# Optimized Design for NB-LDPC-Coded High-Order CPM: Power and Iterative Efficiencies

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## Abstract

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## 1. Introduction

## 2. System Description

## 3. Competitive CPM

#### 3.1. SIR

#### 3.2. Design Criterion

## 4. Code Design and Advantages

#### 4.1. EXIT Technique

#### 4.2. NB-LDPC Code Design

- (1)
- Optimization of degree distribution based on binary parity-matrix using the EXIT technique. The NB-LDPC code has the same Tanner graph and degree distribution property as its corresponding binary representation, except for non-zero values; thus, the EXIT chart was used to explore a binary sparse matrix with acceptable degree distribution. Figure 5 provides the EXIT characteristics of different variable node (VN) and check node (CN) degrees.

- (2)
- Construction of a parity-check matrix with a large girth. After determining the degree distribution, the positions of non-zero elements in binary parity-check matrix ${H}_{b}$ must be ascertained. A girth optimization tool, called progressive edge growth [37], is adopted to avoid small circles and achieve good girth properties when using the BP-like algorithm on the Tanner graph.
- (3)
- Choice of non-zero elements over GF(Q). Generally, this step can be performed by substituting the “1” elements of ${H}_{b}$ with random non-zero elements over GF(Q), which can provide acceptable performance in most cases. Entropy theory, which is the appropriate measure for uncertainty, is introduced to improve the uncertainty or randomness of cycles located at the Tanner graph, which helps obtain a low error rate. First, the cycle-searching algorithm proposed in [38] is applied to search all small circles of ${H}_{b}$ with lengths l = 4, 6, and even 8 if necessary, and record the corresponding positions of non-zero elements in each circle. Next, a general method for constructing the NB-LDPC code is employed to randomly replace all the “1” elements of ${H}_{b}$ with non-zero elements of GF(Q). Eventually, the entropy of each previously recorded circle is calculated and maximized when each element takes various non-zero values over GF(Q), that is:$${E}_{n}=-{\displaystyle \sum _{i=1}^{Q-1}P{r}_{i}}{\mathrm{log}}_{2}P{r}_{i}$$

#### 4.3. Additional Advantages

- (1)
- Each edge of the binary LDPC code in the Tanner graph carries bit messages, but the NB-LDPC code carries Q-ary symbol messages, thus, short girths are avoided in the Tanner graph. This reduces the influence of short girths and stopping set on decoding convergence. Therefore, the BP algorithm becomes closer to the maximum likelihood decoding algorithm. The NB-LDPC code, as an outer code in coded modulation systems, provides an alternative solution in enhancing BER performance in practical applications.
- (2)
- In comparison with the traditional BICM, the interleaver of the NB-LDPC-coded high-order CPM works at the symbol level, which always yields a lower convergence threshold than bit level. This advantage is rather significant in a serial concatenation [22].
- (3)
- As the NB-LDPC code and CPM select the uniform M-ary in the investigated systems, the symbol mapping issue that is likely to result in conversion information loss from bit to symbol may be ignored. This phenomenon usually occurs in the case of $M>q$. Thus, more possible input code symbols exist between the current and next phase states in the trellis diagrams. For an example of the CPM scheme with 8M2RC using h = 1/2, the corresponding transfer diagram of the phase states using Gray and natural mappings is shown in Figure 7.

## 5. Positive Feedback Issue

## 6. Optimization Design for Iterative Efficiency

## 7. Simulation Results

^{−3}, respectively. The schemes with extrinsic information operation always converge to a smaller BER at the turbo cliff region compared to those without extrinsic information operation and it can achieve approximately 0.1 dB advantages. In addition, Figure 13 shows that these optimized schemes using five inner iterations and $\epsilon =1{e}^{-5}$ have fewer average outer iterations with increasing ${\mathrm{E}}_{\mathrm{b}}{/\mathrm{N}}_{0}$ and exhibit negligible BER performance degradation with respect to their original counterparts with a total of 140 iterations (seven inner iterations $\times 20$ outer iterations). These optimized schemes attain a suitable tradeoff of the communication reliability and iterative decoding delay and enhance systematic iterative efficiency.

## 8. Conclusions

_{b}/N

_{0}but also accelerate iterative convergence at medium–high ${\mathrm{E}}_{\mathrm{b}}/{\mathrm{N}}_{0}$, and the value of BER can be reduced by approximately $5\times {10}^{-3}$. An improper iteration match between demodulation and decoding was addressed using the EXIT technique and mutual information to improve the iterative efficiency and attain a suitable tradeoff of the communication reliability and the iterative decoding delay. Finally, simulation results show that the resulting NB-LDPC-coded high-order CPM scheme provides an approximately 3.95 dB coding gain compared to the uncoded CPM and achieves approximately 0.5 and 0.7 dB advantages compared with the candidate schemes. The resulting scheme using the proposed method attains the convergence threshold earlier compared with other competitors and further improves power and iterative efficiencies.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Non-binary low-density parity-check (NB-LDPC)-coded high-order continuous phase modulation (CPM) transmitter and receiver.

**Figure 5.**Variable node (VN) and check node (CN) extrinsic information transfer (EXIT) curves with various degrees under R = 2/3 and ${\mathrm{E}}_{\mathrm{b}}/{\mathrm{N}}_{0}=1\mathrm{dB}$.

**Figure 6.**EXIT chart of the optimized NB-LDPC-coded high-order CPM system with 8M2RC under five inner iterations.

**Figure 7.**Transfer diagram of the phase states of 8M2RC with h = 1/2 using Gray and natural mappings.

**Figure 8.**Iterative convergence of the NB-LDPC code for 8M2RC with h = 1/2 using the proposed and original methods at ${\mathrm{E}}_{\mathrm{b}}/{\mathrm{N}}_{0}=0.4\mathrm{dB}$.

**Figure 9.**Iterative convergence of the NB-LDPC code for 8M2RC with h = 1/2 using the proposed and original methods at ${\mathrm{E}}_{\mathrm{b}}/{\mathrm{N}}_{0}=1.2\mathrm{dB}$.

**Figure 11.**Bit error rate (BER) performance of the optimized NB-LDPC code for 8M2RC with various inner iterations.

**Figure 12.**BER performance of the resulting scheme and other candidate schemes for $\eta =0.5\mathrm{bit}/\mathrm{s}/\mathrm{Hz}$ and R = 2/3, as well as their original counterparts with or without extrinsic information operation. The vertical lines represent their convergence thresholds achieved from the EXIT chart. The interleaver length is 1152 symbols.

**Figure 13.**Average outer iteration comparison of the resulting scheme and other candidate schemes for $\eta =0.5\mathrm{bit}/\mathrm{s}/\mathrm{Hz}$ and R = 2/3, as well as their original counterparts with or without extrinsic information operation.

**Table 1.**${d}_{\mathrm{min}}^{2}$ and ${B}_{99\%}{T}_{b}$ of all candidate CPM schemes subject to constraints $q\le 5$, L < 3, M < 8, and h < 1.

h | 4M1REC | 4M1RC | 8M1REC | 8M1RC |

1/5 | ${B}_{99\%}{T}_{b}=0.65,{d}_{\mathrm{min}}^{2}=0.97$ | ${B}_{99\%}{T}_{b}=1.06,{d}_{\mathrm{min}}^{2}=1.13$ | ${B}_{99\%}{T}_{b}=0.76,{d}_{\mathrm{min}}^{2}=1.46$ | ${B}_{99\%}{T}_{b}=1.15,{d}_{\mathrm{min}}^{2}=1.69$ |

1/4 | ${B}_{99\%}{T}_{b}=0.81,{d}_{\mathrm{min}}^{2}=1.45$ | ${B}_{99\%}{T}_{b}=1.25,{d}_{\mathrm{min}}^{2}=1.66$ | ${B}_{99\%}{T}_{b}=0.91,{d}_{\mathrm{min}}^{2}=2.18$ | ${B}_{99\%}{T}_{b}=1.36,{d}_{\mathrm{min}}^{2}=2.49$ |

1/3 | ${B}_{99\%}{T}_{b}=0.99,{d}_{\mathrm{min}}^{2}=2.35$ | ${B}_{99\%}{T}_{b}=1.49,{d}_{\mathrm{min}}^{2}=2.25$ | ${B}_{99\%}{T}_{b}=1.12,{d}_{\mathrm{min}}^{2}=3.50$ | ${B}_{99\%}{T}_{b}=1.78,{d}_{\mathrm{min}}^{2}=3.33$ |

2/5 | ${B}_{99\%}{T}_{b}=1.09,{d}_{\mathrm{min}}^{2}=3.06$ | ${B}_{99\%}{T}_{b}=1.67,{d}_{\mathrm{min}}^{2}=2.40$ | ${B}_{99\%}{T}_{b}=1.31,{d}_{\mathrm{min}}^{2}=4.60$ | ${B}_{99\%}{T}_{b}=2.09,{d}_{\mathrm{min}}^{2}=3.60$ |

1/2 | ${B}_{99\%}{T}_{b}=1.28,{d}_{\mathrm{min}}^{2}=4.00$ | ${B}_{99\%}{T}_{b}=1.95,{d}_{\mathrm{min}}^{2}=2.24$ | ${B}_{99\%}{T}_{b}=1.54,{d}_{\mathrm{min}}^{2}=6.00$ | ${B}_{99\%}{T}_{b}=2.52,{d}_{\mathrm{min}}^{2}=3.36$ |

3/5 | ${B}_{99\%}{T}_{b}=1.49,{d}_{\mathrm{min}}^{2}=3.50$ | ${B}_{99\%}{T}_{b}=2.26,{d}_{\mathrm{min}}^{2}=2.40$ | ${B}_{99\%}{T}_{b}=1.79,{d}_{\mathrm{min}}^{2}=5.24$ | ${B}_{99\%}{T}_{b}=2.96,{d}_{\mathrm{min}}^{2}=3.60$ |

2/3 | ${B}_{99\%}{T}_{b}=1.57,{d}_{\mathrm{min}}^{2}=3.57$ | ${B}_{99\%}{T}_{b}=2.49,{d}_{\mathrm{min}}^{2}=2.60$ | ${B}_{99\%}{T}_{b}=1.96,{d}_{\mathrm{min}}^{2}=5.37$ | ${B}_{99\%}{T}_{b}=3.31,{d}_{\mathrm{min}}^{2}=3.90$ |

3/4 | ${B}_{99\%}{T}_{b}=1.71,{d}_{\mathrm{min}}^{2}=3.72$ | ${B}_{99\%}{T}_{b}=2.81,{d}_{\mathrm{min}}^{2}=2.90$ | ${B}_{99\%}{T}_{b}=2.16,{d}_{\mathrm{min}}^{2}=5.58$ | ${B}_{99\%}{T}_{b}=3.68,{d}_{\mathrm{min}}^{2}=4.15$ |

4/5 | ${B}_{99\%}{T}_{b}=1.83,{d}_{\mathrm{min}}^{2}=3.84$ | ${B}_{99\%}{T}_{b}=2.98,{d}_{\mathrm{min}}^{2}=3.37$ | ${B}_{99\%}{T}_{b}=2.28,{d}_{\mathrm{min}}^{2}=5.72$ | ${B}_{99\%}{T}_{b}=3.89,{d}_{\mathrm{min}}^{2}=4.28$ |

h | 4M2REC | 4M2RC | 8M2REC | 8M2RC |

1/5 | ${B}_{99\%}{T}_{b}=0.44,{d}_{\mathrm{min}}^{2}=0.64$ | ${B}_{99\%}{T}_{b}=0.56,{d}_{\mathrm{min}}^{2}=0.88$ | ${B}_{99\%}{T}_{b}=0.57,{d}_{\mathrm{min}}^{2}=0.96$ | ${B}_{99\%}{T}_{b}=0.63,{d}_{\mathrm{min}}^{2}=1.32$ |

1/4 | ${B}_{99\%}{T}_{b}=0.53,{d}_{\mathrm{min}}^{2}=0.98$ | ${B}_{99\%}{T}_{b}=0.64,{d}_{\mathrm{min}}^{2}=1.33$ | ${B}_{99\%}{T}_{b}=0.68,{d}_{\mathrm{min}}^{2}=1.48$ | ${B}_{99\%}{T}_{b}=0.75,{d}_{\mathrm{min}}^{2}=1.99$ |

1/3 | ${B}_{99\%}{T}_{b}=0.69,{d}_{\mathrm{min}}^{2}=1.69$ | ${B}_{99\%}{T}_{b}=0.79,{d}_{\mathrm{min}}^{2}=2.16$ | ${B}_{99\%}{T}_{b}=0.87,{d}_{\mathrm{min}}^{2}=2.52$ | ${B}_{99\%}{T}_{b}=0.95,{d}_{\mathrm{min}}^{2}=3.30$ |

2/5 | ${B}_{99\%}{T}_{b}=0.81,{d}_{\mathrm{min}}^{2}=2.35$ | ${B}_{99\%}{T}_{b}=0.91,{d}_{\mathrm{min}}^{2}=2.91$ | ${B}_{99\%}{T}_{b}=1.02,{d}_{\mathrm{min}}^{2}=3.53$ | ${B}_{99\%}{T}_{b}=1.11,{d}_{\mathrm{min}}^{2}=4.36$ |

1/2 | ${B}_{99\%}{T}_{b}=0.99,{d}_{\mathrm{min}}^{2}=3.45$ | ${B}_{99\%}{T}_{b}=1.06,{d}_{\mathrm{min}}^{2}=3.71$ | ${B}_{99\%}{T}_{b}=1.26,{d}_{\mathrm{min}}^{2}=5.18$ | ${B}_{99\%}{T}_{b}=1.32,{d}_{\mathrm{min}}^{2}=5.56$ |

3/5 | ${B}_{99\%}{T}_{b}=1.15,{d}_{\mathrm{min}}^{2}=4.60$ | ${B}_{99\%}{T}_{b}=1.22,{d}_{\mathrm{min}}^{2}=3.75$ | ${B}_{99\%}{T}_{b}=1.49,{d}_{\mathrm{min}}^{2}=6.31$ | ${B}_{99\%}{T}_{b}=1.58,{d}_{\mathrm{min}}^{2}=5.63$ |

2/3 | ${B}_{99\%}{T}_{b}=1.26,{d}_{\mathrm{min}}^{2}=4.00$ | ${B}_{99\%}{T}_{b}=1.33,{d}_{\mathrm{min}}^{2}=3.47$ | ${B}_{99\%}{T}_{b}=1.64,{d}_{\mathrm{min}}^{2}=6.01$ | ${B}_{99\%}{T}_{b}=1.74,{d}_{\mathrm{min}}^{2}=5.20$ |

3/4 | ${B}_{99\%}{T}_{b}=1.39,{d}_{\mathrm{min}}^{2}=4.19$ | ${B}_{99\%}{T}_{b}=1.45,{d}_{\mathrm{min}}^{2}=3.71$ | ${B}_{99\%}{T}_{b}=1.84,{d}_{\mathrm{min}}^{2}=6.28$ | ${B}_{99\%}{T}_{b}=1.93,{d}_{\mathrm{min}}^{2}=5.56$ |

4/5 | ${B}_{99\%}{T}_{b}=1.47,{d}_{\mathrm{min}}^{2}=4.88$ | ${B}_{99\%}{T}_{b}=1.51,{d}_{\mathrm{min}}^{2}=4.11$ | ${B}_{99\%}{T}_{b}=1.95,{d}_{\mathrm{min}}^{2}=6.00$ | ${B}_{99\%}{T}_{b}=2.05,{d}_{\mathrm{min}}^{2}=6.16$ |

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**MDPI and ACS Style**

Xue, R.; Wang, T.; Sun, Y.; Tang, H.
Optimized Design for NB-LDPC-Coded High-Order CPM: Power and Iterative Efficiencies. *Symmetry* **2020**, *12*, 1353.
https://doi.org/10.3390/sym12081353

**AMA Style**

Xue R, Wang T, Sun Y, Tang H.
Optimized Design for NB-LDPC-Coded High-Order CPM: Power and Iterative Efficiencies. *Symmetry*. 2020; 12(8):1353.
https://doi.org/10.3390/sym12081353

**Chicago/Turabian Style**

Xue, Rui, Tong Wang, Yanbo Sun, and Huaiyu Tang.
2020. "Optimized Design for NB-LDPC-Coded High-Order CPM: Power and Iterative Efficiencies" *Symmetry* 12, no. 8: 1353.
https://doi.org/10.3390/sym12081353