# Study of 5G-NR-MIMO Links in the Presence of an Interferer
## 2 Diversity Transmission Using Alamouti MIMO Technique  
Transmit diversity can be obtained by Space-Time Coding (STC) that is designed to achieve maximum spatial diversity. One simple space-time code is the Alamouti code, which is used in most MIMO systems today. This method's significant advantage over other techniques is that it doesn't require CSIT, which reduces complexity while achieving maximum spatial diversity. We present the effects of interfering with the Alamouti STC. To understand these effects in-depth, we will provide a short brief on the known \(2\times 2\) Alamouti STC technique [1]. Figure 1 shows a block diagram of the \(2\times 2\) Alamouti system case. The transmission consists of two transmission antennas, \(TX_{1}\) and \(TX_{2}\). These antennas transmit in the Alamouti STC technique and represent the UE. Next, the signal transmits through a MIMO channel model with four flat fading paths independent and uncorrelated assume. These four paths are represented by complex channel gains \(h_{11}\), \(h_{12}\), \(h_{21}\), \(h_{22}\). The indexes are the indexes of \(TX\) and \(RX\) antennas, respectively, as shown in Table 1[1; 27]. The receive block includes two receive antennas, \(RX_{1}\) and \(RX_{2}\), and an Alamouti space-time decoder combined with MRRC or diversity combining channel state estimation with ML decoding. This receive block is subject to the rules of the base-station (NodeB). \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\mathbf{RX_{1}}\) & \(\mathbf{RX_{2}}\) \\ \hline \(TX_{1}\) & \(h_{11}\) & \(h_{12}\) \\ \(TX_{2}\) & \(h_{21}\) & \(h_{22}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definition of complex channel gain between the transmit and receive antennas. The Alamouti STC operation of the two symbols \(s_{1}\) and \(s_{2}\) is defined in Table 2 and includes two transmit antennas under two-time slots. At some time \(t\), symbols \(s_{1}\) and \(s_{2}\) are transmitted from \(TX_{1}\) and \(TX_{2}\), respectively. At time \(t+T_{s}\), where \(T_{s}\) is the symbol duration, the symbols \(-s_{2}^{*}\) and \(s_{1}^{*}\) are transmitted from \(TX_{1}\) and \(TX_{2}\), respectively [1]. The operator \((\ \cdot\ )^{*}\) represents the complex conjugate. The received signal \(r\) and Additive White Gaussian Noise (AWGN) signal \(n\) are denoted by two indexes, \(r_{k}(l)\) and \(n_{k}(l)\), where \(k=1,2\) denotes the number of received antennas, and \(l=1,2\) denotes the received signal at time \(t\) or \(t+T_{s}\), respectively. In Table 3, for example, \(r_{1}(1)\) denotes the received signal at time \(t\) at \(RX_{1}\). Figure 1 implies that the receiver equations can be described as [1; 27]  
\[ \begin{split} r_{1}(1)&=h_{11}s_{1}+h_{12}s_{2}+n_{ 1}(1)\\ r_{1}(2)&=-h_{11}s_{2}^{*}+h_{12}s_{1}^{*}+n_{1}(2) \\ r_{2}(1)&=h_{21}s_{1}+h_{22}s_{2}+n_{2}(1)\\ r_{2}(2)&=-h_{21}s_{2}^{*}+h_{22}s_{1}^{*}+n_{2}(2) \end{split}  \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\mathbf{t}\) & \(\mathbf{t+T_{s}}\) \\ \hline \(TX_{1}\) & \(s_{1}\) & \(-s_{2}^{*}\) \\ \(TX_{2}\) & \(s_{2}\) & \(s_{1}^{*}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Transmission Alamouti coding operation. Figure 1: Illustration of \(2\times 2\) Alamouti space time coding (STC). \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\mathbf{t}\) & \(\mathbf{t+T_{s}}\) \\ \hline \(RX_{1}\) & \(r_{1}(1)\) & \(r_{1}(2)\) \\ \(n_{1}(1)\) & \(n_{1}(2)\) \\ \hline \(RX_{2}\) & \(r_{2}(1)\) & \(r_{2}(2)\) \\ \(n_{2}(1)\) & \(n_{2}(2)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Received signal \(r\) and additive white Gaussian noise (AWGN) signal \(n\). The combining rules for a \(2\times 2\) Alamouti system case are here defined as [1; 27]  
\[ \begin{split}\tilde{s}_{1}&=h_{11}^{*}r_{1}(1)+h_{12}r _{1}^{*}(2)+h_{21}^{*}r_{2}(1)+h_{22}r_{2}^{*}(2)\\ \tilde{s}_{2}&=h_{12}^{*}r_{1}(1)-h_{11}r_{1}^{*}(2)+h _{22}^{*}r_{2}(1)-h_{21}r_{2}^{*}(2)\end{split}