## 4 Optimization of Channel MIMO Capacity by Combining OSTBC and Polarization-Diversity with XPI
### Diversity Transmission Using OSTBC-MRRC MIMO Technique
\begin{split}&\big{(}h_{x,y}(t_{0})h_{x,x}(t_{0})^{*}+h_{x,y}(t_{1})h _{x,x}(t_{1})^{*}+h_{y,y}(t_{0})h_{y,x}(t_{0})^{*}+h_{y,y}(t_{1})h_{y,x}(t_{1})^ {*}\big{)}\Big{(}A^{2}+D^{2}\Big{)}+\\ &+\left(\begin{array}{c}\left|h_{x,x}(t_{0})\right|^{2}+\left|h_ {x,x}(t_{1})\right|^{2}+\\ +\left|h_{x,y}(t_{0})\right|^{2}+\left|h_{x,y}(t_{1})\right|^{2}+\\ +\left|h_{y,x}(t_{0})\right|^{2}+\left|h_{y,x}(t_{1})\right|^{2}+\\ +\left|h_{y,y}(t_{0})\right|^{2}+\left|h_{y,y}(t_{1})\right|^{2}\end{array} \right)(AB+CD)+\\ &+\left(\begin{array}{c}h_{x,x}(t_{0})h_{y,x}(t_{0})^{*}+h_{y,x}(t_{0})h_{x, x}(t_{0})^{*}+\\ +h_{x,x}(t_{1})h_{y,x}(t_{1})^{*}+h_{y,x}(t_{1})h_{x,x}(t_{1})^{*}+\\ +h_{x,y}(t_{0})h_{y,y}(t_{0})^{*}+h_{y,y}(t_{0})h_{x,y}(t_{0})^{*}+\\ +h_{x,y}(t_{1})h_{y,y}(t_{1})^{*}+h_{y,y}(t_{1})h_{x,y}(t_{1})^{*}\end{array} \right)(AC+DB)+\\ &+2\big{(}h_{x,y}(t_{0})h_{y,x}(t_{0})^{*}+h_{y,y}(t_{0})h_{x,x}(t_{0})^{*}+h_{ x,y}(t_{1})h_{y,x}(t_{1})^{*}+h_{y,y}(t_{1})h_{x,x}(t_{1})^{*}\big{)}AD+\\ &+\big{(}h_{x,x}(t_{0})h_{x,y}(t_{0})^{*}+h_{x,x}(t_{1})h_{x,y}(t_{1})^{*}+h_{ y,x}(t_{0})h_{y,y}(t_{0})^{*}+h_{y,x}(t_{1})h_{y,y}(t_{1})^{*}\big{)}\Big{(}B^{2}+C^{2} \Big{)}+\\ &+2\big{(}h_{x,x}(t_{0})h_{y,y}(t_{0})^{*}+h_{y,x}(t_{0})h_{x,y}(t_{0})^{*}+h_ {x,x}(t_{1})h_{y,y}(t_{1})^{*}+h_{y,x}(t_{1})h_{x,y}(t_{1})^{*}\big{)}BC=0. \end{split}  \tag{21}\]  
Let  
\[ \begin{cases}a=h_{x,y}(t_{0})h_{x,x}(t_{0})^{*}+h_{x,y}(t_{1})h_{x,x}(t_{1})^ {*}+h_{y,y}(t_{0})h_{y,x}(t_{0})^{*}+h_{y,y}(t_{1})h_{y,x}(t_{1})^{*}\\ b=h_{x,y}(t_{0})h_{y,x}(t_{0})^{*}+h_{y,y}(t_{0})h_{x,x}(t_{0})^{*}+h_{x,y}(t _{1})h_{y,x}(t_{1})^{*}+h_{y,y}(t_{1})h_{x,x}(t_{1})^{*}\\ c=h_{x,x}(t_{0})h_{y,x}(t_{0})^{*}+h_{y,x}(t_{0})h_{x,x}(t_{0})^{*}+h_{x,x}(t _{1})h_{y,x}(t_{1})^{*}+h_{y,x}(t_{1})h_{x,x}(t_{1})^{*}+\\ +h_{x,y}(t_{0})h_{y,y}(t_{0})^{*}+h_{y,y}(t_{0})h_{x,y}(t_{0})^{*}+h_{x,y}(t_{1 })h_{y,y}(t_{1})^{*}+h_{y,y}(t_{1})h_{x,y}(t_{1})^{*}\\ d=\left|h_{x,x}(t_{0})\right|^{2}+\left|h_{x,x}(t_{1})\right|^{2}+\left|h_{x,y }(t_{0})\right|^{2}+\left|h_{x,y}(t_{1})\right|^{2}+\\ +\left|h_{y,x}(t_{0})\right|^{2}+\left|h_{y,x}(t_{1})\right|^{2}+\left|h_{y, y}(t_{0})\right|^{2}+\left|h_{y,y}(t_{1})\right|^{2}.\end{cases} \]  
Then, (21) is equivalent to  
\[a\Big{(}A^{2}+D^{2}\Big{)}+d(AB+CD)+c(AC+DB)+2 bAD+a^{*}\Big{(}B^{2}+C^{2} \Big{)}+2b^{*}BC=0. \tag{22}\]  
Now, since \(trace\big{(}H_{TR}(t_{0},t_{1})^{*}H_{TR}(t_{0},t_{1})\big{)}\) is a function of the form of \(f\) from Lemma 2, it follows that it is maximized at one of the exposed points (17). Therefore, in view of (22), over the points of maximum, the columns of \(H_{TR}(t_{0},t_{1})\) are orthogonal if and only if (\(a=0\) and equivalently \(a^{*}=0\)):  
\[h_{x,y}(t_{0})h_{x,x}(t_{0})^{*}+h_{y,y}(t_{0})h_{y,x}(t_{0})^{*}+h_{x,y}(t_{1} )h_{x,x}(t_{1})^{*}+h_{y,y}(t_{1})h_{y,x}(t_{1})^{*}=0. \tag{23}\]  
The third step is an application of conditions with the help of the proposed effective feedback and combining this feedback between the transmitter-receiver according to the architecture described in Section 4. The results of the simulation on the MATLAB/SIMULINK platform in terms of BER and average capacity are depicted in the illustrations Figures 8 and 9, respectively. In this simulation, we create a MIMO channel response engine that changes the channel values in each time slot of an icon. The capacity and BER calculators measure BER performance and capacity performance throughout the duration of the simulation. The last step is to activate the feedback that implements the condition (23). In Figure 8, a comparison was carried out between two graphs representing the BER performance of the two scenarios we have specified. The blue graph represents the BER performance of the communication system under the destructive effect of changing the MIMO response channel in every symbol time slot, without any feedback facility. The red graph represents the BER performance under the same scenario with combining of feedback.