# Intelligent Transmit Antenna Selection Schemes for High-Rate Fully Generalized Spatial Modulation
## 2 System Model of FGSM Transceiver  
FGSM is a high-rate version of SM that allows one, many, or all antennas to transmit the same symbol at the same instant. The SE is directly proportional to \(N_{tx}\) as compared to SM, which improves the SE. This variation does not require that the number of transmit antennas be a power of two. The SE of FGSM is calculated as follows:  
\[\alpha_{FGSM}=\log_{2}(M)+(N_{tx}-1). \tag{2}\]  
Consider the input bits [0 0 1 1] to comprehend FGSM mapping. An example with an SE of 4 bpcu (\(M=4\) and \(N_{tx}=3\)) is used to demonstrate FGSM mapping. As indicated in Table 1, the symbol \(s_{1}\) is identified using the first \(\log_{2}(M)\) bits, i.e., [0 0]. It will be transmitted by the antenna pair (1, 2), which is selected using the next \((N_{tx}-1)\) bits, i.e., [1 1], as illustrated in Table 2. For the given block of bits, the transmit vector generated is given by \(\chi=[-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}j,-\frac{1}{\sqrt{2}}+\frac{1}{ \sqrt{2}}j,0]^{T}\). Consider an alternative set of bits [0 1 0 1], where the symbol \(s_{2}\) is chosen by using the first \(\log_{2}(M)\) bits [0 1], as shown in Table 1. It will be transmitted by antenna 2, which is selected using the next \((N_{tx}-1)\) bits, i.e., [0 1], as illustrated in Table 2. For this set of bits, the vector to be transmitted is \(\chi=[0,-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}j,0]^{T}\). The multipath channel \(\mathbf{H}\) and AWGN \(\mathbf{\eta}\) influence the transmit vector \(\mathbf{\chi}\). At the receiver end, the signal vector \(\mathbf{y}\) is stated as follows:  
\[\underbrace{\mathbf{y}}_{N_{x}\times 1}=\underbrace{\mathbf{H}}_{N_{x}\times N_{tx}} \underbrace{\mathbf{\chi}}_{N_{tx}\times 1}+\underbrace{\mathbf{\eta}}_{N_{rx}\times 1}. \tag{3}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Data** & **Index of Symbols** & **Transmitted Symbols** \\ \hline
00 & \(s_{1}\) & \(-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}j\) \\
01 & \(s_{2}\) & \(-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}j\) \\
10 & \(s_{3}\) & \(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}j\) \\
11 & \(s_{4}\) & \(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}j\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A sample mapping of the constellation bits for \(M=4\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Antenna** **Bits** & **Antenna Index** \\ \hline
00 & 1 \\
01 & 2 \\
10 & 3 \\
11 & 1,2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: A sample mapping of the spatial bits for \(N_{tx}=3\). Consider the MIMO configuration \((N_{tx}=3,N_{rx}=4)\) with \(M=4\). For the single antenna operational configuration explained earlier, the received signal vector could be expressed as follows:  
\[\left[ \begin{array}{c}y_{1}\\ y_{2}\\ y_{3}\\ y_{4}\end{array} \right]=\left[ \begin{array}{ccc}h_{11}&h_{12}&h_{13}\\ h_{21}&h_{22}&h_{23}\\ h_{31}&h_{32}&h_{33}\\ h_{41}&h_{42}&h_{43}\end{array} \right]\left[ \begin{array}{c}0\\ s_{2}\\ 0\end{array} \right]+\left[ \begin{array}{c}\eta_{1}\\ \eta_{2}\\ \eta_{3}\\ \eta_{4}\end{array} \right]. \tag{4}\]  
It comes down to  
\[\left[ \begin{array}{c}y_{1}\\ y_{2}\\ y_{3}\\ y_{4}\end{array} \right]=\left[ \begin{array}{c}h_{12}\\ h_{22}\\ h_{32}\\ h_{42}\end{array} \right]s_{2}+\left[ \begin{array}{c}\eta_{1}\\ \eta_{2}\\ \eta_{3}\\ \eta_{4}\end{array} \right]. \tag{5}\]  
For single antenna operational case, the generalized form of the received signal is  
\[\boldsymbol{y}=\boldsymbol{h}_{j}s_{m}+\boldsymbol{\eta},j\in\{1,2,..N_{tx}\},m\in\{1,2,...M\}. \tag{6}\]  
For the double antenna operational configuration explained earlier, the received signal vector could be indicated as follows:  
\[\left[ \begin{array}{c}y_{1}\\ y_{2}\\ y_{3}\\ y_{4}\end{array} \right]=\left[ \begin{array}{ccc}h_{11}&h_{12}&h_{13}\\ h_{21}&h_{22}&h_{23}\\ h_{31}&h_{32}&h_{33}\\ h_{41}&h_{42}&h_{43}\end{array} \right]\left[ \begin{array}{c}s_{1}\\ s_{1}\\ 0\end{array} \right]+\left[ \begin{array}{c}\eta_{1}\\ \eta_{2}\\ \eta_{3}\\ \eta_{4}\end{array} \right]. \tag{7}\]  
It reduces to  
\[\left[