# Quaternion Codes in MIMO System of Dual-Polarized Antennas +
## 4 Quaternionic Channel Model  
The simultaneous transmission from both ends of a dual-polarized antenna can be regarded as a _hyper signal_ which consists of two complex numbers in two orthogonal polarizations. It propagates through space and received as a hyper signal by the dual-polarized antenna at the receiver end in a given time slot. This hyper signal can be represented as a quaternion [5] which gives us a reason to develop system model in the quaternion domain. An important component of the system model is the channel which we assume to be quaternionic following the line of approach followed in [18]. Therein, we observe that the product in the quaternion domain holds a key of consistent and viable model. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Code Designs** & \(N_{t}=3\) & \(N_{t}=4\) & \(N_{t}=5\) \\ \hline Design 2.1 & \(*\) & \(\xi=2\) & \(*\) \\ Design 2.2 & \(*\) & \(\xi=4\) & \(*\) \\ Design 2.3 & \(*\) & \(\xi=4\) & \(*\) \\ Design 2.4 & \(\xi=4\) & \(\xi=8\) & \(\xi=15\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Decoding Delays. Figure 2: The code rate \(R_{\mathbf{Q}}\) of QODs based on new approach remains stable around 0.5. We first consider a two-input and single-output system (TISO) of dual-polarized antennas. It is necessary to emphasize the role of quaternions which is more recognizable in this case, therefore, we have  
\[\mathbf{R}= \begin{bmatrix}r_{1}\\ r_{2}\end{bmatrix} = \begin{bmatrix}q_{1}&q_{2}\\ q_{3}&q_{4}\end{bmatrix}\begin{bmatrix}h_{1}\\ h_{2}\end{bmatrix} + \begin{bmatrix}n_{1}\\ n_{2}\end{bmatrix} , \tag{18}\]  
where each element in the above construction is a quaternion. Through each antenna in the above TISO system, the transmission of a pair of two complex symbols is encoded in \(q_{1}\) and another pair in \(q_{3}\). This indicates that the above QOD exploits time and space diversities along with polarization diversity. It is worth pointing out that each quaternionic product, for example, \(q_{a}h_{b}\) contains a crucial information about the nature of quaternion domain. If we decompose it for a general quaternionic product then we obtain \(q_{a1}h_{b1}-q_{a2}h_{b2}+j(q_{a1}h_{b2}+q_{a2}h_{b1})\), where \(q_{a}=q_{a1}+jq_{a2}\) and \(h_{b}=h_{b1}+jh_{b2}\). Therefore, we will obtain four complex channel gains for each antenna in a \(2\times 1\) system. Note that we have four complex channel gains between a TISO system of dual-polarized antennas. As this system is equivalent to a MIMO \(4\times 2\) system of single-polarized antennas, therefore, it may appear that it should have eight channel gains in total with two for each link. However, in our proposed model each quaternionic product results in the same number of channel gains. Subsequently, a system model for a MIMO system of dual-polarized antennas can be constructed in the same way for such a system with \(N_{t}\times N_{r}\)dual-polarized antennas  
\[\mathbf{R}_{T\times N_{r}}=\mathbf{Q}_{T\times N_{t}}\mathbf{H}_{N_{t}\times N _{t}}+\mathbf{N}_{T\times N_{t}}, \tag{19}\]  
which transmits symbols in \(T-\)times slots, which are assumed to be points in the quadrature phase shift keying (QPSK) constellation. The channel matrix is \(\mathbf{H}=[h_{\varnothing\sigma}]\), where \(\rho=1,2,\ldots,N_{t}\) and \(\sigma=1,2,\ldots,N_{r}\). The channel is assumed to represent a flat fading channel and the path gain from \(\rho\) transmit dual-polarized antenna to receive dual-polarized antenna \(\sigma\) given by a quaternion \(h_{\rho\sigma}=h_{\rho\sigma 1}+h_{\rho\sigma 2}j\). The complex channel gains, \(h_{\rho\sigma 1}\) and \(h_{\rho\sigma 2}\) incorporate the effects of cross polar scattering and each element of channel gain matrix is a complex Gaussian random variable (RV) with zero mean and unit variance.