Range cell migration correction analysis of one-step and two-step motion compensation for millimeter-wave airborne SAR imaging
© The Author(s) 2016
Received: 13 May 2016
Accepted: 28 October 2016
Published: 10 November 2016
Conventional two-step motion compensation (MOCO) method is widely adapted for airborne synthetic aperture radar (SAR) imaging due to its conciseness combining with the SAR focusing procedure. For two-step MOCO, range-independent compensation is processed before range cell migration correction (RCMC), and the range-dependent phase correction is implemented after RCMC. However, the accuracy of RCMC would be seriously decreased by the residual range-dependent phase, which is a fatal problem for high-resolution millimeter-wave (MMW) SAR imaging. In this paper, an extensive investigation on the RCMC accuracy is provided by establishing an accurate formula expression between the range cell migration error and the residual range-dependent phase error. One-step MOCO-based SAR imaging algorithm is investigated by compensating the range-dependent motion error before RCMC, so the presence of range cell migration error would be significantly suppressed. What is more, a modified azimuth match filtering (AMF) function is given by precise topography and aperture-dependent motion compensation (PTA) method to overcome the residual azimuth-dependent phase error in the azimuth compression stage. Both simulated and real-measured MMW SAR data sets are used to validate the analysis for high-resolution airborne SAR imaging.
KeywordsSynthetic aperture radar (SAR) Motion compensation (MOCO) Millimeter-wave (MMW) One-step MOCO
Motion compensation (MOCO) [1–4] is a crucial operating step for airborne synthetic aperture radar (SAR) [5–7] imaging because the non-ideal movement deviates the radar platform from the predetermined flight trajectory. More importantly, for high-resolution millimeter-wave (MMW)  SAR systems, imaging performance is more sensitive to the envelope and phase of motion errors, so a precise MOCO is essential with the availability of high-precision inertial navigation system (INS) measurement. An efficient two-step MOCO algorithm  is proposed by Moreira and Huang, combining with chirp scaling algorithm (CSA) [9, 10] for airborne SAR imaging. This method is divided into range-independent compensation step and range-dependent compensation step, while the first step is processed to the range compressed data and the second step is processed after range cell migration correction (RCMC). The problem of conventional two-step MOCO processing is also obvious. The residual range-dependent motion error remained after the first step seriously decreases the accuracy of RCMC in two-dimensional wavenumber domain, which presents as a curving range cell migration (RCM) range profile in two-dimensional time domain, and destroys the performance of azimuth pulse compression. Reference  describes the problem above for Omega-k algorithm [12, 13]. In their work, a one-step MOCO method is proposed, but the detailed analysis of the RCMC error is not given. Besides, the original MOCO methods only take range-dependent motion error into account, and the residual azimuth-dependent motion error should also be considered, which is non-ignorable for high-resolution airborne MMW SAR imaging with wide swath. The existed azimuth-dependent MOCO algorithms [14–18] could precisely compensate the azimuth-dependent motion error and modify the azimuth matched filtering function in order to eliminate the influence of azimuth-variant motion error.
Based on the signal model in , we investigate the cause of RCMC error as well as its definite expression deduction for MMW SAR system in this paper. A background assumption is confirmed that the trajectory information is accurately recorded by the INS and the whole motion compensation procedure is processed without autofocus step. The one-step MOCO-based imaging algorithm is investigated, which compensates range-dependent motion error before RCMC in order to suppress the residual envelope and phase error of RCM range profile. Moreover, according to the analytical expression of the residual spatial variant error, an accurate azimuth match filtering (AMF) function is modified by precise topography- and aperture-dependent motion compensation (PTA) , which compensates the residual azimuth-dependent motion errors remained by RCMC. In this paper, the conventional two-step MOCO-based imaging algorithm is introduced for comparison, theoretical analysis to the superiority of one-step MOCO would be adequately verified by simulated and real-measured data experiments.
The whole paper is organized as follows: Section 2 gives the signal and geometry model of the SAR imaging, RCMC accuracy respect to the residual range-dependent error is analyzed as well, and flowcharts of one-step and two-step MOCO-based SAR imaging algorithms are then given for comparison. In Section 3, RCMC error comparison between one-step and two-step MOCO is discussed, and computational burden of both methods is also analyzed in detail. In Section 4, extensive experimental results are given with both simulated and real measured MMW airborne SAR data. Conclusions are given in the last section.
2 One-step MOCO-based SAR imaging algorithm
2.1 SAR imaging and RCMC error analysis
It is worthy to note that 𝜗 0 represents the phase component of RCMC error and 𝜗 1 denotes the envelope error of RCMC, which seriously destroys the imaging performance. 𝜗 2 is the second-order term, which slightly reflects the focusing performance in range, and the effect could be ignored in most cases.
2.2 Flowcharts of one-step and two-step MOCO-based imaging algorithms
As the one-step MOCO-based imaging algorithm is described in detail, we briefly describe the procedure of the two-step MOCO-based imaging algorithm for comparison. For two-step MOCO, the envelope compensation and the first-phase compensation are processed with respect to the reference slant range r s before RCMC, and the second range-dependent MOCO step is processed after RCMC, but serious RCM envelope error is remained at this step. In order to make the comparison more equitable, azimuth-dependent motion error is then compensated by method of PTA.
3 Comparative analysis of one-step and two-step MOCO
3.1 RCMC error comparison between one-step and two-step MOCO
In the previous section, we analyze the RCMC error with respect to the residual motion error Δ R E , but Δ R E is different for the one-step and two-step MOCO. In this subsection, we focus on calculating Δ R E and comparing the RCMC error between the one-step and two-step MOCO.
where r denotes the range bin of target, r s denotes the slant range from the radar to the beam center, δ R a is the azimuth-dependent motion error. It is obvious in (27) that the residual motion error is in proportion to the range between target and scene center, so the error is diffused along range direction.
3.2 Computational burden analysis
In this subsection, the computational burden of the one-step and two-step MOCO-based SAR imaging algorithms is respectively measured by operating number of fast Fourier transform (FFT), inverse fast Fourier transform (IFFT), and complex multiplication for comparison. As shown in Fig. 2, suppose the azimuth and range point numbers are denoted by N a and N r . It needs to be noticed that we analyze the operand by merging adjoining phase terms and without regard of the calculation of PTA operation. For the one-step MOCO-based imaging algorithm, there are 4N a times N r -point FFT/IFFT operators, 2N r times N a -point FFT/IFFT operators, and 5 times N r ×N a -point complex multiplications to obtain a focused imaging. Comparing with the conventional two-step based imaging algorithm, there are 2N a times N r -point FFT/IFFT operators, 2N r times N a -point FFT/IFFT operators, and 5 times N r ×N a -point complex multiplications. It could be found that the one-step MOCO-based imaging algorithm adds 2N r more times N a -point FFT/IFFT operators, which slightly increases the computational burden and exchanges for a better focused imagery.
4 Simulated and real data experiments
4.1 Experiments with simulated data
Pulse repetition frequency
Center closest slant range
Point A coordinate
Point B coordinate
Point C coordinate
Focusing performance comparison between two focusing algorithms
4.2 Experiments with real-measured data
Pulse repetition frequency
Center closest slant range
Focusing performance comparison of points A and B between two focusing algorithms
In order to verify the calculation analysis in Section 2, we record the calculation time for both two-step and one-step MOCO-based SAR imaging algorithms. The computer platform is installed with Windows10 64-bit operating system, Core i7-4720HQ@2.6GHz CPU, 16-GB memory and Matlab with version of R2015a. A block of 16,384×8192 (range ×azimuth) points SAR data is used for test, the whole data is divided into four sub-blocks in azimuth, and the calculation time of the two-step and one-step MOCO-based imaging algorithms are 855.13s and 909.49s, respectively. With the nearly equal computation complexity compared with the two-step MOCO, the one-step MOCO is applicative for practical MMW SAR imaging application.
The conventional two-step MOCO algorithm remains the range-dependent motion error before RCMC, which decreases the accuracy of RCMC in two-dimensional wavenumber domain, inducing serious envelope and phase error to the range profile. In this paper, analytical expressions of these errors are deduced in detail. The one-step MOCO-based imaging algorithm is also investigated to compare with the conventional two-step MOCO-based imaging algorithm, which removes the range-dependent motion error before RCMC, so the RCMC error is significantly suppressed. Simulations and measured MMW data experiments illustrate the outperforms of the one-step MOCO-based SAR imaging algorithm, which verify the analysis in this paper.
The authors thank the anonymous reviewers for their valuable comments to improve the paper quality. This work was supported by the National Natural Science Foundation of China under grant numbers 61301280 and 61301293 and the High-Resolution Earth Observation System Major Special Project Youth Innovation Foundation of China under grant number GFZX04060103.
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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