, and it only requires a single RF chain for data transmission. The mechanism of GSM is similar to SM, in which the data is mapped by antenna indices and the symbol constellation. In addition, the same symbol data is transmitted from the active antennas. Due to the activation of a higher number of transmitting antennas; GSM achieves a higher data rate than the SM, and its achievable data rate is stated as follows [21; 22; 37; 38]:  
\begin{table}
\begin{tabular}{c c c c} \hline
**Block Bits** & **Spatial Bits** & **Data Bits** & **Antenna Index** \\ \hline
00 b1b2 & 00 & b1b2 & 1 \\
01 b1b2 & 01 & b1b2 & 2 \\
10 b1b2 & 10 & b1b2 & 3 \\
11 b1b2 & 11 & b1b2 & 4 \\ \hline \end{tabular}
\end{table}
Table 2: Example of SM mapping for 4 bpcu, \(N_{t}=4\) and 4-QAM modulation.  
\[m_{GSM}=\log_{2}(M)+\left\lfloor\log_{2}\left(\begin{array}{c}N_{t}\\ N_{a}\end{array}\right)\right\rfloor. \tag{7}\]  
The GSM mapping procedure is revealed in Table 3, for 4 bpcu, and is fulfilled by using \(N_{t}=4\) and 4-QAM based on the gray-coded constellation points. Consider the bit stream of [10 00] at any time instant is transmitted through the GSM system. The bit stream is split into two parts where the first group of two bits [10] is opted to select the data symbol from the constellation. The second group of bits [00] is used as the spatial bits that chooses the antenna for symbol transmission from the antenna combination given in Table 3. As such, the resultant transmission vector of the GSM-MIMO system is presented as:  
\[\mathbf{x}_{g}\overset{\Delta}{=}[\begin{array}{cccc}x_{k}&x_{k}&0&0\end{array}]. \tag{8}\]  
In GSM, the same symbol transmission from active antennas in one time instant has no hard bound dictating that the number of transmitting antennas should be a power of two, but an absolute number of transmitting antennas can be employed. The GSM attains a spectral efficiency higher than SM with a remarkably lower number of transmitting antennas, and with a slight increment in transmitter complexity. Furthermore, this complexity gain is due to the number of active antennas. To reach the spectral efficiency similar to the GSM, SM needs a greater number of transmitting antennas than GSM at the same modulation order.