# Joint Design of Polar Coding and Physical Network Coding for Two−User Downlink Non−Orthogonal Multiple Access

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## Abstract

**:**

## 1. Introduction

## 2. Background and System Model

#### 2.1. System Model

#### 2.2. Channel Polarization

## 3. Polar Coded for NOMA

#### 3.1. Proposed PN-DNOMA Scheme

#### 3.2. Information-Theoretic Finite-Length Code Performance

## 4. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PNC | Physical network coding |

PN-DNOMA | A joint PNC for two−user downlink non−orthogonal multiple access |

NOMA | Non−orthogonal multiple access |

SIC | Successive interference cancellation |

PA | Power allocation |

BS | Base station |

PC-SIC | A joint polar decoding and SIC decoding strategy |

NCMA | Network-coded multiple access |

SC | Successive cancellation |

CS-SCL | Cyclic−redundancy−check−aided SC−list |

CSIT | Channel state information at the transmitter |

B-DMC | Binary−input discrete memoryless channel |

SINR | Signal−to−interference−plus−noise ratio |

JSC | Joint successive cancellation |

QAM | Quadrature amplitude modulation |

MF | Matched-filter |

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**Figure 5.**Error performance of two−user PN−DNOMA and PC−SIC with different power allocations at SNR = 18 dB.

**Figure 6.**Error performance of PN−DNOMA and PC−SIC with the optimal power allocation, over two−user downlink NOMA.

**Figure 7.**Effective throughput ${T}_{t}$ achieved by PN−DNOMA and PC−SIC, over two−user downlink NOMA.

j | ${\mathit{x}}_{1}$ | ${\mathit{x}}_{2}$ | ${\mathit{x}}_{\mathit{o}}$ | ${\mathit{h}}_{\mathit{s}}\sqrt{{\mathit{p}}_{1}}{\mathit{r}}_{1}$ | ${\mathit{h}}_{\mathit{s}}\sqrt{{\mathit{p}}_{2}}{\mathit{r}}_{2}$ | S |
---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | ${h}_{s}\sqrt{{p}_{1}}$ | ${h}_{s}\sqrt{{p}_{2}}$ | ${h}_{s}(\sqrt{{P}_{1}}+\sqrt{{P}_{2}})$ |

1 | 1 | 0 | 1 | $-{h}_{s}\sqrt{{p}_{1}}$ | ${h}_{s}\sqrt{{p}_{2}}$ | ${h}_{s}(-\sqrt{{P}_{1}}+\sqrt{{P}_{2}})$ |

2 | 0 | 1 | 1 | ${h}_{s}\sqrt{{p}_{1}}$ | $-{h}_{s}\sqrt{{p}_{2}}$ | ${h}_{s}(\sqrt{{P}_{1}}-\sqrt{{P}_{2}})$ |

3 | 1 | 1 | 0 | $-{h}_{s}\sqrt{{p}_{1}}$ | $-{h}_{s}\sqrt{{p}_{2}}$ | $-{h}_{s}(\sqrt{{P}_{1}}+\sqrt{{P}_{2}})$ |

**Table 2.**Error performance and decoding complexity of PN−DNOMA and PC−SIC over two−user downlink NOMA.

Decoder | ||||
---|---|---|---|---|

SC Decoder | CA−SCL Decoder | |||

PC−SIC | PN−DNOMA | PC−SIC | PN−DNOMA | |

BER: $1.0\times {10}^{-4}$ | 30.1 (dB) | 28.3 (dB) | 25.6 (dB) | 24.5 (dB) |

BER: $1.0\times {10}^{-5}$ | 31.8 (dB) | 30.9 (dB) | 26.7 (dB) | 25.6 (dB) |

Decoding Complexity | $4NlogN$ | $NlogN+2Nlog2N$ | $4LNlogN$ | $LNlogN+2LNlog2N$ |

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**MDPI and ACS Style**

Xie, Z.; Chen, P.; Li, Y. Joint Design of Polar Coding and Physical Network Coding for Two−User Downlink Non−Orthogonal Multiple Access. *Entropy* **2023**, *25*, 233.
https://doi.org/10.3390/e25020233

**AMA Style**

Xie Z, Chen P, Li Y. Joint Design of Polar Coding and Physical Network Coding for Two−User Downlink Non−Orthogonal Multiple Access. *Entropy*. 2023; 25(2):233.
https://doi.org/10.3390/e25020233

**Chicago/Turabian Style**

Xie, Zhaopeng, Pingping Chen, and Yong Li. 2023. "Joint Design of Polar Coding and Physical Network Coding for Two−User Downlink Non−Orthogonal Multiple Access" *Entropy* 25, no. 2: 233.
https://doi.org/10.3390/e25020233