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A frequency scaling (FS) imaging algorithm is proposed for spotlight bistatic SAR data processing. Range cell migration correction (RCMC) is realized through phase multiplication. The proposed algorithm is insensitive to the length of the baseline due to the high precision of the point target (PT) spectrum that we are based on. It is capable of handling bistatic SAR data with a large baseline to range ratio. The algorithms suitable for small and high squint angles are both discussed according to whether the range dependence of the second range compression (SRC) can be neglected or not. Simulated experiments validate the effectiveness of the proposed algorithm.

Bistatic synthetic aperture radar (SAR) imaging has been widely discussed in recent years [

We focus on the well-known tandem configuration here. A frequency scaling (FS) imaging algorithm is proposed in this paper for bistatic SAR imaging in spotlight mode based on the spectrum presented in [

The paper is organized as follows. In Section

Figure

Spotlight bistatic SAR geometry in tandem configuration.

In spotlight mode, suppose that radar transmits the linear frequency-modulated (LFM) pulses, and the echo signal after dechirp on receiver can be written as

Transforming the signal into the range wavenumber domain, in case of large time bandwidth product [

Transforming the signal into the two-dimensional (2D) wavenumber domain based on the IDW concept [

The proposed FS imaging algorithm is based on this equation. Wu et al. deduced an exact analytical expression of the half bistatic angle

The detailed expressions of

In the following, we fit a straight line to the RCM factor

Substituting (

To equalize the RCM of all the ranges to the one of the scene center, resembling the monostatic case [

The multiplication between (

From the comparison between (

Again, transforming the signal into the 2D wavenumber domain, the signal moves to

Then, the inverse frequency scaling function is introduced to correct the second-order range phase error caused by the frequency scaling operation

The following operation is the bulk range shift function for the RCMC:

The range dependence of SRC can be neglected when the bistatic SAR system works with small squint angles, and it can be compensated with the parameters of the scene center. The SRC function is given as

The operations in range are finished after transforming the signal into the

If the azimuth bandwidth is smaller than PRF, the azimuth compression filter can be directly given as

Note that, due to the change of the image in range as is shown in (

However, the PRF is usually smaller than the whole azimuth bandwidth in practice in spotlight mode, especially in spaceborne case; thus the imaging algorithms cannot be directly applied into the whole aperture because of the spectral folding effect. One way to solve the problem is the subaperture method and combining it with the deramping process. The data is divided into several subapertures according to the azimuth time, ensuring that the bandwidth within each subaperture is smaller than PRF. The FS algorithm is then implemented in each subaperture.

In the following, we come to get the high-order phase compensation function to transform the bistatic azimuth phase history into a purely quadratic one as the monostatic case [

To summarize and make the proposed FS algorithm more clearly, the workflows of the proposed FS algorithm are given as follows: (1) divide the bistatic SAR data into subapertures; (2) perform azimuth FFT and implement the frequency scaling by (

Block diagram of tandem bistatic SAR in spotlight mode with small squint angles.

In some cases, the bistatic SAR system works with high squint angles, such as observing the interested object which is squint-depended or inspecting the front situations. The range dependence of SRC has to be taken into account in such cases. Taking (

To eliminate the range dependence of SRC, firstly, a third-order filter term is introduced

The coefficient

Multiply the signal by the nonlinear scaling function which is constructed as

A range FFT is performed to the signal after the multiplication between (

Due to the range dependence of

Substituting (

Making

As can be seen, the range dependence of SRC has been eliminated. The function for range compression is given as

And, then, the phase correction factor is given as

To construct accurate azimuth compression filter, we have to accommodate

An azimuth IFFT is performed to transform the focused data into the complex image domain in the end.

Similarly, we summarize the proposed nonlinear FS algorithm that is suitable for high squint angle bistatic SAR. The corresponding workflows are given as follows: (1) perform azimuth FFT and implement frequency scaling by (

Block diagram of tandem bistatic SAR with high squint angles.

From the above discussion, we can tell that the range history of bistatic SAR is the sum of two square roots, which results in different bistatic spectrum from the monostatic case. The different bistatic spectrum leads to different range cell migration. Thus the conventional FS algorithm cannot be directly utilized into the bistatic case. In this paper, we design a new frequency scaling algorithm which is suitable for bistatic case. The proposed bistatic frequency scaling function is established based on an exact analytical bistatic spectrum to realize range-dependent RCM correction. And the corresponding functions for bistatic FS algorithm are also established, such as the inverse frequency scaling function, the residual video phase correction function, the range migration correction function, and the second range compression function. We construct all these new functions based on the bistatic spectrum. Besides, the range-dependent parameters need to be updated for bistatic case. In addition, a nonlinear frequency scaling algorithm is proposed to deal with the high squint angle bistatic SAR based on the analytical bistatic spectrum. We also establish the functions of the nonlinear FS algorithm for bistatic SAR accordingly. Although these functions for the bistatic FS algorithm are much more complex than the monostatic one, fortunately, we can evaluate the correctness of the algorithm by degenerating the conditions into the monostatic case. In practice, we can choose the suitable algorithm according to the squint angles.

Experiments are carried out to validate the effectiveness of the proposed algorithm. Table

Simulation parameters.

Case I | Case II | Case III | ||||
---|---|---|---|---|---|---|

Range bandwidth | 200 MHz | |||||

PRF | 3000 Hz | |||||

Carrier frequency | 10 GHz | |||||

Platform velocity | 7000 m/s | |||||

Closest distance from the flight track to the scene center | 600 km | |||||

Illumination time | 1.34 s | 1.44 s | 1.61 s | |||

Range to reference target | 611.9 km | 635.7 km | 670.8 km | |||

Length of the baseline | 240 km | 420 km | 600 km | |||

Squint angle | 11.31°(T) | −11.31°(R) | 19.29°(T) | −19.29°(R) | 26.57°(T) | −26.57°(R) |

Quality parameters of impulse-response function for Case I.

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

Resolution (m) | PSLR (dB) | ISLR (dB) | Resolution (m) | PSLR (dB) | ISLR (dB) | |

Reference | 0.75 | −13.2626 | −9.7577 | 1.0625 | −13.2669 | −9.7605 |

Edge | 0.75 | −13.2618 | −9.7502 | 1.0625 | −13.2639 | −9.7520 |

Profile error caused by (

Residual phase error of the SRC for Case I.

The imaging results for Case I. (a) The center target. (b) The edge target.

To show the advantage of the proposed algorithm, we compare it with the MSR and DMO spectra based algorithms. Satisfying focusing quality can be obtained by using all the algorithms for Case I except the DMO based one. For Case II, the baseline increases to 420 km, and the imaging results obtained by using different algorithms are shown in Figures

Contour plots for Case II by using the proposed algorithm. (a) The center target. (b) The edge target.

Contour plots for Case II by using the MSR spectrum based FS algorithm. (a) The center target. (b) The edge target.

Contour plots for Case II by using the DMO spectrum based algorithm. (a) The center target. (b) The edge target.

In the following, we come to Case III with the baseline 600 km; corresponding imaging results by using the algorithms are shown in Figures

Quality parameters of impulse-response function of the center target by using different algorithms for Cases II and III.

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

Resolution (m) | PSLR (dB) | ISLR (dB) | Resolution (m) | PSLR (dB) | ISLR (dB) | |

Case II (the proposed) | 0.75 | −13.2537 | −9.7736 | 1.0625 | −13.2740 | −9.7137 |

Case II (the MSR based) | 0.75 | −13.2464 | −9.7753 | 1.0625 | −13.2728 | −9.7131 |

Case II (the DMO based) | 0.75 | −13.2636 | −9.7792 | 5.8125 | −0.0102 | 5.1913 |

Case III (the proposed) | 0.75 | −13.2346 | −9.7749 | 1.0625 | −13.2725 | −9.7115 |

Case III (the MSR based) | 0.75 | −13.2579 | −9.7751 | 1.0625 | −13.2721 | −9.7087 |

Case III (the DMO based) | 0.75 | −13.2427 | −9.7687 | 5.9375 | −00005 | 8.2978 |

Quality parameters of impulse-response function of the edge target by using different algorithms for Cases II and III.

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

Resolution (m) | PSLR (dB) | ISLR (dB) | Resolution (m) | PSLR (dB) | ISLR (dB) | |

Case II (the proposed) | 0.75 | −13.2648 | −9.7708 | 1.0625 | −13.2669 | −9.7605 |

Case II (the MSR based) | 0.75 | −13.2618 | −9.7542 | 1.375 | −11.1847 | −7.9352 |

Case II (the DMO based) | 0.75 | −13.2365 | −9.7750 | 5.8750 | −0.0087 | 8.2852 |

Case III (the proposed) | 0.75 | −13.2626 | −9.7577 | 1.0625 | −13.2547 | −9.7269 |

Case III (the MSR based) | 0.75 | −13.2583 | −9.7502 | 1.6875 | −4.0652 | −2.9469 |

Case III (the DMO based) | 0.75 | −13.2398 | −9.7722 | 6.0625 | −00004 | 8.8628 |

Contour plots for Case III by using the proposed algorithm. (a) The center target. (b) The edge target.

Contour plots for Case III by using the MSR spectrum based algorithm. (a) The center target. (b) The edge target.

Contour plots for Case III by using the DMO spectrum based algorithm. (a) The center target. (b) The edge target.

We can tell that the proposed algorithm is capable of dealing with spotlight bistatic SAR data with a large baseline. Note that, for the MSR spectrum based algorithm, we only expand the Taylor series expansion up to the third term for comparison here. For better focusing quality of the MSR based algorithm, higher orders can be expanded.

In the following, we take a look at the case with high squint angels. The main parameters are shown in Table

Simulation parameters with high squint angles.

Wavelength | Pulse duration | Transmitted bandwidth | Platform velocity | PRF | Doppler bandwidth | Reference distance |
---|---|---|---|---|---|---|

0.03 m | 30 us | 250 MHz | 110 m/s | 600 Hz | 220 Hz | 8000 m |

If the parameters of the scene center are still chosen to compensate the range-dependent SRC, the residual phase error will be larger than

Quality parameters of impulse-response function for the configuration with high squint angles.

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

Resolution (m) | PSLR (dB) | ISLR (dB) | Resolution (m) | PSLR (dB) | ISLR (dB) | |

Algorithm for high squint angles | 0.6 | −13.2751 | −9.6878 | 0.5 | −13.2431 | −9.7215 |

Algorithm for small squint angles | 1.9125 | −3.4346 | −3.0418 | 0.7813 | −9.2565 | −6.7511 |

The MSR based algorithm | 0.6 | −13.2431 | −9.7331 | 0.7188 | −9.7331 | −8.2379 |

Residual phase error with high squint angles.

Contour plots with the proposed algorithm suitable for small squint angles. (a) The center target. (b) The edge target.

Contour plots with the proposed algorithm suitable for high squint angles. (a) The center target. (b) The edge target.

Contour plots with the MSR spectrum based algorithm. (a) The center target. (b) The edge target.

In practice, when the data is obtained, we can determine the SRC to test the amplitude of the residual phase error and choose the proper imaging algorithm subsequently.

An FS algorithm suitable for tandem bistatic SAR in spotlight mode is proposed, which is insensitive to the baseline to range ratio. The subaperture approach and the deramping process are combined to handle the problem of the azimuth spectral folding effect like the monostatic case. Ideal focusing results are obtained in the frequency domain without interpolation. A nonlinear FS algorithm is also discussed to deal with the tandem bistatic data with high squint angles, in which situation the range dependence of SRC must be taken into consideration. Satisfactory results are obtained with simulated experiments.

However, we have to note that the imaging algorithm presented is based on ideal conditions without considering motion errors, which cannot be avoided in practice [

Here, we give the detailed formations of some parameters. Wu et al. deduced an exact expression for tandem bistatic SAR [

Substituting it into (

In (

As can be seen, the expressions seem rather complicated due to the complex formation of the half bistatic angle; however, they will turn into the familiar ones under the monostatic condition when

The authors declare that they have no competing interests.

This work was supported by the Natural Science Foundation of China (Grant 61222108) and the Fundamental Research Funds for the Central Universities (Grant GK201603089).