# Pilot-Based Adaptive Channel Estimation for Underwater Spatial Modulation Technologies
## 2 Spatial Modulation Technologies
### Spatial Modulation  
According to the current survey, the conception of SMTs is thought of as a hot topic among the top innovative researchers in the past decade. The easiest and most trivial version of SMT is the SSK [32; 33; 34], where information is embedded employing a transmit antenna index instead of using the constant amplitude/phase constellation symbols. SMTs do not require amplitude/phase elements to produce the data constellation symbols (for instance, RF chains are not required), which eventually relaxes the hardware need in the transmitter side, and in the computational complexity of the receiver-side detection. The achievable data rate of the SSK can be written as [32; 33; 34]:  
\[m_{SSK}=\log_{2}(N_{t}), \tag{1}\]  
where \(N_{t}\) is the number of transmitting antennas.  
The SM is a transmission technology that has been introduced for MIMO communication systems. The idea behind SM is to increase the total spectral efficiency so that the data is conveyed by both spatial antenna positions and amplitude/phase-modulation schemes. The spectral efficiency of SM increases in proportion to the base-two logarithm of the number of transmitting antennas. In SM, there is no need for the synchronization between the transmitting antennas because, at transmission instant, just one transmitting antenna from all transmitting antennas is active, while the others are off (only single RF chain is needed at transmitter side). Moreover, ICI is averted at the receiver, which results in low receiver complexity. At the receiver, the SM is used to estimate both transmitting antenna number and the sent symbol; after that, the spatial demodulator is used to restore the original data bits [18]. In the SM, input binary bits are split into two sets--the index sets and the constellation sets. The index of an active transmit antenna chosen by the set of \(\log_{2}(N_{t})\) bits and the second set of bits is \(\log_{2}(M)\) chooses the transmit symbol from M-ary signal constellation. In the SM system, the number of transmitted data bits \(m\) can be varied by signal and spatial modulation. For example, in the SM system, four bits can be transmitted two ways to achieve the same spectral efficiency; one is using two transmitting antennas and 8-QAM modulation, and in the second, four transmitting antennas and 4-QAM can be used. Generally, the spectral efficiency of SM is given as follows [12].  
\[m_{SM}=\log_{2}(M)+\log_{2}(N_{t}). \tag{2}\]  
Figure 1: SMTs System Model.  
SM maps a random stream of \(m\) bits at different symbol intervals into \(N_{t}\)-dimensional signal as:  
\[\mathbf{x}=[x_{1},x_{2},...x_{N_{t}}]^{T}. \tag{3}\]  
Only one of \(\mathbf{x}_{g}\) that is active in \(\mathbf{x}\) is nonzero, where \(g\) represents the index of active antenna. Then, at the \(k^{th}\) symbol interval, at the SM-MIMO transmitter the resultant output transmit vector can be represented as:  
\[\mathbf{x}_{g}\stackrel{{\Delta}}{{=}}[0...x_{k}...0]^{T}, \tag{4}\]  
where \(x_{k}\) denotes the \(k^{th}\) symbol from the M-ary constellation diagram. Over this symbol duration, the other antennas will not be active. The symbol \(x_{k}\) is the transmitted data symbol over an \(N_{r}\times N_{t}\) MIMO channel from antenna \(g\).  
The SM mapping procedure is revealed in Table 2, for 4 bpcu, where \(N_{t}=4\) and 4-QAM based on the gray-coded constellation points. Consider that the bit stream of [10 11] at any time instant is transmitted through the SM 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 [11] is used as the spatial bits that chooses the antenna for symbol transmission from the antenna combination given in Table 2. As a result, the transmit vector for SM-MIMO system becomes:  
\[\mathbf{x}_{g}\stackrel{{\Delta}}{{=}}[\begin{array}{cccc}0&0& x_{k}&0\end{array}]. \tag{5}\]