## 2 Channel Modeling of Dual-Polarization for MIMO-TITO Systems  
Many complex definitions, parameters, and characteristics of DP channels exist under different physical assumptions. A few basic definitions are needed to define a communication model that includes all these assumptions and complex parameters. For example, in practical antenna issues, the cross-polarization transmission-receiving (e.g., transmission signals from a vertically-polarized \(T_{x}\) antenna, to a horizontally-polarized \(R_{x}\) antenna) is a very significant definition. The direct meaning is perfect isolation between the orthogonal feeds at the antenna. The practical measurement of the extent of the depolarization in a wireless channel is defined as Cross-Polar Discrimination (XPD). Another example of a complex parameter is the XPI mechanism. The XPI describes the insufficient measure of energy given specific polarization and the relative part of the energy that decouples into the orthogonal polarization. In the case of wireless-communication systems based on mmWave frequency bands and orthogonal polarization issues, the propagation waves that pass through the wireless media are very vulnerable. Under the assumption of flat-fading channels that includes destructive effects such as scatters, fading, interference, and additional weather effects, in the perspective of polarization related to the propagation channel, these effects are reflected and expressed as XPR. Both definitions, the XPI and the XPR, are combined and exist in the cross-polarization issue to global XPD. In general, high values of XPD would indicate a higher level of separability between the two states of polarization, and such channels are amenable to polarization multiplexing techniques. On the other hand, channels with lower values of XPD would indicate significant cross-coupling between the two states of polarization and encourage MIMO-diversity techniques. In this section, we describe a general MIMO-TITO channel modeling under rain or scatter distortion. To simplify the channel model, we consider the 2 \(\times\) 2 MIMO-TITO assumption, flat-fading channels, and outdoor-to-outdoor scenario. This general MIMO-TITO channel model expresses the media between transmitter-receiver wireless communication systems based on an array of orthogonal dual-polarization antennas. The array antennas at the transmitter includes one vertical-linear polarization and one horizontal-linear polarization. On the receiver side, we assume a symmetric array antenna as at the transmitter side with co-polarization at the transmitter-receiver antennas. This architecture creates a two LOS environment based on orthogonal polarization separability conditions at the wireless channel that ideally offers a much better separation between channels. Under ideal conditions on the channel, we assume Ricean distribution with NLOS components close to zero. However, as we mentioned, in the case of present destructive effects, the generally MIMO-TITO channel modeling included NLOS Rayleigh components. The general communication MIMO-TITO model combining components of LOS and components of NLOS is described as:  
\[Y=\sqrt{P}HS+Z \tag{1}\]  
where \(P\) is the total transmission power, \(S\) is the symbol-matrix transmission, \(Z\) is the independent complex Gaussian random variable noise, and \(H\) is the MIMO channel response matrix under Quasistatic Flat-Ricean-Rayleigh-Fading assumptions, given by:  
\[H=\underbrace{\sqrt{\frac{k}{1+k}}\cdot R\cdot \begin{bmatrix}\exp(j\Psi)&0\\ 0&\exp(j\Psi)\end{bmatrix} }_{LOS}\cdot T+\underbrace{\sqrt{\frac{1}{1+k}}\cdot R \cdot H_{medium}\cdot T}_{NLOS} \tag{2}\]  
where \(\Psi\) is the orientation angle between the transmitter-receiver antennas, and \(k\) is the Ricean-Fading distribution factor characterized. \(H_{medium}\) is the MIMO channel response that describes the NLOS components, \(T\) and \(R\), are the coupling antenna matrices at the transmitter and the receiver sides, respectively, and will be defined going forward.