# Focusing High-Resolution Airborne SAR with Topography Variations Using an Extended BPA Based on a Time/Frequency Rotation Principle

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## Abstract

**:**

## 1. Introduction

## 2. Modeling and Analysis

#### 2.1. Modeling of Airborne SAR

#### 2.2. Problem

- (1)
- Three-dimensional (3-D) variations in time-domain. Clearly, the general range history in Equation (1) was a vector variable function, which means that one should consider the changes in range, azimuth, and height directions, when designing the focusing approach in the time-domain. An accurate equation indicates the good performance of the proposed focusing method, or it would lead to a great deterioration in the final imaging result, including the IRW, the peak side-lobe ratio (PSLR), and the integration side-lobe ratio (ISLR).
- (2)
- 3-D variations in the frequency-domain. Because of the complex composition terms in Equation (1), it is difficult to derive the 2-D spectrum, even when using the method of the series reversion (MSR). Moreover, with the complex range history, the focused approach designed in the 2-D frequency-domain would be more complicated than those in the time-domain. Thus, the process directly applied in frequency-domain was not the best choice for an efficient algorithm design for high-resolution airborne SAR in this work.

## 3. Imaging Algorithm

#### 3.1. Frequency-Domain Processing

#### 3.2. Time-Domain Processing

#### 3.3. Flowchart of the EBP Algorithm

- Azimuth Fourier transform (FT). Apply an azimuth FT on the raw data;
- Deramping processing; the first step of the frequency-domain processing. Modulate the raw data with Equation (4);
- Equivalent Azimuth IFT; the second step of the frequency-domain processing. Apply the transformation in the data from the $\eta $ and ${f}_{\eta}$ coordinate, to the ${\eta}^{\prime}$ and ${f}_{\eta}^{\prime}$ coordinate;
- Residual phase compensation; the final step of the frequency-domain processing. Compensate the data with Equation (7) in the new azimuth frequency-domain;
- Azimuth IFT. Transform the data from the ${f}_{\eta}^{\prime}$ domain to ${\eta}^{\prime}$ domain;
- Second phase compensation. Compensate the data with Equation (9) in the new azimuth time-domain;
- Range FT. Apply a range of FT on the renovated data. Then, the range compression will be implemented in the range frequency-domain;
- Range compression. Compress the data in the range frequency-domain with Equation (11);
- Range IFT. Transform the data from the ${t}_{r}$ domain to ${f}_{r}$ domain;
- Selection of samples. The important sampling points are selected appropriately, and the premise is that the echo signal will not alias the azimuth frequency-domain. In this way, the computational burden of the BP algorithm will be reduced;
- BP processing. Calculate the accurate slant range from each azimuth position to the targets, with the combination of the airborne acquisition scenarios and the DEM data. The other substeps in this step are similar to those of the classical BP processing.

## 4. Simulation Results

#### 4.1. Spot Matrix Simulation Results

#### Case I

#### Case II

#### 4.2. Scene Simulation Results

## 5. Discussion

#### 5.1. Parameter Selection

#### 5.2. Computational Burden

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Quadratic phase errors (QPEs) introduced by topography variations in different domains: (

**a**) QPEs introduced by topography variations in range/height domain; (

**b**) QPEs introduced by topography variations in azimuth/height domain; (

**c**) QPEs introduced by topography variations in the resolution/height domain; (

**d**) QPEs introduced by topography variations in the squint angle/height domain.

**Figure 3.**PEs introduced by topography variations in different domains: (

**a**) PEs introduced by topography variations in the range/height domain; (

**b**) PEs introduced by topography variations in the azimuth/height domain.

**Figure 4.**Steps of the frequency-domain processing using the time-frequency diagrams (TFDs). (

**a**) Echo signal and reference function ${H}_{1}$ in the range/Doppler domain; (

**b**) signal after deramping; (

**c**) signal after inverse Fourier transform (IFT) with new IFT kernel and residual phase compensation function ${H}_{2}$; (

**d**) signal after the residual phase compensation.

**Figure 6.**The flowchart of the EBP algorithm. In this flowchart, frequency-domain processing and time-domain processing are included.

**Figure 7.**Distribution of targets in the illuminated spot matrix which contained nine-point targets.

**Figure 8.**Comparative results by proposed algorithm and CSA. (Left to right) Targets PT1, PT5, and PT9, respectively. (

**a**) Contour plot of targets (PT1, PT5, and PT9) processed by the EBP algorithm; (

**b**) Contour plot of targets (PT1, PT5, and PT9) processed by CSA.

**Figure 9.**Motion errors and simulation results: (

**a**) motion errors extracted from the airborne inertial navigation system; (

**b**) focused result of the simulation spot matrix by the proposed algorithm.

**Figure 10.**Simulation results: (

**a**) comparative results of azimuth profiles of PT1 by PMA and the proposed algorithm; (

**b**) comparative results of azimuth profiles of PT5 by PMA and the proposed algorithm; (

**c**) comparative results of azimuth profiles of PT9 by PMA and the proposed algorithm.

**Figure 11.**Scene simulation data results processed by different algorithms. (

**a**) Scene simulation data results processed by traditional BPA. (

**b**) Scene simulation data results processed by the EBP algorithm. (

**c**) Scene simulation data results processed by the reference algorithm.

Parameters | Value |
---|---|

Carrier frequency | 9.65 GHz |

Pulse duration | 2.5 μs |

Pulse Bandwidth | 300 MHz |

Sampling frequency | 420 MHz |

Reference slant range | 25.0 km |

Altitude | 8.0 km |

Velocity vector | (0, 120, 0) m/s |

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

Proposed algorithm | PT1 | 0.687 | −13.22 | −10.03 |

PT5 | 0.681 | −13.27 | −10.05 | |

PT9 | 0.679 | −13.24 | −10.02 | |

CSA | PT1 | 0.795 | −7.23 | −6.65 |

PT5 | 0.682 | −13.25 | −10.06 | |

PT9 | 0.813 | −7.14 | −6.37 |

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

Proposed algorithm | PT1 | 0.474 | −13.27 | −10.07 |

PT5 | 0.472 | −13.31 | −10.04 | |

PT9 | 0.471 | −13.30 | −10.01 | |

PMA | PT1 | 0.613 | −4.93 | −4.64 |

PT5 | 0.423 | −13.32 | −10.06 | |

PT9 | 0.546 | −10.35 | −7.58 |

Data Size in Azimuth (×10^{3}) | 1 | 2 | 4 | 8 | 16 |
---|---|---|---|---|---|

BPA/CSA | 150.30 | 290.11 | 560.73 | 1084.92 | 2101.48 |

Proposed/CSA | 12.07 | 22.15 | 41.64 | 79.39 | 152.59 |

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## Share and Cite

**MDPI and ACS Style**

Lin, C.; Tang, S.; Zhang, L.; Guo, P.
Focusing High-Resolution Airborne SAR with Topography Variations Using an Extended BPA Based on a Time/Frequency Rotation Principle. *Remote Sens.* **2018**, *10*, 1275.
https://doi.org/10.3390/rs10081275

**AMA Style**

Lin C, Tang S, Zhang L, Guo P.
Focusing High-Resolution Airborne SAR with Topography Variations Using an Extended BPA Based on a Time/Frequency Rotation Principle. *Remote Sensing*. 2018; 10(8):1275.
https://doi.org/10.3390/rs10081275

**Chicago/Turabian Style**

Lin, Chunhui, Shiyang Tang, Linrang Zhang, and Ping Guo.
2018. "Focusing High-Resolution Airborne SAR with Topography Variations Using an Extended BPA Based on a Time/Frequency Rotation Principle" *Remote Sensing* 10, no. 8: 1275.
https://doi.org/10.3390/rs10081275