# Mapping Matrix Design and Improved Belief Propagation Decoding Algorithm for Rate-Compatible Modulation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Original Rate Compatible Modulation

- The mapping matrix should be regular in rows.
- The mapping matrix should be as regular as possible in columns.
- The weight set is able to create diverse symbol values.

## 3. Mapping Matrix Construction Method

## 4. Improved Belief Propagation Algorithm

#### 4.1. Introduction of Improved Belief Propagation

#### 4.2. Analysis of Error Probability of Soft Information

## 5. Simulation

#### 5.1. Threshold of Soft Information

#### 5.2. Performance Evaluation

#### 5.3. Comparison of Complexity

#### 5.4. Sensitivity to SNR Estimation Error

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Goeckel, D.L. Adaptive coding for time-varying channels using outdated fading estimates. IEEE Trans. Commun.
**1999**, 47, 844–855. [Google Scholar] [CrossRef] - Song, K.B.; Ekbal, A.; Chung, S.T.; Cioffi, J.M. Adaptive modulation and coding (AMC) for bit-interleaved coded OFDM (BIC-OFDM). IEEE Int. Conf. Commun.
**2006**, 6, 3197–3201. [Google Scholar] - Tong, J.; Ping, L.; Ma, X. Superposition Coded Modulation With Peak-Power Limitation. IEEE Trans. Inf. Theory
**2009**, 55, 2562–2576. [Google Scholar] [CrossRef] [Green Version] - Shokrollahi, A. Raptor codes. IEEE Trans. Inf. Theory
**2006**, 52, 2551–2567. [Google Scholar] [CrossRef] - Luby, M. LT codes. In Proceedings of the Symposium on Foundations of Computer Science, Vancouver, BC, Canada, 16–19 November 2002; pp. 271–280. [Google Scholar]
- Perry, J.; Iannucci, P.A.; Fleming, K.E.; Balakrishnan, H.; Shah, D. Spinal codes. ACM SIGCOMM Comput. Commun. Rev.
**2012**, 42, 49–60. [Google Scholar] [CrossRef] - Shirvanimoghaddam, M.; Li, Y.; Vucetic, B. Near-Capacity Adaptive Analog Fountain Codes for Wireless Channels. IEEE Commun. Lett.
**2013**, 17, 2241–2244. [Google Scholar] [CrossRef] [Green Version] - Cui, H.; Luo, C.; Tan, K.; Wu, F.; Chen, C.W. Seamless rate adaptation for wireless networking. In Proceedings of the International Symposium on Modeling Analysis and Simulation of Wireless and Mobile Systems, MSWIM 2011, Miami, FL, USA, 31 October–4 November 2011; pp. 437–446. [Google Scholar]
- Rao, W.; Dong, Y.; Lu, F.; Wang, S. Log-likelihood ratio algorithm for rate-compatible modulation. In Proceedings of the IEEE International Symposium on Circuits and Systems, Beijing, China, 19–23 May 2013; pp. 1938–1941. [Google Scholar]
- Cui, H.; Luo, C.; Wu, J.; Chen, C.W.; Wu, F. Compressive Coded Modulation for Seamless Rate Adaptation. IEEE Trans. Wirel. Commun.
**2013**, 12, 4892–4904. [Google Scholar] [CrossRef] [Green Version] - Lu, F.; Dong, Y.; Rao, W.; Chen, C.W. Low Complexity Decoding Algorithms for Rate Compatible Modulation. IEEE Access
**2018**, 6, 31417–31429. [Google Scholar] [CrossRef] - Lu, F.; Dong, Y.; Rao, W. A Parallel Belief Propagation Decoding Algorithm for Rate Compatible Modulation. IEEE Commun. Lett.
**2017**, 21, 1735–1738. [Google Scholar] [CrossRef] - Wu, J.; Teng, Z.; Cui, H.; Luo, C.; Huang, X.; Chen, H.H. Arithmetic-BICM for Seamless Rate Adaptation for Wireless Communication Systems. IEEE Syst. J.
**2016**, 10, 228–239. [Google Scholar] [CrossRef] - Duan, R.; Liu, R.; Shirvanimoghaddam, M.; Li, Y.; Chen, C.W. A Low PAPR Constellation Mapping Scheme for Rate Compatible Modulation. IEEE Commun. Lett.
**2016**, 20, 256–259. [Google Scholar] [CrossRef] - Rao, W.; Dong, Y.; Chen, S.; Lu, F.; Wang, S. An Efficient Rate Compatible Modulation with Variable Weight Sets. IEEE Access
**2018**, 6, 5064–5074. [Google Scholar] [CrossRef] - Wang, M.; Wu, J.; Yu, W.; Wang, H.; Li, J.; Shi, J.; Luo, C. Efficient coding modulation and seamless rate adaptation for visible light communications. IEEE Wirel. Commun.
**2015**, 22, 86–93. [Google Scholar] [CrossRef] - Shirvanimoghaddam, M.; Li, Y.; Dohler, M.; Vucetic, B.; Feng, S. Probabilistic Rateless Multiple Access for Machine-to-Machine Communication. IEEE Trans. Wirel. Commun.
**2015**, 14, 6815–6826. [Google Scholar] [CrossRef] - Gallager, R. Low-density parity-check codes. IRE Trans. Inf. Theory
**1962**, 8, 21–28. [Google Scholar] [CrossRef] [Green Version] - Kou, Y.; Lin, S.; Fossorier, M.P.C. Low-density parity-check codes based on finite geometries: A rediscovery and new results. IEEE Trans. Inform. Theory
**2001**, 47, 2711–2736. [Google Scholar] [CrossRef] - Fan, J.L. Array codes as LDPC codes. In Constrained Coding and Soft Iterative Decoding; Springer: Berlin, Germany, 2001; pp. 195–203. [Google Scholar]
- Tanner, R.M.; Sridhara, D.; Fuja, T. A class of group-structured LDPC codes. In Proceedings of the International Symposium on Communication Theory Applications, Ambleside, UK, 15–20 July 2001. [Google Scholar]
- Wang, Y.; Yedidia, J.S.; Draper, S.C. Construction of high-girth QC-LDPC codes. In Proceedings of the International Symposium on Turbo Codes and Related Topics, Lausanne, Switzerland, 1–5 September 2008; pp. 180–185. [Google Scholar]
- Baron, D.; Sarvotham, S.; Baraniuk, R.G. Bayesian Compressive Sensing Via Belief Propagation. IEEE Trans. Signal Process.
**2009**, 58, 269–280. [Google Scholar] [CrossRef] - Edalat, F.; Edalat, F.; Katabi, D.; Sodini, C.G. Frequency-aware rate adaptation and MAC protocols. In Proceedings of the International Conference on Mobile Computing and NETWORKING, Beijing, China, 20–25 September 2009; pp. 193–204. [Google Scholar]

**Figure 2.**Comparison between original receiver model and improved receiver model. (

**a**) Original receiver model. (

**b**) Improved receiver model.

SNR(dB) | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |

Threshold | 8.06 | 7.57 | 7.35 | 6.90 | 6.81 | 6.5 | 6.34 | 6.17 | 6.00 |

SNR(dB) | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |

Threshold | 5.91 | 5.80 | 5.33 | 5.14 | 4.67 | 4.47 | 4.25 | 4.13 | 4.04 |

SNR(dB) | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | |

Threshold | 3.91 | 3.75 | 3.55 | 3.46 | 3.40 | 3.24 | 3.10 | 3.07 |

Name | Weight Set | L | Type of Symbol |
---|---|---|---|

RCM1 | $\left(\right)$ | 8 | 13 |

RCM2 | $\left(\right)$ | 8 | 23 |

RCM3 | $\left(\right)$ | 8 | 31 |

SNR(dB) | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |

Number | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 |

SNR(dB) | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |

Number | 7 | 7 | 7 | 5 | 5 | 5 | 3 | 3 | 3 |

SNR(dB) | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | |

Number | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 |

SNR(dB) | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |

Number | 2650 | 2100 | 1280 | 1100 | 980 | 770 | 710 | 450 | 350 |

SNR(dB) | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |

Number | 300 | 290 | 260 | 240 | 230 | 200 | 190 | 170 | 170 |

SNR(dB) | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | |

Number | 170 | 150 | 150 | 150 | 150 | 150 | 150 | 140 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhu, J.; Pan, Z.; Liu, W.; Lei, J.; Li, W.
Mapping Matrix Design and Improved Belief Propagation Decoding Algorithm for Rate-Compatible Modulation. *Electronics* **2019**, *8*, 307.
https://doi.org/10.3390/electronics8030307

**AMA Style**

Zhu J, Pan Z, Liu W, Lei J, Li W.
Mapping Matrix Design and Improved Belief Propagation Decoding Algorithm for Rate-Compatible Modulation. *Electronics*. 2019; 8(3):307.
https://doi.org/10.3390/electronics8030307

**Chicago/Turabian Style**

Zhu, Jinkun, Zhipeng Pan, Wei Liu, Jing Lei, and Wei Li.
2019. "Mapping Matrix Design and Improved Belief Propagation Decoding Algorithm for Rate-Compatible Modulation" *Electronics* 8, no. 3: 307.
https://doi.org/10.3390/electronics8030307