# A Sparse Signal Reconstruction Method Based on Improved Double Chains Quantum Genetic Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Data Models of the Redundant Dictionary and FOMP Algorithm

#### 2.1. Redundant Dictionary

#### 2.2. OMP

#### 2.3. The Proposed FOMP Algorithm

## 3. IDCQGA-Based FOMP Algorithm

#### 3.1. The Principle of DCQGA

#### 3.1.1. Double Chains Qubit Encoding

#### 3.1.2. Quantum Rotation Gate Updating

#### 3.1.3. Quantum Chromosome Mutation

#### 3.2. The Proposed IDCQGA

#### 3.2.1. High Density Qubit Encoding

#### 3.2.2. Adaptive Step Size for Updating

#### 3.2.3. Quantum $\pi /6$-Gate for Mutation

#### 3.3. FOMP Algorithm Combined with IDCQGA

## 4. Simulation Results and Analysis

#### 4.1. Experiment 1 and Analysis: Performance of the IDCQGA

#### 4.2. Experiment 2 and Analysis: Performance of the OMP Based on IDCQGA

#### 4.3. Experiment 3 and Analysis: Performance of the FOMP Based on IDCQGA

#### 4.4. Experiment 4 and Analysis: The Applicability of the Proposed Algorithms for Radar Signals

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagrams of the coding space. (

**a**) The traditional coding space; (

**b**) the improved coding space when k = 1 and k = 2.

**Figure 2.**Shaffer’s F6 function. (

**a**) Three-dimensional surface; (

**b**) profile of y=0; (

**c**) profile of x=0.

**Figure 5.**The original frame signal and the reconstruction frame signal using 100 atoms. (

**a**) The original frame signal; (

**b**) the original frequency frame signal; (

**c**) the reconstruction frame signal; (

**d**) the reconstruction frequency frame signal.

**Figure 6.**The residual frame signal and residual frequency after the reconstruction. (

**a**) The residual frame signal; (

**b**) the residual frequency after the reconstruction

**Figure 7.**The original speech signal and the reconstructed speech signal using the IDCQGA-based OMP algorithm. (

**a**) The original speech signal; (

**b**) the reconstructed speech signal.

**Figure 8.**Performance of the OMP based on IDCQGA. (

**a**) The relationship between the fidelity and the iterations; (

**b**) the relationship between the RMSE and the number of atoms.

**Figure 10.**The original signal, the noisy signal and the reconstructed signal of the radar emitter signals. (

**a**) The original signal of CON; (

**b**) the original signal of LFM; (

**c**) the original signal of BPSK; (

**d**) the original signal of BFSK; (

**e**) the noisy signal of CON; (

**f**) the noisy signal of LFM; (

**g**) the noisy signal of BPSK; (

**h**) the noisy signal of BFSK; (

**i**) the reconstructed signal of CON; (

**j**) the reconstructed signal of LFM; (

**k**) the reconstructed signal of BPSK; (

**l**) the reconstructed signal of BFSK.

**Table 1.**Parameters setting of the improved double chains quantum genetic algorithm (IDCQGA) and other algorithms.

Algorithm | Population | Bits of | Encoding | Crossover | Mutation | Initial Rotation | Evolutionary |
---|---|---|---|---|---|---|---|

Size | Gene | Method | Probability | Probability | Angle | Generation | |

PSO | 50 | - | - | - | - | - | 200 |

GA | 50 | 100 | Binary | 0.7 | 0.05 | - | 200 |

QGA | 10 | 100 | qubit | - | - | - | 200 |

DCQGA | 10 | 2 | Double chains | - | 0.05 | $0.01\pi $ | 200 |

IDCQGA | 10 | 2 | High density | - | 0.05 | - | 200 |

Algorithm | x | y | Best Result | Convergent Generations |
---|---|---|---|---|

PSO | 2.7583 | 5.01376 | 0.98327 | 105 |

GA | 8.0283 | 4.81177 | 0.91128 | 9 |

QGA | 2.8985 | 5.5678 | 0.96278 | 1 |

DCQGA | −2.6202 | −1.7276 | 0.99028 | 40 |

IDCQGA | 1.51263 | 1.39374 | 0.99793 | 22 |

Algorithm | Best Result | Worst Result | Average Result | Number of Convergence |
---|---|---|---|---|

PSO | 0.9901 | 0.8925 | 0.9512 | 3 |

GA | 0.9604 | 0.6545 | 0.8535 | 0 |

QGA | 0.9628 | 0.8711 | 0.9402 | 0 |

DCQGA | 0.9903 | 0.8666 | 0.9687 | 6 |

IDCQGA | 1 | 0.9902 | 0.9946 | 10 |

SNR | CON | LFM | BPSK | BFSK | |
---|---|---|---|---|---|

5dB | Noisy signal | 0.6545 | 0.6531 | 0.6572 | 0.6496 |

Reconstructed signal | 0.0530 | 0.0335 | 0.0402 | 0.0545 | |

10dB | Noisy signal | 0.5263 | 0.0372 | 0.0369 | 0.5302 |

Reconstructed signal | 0.0324 | 0.0372 | 0.0369 | 0.0357 | |

15dB | Noisy signal | 0.4179 | 0.3986 | 0.3659 | 0.3982 |

Reconstructed signal | 0.0213 | 0.0257 | 0.0315 | 0.0268 | |

20dB | Noisy signal | 0.3345 | 0.2856 | 0.3549 | 0.3058 |

Reconstructed signal | 0.0197 | 0.0190 | 0.0204 | 0.0173 |

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**MDPI and ACS Style**

Guo, Q.; Ruan, G.; Wan, J.
A Sparse Signal Reconstruction Method Based on Improved Double Chains Quantum Genetic Algorithm. *Symmetry* **2017**, *9*, 178.
https://doi.org/10.3390/sym9090178

**AMA Style**

Guo Q, Ruan G, Wan J.
A Sparse Signal Reconstruction Method Based on Improved Double Chains Quantum Genetic Algorithm. *Symmetry*. 2017; 9(9):178.
https://doi.org/10.3390/sym9090178

**Chicago/Turabian Style**

Guo, Qiang, Guoqing Ruan, and Jian Wan.
2017. "A Sparse Signal Reconstruction Method Based on Improved Double Chains Quantum Genetic Algorithm" *Symmetry* 9, no. 9: 178.
https://doi.org/10.3390/sym9090178