## 4 Optimization of Channel MIMO Capacity by Combining OSTBC and Polarization-Diversity with XPI
### Diversity Transmission Using OSTBC-MRRC MIMO Technique
Let  
\[f(\theta,\phi)=trace(H^{*}_{TR}H_{TR})=\] \[=2\big{(}h_{x,x}\big{(}B\delta_{x,y}+A\big{)}+h_{y,y}\big{(}D\delta_ {y,x}+C\big{)}\big{)}\cdot\Big{(}h^{*}_{x,x}\big{(}B\delta^{*}_{x,y}+A\big{)}+h^ {*}_{y,y}\big{(}D\delta^{*}_{y,x}+C\big{)}\Big{)}+\] \[+2\big{(}h_{x,x}\big{(}A\delta_{x,y}+B\big{)}+h_{y,y}\big{(}C \delta_{y,x}+D\big{)}\big{)}\cdot\Big{(}h^{*}_{x,x}\big{(}A\delta^{*}_{x,y}+B \big{)}+h^{*}_{y,y}\big{(}C\delta^{*}_{y,x}+D\big{)}\Big{)}+\] \[+2\big{(}h_{y,y}\big{(}B\delta_{y,x}+A\big{)}+h_{x,x}\big{(}C \delta_{x,y}+C\big{)}\big{)}\cdot\Big{(}h^{*}_{y,y}\big{(}B\delta^{*}_{y,x}+A \big{)}+h^{*}_{x,x}\big{(}D\delta^{*}_{x,y}+C\big{)}\Big{)}+\] \[+2\big{(}h_{y,y}\big{(}A\delta_{y,x}+B\big{)}+h_{x,x}\big{(}C \delta_{x,y}+D\big{)}\big{)}\cdot\Big{(}h^{*}_{y,y}\big{(}A\delta^{*}_{y,x}+B \big{)}+h^{*}_{x,x}\big{(}C\delta^{*}_{x,y}+D\big{)}\Big{)}=\] \[=2\big{|}h_{x,x}\big{(}B\delta_{x,y}+A\big{)}+h_{y,y}\big{(}D \delta_{y,x}+C\big{)}\big{|}^{2}+ \tag{16}\] \[+2\big{|}h_{x,x}\big{(}A\delta_{x,y}+B\big{)}+h_{y,y}\big{(}C \delta_{y,x}+D\big{)}\big{|}^{2}+\] \[+2\big{|}h_{y,y}\big{(}B\delta_{y,x}+A\big{)}+h_{x,x}\big{(}D \delta_{x,y}+C\big{)}\big{|}^{2}+\] \[=2\Bigg{\|}\Bigg{[} \begin{array}{c}h_{x,x}\big{(}B\delta_{x,y}+ A\big{)}+h_{y,y}\big{(}D\delta_{y,x}+C\big{)}\\ h_{x,y}\big{(}A\delta_{x,y}+B\big{)}+h_{y,y}\big{(}C\delta_{y,x}+D\big{)}\\ h_{y,y}\big{(}B\delta_{y,x}+A\big{)}+h_{x,x}\big{(}D\delta_{x,y}+C\big{)}\\ h_{y,y}\big{(}A\delta_{y,x}+B\big{)}+h_{x,x}\big{(}C\delta_{x,y}+D\big{)}\\ \end{array} \Bigg{]}\Bigg{\|}^{2}.\]  
Then, in order to maximize the capacity (as well as minimizing the BER), we need to maximize \(f\). Now, we will show that \(f\) is a convex function, as a function of \(A,B,C,D\) over the region \(0\leq A,B,C,D\leq 1\). Lemma 1: _Let \(g:\mathbb{C}^{m}\to\mathbb{R}\) be defined by \(g(v)=\|v\|^{2}\). Then, \(g\) is convex._  
Proof: Let \(v_{1},v_{2}\in\mathbb{C}^{m}\) and let \(0\leq\lambda\leq 1\). We need to show that \(g(\lambda v_{1}+(1-\lambda)v_{2})\leq\lambda g(v_{1})+(1-\lambda)g(v_{2})\). Now,  
\[g(\lambda v_{1}+(1-\lambda)v_{2})=\|\lambda v_{1}+(1-\lambda)v_{ 2}\|^{2}\] \[=\lambda^{2}\|v_{1}\|^{2}+2\lambda(1-\lambda)\Re(\langle v_{1},v_ {2}\rangle)+(1-\lambda)^{2}\|v_{2}\|^{2}\] \[\leq\lambda^{2}\|v_{1}\|^{2}+2\lambda(1-\lambda)\|v_{1}\|\|v_{2} \|+(1-\lambda)^{2}\|v_{2}\|^{2},\]  
where in the last passage, we used Cauchy-Swartz inequality. Since  
\[\lambda g(v_{1})+(1-\lambda)g(v_{2})=\lambda\|v_{1}\|^{2}+(1-\lambda)\|v_{2}\| ^{2},\]  
it is therefore sufficient to prove that:  
\[\lambda^{2}\|v_{1}\|^{2}+2\lambda(1-\lambda)\|v_{1}\|\|v_{2}\|+(1-\lambda)^{2} \|v_{2}\|^{2}\leq\lambda\|v_{1}\|^{2}+(1-\lambda)\|v_{2}\|^{2}.\]  
Indeed,  
\[\lambda^{2}\|v_{1}\|^{2}+2\lambda(1-\lambda)\|v_{1}\|\|v_{2}  
**Lemma 2**.: _Let \(h:\mathbb{R}^{n}\to\mathbb{C}^{m}\) denote any linear transformation and let \(g:\mathbb{C}^{m}\to\mathbb{R}\) be defined by \(g(v)=\|v\|^{2}\). Let \(f:\mathbb{R}^{n}\to\mathbb{R}\) be defined by \(f(x)=g(h(x))\). Then, \(f\) is convex._  
Proof.: Let \(x_{1},x_{2}\in\mathbb{R}^{n}\) and let \(0\leq\lambda\leq 1\). Let \(v_{1}=h(x_{1})\) and \(v_{2}=h(x_{2})\). Then,  
\[f(\lambda x_{1}+(1-\lambda)x_{2})=g(h(\lambda x_{1}+(1-\lambda)x _{2}))\] \[=g(\lambda h(x_{1})+(1-\lambda)h(x_{2}))\] \[=g(\lambda v_{1}+(1-\lambda)v_{2})\] \[\leq\lambda g(v_{1})+(1-\lambda)g(v_{2})\] \[=\lambda g(h(x_{1}))+(1-\lambda)g(h(x_{2}))\] \[=\lambda^{2}\|v_{1}\|^{2}+2\lambda(1-\lambda)\Re(\langle v_{1},v_ {2}\rangle)+(1-\lambda)^{2}\|v_{2}\|^{2}\]0  
where we used the linearity of \(h\) and Lemma 1. We therefore conclude that the function \(f\) given by (16) is a convex function as a function of \(A,B,C,D\) over the region \(0\leq A,B,C,D\leq 1\). By a well known theorem, the maximum of \(f\) is achieved on the exposed points of the region; that is, on one of the 16 nodes \(A,B,C,D\in\{0,1\}\). However, since here \(A,B,C,D\) depend on \(\theta,\phi\), we have only 4 cases to consider:  
\[