#
Quaternion Codes in MIMO System of Dual-Polarized Antennas^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- We propose a new class of QODs based on Liang mechanism [3] with stable code rate as the number of transmit antennas increases.
- The class is shown as best suitable in describing point-to-point communication among dual-polarized antennas.
- The proposed decoder is shown to provide linear decoding solution for all STBCs obtained from QODs.
- A brief performance analysis is carried out for all obtained QODs.

## 2. Quaternion Orthogonal Designs

**Definition**

**1.**

**Definition**

**2.**

#### 2.1. Symmetric-Paired Design 1: (Square QODs)

**Theorem**

**1.**

**Example**

**1.**

#### 2.2. Symmetric-Paired Design 2: (Non-Square QODs)

**Theorem**

**2.**

**Example**

**2.**

#### 2.3. Symmetric-Paired Design 3: (Non-Square QODs)

**Theorem**

**3.**

**Example**

**3.**

**Theorem**

**4.**

**Remark**

**1.**

#### 2.4. Maximal Rate QODs for General Configuration of Dual-Polarized Antennas

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

## 3. Comparative Analysis of the Construction Techniques

#### 3.1. Code Rates

**Theorem**

**5.**

#### 3.2. Coding & Decoding Delays

## 4. Quaternionic Channel Model

#### Linear and Decoupled ML Decoder

**Theorem**

**6.**

**Corollary**

**1.**

## 5. Quasi Quaternion Orthogonal Designs

## 6. Simulation and Results

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

H | Horizontal |

V | Vertical |

STBC | Space Time Block Codes |

COD | Complex orthogonal designs |

QOD | Quaternion orthogonal designs |

3G | Third generation |

5G | Fifth generation |

MIMO | Multiple-input multiple-output |

SISO | Single-input single-output |

TISO | Two-input single-output |

LAN | Local Area Network |

MISO | Multiple-input single-output |

OSTPBC | Orthogonal space time polarization block code |

QPSK | Quadrature phase shift keying |

PAPR | Peak-to-average power ratio |

FLOPs | Floating point operations |

BER | Bit error rate |

SNR | Signal-to-noise ratio |

ML | Maximum Likelihood |

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**Figure 1.**Quaternionic Nomenclature: Two symmetric-paired complex orthogonal designs (CODs) $\mathbf{A}$ and $\mathbf{B}$ generate a quaternion orthogonal design (QOD) $\mathbf{Q}$, which gives rise to different quasi-codes ${\mathbf{C}}_{\mathbf{Q}}$ and ${\mathbf{C}}_{\mathbf{q}}$ with linear and decoupled decoders.

**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.

**Figure 4.**BER vs. SNR performance of ${\mathbf{Q}}_{1},{\mathbf{Q}}_{2},{\mathbf{Q}}_{3},{\mathbf{Q}}_{4},{\mathbf{Q}}_{5}$ & ${\mathbf{Q}}_{6}$ for two receive dual-polarized antenna.

**Figure 5.**BER vs. SNR performance of ${\mathbf{Q}}_{4}$ for one, two and three receive dual-polarized antenna.

Code Designs | ${\mathit{N}}_{\mathit{t}}=3$ | ${\mathit{N}}_{\mathit{t}}=4$ | ${\mathit{N}}_{\mathit{t}}=5$ |
---|---|---|---|

Design 2.1 | ∗ | $\xi =2$ | ∗ |

Design 2.2 | ∗ | $\xi =4$ | ∗ |

Design 2.3 | ∗ | $\xi =4$ | ∗ |

Design 2.4 | $\xi =4$ | $\xi =8$ | $\xi =15$ |

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**MDPI and ACS Style**

Ali, S.; Qureshi, S.S.; Hassan, S.A.
Quaternion Codes in MIMO System of Dual-Polarized Antennas. *Appl. Sci.* **2021**, *11*, 3131.
https://doi.org/10.3390/app11073131

**AMA Style**

Ali S, Qureshi SS, Hassan SA.
Quaternion Codes in MIMO System of Dual-Polarized Antennas. *Applied Sciences*. 2021; 11(7):3131.
https://doi.org/10.3390/app11073131

**Chicago/Turabian Style**

Ali, Sajid, Sara Shakil Qureshi, and Syed Ali Hassan.
2021. "Quaternion Codes in MIMO System of Dual-Polarized Antennas" *Applied Sciences* 11, no. 7: 3131.
https://doi.org/10.3390/app11073131