# Modeling and Precise Processing for Spaceborne Transmitter/Missile-Borne Receiver SAR Signals

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## Abstract

**:**

## 1. Introduction

- (1)
- The large-maneuver and receiving-only features of the missile-borne receiver reduce the detection probability. The corresponding anti-interference and anti-interception are greatly improved in the ST/MR SAR system.
- (2)
- The radar antenna, installed on the missile-borne platform, is used to receive the echo signals without a transmitting function, which indicates that a large power device can be avoided in the missile system to greatly save space and costs.
- (3)
- Spaceborne transmitter has the characteristics of wide coverage, unrestricted geography, net flexibility, and a high revisiting-frequency, which could provide some degrees of freedom (DOFs) in target area selection, flight trajectories, and number of missile-borne receivers.
- (4)
- By setting the transmitter and receiver on different platforms with a special style, the non-complete coupling areas between iso-range and iso-Doppler lines could be formed in front of missile, which means that the missile has the capability of forwarding-looking imaging.

## 2. Modeling

#### 2.1. Geometric Model

#### 2.2. Flight-Path Constraint

#### 2.3. Flight-Path Constraint

Algorithm 1. Basic procedure. |

1: Fixed input parameters: ${\mathit{r}}_{t}\left(0,A\right)$, ${\mathit{r}}_{r}\left(0,A\right)$, ${\mathit{v}}_{t}$, ${\mathit{v}}_{r}$, ${\mathit{a}}_{r}$, ${T}_{start}$, $c$ 2: Initialization (i=0): ${T}_{end\left(0\right)}={T}_{start}+\left|\mathit{r}\left({T}_{start},A\right)\right|/c$ 3: Iterations (i≥1): 4: Calculate the delay: ${\tau}_{i-1}={T}_{end\left(i-1\right)}-{T}_{start}$ 5: Calculate the slant range of the receiver: $\left|{\mathit{r}}_{r}\left({T}_{end\left(i-1\right)},A\right)\right|=\left|{\mathit{r}}_{r}\left(0,A\right)-{\mathit{v}}_{r}{T}_{end\left(i-1\right)}-{\mathit{a}}_{r}{T}_{end\left(i-1\right)}^{2}/2\right|$ 6: If $\left|{\mathit{r}}_{t}\left({T}_{start},A\right)\right|+\left|{\mathit{r}}_{r}\left({T}_{end\left(i-1\right)},A\right)\right|-{\tau}_{i-1}\xb7c\le \mathsf{\Delta}$, return $\tau ={\tau}_{i-1}$ and stop. 7: Calculate the time of the receiver: ${T}_{end\left(i\right)}={T}_{start}+\left\{\left|{\mathit{r}}_{t}\left({T}_{start},A\right)\right|+\left|{\mathit{r}}_{r}\left({T}_{end\left(i-1\right)},A\right)\right|\right\}/c$ 8: Set $i=i+1$. |

## 3. Imaging Approach

- ➢
- Cross-couplings. Traditionally, the cross-coupling phases in the 2-D spectrum derived by using the principle of stationary point (POSP) and MSR are divided into two parts, i.e., the range- and azimuth-dependent ones, and they are eliminated separately, which would introduce errors in the final image. The 2-D scaling algorithm in this work can avoid this division and make the processing more accurate than that of the traditional methods.
- ➢
- Spatial variations. Generally, the range, azimuth, and cross-coupling spatial variations in the 2-D spectrum are difficult to be removed
**simultaneously**and**entirely**because of the complex phase expressions. In the 2-D scaling algorithm, two perturbation functions performed in the hybrid domains can eliminate the range, azimuth, and cross-coupling spatially variant terms**simultaneously**and**entirely**, which can avoid the defect of the traditional methods that the range and cross-coupling spatial variations cannot be removed.

#### 3.1. 2-D Scaling in Range Frequency/Azimuth Time Domain

_{r}and azimuth time η), is firstly designed to weaken the spatially variant phases and also provides many more DOFs for the echo signal, i.e.,

#### 3.2. Distortion Minimization

- ➢
- ➢
- The expression of the azimuth bandwidth differs from the usual representation based on the linear path because the rate of the Doppler frequency modulation (FM) ${\kappa}_{2}^{C}$ is greatly affected by the motion parameter vectors ${\mathit{a}}_{r}$. Thus, possible azimuth bandwidth widening should be considered in this work [47,48].
- ➢
- After multiplying Equation (6) with echoes, the rate of the Doppler FM becomes ${\kappa}_{2}^{C}+\zeta $, which means that the azimuth bandwidth may display a big change. Thus, the possible spectral distortion should be eliminated before the azimuth FT of the signal.

#### 3.3. 2-D Scaling in Range Frequency/Azimuth Frequency Domain

_{2}(A), β

_{3}(A), and β

_{4}(A), but with different formulae. Making use of the similarities and differences between Equations (13) and (14), we can obtain the following six equations with six unknowns ξ

_{i}, ω

_{i}(i = 2, 3, 4), i.e.,

**simultaneously**and

**entirely**by using the 2-D scaling algorithm, which indicates that the proposed approach could be easy and efficient to implement for the ST/MR SAR data. Next, some discussions are given to facilitate a better understanding of the proposed method.

## 4. Simulation Results

#### 4.1. Case I

#### 4.2. Case II

## 5. Discussion

#### 5.1. 2-D Range Compression

#### 5.2. Computational Load

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A

Coefficient | Transmitter | Coefficients | Receiver |
---|---|---|---|

${\sigma}_{0}$ | $\sqrt{\langle {\mathit{r}}_{t}\left(0,A\right),{\mathit{r}}_{t}\left(0,A\right)\rangle}$ | ${\nu}_{0}$ | $\sqrt{\langle {\mathit{r}}_{r}\left(0,A\right),{\mathit{r}}_{r}\left(0,A\right)\rangle}$ |

${\sigma}_{1}$ | ${\sigma}_{0}^{-1}\xb7\left[-\langle {\mathit{r}}_{t}\left(0,A\right),{\mathit{v}}_{t}\rangle \right]$ | ${\nu}_{1}$ | ${\nu}_{0}^{-1}\xb7\left[-\langle {\mathit{r}}_{r}\left(0,A\right),{\mathit{v}}_{r}\rangle \right]$ |

${\sigma}_{2}$ | ${\sigma}_{0}^{-1}\xb7\left[\langle {\mathit{v}}_{t},{\mathit{v}}_{t}\rangle -{\sigma}_{1}{}^{2}\right]$ | ${\nu}_{2}$ | ${\nu}_{0}^{-1}\xb7\left[\left(-\langle {\mathit{r}}_{r}\left(0,A\right),{\mathit{a}}_{r}\rangle +\langle {\mathit{v}}_{r},{\mathit{v}}_{r}\rangle \right)-{\nu}_{1}{}^{2}\right]$ |

${\sigma}_{3}$ | ${\sigma}_{0}^{-1}\xb7[-3{\sigma}_{1}{\sigma}_{2}]$ | ${\nu}_{3}$ | ${\nu}_{0}^{-1}\xb7\left[\left(3\langle {\mathit{v}}_{r},{\mathit{a}}_{r}\rangle \right)-3{\nu}_{1}{\nu}_{2}\right]$ |

${\sigma}_{4}$ | ${\sigma}_{0}^{-1}\xb7\left[-3{\sigma}_{2}^{2}-4{\sigma}_{1}{\sigma}_{3}\right]$ | ${\nu}_{4}$ | ${\nu}_{0}^{-1}\xb7\left[\left(3\langle {\mathit{a}}_{r},{\mathit{a}}_{r}\rangle \right)-3{\nu}_{2}^{2}-4{\nu}_{1}{\nu}_{3}\right]$ |

## Appendix B

## Appendix C

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**Figure 3.**Simulation experiment. (

**a**) Simulation scene. (

**b**) Simulation result by parameters of Receiver 1. (

**c**) Simulation result by parameters of Receiver 2.

**Figure 7.**Simulation results. (

**a**) Proposed model. (

**b**) Traditional non-‘Stop-Go’ model. (

**c**) Traditional ‘Stop-Go’ model.

**Figure 8.**2-D impulse response of the reference target. (

**a**) Proposed model. (

**b**) Traditional non-‘Stop-Go’ model. (

**c**) Traditional ‘Stop-Go’ model.

Platform | Transmitter | Receiver 1 | Receiver 2 | |
---|---|---|---|---|

Parameter | ||||

Radar Position at ACM | (0, 1000, 500) km | (12.5, −4, 25) km | (−6, −4, 30) km | |

Velocity Vector | (0, 7600, 0) m/s | (0, 1000, 0) m/s | (−100, −1200, 800) m/s | |

Acceleration Vector | / | (10, −30, −50) m/s^{2} | (50, −60, −35) m/s^{2} | |

Elevation Angle | 84.1° | 45.8° | 32.3° |

Platform | Transmitter | Receiver | |
---|---|---|---|

Parameter | |||

Radar Position at ACM | (0, 0, 754) km | (132, 0, 15) km | |

Velocity Vector | (0, 7600, 0) m/s | (300, 1200, −200) m/s | |

Acceleration Vector | / | (30, 54, −26) m/s^{2} | |

Elevation Angle | 74.1° | 44.6° | |

Carrier Frequency | 9.65 GHz | ||

Pulse Width | 20 μs | ||

Pulse Bandwidth | 200 MHz | ||

Sampling Frequency | 300 MHz | ||

Azimuth Resolution | 0.823 m |

Platform | Transmitter | Receiver | |
---|---|---|---|

Parameter | |||

Radar Position at ACM | (0, 0, 510) km | (112, −78, 25) km | |

Velocity Vector | (0, 7600, 0) m/s | (−170, 800, −640) m/s | |

Acceleration Vector | / | (13, −34, −68) m/s^{2} | |

Elevation Angle | 81.3° | 27.7° | |

Carrier Frequency | 9.65 GHz | ||

Pulse Width | 20 μs | ||

Pulse Bandwidth | 240 MHz | ||

Sampling Frequency | 360 MHz | ||

Azimuth Resolution | 0.716 |

Range | Azimuth | ||||||
---|---|---|---|---|---|---|---|

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

Proposed | PT1 | 0.626 | −13.21 | −9.97 | 0.723 | −13.09 | −9.91 |

PT3 | 0.625 | −13.19 | −10.01 | 0.712 | −13.04 | −9.84 | |

NCSA | PT1 | 0.625 | −13.20 | −9.99 | 0.983 | −7.04 | −7.36 |

PT3 | 0.627 | −13.18 | −9.98 | 0.965 | −6.83 | −7.28 | |

GPFA | PT1 | 0.626 | −13.20 | −10.02 | 1.124 | −10.12 | −8.07 |

PT3 | 0.628 | −13.19 | −9.99 | 1.059 | −12.25 | −6.63 |

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

CSA/Proposed | 0.793 | 0.794 | 0.794 | 0.795 | 0.795 |

BPA/Proposed | 35.59 | 91.25 | 194.81 | 387.78 | 750.61 |

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## Share and Cite

**MDPI and ACS Style**

Tang, S.; Guo, P.; Zhang, L.; Lin, C.
Modeling and Precise Processing for Spaceborne Transmitter/Missile-Borne Receiver SAR Signals. *Remote Sens.* **2019**, *11*, 346.
https://doi.org/10.3390/rs11030346

**AMA Style**

Tang S, Guo P, Zhang L, Lin C.
Modeling and Precise Processing for Spaceborne Transmitter/Missile-Borne Receiver SAR Signals. *Remote Sensing*. 2019; 11(3):346.
https://doi.org/10.3390/rs11030346

**Chicago/Turabian Style**

Tang, Shiyang, Ping Guo, Linrang Zhang, and Chunhui Lin.
2019. "Modeling and Precise Processing for Spaceborne Transmitter/Missile-Borne Receiver SAR Signals" *Remote Sensing* 11, no. 3: 346.
https://doi.org/10.3390/rs11030346