# Spectral-Based SPD Matrix Representation for Signal Detection Using a Deep Neutral Network

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Spectral Image-Based Signal Detection with a Deep Neural Network

#### 2.1. GoogLeNet

#### 2.2. ResNet

## 3. Spectral-Based SPD Matrix for Signal Detection with a Deep Neural Network

#### 3.1. Spectral-Based SPD Matrix Construction

#### 3.1.1. SPD Matrix Construction Method Based on Spectrum Transformation

#### 3.1.2. SPD Matrix Construction Method Based on Spectrum Covariance

#### 3.2. SPDnet

## 4. Results

#### 4.1. Experimental Analysis of Simulation Data

#### 4.1.1. Comparison with Convolutional Neural Networks

#### 4.1.2. Comparison with Convolutional Neural Networks

#### 4.2. Experimental Analysis of Semi-Physical Simulation Data

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Nandi, A.; Azzouz, E. Algorithms for automatic modulation recognition of communication signals. IEEE Trans. Commun.
**1998**, 46, 431–436. [Google Scholar] [CrossRef] - Hsue, S.Z.; Soliman, S.S. Automatic modulation classification using zero crossing. IEEE Proc.
**1990**, 137, 459–464. [Google Scholar] [CrossRef] - Hameed, F.; Dobre, O.A.; Popescu, D. On the likelihood-based approach to modulation classification. IEEE Trans. Wirel. Commun.
**2009**, 8, 5884–5892. [Google Scholar] [CrossRef] - Reichert, J. Automatic classification of communication signals using higher order statistics. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Philadelphia, PA, USA, 23–26 March 1992; pp. 221–224. [Google Scholar]
- Pennec, X.; Fillard, P.; Ayache, N. A Riemannian Framework for Tensor Computing. Int. J. Comput. Vis.
**2006**, 66, 41–66. [Google Scholar] [CrossRef] [Green Version] - Huang, Z.; Wang, R.; Li, X.; Liu, W.; Shan, S.; Van Gool, L.; Chen, X. Geometry-Aware Similarity Learning on SPD Manifolds for Visual Recognition. IEEE Trans. Circuits Syst. Video Technol.
**2018**, 28, 2513–2523. [Google Scholar] [CrossRef] [Green Version] - Harandi, M.; Salzmann, M.; Hartley, R. Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods. IEEE Trans. Pattern Anal. Mach. Intell.
**2018**, 40, 48–62. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ionescu, C.; Vantzos, O.; Sminchisescu, C. Matrix backpropagation for deep networks with structured layers. In Proceedings of the IEEE International Conference on Computer Vision, Santiago, Chile, 13–16 December 2015; pp. 2965–2973. [Google Scholar]
- Herath, S.; Harandi, M.; Porikli, F. Learning an invariant hilbert space for domain adaptation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Honolulu, HI, USA, 21–26 July 2017; pp. 3845–3854. [Google Scholar]
- Zhang, J.; Wang, L.; Zhou, L.; Li, W. Exploiting structure sparsity for covariance-based visual representation. arXiv
**2016**, arXiv:1610.08619. [Google Scholar] - Zhou, L.; Wang, L.; Zhang, J.; Shi, Y.; Gao, Y. Revisiting metric learning for SPD matrix based visual representation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Honolulu, HI, USA, 21–26 July 2017; pp. 3241–3249. [Google Scholar]
- Brooks, D.A.; Schwander, O.; Barbaresco, F.; Schneider, J.Y.; Cord, M. Exploring complex time-series representations for Riemannian machine learning of radar data. In Proceedings of the 2019 IEEE International Conference on Acoustics, Speech and Signal Processing, Brighton, UK, 12–17 May 2019; pp. 3672–3676. [Google Scholar]
- Hua, X.; Cheng, Y.; Wang, H.; Qin, Y. Robust Covariance Estimators Based on Information Divergences and Riemannian Manifold. Entropy
**2018**, 20, 219. [Google Scholar] [CrossRef] [Green Version] - Hua, X.; Fan, H.; Cheng, Y.; Wang, H.; Qin, Y. Information Geometry for Radar Target Detection with Total Jensen–Bregman Divergence. Entropy
**2018**, 20, 256. [Google Scholar] [CrossRef] [Green Version] - Wong, K.; Zhang, J.; Jiang, H. Multi-sensor signal processing on a PSD matrix manifold. In Proceedings of the 2016 IEEE Sensor Array and Multichannel Signal Processing Workshop, Rio de Janeiro, Brazil, 10–13 July 2016; pp. 1–5. [Google Scholar]
- Hua, X.; Shi, Y.; Zeng, Y.; Chen, C.; Lu, W.; Cheng, Y.; Wang, H. A divergence mean-based geometric detector with a pre-processing procedure. Measurement
**2019**, 131, 640–646. [Google Scholar] [CrossRef] - Hua, X.; Cheng, Y.; Wang, H.; Qin, Y.; Li, Y. Geometric means and medians with applications to target detection. IET Signal Process.
**2017**, 11, 711–720. [Google Scholar] [CrossRef] - Fathy, M.E.; Alavi, A.; Chellappa, R. Discriminative Log-Euclidean Feature Learning for Sparse Representation-Based Recognition of Faces from Videos. In Proceedings of the International Joint Conferences on Artificial Intelligence, IJCAI, New York, NY, USA, 9–15 July 2016; pp. 3359–3367. [Google Scholar]
- Zhang, S.; Kasiviswanathan, S.; Yuen, P.C.; Harandi, M. Online dictionary learning on symmetric positive definite manifolds with vision applications. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, AAAI, Austin, TX, USA, 25–30 January 2015. [Google Scholar]
- Huang, Z.; Van Gool, L. A Riemannian Network for SPD Matrix Learning. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, AAAI, San Francisco, CA, USA, 4–9 February 2017. [Google Scholar]
- Dong, Z.; Jia, S.; Zhang, C.; Pei, M.; Wu, Y. Deep manifold learning of symmetric positive definite matrices with application to face recognition. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, AAAI, San Francisco, CA, USA, 4–9 February 2017. [Google Scholar]
- Zhang, T.; Zheng, W.; Cui, Z.; Zong, Y.; Li, C.; Zhou, X.; Yang, J. Deep Manifold-to-Manifold Transforming Network for Skeleton-based Action Recognition. IEEE Trans. Multimed.
**2020**, 1. [Google Scholar] [CrossRef] - Gao, Z.; Wu, Y.; Bu, X.; Yu, T.; Yuan, J.; Jia, Y. Learning a robust representation via a deep network on symmetric positive definite manifolds. Pattern Recognit.
**2019**, 92, 1–12. [Google Scholar] [CrossRef] [Green Version] - Bronstein, M.M.; Bruna, J.; LeCun, Y.; Szlam, A.; VanderGheynst, P. Geometric Deep Learning: Going beyond Euclidean data. IEEE Signal Process. Mag.
**2017**, 34, 18–42. [Google Scholar] [CrossRef] [Green Version] - Krizhevsky, A.; Sutskever, I.; Hinton, G.E. Pdf ImageNet classification with deep convolutional neural networks. Commun. ACM
**2017**, 60, 84–90. [Google Scholar] [CrossRef] - Szegedy, C.; Liu, W.; Jia, Y.; Sermanet, P.; Reed, S.; Anguelov, D.; Erhan, D.; Vanhoucke, V.; Rabinovich, A. Going deeper with convolutions. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, IEEE, Boston, MA, USA, 8–12 June 2015; pp. 1–9. [Google Scholar]
- He, K.; Zhang, X.; Ren, S.; Sun, J. Deep residual learning for image recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Las Vegas, NV, USA, 26 June–1 July 2016; pp. 770–778. [Google Scholar]
- Wang, R.; Guo, H.; Davis, L.S.; Dai, Q. Covariance discriminative learning: A natural and efficient approach to image set classification. In Proceedings of the 2012 IEEE Conference on Computer Vision and Pattern Recognition, Providence, RI, USA, 18–20 June 2012; pp. 2496–2503. [Google Scholar]
- The McMaster IPIX Radar Sea Clutter Database. Available online: http://soma.ece.mcmaster.ca/ipix/ (accessed on 20 April 2020).

**Figure 1.**Measurement comparison of the difference between A and B on a manifold of the SPD (symmetric positive definite) matrix.

**Figure 5.**Colour maps of the time-frequency spectrum used in this study. (

**a**) The time-frequency spectrum colour map of clutter; (

**b**) The colour map of the time-frequency spectrum of target signal interfered by clutter. The SCR in it is −13 dB. And the trace of the target signal can be found in the lower part of the figure, but it is blurred by clutter.

**Figure 6.**Detection probability curves of different models. The false alarm probability of all the models is controlled to be less than ${10}^{-4}$. “SPD-threshold” represents the spectral transformation SPD matrix network, and “SPD-cov” represents the spectral covariance SPD matrix network.

**Figure 7.**Detection probability curves under different learning rates. (

**a**) reveals the general trends of three curves; (

**b**) reveals the differences in the three curves when the detection probability is approximately 80%.

**Figure 8.**Detection probability curves under different weight decays. (

**a**) reveals the general trends of three curves; (

**b**) reveals the differences in the three curves when the detection probability is approximately 80%.

**Figure 9.**Detection probability curves of the spectral-based SPD matrix signal detection method under different network layers.

**Figure 10.**Time-frequency spectrum colour map based on IPIX sea clutter. Different from the clutter based on K-distribution simulation, the distribution of IPIX sea clutter in time and frequency is obviously uneven. (

**a**) The time-frequency spectrum colour map of clutter; (

**b**) The colour map of the time-frequency spectrum of target signal interfered by clutter. The SCR in it is −13 dB. And the trace of the target signal can be found in the lower part of the figure, but it is blurred by clutter.

**Figure 11.**Detection probability curves of different models. The false alarm probability of all the models is controlled to be less than ${10}^{-4}$. “SPD-threshold” represents the spectral transformation SPD matrix network.

Model | Spectral Transformation SPD Matrix Network | Spectral Covariance SPD Matrix Network | GoogLeNet with Time-Frequency Spectra | ResNet50 with Time-Frequency Spectra | |
---|---|---|---|---|---|

SCR(dB) | |||||

−5 | Less than 0.01 | Less than 0.01 | Less than 0.01 | Less than 0.01 | |

−10 | Less than 0.01 | Less than 0.01 | Less than 0.01 | Less than 0.01 | |

−15 | Less than 0.01 | 0.23 | 1.12 | 0.50 | |

−20 | 2.47 | 2.22 | 32.97 | 25.95 | |

−25 | 18.80 | 16.65 | 99.96 | 39.01 | |

−30 | 29.80 | 14.00 | 99.98 | 49.15 |

Model | Spectral Transformation SPD Matrix Network | Spectral Covariance SPD Matrix Network | GoogLeNet with Time-Frequency Spectra | ResNet50 with Time-Frequency Spectra |
---|---|---|---|---|

Total Training Time(Min) | 67.6 | 69.5 | 2068.0 | 7083.3 |

Total Number of Epochs | 500 | 500 | 2000 | 2000 |

The Average Time per 100 Epochs(Min) | 13.5 | 13.9 | 103.4 | 354.2 |

Model | Learning Rate 0.01 | Learning Rate 0.001 | Learning Rate 0.0001 | |
---|---|---|---|---|

SCR(dB) | ||||

−5 | Less than 0.01 | Less than 0.01 | Less than 0.01 | |

−10 | Less than 0.01 | Less than 0.01 | Less than 0.01 | |

−15 | Less than 0.01 | Less than 0.01 | Less than 0.01 | |

−20 | 8.02 | 2.50 | 12.42 | |

−25 | 15.51 | 19.44 | 33.00 | |

−30 | 26.38 | 29.98 | 37.00 |

Model | Weight Decay 0.005 | Weight Decay 0.0005 | Weight Decay 0.00005 | |
---|---|---|---|---|

SCR(dB) | ||||

−5 | Less than 0.01 | Less than 0.01 | Less than 0.01 | |

−10 | Less than 0.01 | Less than 0.01 | Less than 0.01 | |

−15 | Less than 0.01 | Less than 0.01 | Less than 0.01 | |

−20 | 9.94 | 2.50 | 9.93 | |

−25 | 23.36 | 19.40 | 23.5 | |

−30 | 27.28 | 29.98 | 27.28 |

Model | 6 Layers | 8 Layers | 10 Layers | |
---|---|---|---|---|

SCR(dB) | ||||

−5 | Less than 0.01 | Less than 0.01 | Less than 0.01 | |

−10 | Less than 0.01 | Less than 0.01 | Less than 0.01 | |

−15 | 0.08 | Less than 0.01 | Less than 0.01 | |

−20 | 4.82 | 2.51 | 6.13 | |

−25 | 26.33 | 19.44 | 24.08 | |

−30 | 28.10 | 30.10 | 28.42 |

Number | Name |
---|---|

1 | 19980223_171533_ANTSTEP |

2 | 19980223_171811_ANTSTEP |

3 | 19980223_172059_ANTSTEP |

4 | 19980223_172410_ANTSTEP |

5 | 19980223_172650_ANTSTEP |

6 | 19980223_184853_ANTSTEP |

7 | 19980223_185157_ANTSTEP |

Pulse Repetition Frequency | Carrier Frequency | The Length of the Pulse | Range Resolution | Polarization Mode |
---|---|---|---|---|

1000 Hz | 9.39 GHz | 60,000 | 3 m | HH |

Model | Spectral Transformation SPD Matrix Network | Spectral Covariance SPD Matrix Network | GoogLeNet with Time-Frequency Spectra | ResNet50 with Time-Frequency Spectra | |
---|---|---|---|---|---|

SCR(dB) | |||||

−5 | Less than 0.01 | 0.25 | Less than 0.01 | Less than 0.01 | |

−10 | Less than 0.01 | 0.42 | Less than 0.01 | Less than 0.01 | |

−15 | 0.08 | 2.17 | 0.15 | Less than 0.01 | |

−20 | 6.90 | 13.92 | 14.49 | 10.52 |

© 2020 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**

Wang, J.; Hua, X.; Zeng, X.
Spectral-Based SPD Matrix Representation for Signal Detection Using a Deep Neutral Network. *Entropy* **2020**, *22*, 585.
https://doi.org/10.3390/e22050585

**AMA Style**

Wang J, Hua X, Zeng X.
Spectral-Based SPD Matrix Representation for Signal Detection Using a Deep Neutral Network. *Entropy*. 2020; 22(5):585.
https://doi.org/10.3390/e22050585

**Chicago/Turabian Style**

Wang, Jiangyi, Xiaoqiang Hua, and Xinwu Zeng.
2020. "Spectral-Based SPD Matrix Representation for Signal Detection Using a Deep Neutral Network" *Entropy* 22, no. 5: 585.
https://doi.org/10.3390/e22050585