## 2 Channel Modeling of Dual-Polarization for MIMO-TITO Systems
Suppose that \(\Psi=0\), the orientation matrix that describes the LOS components will be equal to the identity matrix, \(I\). Our assumption in this article is that with the presence of the destructive effect, for example, rain distortion, the Ricean-Fading distribution factor, \(k\), is equal to zero, and \(H_{medium}\) becomes the MIMO channel response channel under Rayleigh-Fading assumptions, which is defined by:  
\[H_{medium}= \begin{bmatrix}h_{xx}&h_{xy}\\ h_{yx}&h_{yy}\end{bmatrix}  \tag{3}\]  
The components, \(h_{xy},h_{yx}\) describe the cross-dual-polarization paths between the vertically polarized transmission antenna to the horizontally polarized receiver antenna and the horizontally polarized transmission antenna to the vertically polarized receiver antenna, respectively. The components, \(h_{xx},h_{yy}\) define the co-polarization paths between the vertically polarized transmission antenna to the vertically polarized receiver antenna and the horizontally polarized transmission antenna to the horizontally polarized receiver antenna, respectively. To describe the polarization phenomenon in these channels, we will define basic parameters and break down the general channel response matrix into several matricial components based on these parameters. A cross-polar antenna pattern with finite XPI can be defined by a coupling antenna matrix at the transmitter side and also at the receiver side as:  
\[T= \begin{bmatrix}\sqrt{1-\frac{1}{XPI_{T}}}&\sqrt{\frac{1}{XPI_{T}}}\\ \sqrt{\frac{1}{XPI_{T}}}&\sqrt{1-\frac{1}{XPI_{T}}}\end{bmatrix} \] \[and,\] \[R= \begin{bmatrix}\sqrt{1-\frac{1}{XPI_{R}}}&\sqrt{\frac{1}{XPI_{R}} }\\ \sqrt{\frac{1}{XPI_{R}}}&\sqrt{1-\frac{1}{XPI_{R}}}\end{bmatrix} \]  
where the scalar XPI, can obtain the values between \(1\leq XPI\leq\infty\). When \(XPI=\infty\), this value represents perfect polar power isolation. This mean that in an antenna defined with vertical polarization, all the signal's power is projected through the vertical polarization direction without leakage towards the horizontal polarization. When the antenna is defined as horizontal polarization, all the signal's power is projected through the horizontal polarization direction without leakage towards the vertical polarization. If we suppose that XPI is close to 1, the transmitted or received signal power is divided evenly between the vertical and horizontal polarization in the vertical antenna. The same division process also occurs in the horizontal antenna. This situation yields DP with a cross-polarization MIMO model. \(XPI_{T}\) and \(XPI_{R}\) are defined at the transmitter and receiver antennas, respectively. \(\delta_{xy}\), and \(\delta_{yx}\) is the complex factor cross-polarization that is created from the rain cases or other destructive effects and defined as:  
\[\delta_{xy}=\frac{h_{xy}}{h_{xx}}\] \[and,\] \[\delta_{yx}=\frac{h_{yx}}{h_{yy}}\]  
The XPR is a real value that describes the ratio of the channels norms (i.e., the gain on the channel \(i,j\) represented by \(\left|h_{i=x,j=y}\right|^{2}\)), denoted as:  
\[XPR_{x}=\frac{1}{\left|\delta_{xy}\right|^{2}}\] \[and,\] \[XPR_{y}=\frac{1}{\left|\delta_{yx}\right|^{2}}\]  
The new \(H_{medium}\), in relation to the \(H_{medium}\) in (2), is:  
\[H_{medium}= \begin{bmatrix}h_{xx}&\delta_{x,y}h_{xx}\\ \delta_{yx}h_{yy}&h_{yy}\end{bmatrix}  \tag{4}\]  
Rewriting  
\[ \begin{cases}\frac{1}{XPI_{T}}=\alpha,\frac{1}{XPI_{R}}=\beta\\ \sqrt{\alpha}=\sin(\theta),\sqrt{\beta}=\sin(\phi)\\ \sqrt{1-\alpha}=\cos(\theta),\sqrt{1-\beta}=\cos(\phi)\end{cases}  \tag{5}\]  
where  
\[H_{TR} =\left(\sqrt{\frac{1}{1+k}}\right)RH_{medium}T=\] \[=\left(\sqrt{\frac{1}{1+k}} \begin{bmatrix}\cos(\theta)&\sin( \theta)\\ \sin(\theta)&\cos(\theta)\end{bmatrix}\begin{bmatrix}h_{xx}&\delta_{x,y}h_{xx} \\ \delta_{yx}h_{yy}&h_{yy}\end{bmatrix}\begin{bmatrix}\cos(\phi)&\sin(\phi)\\ \sin(\phi)&\cos(\phi)\end{bmatrix}