# Radar Detection of Fluctuating Targets under Heavy-Tailed Clutter Using Track-Before-Detect

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- In order to limit complexity while still retaining the benefits of DP–TBD, we resort to a two-stage detection process with different resolution cells.
- For typical non-Gaussian distributed clutter (K-distribution) and a typical target amplitude fluctuation model (Swerling 1), the DP–TBD algorithm based on prior information is proposed. By using the likelihood ratio merit function in DP integration, the performance loss produced by the “heavy-tailed” clutter measurements can be reduced.
- An efficient but accurate approximation method is proposed to reduce the complexity of evaluating the merit function.

## 2. Models and Notations

#### 2.1. Kinematic Model

#### 2.2. Measurement-Based Model

#### 2.3. K-Distributed Clutter Model

## 3. Development of the Proposed Strategies

#### 3.1. Two-Stage Detection Approach

#### 3.2. Derivation and Implementation of the Merit Function

## 4. Simulation

#### 4.1. Performance Analysis

#### 4.2. Computational Complexity Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Barniv, Y.; Kella, O. Dynamic programming solution for detecting dim moving targets. IEEE Trans. Aerosp. Electron. Syst.
**1985**, 21, 144–156. [Google Scholar] [CrossRef] - Barniv, Y.; Kella, O. Dynamic programming solution for detecting dim moving targets part II: Analysis. IEEE Trans. Aerosp. Electron. Syst.
**1987**, AES-23, 776–788. [Google Scholar] [CrossRef] - Buzzi, S.; Lops, M.; Venturino, L.; Ferri, M. Detection of an unknown number of targets via track-before-detect procedures. In Proceedings of the 2007 IEEE Radar Conference, Boston, MA, USA, 17–20 April 2007; pp. 180–185. [Google Scholar]
- Buzzi, S.; Lops, M.; Venturino, L.; Ferri, M. Track-before-detect procedures in a multi-target environment. IEEE Trans. Aerosp. Electron. Syst.
**2008**, 44, 1135–1150. [Google Scholar] [CrossRef] - Yi, W.; Morelande, M.R.; Kong, L.; Yang, J. An efficient multi-frame track-before-detect algorithm for multi-target tracking. IEEE J. Sel. Top. Signal Process.
**2013**, 7, 421–434. [Google Scholar] [CrossRef] - Grossi, E.; Lops, M.; Venturino, L. A track-before-detect procedure for sparse data. In Proceedings of the 2012 IEEE Statistical Signal Processing Workshop (SSP), Ann Arbor, MI, USA, 5–8 August 2012; pp. 772–775. [Google Scholar]
- Grossi, E.; Lops, M.; Venturino, L. A novel track-before-detect procedure for multi-frame detection in radar systems. In Proceedings of the 2013 IEEE Radar Conference (RadarCon13), Ottawa, ON, Canada, 29 April–3 May 2013; pp. 1–6. [Google Scholar]
- Jiang, H.; Yi, W.; Cui, G.; Kong, L.; Yang, X. Track-before-detect strategy for HRR radars. In Proceedings of the 2015 IEEE International Radar Conference, RadarCon 2015, Arlington, VA, USA, 10–15 May 2015; Institute of Electrical and Electronics Engineers Inc.: Arlington, VA, USA, 2015; pp. 362–367. [Google Scholar]
- Jiang, H.; Yi, W.; Kong, L.; Yang, X.; He, B. Radar detection of Swerling 3 target in G0-distributed clutter via track-before-detect. In Proceedings of the 2016 IEEE Radar Conference, RadarConf 2016, Philadelphia, PA, USA, 2–6 May 2016; Institute of Electrical and Electronics Engineers Inc.: Philadelphia, PA, USA, 2016. [Google Scholar]
- Ciuonzo, D.; Maio, A.D.; Orlando, D. On the statistical invariance for adaptive radar detection in partially homogeneous disturbance plus structured interference. IEEE Trans. Signal Process.
**2016**, 65, 1222–1234. [Google Scholar] [CrossRef] - Ciuonzo, D.; Orlando, D.; Pallotta, L. On the maximal invariant statistic for adaptive radar detection in partially homogeneous disturbance with persymmetric covariance. IEEE Signal Process. Lett.
**2016**, 23, 1830–1834. [Google Scholar] [CrossRef] - Kraut, S.; Scharf, L.L. The CFAR adaptive subspace detector is a scale-invariant GLRT. IEEE Trans. Signal Process.
**1999**, 47, 2538–2541. [Google Scholar] [CrossRef] [Green Version] - Buzzi, S.; Lops, M.; Venturino, L. Track-before-detect procedures for early detection of moving target from airborne radars. IEEE Trans. Aerosp. Electron. Syst.
**2005**, 41, 937–954. [Google Scholar] [CrossRef] - Ebenezer, S.P.; Papandreou-Suppappola, A. Generalized recursive track-before-detect with proposal partitioning for tracking varying number of multiple targets in low SNR. IEEE Trans. Signal Process.
**2016**, 64, 2819–2834. [Google Scholar] [CrossRef] - Zheng, D.; Wang, S.; Qin, X. A dynamic programming track-before-detect algorithm based on local linearization for non-Gaussian clutter background. Chin. J. Electron.
**2016**, 25, 583–590. [Google Scholar] [CrossRef] - Zhou, J.; Chen, D.; Sun, D. K distribution sea clutter modeling and simulation based on ZMNL. In Proceedings of the International Conference on Intelligent Computation Technology and Automation, Nanchang, China, 19 May 2016; pp. 506–509. [Google Scholar]
- Aprile, A.; Grossi, E.; Lops, M.; Venturino, L. Track-before-detect for sea clutter rejection: Tests with real data. IEEE Trans. Aerosp. Electron. Syst.
**2016**, 52, 1035–1045. [Google Scholar] [CrossRef] - Berry, P.; Venkataraman, K.; Rosenberg, L. Adaptive detection of low-observable targets in correlated sea clutter using Bayesian track-before-detect. In Proceedings of the 2017 IEEE Radar Conference (RadarConf), Seattle, WA, USA, 8 June 2017; pp. 0398–0403. [Google Scholar]
- Swerling, P. Probability of detection for fluctuating targets. Inf. Theory IRE Trans.
**1960**, 6, 269–308. [Google Scholar] [CrossRef] - Yi, W.; Jiang, H.; Kirubarajan, T.; Kong, L.; Yang, X. Track-before-detect strategies for radar detection in G0-distributed clutter. IEEE Trans. Aerosp. Electron. Syst.
**2017**, 53, 2516–2533. [Google Scholar] [CrossRef] - Brekke, E.; Hallingstad, O.; Glattetre, J. Tracking small targets in heavy-tailed clutter using amplitude information. IEEE J. Ocean. Eng.
**2010**, 35, 314–329. [Google Scholar] [CrossRef] - Abraham, D.A.; Lyons, A.P. Novel physical interpretations of K-distributed reverberation. Ocean. Eng. IEEE J.
**2002**, 27, 800–813. [Google Scholar] [CrossRef]

**Figure 2.**K-distribution (

**a**) probability density functions (PDFs) of K and Rayleigh distribution for various shape and scale parameters; (

**b**) K-distributed clutter including real part and imaginary part.

**Figure 5.**Histogram of generation data and theory PDF data with signal-to-clutter ratio (SCR) = 15 dB (

**a**) $\alpha =2\text{}\mathrm{and}\text{}\beta =1$; (

**b**) $\alpha =3\text{}\mathrm{and}\text{}\beta =2$; (

**c**) $\alpha =5\text{}\mathrm{and}\text{}\beta =2$; (

**d**) $\alpha =10\text{}\mathrm{and}\text{}\beta =5$.

**Figure 6.**Performance and root-mean square error (RMSE) comparison of different DP–TBD integration method with $\alpha =0.5\text{}\mathrm{and}\text{}\beta =1$ against signal-to-noise ratios (SNRs) from 2 dB to 20 dB. (

**a**) The detection probability ${P}_{d}$; (

**b**) the RMSE on estimated position.

**Figure 7.**Performance comparison of DP–TBD integration method (red solid line with diamond) and the proposed method in this paper (blue solid line with cross) for K-distributed clutter and a Swerling 1 target (

**a**) $\alpha =2\text{}\mathrm{and}\text{}\beta =2$; (

**b**) $\alpha =5\text{}\mathrm{and}\text{}\beta =2$; (

**c**) $\alpha =10\text{}\mathrm{and}\text{}\beta =2$; (

**d**) $\alpha =50\text{}\mathrm{and}\text{}\beta =2$.

**Figure 8.**Performance comparison of DP–TBD integration method. (

**a**) Performance with different number of frames N = 4, N = 6 and N = 8; (

**b**) performance with different number of state transitions q = 4 and q = 9.

Stage 1 | |
---|---|

Mearsurement: | get ${{z}_{n}}^{\prime}\left(i,j\right),1\le i\le {M}_{r}^{\prime},1\le j\le {M}_{\theta}^{\prime},for\text{}n=1,\dots N$ |

Integration: | $V{\left({s}_{n}\right)}^{\prime}=\sqrt{I{\left({s}_{n}\right)}^{\prime}}+\underset{{s}_{n-1}\in \tau \left({s}_{n-1}\right)}{\mathrm{max}}\left[V{\left({s}_{n-1}\right)}^{\prime}\right],for\text{}n=1,\dots N$ integration calculates under the condition of low grid resolution |

Determination: | $V{\left({s}_{N}\right)}^{\prime}>{\gamma}_{1}$ |

Stage 2 | |

Mearsurement: | get ${z}_{n}\left(i,j\right),1\le i\le {M}_{r},1\le j\le {M}_{\theta},for\text{}n=1,\dots N$ |

Integration: | $V\left({s}_{n}\right)=\sqrt{I\left({s}_{n}\right)}+\underset{{s}_{n-1}\in \tau \left({s}_{n-1}\right)}{\mathrm{max}}\left[V\left({s}_{n-1}\right)\right],for\text{}n=1,\dots N$ integration concentrates on the part of states which are indicated by stage 1 |

Determination: | $V\left({s}_{N}\right)>{\gamma}_{2}$ |

Backtracking: | ${\widehat{S}}_{N}=\left\{{\widehat{s}}_{1},\dots ,{\widehat{s}}_{N}\right\}=\mathrm{arg}\underset{{s}_{n-1}\in \tau \left({s}_{n}\right)}{\mathrm{max}}\left[V\left({s}_{n-1}\right)\right],for\text{}n=N,\dots 1$ |

Parameters | ${\mathit{M}}_{\mathit{r}}\times {\mathit{M}}_{\mathit{\theta}}=180\times 90$ | ${\mathit{M}}_{\mathit{r}}^{\prime}\times {\mathit{M}}_{\mathit{\theta}}^{\prime}=90\times 45$ | ${\mathit{M}}_{\mathit{r}}^{\prime}\times {\mathit{M}}_{\mathit{\theta}}^{\prime}=60\times 30$ |
---|---|---|---|

q = 4, N = 6 | 308 ms | 224 ms | 146 ms |

q = 9, N = 6 | 935 ms | 684 ms | 370 ms |

© 2018 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**

Gao, J.; Du, J.; Wang, W.
Radar Detection of Fluctuating Targets under Heavy-Tailed Clutter Using Track-Before-Detect. *Sensors* **2018**, *18*, 2241.
https://doi.org/10.3390/s18072241

**AMA Style**

Gao J, Du J, Wang W.
Radar Detection of Fluctuating Targets under Heavy-Tailed Clutter Using Track-Before-Detect. *Sensors*. 2018; 18(7):2241.
https://doi.org/10.3390/s18072241

**Chicago/Turabian Style**

Gao, Jie, Jinsong Du, and Wei Wang.
2018. "Radar Detection of Fluctuating Targets under Heavy-Tailed Clutter Using Track-Before-Detect" *Sensors* 18, no. 7: 2241.
https://doi.org/10.3390/s18072241