# Radar Imaging of Non-Uniformly Rotating Targets via a Novel Approach for Multi-Component AM-FM Signal Parameter Estimation

## Abstract

**:**

## 1. Introduction

## 2. Signal Model

_{p}, y

_{p}, z

_{p}) from the rotating center O to the position of point P.

## 3. Modified Version of Chirplet Decomposition Based on IHAF

#### 3.1. Principle of Modified Version of Chirplet Decomposition

**Figure 2.**Comparison between Chirplet atom and modified version of Chirplet atom. (

**a**) Time series for the Chirplet atom; (

**b**) WVD for the Chirplet atom; (

**c**) Time series for the modified version of Chirplet atom; (

**d**) WVD for the modified version of Chirplet atom.

#### 3.2. Modified Version of Chirplet Decomposition Based on IHAF

_{k}for the modified version of Chirplet atom can be readily obtained as:

_{k}for the signal component with maximum energy should be estimated as follows:

_{k}is estimated, the other parameters can be estimated by the Dechirp technique and Fourier transform. The parameters for other modified version of Chirplet components can be estimated combined with the CLEAN technique [28].

#### 3.3. Numerical Example

Components (k) | D_{k} | σ_{k} | t_{k} | ω_{k} | β_{k} | γ_{k} |
---|---|---|---|---|---|---|

1 | 4 | 40 | 18 | 0.4 | 5×10^{−3} | 1×10^{−5} |

2 | 4 | 60 | 50 | 0.8 | −5×10^{−3} | −2×10^{−5} |

**Figure 3.**Results of the numerical example. (

**a**) Simulated signal; (

**b**) HAF for the signal; (

**c**) IHAF with lags ${m}_{1}\in [1:10]$ and ${m}_{2}\in [11:20]$ ; (

**d**) IHAF with lags ${m}_{1}\in [1:20]$ and ${m}_{2}\in [21:40]$ .

## 4. Radar Imaging Based on Modified Version of Chirplet Decomposition

## 5. Radar Imaging Results

#### 5.1. Simulated Data

_{0}= 5.52 GHz, the bandwidth is B = 400 Hz, the pulse width is 25.6 µs. After motion compensation, it is assumed that the target is rotating with equal changing acceleration, and the rotating parameters are as follows: the initial velocity is 0.021 rad/s, the acceleration is 0.015 rad/s

^{2}, and the acceleration rate is 1.6 rad/s

^{3}. Figure 5 shows the simulated target model, and it consists of 193 scatterers.

**Figure 7.**Time frequency representations for the received signal in a range bin. (

**a**) WVD for the original signal; (

**b**) WVD for two LFM signal components; (

**c**) WVD for two Chirplet components; (

**d**) WVD for two modified version of Chirplet components.

**Figure 8.**Radar images based on LFM signal model. (

**a**) Radar image at time t = 0.17 s; (

**b**) Radar image at time t = 0.22 s.

**Figure 9.**Radar image based on Chirplet decomposition algorithm. (

**a**) Radar image at time t = 0.17 s; (

**b**) Radar image at time t = 0.22 s.

**Figure 10.**Radar image based on modified version of Chirplet decomposition algorithm proposed in [31]. (

**a**) Radar image at time t = 0.17 s; (

**b**) Radar image at time t = 0.22 s.

**Figure 11.**Radar image based on modified version of Chirplet decomposition algorithm proposed in this paper. (

**a**) Radar image at time t = 0.17 s; (

**b**) Radar image at time t = 0.22 s.

#### 5.2. Real Data

**Figure 14.**Time frequency representations for the received signal in a range bin. (

**a**) WVD for the original signal; (

**b**) WVD for the LFM signal component; (

**c**) WVD for the Chirplet component; (

**d**) WVD for the modified version of Chirplet component.

**Figure 15.**Radar images based on LFM signal model. (

**a**) Radar image at time t = 1.01 s; (

**b**) Radar image at time t = 1.23 s.

**Figure 16.**Radar images based on Chirplet decomposition algorithm. (

**a**) Radar image at time t = 1.01 s; (

**b**) Radar image at time t = 1.23 s.

**Figure 17.**Radar image based on modified version of Chirplet decomposition algorithm proposed in [31]. (

**a**) Radar image at time t = 1.01 s; (

**b**) Radar image at time t = 1.23 s.

**Figure 18.**Radar images based on modified version of Chirplet decomposition algorithm proposed in this paper. (

**a**) Radar image at time t = 1.01 s; (

**b**) Radar image at time t = 1.23 s.

## 6. Conclusions

## Acknowledgements

## Nomenclature

O | Rotating center of the target |

r | Unit vector of the RLOS |

× | Outer product |

● | Inner product |

P | Random scatterer on the target |

λ | Wavelength |

Ω | Synthetic vector for the angular velocity of the rotating target |

K | Polynomial phase order |

α_{0} | Constant term |

R_{0} | Initial distance from radar to the target center |

t_{0} | Initial time |

Q | Number of scatterers in a range cell |

t_{k} | Time center of the modified version of Chirplet atom |

ω_{k} | Frequency center of the modified version of Chirplet atom |

β_{k} | Chirp rate of the modified version of Chirplet atom |

γ_{k} | Curvature of the modified version of Chirplet atom |

σ_{k} | Width for the modified version of Chirplet atom |

C_{k} | Weighted coefficient |

(⋅)^{∗} | Conjugate |

m_{1} | Time lags |

m_{2} | Time lags |

## Conflicts of Interest

## References

- Berizzi, F.; Mese, E.D.; Diani, M.; Martorella, M. High-resolution ISAR imaging of maneuvering targets by means of the range instantaneous Doppler technique: Modeling and performance analysis. IEEE Trans. Image Process.
**2001**, 10, 1880–1890. [Google Scholar] [CrossRef] [PubMed] - Zheng, J.B.; Su, T.; Zhu, W.T.; Liu, Q.H.; Zhang, L.; Zhu, T.W. Fast parameter estimation algorithm for cubic phase signal based on quantifying effects of Doppler frequency shift. Prog. Electromagn. Res.
**2013**, 142, 57–74. [Google Scholar] [CrossRef] - Bao, Z.; Wang, G.Y.; Luo, L. Inverse synthetic aperture radar imaging of maneuvering targets. Opt. Eng.
**1998**, 37, 1582–1588. [Google Scholar] [CrossRef] - Li, Z.; Narayanan, R.M. Manoeuvring target motion parameter estimation for ISAR image fusion. IET Signal Process.
**2008**, 2, 325–334. [Google Scholar] [CrossRef] - Chen, V.C.; Miceli, W.J. Time-varying spectral analysis for radar imaging of maneuvering targets. IEE Proc. Radar Sonar Navig.
**1998**, 145, 262–268. [Google Scholar] [CrossRef] - Li, G.; Zhang, H.; Wang, X.Q.; Xia, X.G. ISAR 2-D imaging of uniformly rotating targets via matching pursuit. IEEE Trans. AES
**2012**, 48, 1838–1846. [Google Scholar] - Walker, J.L. Range-Doppler imaging of rotating objects. IEEE Trans. AES
**1980**, 16, 23–52. [Google Scholar] - Sanchez, M.P.; Guasp, M.R.; Antonino-Daviu, J.A.; Folch, J.R.; Cruz, J.P.; Panadero, R.P. Instantaneous frequency of the left sideband harmonic during the start-up transient: A new method for diagnosis of broken bars. IEEE Trans. Ind. Electron.
**2009**, 56, 4557–4570. [Google Scholar] [CrossRef] - Santos, F.V.; Guasp, M.R.; Henao, H.; Sanchez, M.P.; Panadero, R.P. Diagnosis of rotor and stator asymmetries in wound-rotor induction machines under nonstationary operation through the instantaneous frequency. IEEE Trans. Ind. Electron.
**2014**, 61, 4947–4959. [Google Scholar] [CrossRef] - Cohen, L. Time-Frequency distributions——A review. IEEE Proc.
**1989**, 77, 941–981. [Google Scholar] [CrossRef] - Wang, Y.; Jiang, Y.C. ISAR imaging of maneuvering target based on the L-Class of fourth order complex-Lag PWVD. IEEE Trans. Geosci. Remote Sens.
**2010**, 48, 1518–1527. [Google Scholar] [CrossRef] - Stankovic, L. L-class of time-frequency distributions. IEEE Signal Process. Lett.
**1996**, 3, 22–25. [Google Scholar] [CrossRef] - Stankovic, L.; Thayaparan, T.; Dakovic, M. Signal decomposition by using the S-Method with application to the analysis of HF radar signals in sea-clutter. IEEE Trans. Signal Process.
**2006**, 54, 4332–4342. [Google Scholar] [CrossRef] - Wang, Y.; Jiang, Y.C. Inverse synthetic aperture radar imaging of three-dimensional rotation target based on two-order match Fourier transform. IET Signal Process.
**2012**, 6, 159–169. [Google Scholar] [CrossRef] - Bai, X.; Tao, R.; Wang, Z.J.; Wang, Y. ISAR imaging of a ship target based on parameter estimation of multicomponent quadratic frequency-modulated signals. IEEE Trans. Geosci. Remote Sens.
**2014**, 52, 1418–1429. [Google Scholar] [CrossRef] - Zheng, J.B.; Su, T.; Zhu, W.T.; Liu, Q.H. ISAR imaging of targets with complex motions based on the keystone time-chirp rate distribution. IEEE Geosci. Remote Sens. Lett.
**2014**, 11, 1275–1279. [Google Scholar] [CrossRef] - Zheng, J.B.; Su, T.; Zhang, L.; Zhu, W.T.; Liu, Q.H. ISAR imaging of targets with complex motion based on the chirp rate-quadratic chirp rate distribution. IEEE Trans. Geosci. Remote Sens.
**2014**, 52, 7276–7289. [Google Scholar] [CrossRef] - Wang, Y. Inverse synthetic aperture radar imaging of manoeuvring target based on range-instantaneous-Doppler and range-instantaneous-chirp-rate algorithms. IET Radar Sonar Navig.
**2012**, 6, 921–928. [Google Scholar] [CrossRef] - Bultan, A. A four-parameter atomic decomposition of chirplets. IEEE Trans. Signal Process.
**1999**, 47, 731–745. [Google Scholar] [CrossRef] - Yin, Q.Y.; Qian, S.; Feng, A.G. A fast refinement for adaptive Gaussian chirplet decomposition. IEEE Trans. Signal Process.
**2002**, 50, 1298–1306. [Google Scholar] [CrossRef] - Greenberg, J.M.; Wang, Z.S.; Li, J. New approaches for chirplet approximation. IEEE Trans. Signal Process.
**2007**, 55, 734–741. [Google Scholar] [CrossRef] - Wang, Y.; Jiang, Y.C. ISAR imaging for three-dimensional rotation targets based on adaptive Chirplet decomposition. Multidimens. Syst. Signal Process.
**2010**, 21, 59–71. [Google Scholar] [CrossRef] - Angrisani, L.; D’Arco, M. A measurement method based on a modified version of the Chirplet transform for instantaneous frequency estimation. IEEE Trans. Instrument. Meas.
**2002**, 51, 704–711. [Google Scholar] [CrossRef] - Yang, Y.; Zhang, W.M.; Peng, Z.K.; Meng, G. Multicomponent signal analysis based on polynomial Chirplet transform. IEEE Trans. Ind. Electron.
**2013**, 60, 3948–3956. [Google Scholar] [CrossRef] - Wang, Y.; Jiang, Y.C. Modified adaptive Chirplet decomposition with application in ISAR imaging of maneuvering targets. EURASIP J. Adv. Signal Process. 2008. [CrossRef]
- Yang, Y.; Peng, Z.K.; Meng, G.; Zhang, W.M. Spline-kernelled Chirplet transform for the analysis of signals with time-varying frequency and its application. IEEE Trans. Ind. Electron.
**2012**, 59, 1612–1621. [Google Scholar] [CrossRef] - Barbarossa, S.; Petrone, V. Analysis of polynomial-phase signals by the integrated generalized ambiguity function. IEEE Trans. Signal Process.
**1997**, 45, 316–327. [Google Scholar] [CrossRef] - Gough, P.T. A fast spectral estimation algorithm based on the FFT. IEEE Trans. Signal Process.
**1994**, 42, 1317–1322. [Google Scholar] [CrossRef] - Wang, P.; Li, H.B.; Djurovic, I.; Himed, B. Integrated cubic phase function for linear FM signal analysis. IEEE Trans. Aerosp. Electron. Syst.
**2010**, 46, 963–977. [Google Scholar] [CrossRef] - Wang, Y.; Jiang, Y.C. Approach for high resolution inverse synthetic aperture radar imaging of ship target with complex motion. IET Signal Process.
**2013**, 7, 146–157. [Google Scholar] [CrossRef] - Wang, Y.; Zhao, B.; Jiang, Y.C. Inverse synthetic aperture radar imaging of targets with complex motion based on cubic Chirplet decomposition. IET Signal Process.
**2015**. accepted. [Google Scholar] - Wang, J.F.; Liu, X.Z. Improved global range alignment for ISAR. IEEE Trans. Aerosp. Electron. Syst.
**2007**, 43, 1070–1075. [Google Scholar] [CrossRef]

© 2015 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 license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, Y.
Radar Imaging of Non-Uniformly Rotating Targets via a Novel Approach for Multi-Component AM-FM Signal Parameter Estimation. *Sensors* **2015**, *15*, 6905-6923.
https://doi.org/10.3390/s150306905

**AMA Style**

Wang Y.
Radar Imaging of Non-Uniformly Rotating Targets via a Novel Approach for Multi-Component AM-FM Signal Parameter Estimation. *Sensors*. 2015; 15(3):6905-6923.
https://doi.org/10.3390/s150306905

**Chicago/Turabian Style**

Wang, Yong.
2015. "Radar Imaging of Non-Uniformly Rotating Targets via a Novel Approach for Multi-Component AM-FM Signal Parameter Estimation" *Sensors* 15, no. 3: 6905-6923.
https://doi.org/10.3390/s150306905