# A Generalized Spatial Modulation System Using Massive MIMO Space Time Coding Antenna Grouping
## 2 System Model and Analysis
### GSM System Model  
One of the great challenges for massive MIMO to achieve a high data rate. The GSM system model can achieve a high data rate by using a set of transmit antennas to send the same constellation symbol. The proposed system is explained in Figure 1, showing that a group of antennas is formed and uses the same constellation point. \(N_{t}\) refers to the total number of antennas available at transmitter and \(N_{u}\) is the number of active transmit antennas. However, the set of antenna combinations that will be used for transmission must be a power of two.  
The proposed GSM system model is designed for \(N_{t}\) = 5 and \(N_{u}\) = 2.  
Thus, the possible antenna combinations \(N_{c}\) = 2\({}^{m}\), \(m_{l}\) = \(\left[\begin{matrix}N_{t}\\ N_{u}\end{matrix}\right]\)  
The mapping data below are considered in [22] to map the incoming data bits to a spatial symbol and a data symbol, which is used before for Quadrature Phase Shift Keying (QPSK) as illustrated in Table 1.  
In the example of the grouped bits \(g(n)=[0\ 0\ 0\ 0]\), the first three bits \(000\) refer to the antenna combination (1,2) and the rest bit \(0\) refers to the transmitted symbol \(-1\). So, if \(g(n)=[1\ 0\ 1\ 1]\), the antenna combination will be (2,4) and the transmitted symbol will be \(+1\).  
The GSM model is illustrated in Figure 1. The first \(ml\) bits are mapped to be used for the set of antenna combinations and the set of the remaining bits \(ms\) are modulated using M-QAM modulation, where \(M=2^{ms}\).  
In general, the total number of bits that will be transmitted using the GSM is given by [22]:  
\[m=ms+ml=\log_{2}(N_{t}/N_{u})+\log_{2}M \tag{1}\]  
The modulated signal for GSM is transmitted over a MIMO selective Rayleigh fading channel, H. The channel matrix H has the dimension size of \(N_{r}\times N_{t}\), where \(N_{r}\) and \(N_{t}\) are, respectively, the number of the receiving and transmitting antennas.  
The input data stream q(n) is grouped firstly using data grouping and then the output g(n) is mapped using GSM mapping (Table 1).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Group of Bits** & **Set of Antennas (\(l\))** & **Transmitted Symbol (s)** \\ \hline
0001 & (1,2) & \(+1\) \\ \hline
0000 & (1,2) & \(-1\) \\ \hline
0011 & (1,3) & \(+1\) \\ \hline
0010 & (1,3) & \(-1\) \\ \hline
0101 & (1,4) & \(+1\) \\ \hline
0100 & (1,4) & \(-1\) \\ \hline
0111 & (1,5) & \(+1\) \\ \hline
0110 & (1,5) & \(-1\) \\ \hline
1001 & (2,3) & \(+1\) \\ \hline
1000 & (2,3) & \(-1\) \\ \hline
1011 & (2,4) & \(+1\) \\ \hline
1010 & (2,4) & \(-1\) \\ \hline
1101 & (3,5) & \(+1\) \\ \hline
1100 & (3,5) & \(-1\) \\ \hline
1111 & (4,5) & \(+1\) \\ \hline
1110 & (4,5) & \(-1\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: GSM mapping table for \(N_{t}=5,N_{u}=2\)[22].  
Figure 1: GSM (generalized spatial modulation) system model.  
The received signal at any given time  
\[\textbf{Y=}h^{{}^{\prime}}_{l,s}+\textbf{\mu} \tag{2}\]  
where \(s\) is the symbol transmitted through the set of antenna combination, \(s\in\) M-QAM \(l=[l_{1}\), \(l_{2}\),......., \(l_{Nu}]\in\varnothing\), \(l_{n}\) refers to the index of \(n^{th}\) antenna in the set of antennas, \(l\) and \(\varnothing\) refer to the set of antenna combination, and \(\mu\) is an Additive White Gaussian Noise (AWGN) vector with zero-mean.  
At the receiver, the ML algorithm is used to decode the spatial symbol and the data symbol as follows  
\[[l,s]=\arg\min\sum_{i=1}^{Nr}\abs{y_{i}-h_{l,s}s}^{2} \tag{3}\]  
where  
\[Py(Y|s,l,H)=\frac{1}{(\pi\sigma^{2})}exp(-\frac{Y-h_{l,s}^{2}}{\sigma^{2}}) \tag{4}\]  
Here, \(Py(Y|s,l,H)\) is the probability density function (PDF) of \(Y\) conditional on \([l,s]\) and \(H\), and \({}^{2}\) is the Frobenius norm [22].