- Open Access
Focusing high-squint and large-baseline one-stationary bistatic SAR data using keystone transform and enhanced nonlinear chirp scaling based on an ellipse model
© The Author(s). 2017
Received: 16 December 2016
Accepted: 24 April 2017
Published: 8 May 2017
This paper deals with the imaging problem for one-stationary bistatic synthetic aperture radar (BiSAR) with high-squint, large-baseline configuration. In this bistatic configuration, accurate focusing of BiSAR data is a difficult issue due to the relatively large range cell migration (RCM), severe range-azimuth coupling, and inherent azimuth-geometric variance. To circumvent these issues, an enhanced azimuth nonlinear chirp scaling (NLCS) algorithm based on an ellipse model is investigated in this paper. In the range processing, a method combining deramp operation and keystone transform (KT) is adopted to remove linear RCM completely and mitigate range-azimuth cross-coupling. In the azimuth focusing, an ellipse model is established to analyze and depict the characteristic of azimuth-variant Doppler phase. Based on the new model, an enhanced azimuth NLCS algorithm is derived to focus one-stationary BiSAR data. Simulating results exhibited at the end of this paper validate the effectiveness of the proposed algorithm.
Bistatic synthetic aperture radar (BiSAR) system is operated with separated transmitter and receiver platforms which offers particular advantages like flexible configuration, reduced cost, strong hiding performance, and forward-looking imaging ability when it is compared with traditional monostatic SAR [1–5]. Owing to these benefits, BiSAR has raised increasing concerns in SAR research community in the last decade.
One-stationary BiSAR, where the transmitter or the receiver is stationary, is a special configuration of general BiSAR that is relatively easy to be constructed and deployed. This kind of BiSAR is of great value to remote sensing applications, as it allows small and light-weighted unmanned aerial vehicles or in-orbit SAR satellite to produce bistatic images [6, 7]. The one-stationary BiSAR could be used to image the target area and exploit multi-dimensional information, such as region monitoring, resolution enhancement, and ground moving target detection and imaging, which makes broad application prospects in both civilian and military fields [8, 9].
Currently, many imaging algorithms have been proposed for one-stationary BiSAR. In [10, 11], nonlinear chirp scaling (NLCS) algorithm has been applied to focus one-stationary BiSAR data, where a curve fitting method is used to generate a perturbation function to equalize the azimuth-variant frequency modulation (FM) rate. In [12–14], NLCS algorithm is combined with keystone transform (KT) to handle one-stationary BiSAR data. In these methods, KT is used to eliminate the linear range-azimuth coupling that increases with the squint angle, and then, numerical integral is utilized in the azimuth focusing to generate the perturbation function. Inverse scaled Fourier transform (ISFT) can also handle one-stationary BiSAR data [15, 16]. This kind of algorithm uses chirp multiplications and FFTs in the frequency domain to achieve the 2-D spatial variance correction which avoids the use of interpolation. Furthermore, omega-k algorithm is also applied to the image formation of one-stationary BiSAR [17, 18]. In this algorithm, 2-D Stolt transformation is used to deal with the spatial variance, but the Stolt interpolation makes the method quite time-consuming.
In summary, NLCS is an excellent algorithm proposed in recent years for image formation of one-stationary BiSAR, which was originally adopted for monostatic SAR to equalize the azimuth-variant Doppler FM rate . In , this algorithm has been applied to focus BiSAR data. However, the inherent geometric variance of BiSAR has not been taken into account, and thus, the scene size of the final image is restricted. In , the NLCS has been extended for BiSAR with a model of range offset to improve the azimuth equalization. With the increase of baseline, however, the range offset model is becoming inaccurate, which leads to deterioration of the imaging performance. To overcome this issue, numerical methods for NLCS are utilized in [10–14] to generate the perturbation function, but these methods need a large amount of computation, and also, their applicability is limited.
In this paper, an ellipse model is established to reveal the azimuth-variant characteristic of slant ranges for BiSAR, and then, an enhanced NLCS algorithm based on the new model is proposed to focus one-stationary BiSAR data with high-squint, large-baseline configuration. In this algorithm, deramp operation is used first to remove the range walk and the Doppler ambiguity of the echo, and then, KT is utilized to eliminate the residual linear RCM. After that, bulk range cell migration correction (RCMC) and second range compression (SRC) are carried out to compensate the high-order RCM and range compression terms. Following that, an azimuth ellipse model is constructed to reveal the azimuth-variant characteristic of BiSAR. Based on the new model, an enhanced azimuth NLCS is derived to focus one-stationary BiSAR data at last.
The rest of this paper is organized as follows. Section 2 gives the geometric signal model of BiSAR with stationary transmitter. Section 3 presents the range processing by deramp operation, KT, and bulk RCMC. In Section 4, the azimuth-variant characteristic of BiSAR is analyzed based on a new azimuth model, and then, the enhanced azimuth NLCS is derived. Simulation results of the proposed algorithm are given in Section 5. Finally, conclusions are provided in Section 6.
2 Geometry and signal model
The linear term in (5) represents the linear RCM, and the high-order terms represent the high-order RCMs, respectively.
where τ is the range fast time and w r(·) and w a(·) imply the range and azimuth envelopes, respectively. T a is the synthetic aperture time, f c is the carrier frequency, K r is the range FM rate, and c represents the speed of light.
3 Range processing
In this section, we first quantitatively analyze the ratio of linear RCM component to high-order RCM components with varying squint angles to confirm that the linear RCM component dominates the total RCM in high-squint BiSAR. Then, based on that point, we used a method combining deramp operation and keystone transform to remove the linear RCM completely. In this method, deramp operation is utilized to eliminate the range walk and the Doppler ambiguity of the echo, and then, KT is used to remove the residual linear RCM left by deramp operation. After that, bulk RCMC and SRC are performed to compensate the high-order RCM and range compression terms. At last, analyses are conducted to validate the effectiveness of the range processing.
3.1 RCM proportion analysis
Stationary parameters of BiSAR
Beam center slant range
Pulse repetition frequency
Synthetic aperture time
Bistatic bsaeline range
3.2 Algorithm derivation
Based on the discussion aforementioned, we can affirm that in high-squint BiSAR, the linear RCM component takes the dominant part of the total RCM, while the high-order RCM is extremely small. When the squint angle is approaching 70°, for instance, the high-order RCM at the edge in azimuth is only about 3.0 m, while the ratio of linear RCM to high-order one is over 1500 times. Accordingly, the linear RCMC operation is particularly significant in the entire RCMC for high-squint BiSAR. Once the linear RCM is removed, only a small amount of high-order RCM will remain, and it is much easier to be corrected. Therefore, we apply the combination of deramp operation and KT to remove the linear RCM and then a bulk RCMC to correct the high-order RCM.
In (14), the second term denotes the residual linear RCM and it is azimuth-variant. Thus, after deramp operation, only the linear RCMs of the central targets in azimuth have been fully removed, while the residual linear RCMs of the noncentral targets in azimuth still exist and cannot be ignored in high-squint BiSAR which will be discussed in Section 3.3.
where t m is the new azimuth slow time after KT.
In (16), φ 0 is the azimuth modulation term, φ 1 is the range position term, and φ 2 and φ n denote the SRC and high-order range-azimuth coupling terms, respectively.
where ΔZ 1 denotes the migration error caused by bulk RCMC. In high-squint BiSAR, this error is extremely small and can be neglected; the detailed analysis is given in Section 3.3.
Thus, we draw a conclusion that the bistatic range histories of echoes with the same R total(0;r c,t c) in the coordinate plane have been shifted into a same range cell after RCMC.
where the high-order filter is usually kept up to third-order term.
Inspecting (28), the Doppler FM rate is determined by the slant range at the beam center crossing time, r Rc. According to (22), however, the slant ranges of echoes in a same range cell after RCMC are different, and so are the Doppler FM rates, which must be equalized before azimuth compression. The detailed derivation of equalization will be discussed in Section 4.
3.3 Algorithm validation
To validate the effectiveness of the range processing mentioned above, analyses on the bistatic range histories of the echoes after deramp operation, KT, and bulk RCMC are performed based on BiSAR parameters in Table 1, respectively. We assume five targets placed at the iso-range line of R total(0; r c, t c) in the coordinate plane, and the azimuth interval between them is 550 m.
4 Enhanced azimuth NLCS algorithm
According to (28), the effectiveness of azimuth NLCS for BiSAR data is determined by the azimuth model of slant ranges at the beam center crossing time. However, the azimuth model of monostatic SAR is directly applied to focus BiSAR data in , which makes the imaging algorithm suffer limitations of applicability, especially when the baseline between the transmitter and the receiver is large.
In this section, we first analyze the azimuth dependency of slant ranges for monostatic SAR by a circle model. Then, an ellipse model is established to reveal the azimuth-variant characteristic of slant ranges for BiSAR. Based on the new model, coefficients of enhanced azimuth NLCS are derived to focus one-stationary BiSAR data. At last, the processing errors and limitations of scene size in azimuth of the new algorithm are both analyzed.
4.1 Circle model for monostatic SAR
In the data space, the slant ranges of A and B are different and so are their Doppler FM rates according to (28), as seen in Fig. 8b. After RCMC, their range histories have been shifted into a same range cell, but their Doppler FM rates are still dependent on r cA and r cB , respectively, as seen in Fig. 8c. Therefore, we have to figure out the relationship between r cA and r cB , which is the key to azimuth NLCS processing.
when target B lies close to the reference target A.
4.2 Ellipse model for one-stationary BiSAR
In BiSAR configuration, owing to the separation of the transmitter and the receiver, the circle model aforementioned does not hold anymore. Thus, to reveal the azimuth-variant characteristic of Doppler FM rate for BiSAR, a new azimuth geometric model should be established.
where r OB is distance from coordinate origin O to the target B, which implies the slant range of target B after RCMC operations.
where e = c/a is the eccentricity of the ellipse. It is obvious that (29) is a special case of (33) when the eccentricity e is set to zero and a equals to r RcA , which means that the ellipse model degrades to a circle model.
Notice that the derived result in  for BiSAR is exactly the same as in (29) for monostatic SAR, which means that the monostatic model is directly applied to BiSAR case in . This will result in severe azimuth processing error, especially when focusing large-baseline BiSAR data.
4.3 Azimuth equalization and compression operation
In this subsection, we derive an enhanced azimuth NLCS to focus one-stationary BiSAR data based on the result of the ellipse model established in Section 4.2.
where K aA is the Doppler FM rate of the reference target A, and K s is the coefficient of azimuth-variant component of the Doppler FM rate.
where the parameter p is the coefficient of the perturbation function.
4.4 QPE and scene size analysis
Inspecting (34), the first-order expansion is adopted to approximate the Doppler FM rate, and this will cause an inevitable error along azimuth direction which is called the quadratic phase error (QPE). In this subsection, the QPEs along azimuth direction based on the expressions of Doppler FM rate in (34), [20, 21] are analyzed, respectively.
which should not exceed the threshold value π/4 for the validity of azimuth NLCS.
5 Simulation results
To demonstrate the effectiveness of the enhanced azimuth NLCS algorithm proposed by this paper, two experiments with simulated data are carried out in this section. Both the two simulations involve airborne configurations. The first simulation is carried out with a typical airplane self-navigation configuration, which processes a relatively large imaging scene. The second one is operated with an airplane self-landing configuration, which involves high resolutions in both range and azimuth.
Case I: the imaging scene is set to be 2.0 km in range and 2.2 km in azimuth, and this simulation chooses an array of 5 × 5 point targets with an interval of 500.0 m in range and 550.0 m in azimuth. The theoretical range resolution is 2.0 m, and the azimuth resolution is 1.7 m. The simulation parameters for this experiment are listed in Table 1.
Performance parameters in azimuth for case I
Traditional algorithm in 
Proposed algorithm in this paper
Stationary parameters of BiSAR for case II
Beam center slant range
Pulse repetition frequency
Synthetic aperture time
Bistatic bsaeline range
Performance parameters in azimuth for case II
Traditional algorithm in 
Proposed algorithm in this paper
This paper proposes an enhanced azimuth NLCS algorithm based on an ellipse model to focus one-stationary BiSAR data with high-squint, large-baseline configuration. In the range processing, a method combining deramp operation and KT is adopted to remove linear RCM completely and mitigate range-azimuth cross-coupling. After that, an ellipse model is proposed to analyze the azimuth-variant characteristic of Doppler phase of one-stationary BiSAR data. Based on the new model, an enhanced azimuth NLCS is derived to handle one-stationary BiSAR data. Compared with the traditional algorithm, better imaging performance can be achieved by the proposed algorithm in this paper. Additionally, the ellipse model and the enhanced NLCS algorithm proposed by this paper also show potential for other bistatic cases like BiSAR with nonparallel tracks, forward-looking BiSAR, and even general BiSAR.
The authors would like to thank all the anonymous reviewers for their hard work.
This work was supported by the National Natural Science Foundation of China (61301248, 61271214), and by the Chinese Innovation Foundation of Aerospace Science and Technology.
HZ and SZ conceived and designed the research. HZ, SZ, and JH performed the experiments. HZ and SZ wrote the manuscript. HZ, SZ, and MS reviewed and edited the manuscript. All authors read and approved the manuscript.
The authors declare that they have no competing interests.
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