# A Generalized Spatial Modulation System Using Massive MIMO Space Time Coding Antenna Grouping
## 2 System Model and Analysis
### GSM-STC Model  
In order to achieve a high data rate and diversity, GSM-STC is designed. The GSM-STC will use two subgroups of antennas at the transmitter using the combination method [23] to achieve diversity with a high data rate and low bit error rate. Two groups will be added in the transmitter and each group has \(N_{t}\) total transmitting antenna and \(N_{u}\) number of active antennas per group. The transmitting symbols will be during two time slots, \(t\) and \(t+\tau\), as seen in Figure 2.  
In this system, two subgroups of antennas are proposed at the transmitter, and each group has \(N_{t}\) antenna and \(N_{u}\) active antennas. At the receiver, we have \(N_{r}\) receiving antennas with the same number of active antennas at transmitter.  
Let \(S=\left[s_{0},s_{1}\right]^{T}\) be the un-coded M-QAM information symbol vector that will be transmitted over two sequential OFDM symbols, so the Alamouti scheme generates the following code word matrix:  
\[X=\begin{array}{cc}s_{0}&-s_{1}^{*}\\ s_{1}&s_{0}^{*}\end{array} \tag{5}\]  
The subgroup antennas are illustrated in Figure 2. At time t, the first group will transmit symbol \(s_{0}\) and the second group will transmit symbol \(s_{1}\). At time (\(t+\tau\)), the first group will transmit symbol \(-s_{1}^{*}\) and the second group will transmit symbol \(s_{0}^{*}\).  
At the transmitted time \(t\), active antennas from first group send \(s_{0}\) symbols and \(s_{1}\) will be transmitted from second group at same time. At time \(+\tau\), active antennas from first group send \(-s1^{*}\)  
Figure 2: GSM-STC (space time coding technique) system model.  
symbols and \(s0^{*}\) will be transmitted from second group at same time too. Table 2 shows that the encoding and transmission sequence of the information symbols for this configuration is identical to the case single receiver. In Table 2, \(h_{m,n}\) indicates the channel between transmitter and receiver, where \(n\) is the number of Rx antenna, \(n=1\), 2, 3, 4, and \(m\) is the number of Tx antenna, \(m=1\), 2, 3... 10.  
\(r_{0},r_{1},\ r_{2},\ r_{3},\ r_{4},\ r_{5},\ r_{6}\)_and \(r_{7}\)_ are the complex random variables at the receiver, including noise and interference. The following two signals are built from a combiner that will be sent to the ML decoder as  
\[s_{0}=h_{0}^{*}r_{0}+h_{1}r_{1}^{*}+h_{2}^{*}r_{2}+h_{3}r_{3}^{*}+h_{4}^{*}r_{4} +h_{5}r_{5}^{*}+h_{6}^{*}r_{6}+h_{7}r_{7}^{*} \tag{6}\]  
\[s_{1}=h_{1}^{*}r_{0}-h_{0}r_{1}^{*}+h_{3}^{*}r_{2}-h_{2}r_{3}^{*}+h_{5}^{*}r_{4 }-h_{4}r_{5}^{*}+h_{7}^{*}r_{6}-h_{6}r_{7}^{*} \tag{7}\]  
Therefore, the resulting diversity order from the new two-branch transmit diversity scheme with two receivers is equal to that of the four-branch multi-rate resource control (MRRC) scheme. It is important to highlight that the combined signals from the two receive antennas and the combined signals from each receive antenna are simply added, i.e., the combining scheme is more identical to the case with a single receive antenna. One can draw a concern that, if we use two transmit and M receive antennas, one of the transmitting antennas can use the combiner for each receive antenna and then easily add the combined signals from all the receive antennas to obtain the same diversity order as 2M-branch MRRC. Finally, when using two antennas at the transmitter side, the scheme doubles the diversity order of systems with one transmit and multiple receive antennas.