# Quaternion Codes in MIMO System of Dual-Polarized Antennas +
## 6 Simulation and Results  
To evaluate the performance and diversity gains, we employ QODs, that is, \(\mathbf{Q}_{1}-\mathbf{Q}_{6}\), corresponding to single and dual-polarized receive antenna configurations. We obtain their bit-error-rate (BER) curves against signal-to-noise (SNR) ratios. For simulations, quadrature phase shift keying is used. The receivers are aware of the channel coefficients and uniformwhite noise is added in each polarization. We have utilized the basic concepts of pure quaternion algebra, the matrix computations and complex domain calculations to optimize the overall system model in MATLAB. The codes constructed using the Liang based construction technique provides a lower number of complex receivers. For the codes \(\mathbf{Q}_{1}-\mathbf{Q}_{6}\), in Figure 3, it is clear that these codes have linear and decoupled decoding at the receiving end due to the use of the specific construction technique using the dual-polarized antennas and the quaternionic channel model. This has been possible due to the quaternionic channel exploiting the polarization diversity independently using the polar as well as cross-polar scattering between the dual-polarized antennas. The codes \(\mathbf{Q}_{1},\mathbf{Q}_{2}\), \(\mathbf{Q}_{3}\) and \(\mathbf{Q}_{5}\) are transmitted using four dual-polarized antennas at the transmitting end and have the code rates of \(\frac{3}{2}\), \(\frac{1}{2}\), \(\frac{3}{4}\) and \(\frac{3}{4}\), respectively. These codes seem to show similar curves but they differ in their intrinsic behaviors based on the number of timeslots used and the symbols transmitted per timeslot. The codes \(\mathbf{Q}_{2}\) and \(\mathbf{Q}_{3}\) show similar performance as \(\mathbf{Q}_{1}\), but they will experience the issue of peak-to-average power ratio (\(PAPR\)), where switching on and off of one of the dual-polarized antenna in every timeslot is a tradeoff. Also, \(\mathbf{Q}_{5}\) shows competitive response in comparison to \(\mathbf{Q}_{1}\) by transmitting double the number of complex symbols in every timeslot, however, this has been achieved at a compromise on the increased number of timeslots used in \(\mathbf{Q}_{5}\). Thus, it takes longer time to decode the complete code block in case of \(\mathbf{Q}_{5}\) with increased number of complex symbols when compared to \(\mathbf{Q}_{1}\), where the receiver is able to decode the original information faster from the received code block. While, the codes \(\mathbf{Q}_{4}\) and \(\mathbf{Q}_{6}\) use three and four dual-polarized antennas in transmission which have codes rate \(\frac{3}{4}\) and \(\frac{3}{8}\), respectively. These codes demonstrate that the proposed scheme is successful with odd number of transmit dual-polarized antennas also. Code \(\mathbf{Q}_{6}\) is a unique presentation of pure quaternion code designs that favors the utility of this approach in massive MIMO systems in future wireless communication systems. We can see that the proposed code construction technique has no restrictions on the number of transmit and receive antennas. This is demonstrated in the Figure 3 where both even and odd number of dual-polarized antennas are used to transmit the codes. The effects of increasing the receiver diversity are visible in the Figure 4. It clearly shows that the receiver diversity has positive impact on the diversity gains. Use of the quaternionic channel exploits the polarization diversity and promises decoupled decoder for any number of receive dual-polarized antennas. In comparison to the work done int he  
Figure 3: BER vs. SNR performance of \(\mathbf{Q}_{1},\mathbf{Q}_{2},\mathbf{Q}_{3},\mathbf{Q}_{4},\mathbf{Q}_{5}\) & \(\mathbf{Q}_{6}\) for single receive dual-polarized antenna. past, the proposed design does not compromise the code rates when the number of transmit antennas are increased. Such an increase in the code rates with higher diversity gains are the requirements of the future MIMO systems to support greater channel utilization and efficiency.