## 4 Optimization of Channel MIMO Capacity by Combining OSTBC and Polarization-Diversity with XPI
### Diversity Transmission Using OSTBC-MRRC MIMO Technique  
To understand the OSTBC operation in depth, we will briefly explain the known \(2\times 2\) OSTBC technique [25]. The transmission consists of two transmission antennas, \(TX_{1}\) and \(TX_{2}\). Next, the signal transmits through a MIMO channel model with four flat fading paths independent and uncorrelated. 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 3[24,25]. The receive block includes two receive antennas, \(RX_{1}\) and \(RX_{2}\), and an OSTBC decoder combined with MRRC or diversity combining channel state estimation with Maximum-Likelihood (ML) decoding. The OSTBC operation of the two symbols \(s_{1}\) and \(s_{2}\) is defined in Table 4 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 [25]. The operator \((\ \cdot\ )^{*}\) represents the complex conjugate. Figure 4: Dual Polarization-OSTBC-MIMO Scheme. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(RX_{1}\) & \(RX_{2}\) \\ \hline \(TX_{1}\) & \(h_{11}\) & \(h_{12}\) \\ \(TX_{2}\) & \(h_{21}\) & \(h_{22}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Definition of complex channel gain between the transmit and receive antennas. The received signal \(Y\) and Additive White Gaussian Noise (AWGN) signal \(Z\) are denoted by two indexes, \(Y_{k}(l)\) and \(Z_{k}(l)\), 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 5, for example, \(Y_{1}(1)\) denotes the received signal at time \(t\) at \(RX_{1}\). The receiver equations are [24; 25]  
\[ \begin{split} Y_{1}(1)&=h_{11}s_{1}+h_{12}s_{2}+Z_{ 1}(1)\\ Y_{1}(2)&=-h_{11}s_{2}^{*}+h_{12}s_{1}^{*}+Z_{1}(2) \\ Y_{2}(1)&=h_{21}s_{1}+h_{22}s_{2}+Z_{2}(1) \\ Y_{2}(2)&=-h_{21}s_{2}^{*}+h_{22}s_{1}^{*}+Z_{2}(2) \\ \end{split}  \tag{8}\]  
The combining rules for a \(2\times 2\) OSTBC system are [24; 25]  
\[ \begin{split}&\tilde{s}_{1}=h_{11}^{*}Y_{1}(1)+h_{12}Y_{1}^{*}(2)+h_{ 21}^{*}Y_{2}(1)+h_{22}Y_{2}^{*}(2)\\ &\tilde{s}_{2}=h_{12}^{*}Y_{1}(1)-h_{11}Y_{1}^{*}(2)+h_{22}^{*}Y _{2}(1)-h_{21}Y_{2}^{*}(2)\\ \end{split}  \tag{9}\]  
Substituting (8) into (9) yields [24]  
\[\tilde{s}_{1}=\quad\left(\sum_{i=1}^{2}\sum_{j=1}^{2}\lvert h_{ij}\rvert^{2} \right)s_{1}+h_{11}^{*}z_{1}(1)+h_{12}z_{1}^{*}(2)+h_{21}^{*}z_{2}(1)+h_{22}z_ {2}^{*}(2)\]  
and  
\[\tilde{s}_{2}=\quad\left(\sum_{i=1}^{2}\sum_{j=1}^{2}\lvert h_{ij}\rvert^{2} \right)s_{2}+h_{12}^{*}z_{1}(1)-h_{11}z_{1}^{*}(2)+h_{22}^{*}z_{2}(1)-h_{21}z_ {2}^{*}(2) \tag{10}\]  
The estimation of each symbol \(s_{1}\) and \(s_{2}\) is multiplied by the norm of each complex channel gain without the other symbol's presence. Finally, the last receive step is the ML estimate. The transmitted symbol is estimated as follows [24]:  
\[ \begin{split}&\tilde{s}_{1}=argmin_{s_{1}}[\lvert\tilde{s}_{1}-s_{1} \rvert^{2}]\\ &\tilde{s}_{2}=argmin_{s_{2}}[\lvert\tilde{s}_{2}-s_{2}\rvert^{2 }]\\ \end{split}  \tag{11}\]  
Another means to display Equation (8) is through matrices. After mathematical operations of conjugation, the model will be [24; 25]  
\[Y= \begin{bmatrix}Y_{1}(1)\\ Y_{1}^{*}(2)\\ Y_{2}(1)\\ Y_{2}^{*}(2)\end{bmatrix} = \begin{bmatrix}h_{11}&h_{12}\\ h_{12}^{*}&-h_{11}^{*}\\ h_{21}&h_{22}\\ h_{22}^{*}&-h_{21}^{*}\end{bmatrix}\begin{bmatrix}s_{1}\\ s_{1}\\ s_{2}\end{bmatrix} + \begin{bmatrix}z_{1}(1)\\ z_{1}^{*}(2)\\ z_{2}(1)\\ z_{2}^{*}(2)\end{bmatrix}  \tag{12}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(t\) & \(t+T_{s}\) \\ \hline \(TX_{1}\) & \(s_{1}\) & \(-s_{2}^{*}\) \\ \(TX_{2}\) & \(s_{2}\) & \(s_{1}^{*}\) \\ \hline \hline \end{tabular}
\end{table}