# Fast Factorized Backprojection Algorithm in Orthogonal Elliptical Coordinate System for Ocean Scenes Imaging Using Geosynchronous Spaceborne–Airborne VHF UWB Bistatic SAR

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## Abstract

**:**

## 1. Introduction

## 2. Imaging Geometry and Signal Model

^{7}m from the transmitter to the target ${P}_{0}\left({x}_{0},{y}_{0},0\right)$. On the other hand, the altitude of the UAV typically ranges from the order of 10

^{2}m to 10

^{4}m, resulting in the receiver being near the target. The transmitting slant range is therefore much larger than the receiving slant range, which is a unique geometric configuration for the GEO-UAV BiSAR imaging. In general, the transmitter transmits a linear frequency-modulated (LFM) signal $p\left(\tau \right)$, which is then received by the receiver after the reflection through the target P

_{0}. After the range compression, the received echo signal becomes

_{0}, ${p}_{r}\left(\cdot \right)$ denotes the range-compressed pulse envelop, B denotes the signal bandwidth, f

_{c}denotes the central frequency, and c

_{0}represents the speed of light.

## 3. Bistatic FFBP Algorithm in OEP Coordinate System

#### 3.1. Subaperture Imaging

#### 3.2. Sample Requirements

#### 3.3. Superiority of Subimages in OEP Coordinate System

#### 3.4. Implementation Process

#### 3.5. Computational Burden

## 4. Experimental Results and Performance Analysis

#### 4.1. Experimental Results of Point Targets

^{7}m in the imaging scene at the center of the synthetic aperture, with a slant range of approximately 3.8 × 10

^{7}. The GEO satellite operates in the spotlight mode, and it is assumed that the transmitter is always beam-synchronized with the receiver due to the large beam coverage of the GEO satellite.

^{2}, and the effects of the radar frequency interferometer (RFI), Gaussian white noise, and electromagnetic wave propagation loss are not considered. To objectively evaluate the performance of the imaging algorithms, no window function is used to suppress the side flaps of the imaging results of the point targets.

#### 4.2. Experimental Results of Natural Scene

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Subaperture imaging geometry of the GEO-UAV BiSAR in the orthogonal elliptical coordinate system. (

**a**) The kth subaperture and subimage grid; (

**b**) The kth OEP coordinate system.

**Figure 5.**Analysis of the angular dimension sampling requirement of the subimages in both EP and OEP systems for the same imaging scene.

**Figure 6.**Two-dimensional (2D) spectrum comparison of the imaging result of the single ideal point target with the different subaperture lengths n. (

**a**) Two-dimensional (2D) spectrum for n = 256 in the EP system; (

**b**) Two-dimensional (2D) spectrum for n = 512 in the EP system; (

**c**) Two-dimensional (2D) spectrum for n = 1024 in the EP system; (

**d**) Two-dimensional (2D) spectrum for n = 256 in the OEP system; (

**e**) Two-dimensional (2D) spectrum for n = 512 in the OEP system; (

**f**) Two-dimensional (2D) spectrum for n = 1024 in the OEP system.

**Figure 8.**Variation of the speed-up factor. (

**a**) With respect to the fusion time M; (

**b**) With respect to the fusion aperture n.

**Figure 9.**The experimental scene for the GEO-UAV UWB BiSAR imaging. (

**a**) The imaging geometry; (

**b**) The distribution of the point targets.

**Figure 10.**Imaging results of the point targets. (

**a**) Bistatic BP algorithm; (

**b**) Bistatic EP FFBP algorithm; (

**c**) Proposed bistatic FFBP algorithm.

**Figure 11.**The counter plots of the impulse response of the selected point targets. (

**a**) The target A focused by the bistatic BP algorithm; (

**b**) The target A focused by the bistatic EP FFBP algorithm; (

**c**) The target A focused by the proposed bistatic FFBP algorithm; (

**d**) The target B focused by the bistatic BP algorithm; (

**e**) The target B focused by the bistatic EP FFBP algorithm; (

**f**) The target B focused by the proposed bistatic FFBP algorithm; (

**g**) The target C focused by the bistatic BP algorithm; (

**h**) The target C focused by the bistatic EP FFBP algorithm; (

**i**) The target C focused by the proposed bistatic FFBP algorithm.

**Figure 12.**The profiles of the impulse response of the selected point targets. (

**a**) Azimuthal profile of the target A; (

**b**) Range profile of the target A; (

**c**) Azimuthal profile of the target B; (

**d**) Range profile of the target B; (

**e**) Azimuthal profile of the target C; (

**f**) Range profile of the target C.

**Figure 13.**The natural ocean scene and its echo signal generated by the TBT algorithm for the GEO-UAV VHF UWB BiSAR imaging. (

**a**) The natural ocean scene; (

**b**) Amplitude of the echo signal; (

**c**) Phase of the echo signal.

**Figure 14.**Reconstructed SAR image obtained by the different algorithms. (

**a**) The bistatic BP algorithm; (

**b**) The bistatic EP FFBP algorithm; (

**c**) The proposed bistatic FFBP algorithm.

**Figure 15.**The selected ship targets and their profiles of the imaging results obtained by the different algorithms. (

**a**) Ship target A; (

**b**) Ship target B; (

**c**) Ship target C; (

**d**) Azimuthal profile of the ship target A; (

**e**) Azimuthal profile of the ship target B; (

**f**) Azimuthal profile of the ship target C; (

**g**) Range profile of the ship target A; (

**h**) Range profile of the ship target B; (

**i**) Range profile of the ship target C.

**Table 1.**Two-dimensional (2D) spectrum angular width of the single ideal point target with the different subaperture lengths n.

Subaperture Length | Bistatic EP FFBP (Sampling Point) | Proposed Bistatic FFBP (Sampling Point) | Upgrade Factor |
---|---|---|---|

256 | 25 | 20 | 20% |

512 | 61 | 46 | 25% |

1024 | 96 | 80 | 17% |

Parameters | Values | Parameters | Values | |
---|---|---|---|---|

BiSAR System | Center frequency | 350 MHz | Signal bandwidth | 200 MHz |

Pulse duration | 1 µs | Pulse repetition frequency | 500 Hz | |

Sampling frequency | 220 MHz | Synthetic aperture time | 3.66 s | |

GEO Satellite | Orbital semi-major axis | 42,164 km | Orbital eccentricity | 0.005 |

Orbital inclination | 57° | Perigee argument | 90° | |

Initial coordinates | (1.5, −3.5, 0.25) × 10^{7} m | Normal velocity | 1424.3 m/s | |

UAV | Height | 500 m | Normal velocity | 300 m/s |

Initial coordinates | (0, 0, 500) m |

IRW (m) | PSLR (dB) | ISLR (dB) | |||||
---|---|---|---|---|---|---|---|

Azimuth | Range | Azimuth | Range | Azimuth | Range | ||

Target A | Bistatic BP | 0.96 | 0.68 | −15.31 | −15.49 | −12.05 | −13.35 |

Bistatic EP FFBP | 0.97 | 0.73 | −15.16 | −18.66 | −12.03 | −16.79 | |

Proposed bistatic FFBP | 0.98 | 0.74 | −15.29 | −18.65 | −11.90 | −16.68 | |

Target B | Bistatic BP | 0.95 | 0.69 | −15.37 | −15.48 | −12.01 | −13.32 |

Bistatic EP FFBP | 0.96 | 0.74 | −15.16 | −18.59 | −11.89 | −16.78 | |

Proposed bistatic FFBP | 0.97 | 0.74 | −14.23 | −18.58 | −11.61 | −16.68 | |

Target C | Bistatic BP | 0.95 | 0.69 | −15.40 | −15.51 | −11.85 | −13.34 |

Bistatic EP FFBP | 0.95 | 0.74 | −15.28 | −18.66 | −11.84 | −16.80 | |

Proposed bistatic FFBP | 0.96 | 0.73 | −15.89 | −18.72 | −12.02 | −16.71 |

**Table 4.**Average imaging time of the bistatic BP algorithm, bistatic EP FFBP algorithm and proposed bistatic FFBP algorithm for the imaging scene with the different sizes.

Imaging Scene Size (Azimuth × Range) | Bistatic BP | Bistatic EP FFBP | Proposed Bistatic FFBP | Speed-Up Factor (BP/EP FFBP) |
---|---|---|---|---|

(100 × 100) m | 12.69 s | 7.67 s | 6.82 s | 1.86/1.12 |

(300 × 300) m | 84.13 s | 19.63 s | 15.88 s | 5.30/1.24 |

(500 × 500) m | 219.53 s | 36.58 s | 29.11 s | 7.61/1.26 |

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## Share and Cite

**MDPI and ACS Style**

Hu, X.; Xie, H.; Zhang, L.; Hu, J.; He, J.; Yi, S.; Jiang, H.; Xie, K.
Fast Factorized Backprojection Algorithm in Orthogonal Elliptical Coordinate System for Ocean Scenes Imaging Using Geosynchronous Spaceborne–Airborne VHF UWB Bistatic SAR. *Remote Sens.* **2023**, *15*, 2215.
https://doi.org/10.3390/rs15082215

**AMA Style**

Hu X, Xie H, Zhang L, Hu J, He J, Yi S, Jiang H, Xie K.
Fast Factorized Backprojection Algorithm in Orthogonal Elliptical Coordinate System for Ocean Scenes Imaging Using Geosynchronous Spaceborne–Airborne VHF UWB Bistatic SAR. *Remote Sensing*. 2023; 15(8):2215.
https://doi.org/10.3390/rs15082215

**Chicago/Turabian Style**

Hu, Xiao, Hongtu Xie, Lin Zhang, Jun Hu, Jinfeng He, Shiliang Yi, Hejun Jiang, and Kai Xie.
2023. "Fast Factorized Backprojection Algorithm in Orthogonal Elliptical Coordinate System for Ocean Scenes Imaging Using Geosynchronous Spaceborne–Airborne VHF UWB Bistatic SAR" *Remote Sensing* 15, no. 8: 2215.
https://doi.org/10.3390/rs15082215