Method Of Series Reversion Research Articles

An Extended Omega-K Algorithm for Automotive SAR with Curved Path

Automotive millimeter-wave (MMW) synthetic aperture radar (SAR) systems can achieve high-resolution images of detection areas, providing environmental perceptions that facilitate intelligent driving. However, curved path is inevitable in complex urban road environments. Non-uniform spatial sampling, brought about by curved path, leads to cross-coupling and spatial variation deteriorates greatly, significantly impacting the imaging results. To deal with these issues, we developed an Extended Omega-K Algorithm (EOKA) for an automotive SAR with a curved path. First, an equivalent range model was constructed based on the relationship between the range history and Doppler frequency. Then, using azimuth time mapping, the echo data was reconstructed with a form similar to that of a uniform linear case. As a result, an analytical two-dimensional (2D) spectrum was easily derived without using of the method of series reversion (MSR) that could be exploited for EOKA. The results from the parking lot, open road, and obstacle experimental scenes demonstrate the performance and feasibility of an MMW SAR for environmental perception.

An Improved NLCS Algorithm Based on Series Reversion and Elliptical Model Using Geosynchronous Spaceborne–Airborne UHF UWB Bistatic SAR for Oceanic Scene Imaging

Geosynchronous spaceborne–airborne (GEO-SA) ultra-high-frequency ultra-wideband bistatic synthetic aperture radar (UHF UWB BiSAR) provides high-precision images for marine and polar environments, which are pivotal in glacier monitoring and sea ice thickness measurement for polar ocean mapping and navigation. Contrasting with traditional high-frequency BiSAR, it faces unique challenges, such as the considerable spatial variability, significant range–azimuth coupling, and vast volumes of echo data, which impede high-resolution image reconstruction. This paper presents an improved bistatic nonlinear chirp scaling (NLCS) algorithm for imaging oceanic scenes with GEO-SA UHF UWB BiSAR. This methodology extends the two-dimensional (2-D) spectrum up to the sixth order via the method of series reversion (MSR) to meet accuracy demands and then employs an elliptical model to elucidate the alterations in the azimuth frequency modulation (FM) rate mismatch. Initially, the imaging geometry and signal model are introduced, and then a separation of bistatic slant ranges based on the configuration is proposed. In addition, during range processing, after eliminating linear range cell migration (RCM), the derivation process for the sixth-order 2-D spectrum is detailed and an improved filter is applied to correct the high-order RCM. Finally, during azimuth processing, the causes of the FM rate mismatch are analyzed, a cubic perturbation function derived from the elliptical model is used for FM rate equalization, and a unified sixth-order filter is applied to complete the azimuth compression. Experimental results with point targets and natural oceanic scenes validate the outstanding efficacy of the proposed NLCS algorithm, particularly in imaging quality enhancements for GEO-SA UHF UWB BiSAR.

Focus Improvement of Spaceborne-Missile Bistatic SAR Data Using the Modified NLCS Algorithm Based on the Method of Series Reversion

The speed and direction of a missile shifts sharply in the dive phase, making the azimuth frequency modulation (FM) rate change with the azimuthal position, leading to azimuth ambiguities and image distortion. To solve this problem, a modified nonlinear chirp scaling (NLCS) algorithm was adopted to compensate for the azimuth FM rate. First, the geometric configuration and echo signal model of the spaceborne missile bistatic synthetic aperture radar (SAR) were built, and then the Doppler frequency correction was performed, and the 2-D spectrum of the signal was derived by the method of series reversion. Next, range migration correction and range compression were finished in the 2-D frequency domain. Following this, a modified NLCS algorithm was proposed to solve the space variance of Doppler phase problem. After compensating for the azimuth FM rate, the azimuth compression focusing was completed and the imaging result was obtained. Finally, by comparing the calculation amount, imaging effect, and performance index with the traditional NLCS algorithm, it can be concluded that the algorithm reduced the calculation amount by 1.0128 × 108 floating point operations per second (FLOPs) compared with the traditional NLCS algorithm, and the azimuth focusing effect of the edge point was greatly improved. Its resolution, peak sidelobe ratio (PSLR), and integrated sidelobe ratio (ISLR) were improved by 0.87 m, 3.32 dB, and 1.79 dB, respectively, which proved the effectiveness and feasibility of this method.

A chirp scaling algorithm based on the method of series reversion for bistatic SAR

For the problem that the existing chirp scaling algorithm based on the method of series reversion (MSRCSA) needs to approximate the range cell migration (RCM) into a linear term of range and causes a larger range cell migration correction (RCMC) error, we propose an improved MSRCSA, which is able to focus azimuth-variation bistatic SAR. Firstly, to avoid the linear approximation for RCM, this paper derives a new curvature factor varying with the azimuth frequency and range, utilizing to describe RCM without any approximation. Secondly, since the new curvature factor is different with the traditional curvature factor varying only with the azimuth frequency, a new scaling function is given to modulate the echo signal so that the required range-variant RCMC shift can be implemented by using phase multiplies. Thirdly, the proposed MSRCSA is implemented by utilizing three phase multiplications and four fast Fourier transforms (FFTs). Finally, the good performance of the proposed MSRCSA is examined by comparing the imaging result of the proposed MSRCSA with that of the existing MSRCSA.

A General Method of Series Reversion for Synthetic Aperture Radar Imaging

Synthetic aperture radar (SAR) imaging usually needs to be converted between the time domain and the frequency domain, in which the solution of stationary phase point determines the accuracy of time–frequency conversion and the final image quality. Theoretically, the stationary phase point can be accurately solved by the method of series reversion (MSR) in an arbitrary configuration (e.g., bistatic and/or nonlinear trajectory). However, the existing methods based on MSR are proposed based on the assumption of specific signal form. In other words, it is necessary to reuse MSR to derive the time–frequency conversion for different signal forms, which brings great inconvenience to engineering applications. In this article, we aim to propose a general method based on MSR. Based on this method, the accurate time–frequency conversion can be derived by simply arranging any signal to the standard form specified in this article first and then simply replacing the variables.

Azimuth-Invariant Motion Compensation and Imaging Chirp Scaling Algorithm for Multiple-Receiver Synthetic Aperture Sonar

To improve the imaging quality of conventional imaging algorithms without motion compensation (MOCO) and the efficiency of point-by-point MOCO algorithms for multiple-receiver synthetic aperture sonar (SAS) with azimuth-invariant six-degree of freedom (DOF) motion errors, an azimuth-invariant MOCO and imaging chirp scaling (CS) algorithm is presented in this paper. Taylor series approximation is used to process the range history in double square root form, while considering the fourth-order and inner terms with respect to the azimuth time. Using the method of series reversion and Fourier transform (FT) properties, the analytical two-dimensional frequency spectrum of the point target response of each receiver is derived. On this basis, the azimuth-invariant MOCO and imaging CS algorithm is proposed for six-DOF motion error compensation and multiple-receiver SAS imaging. Because it considers the azimuth-invariant six-DOF motion errors, the proposed algorithm has better imaging quality than conventional imaging algorithms without MOCO. Additionally, it has a significantly higher efficiency than point-by-point MOCO algorithms because the CS algorithm is a fast FT-based algorithm and does not require interpolation processing. The imaging simulation and experimental results verified the effectiveness and efficiency of the proposed algorithm.

Bistatic SAR Moving Targets Refocus Based on Polar Formatted Phase Error Analysis and Clutter Separation

The refocus process of moving targets in bistatic synthetic aperture radar (SAR) imagery obtained by polar format algorithm (PFA) is discussed in this paper. The analytical expression of the motion phase error in the two-dimensional frequency domain is deduced based on the method of series reversion. The derived phase error equation is utilized to formulate a dictionary, and a sparse-based iteration processing combined with a subtraction operation is proposed to implement the separation of clutter and refocusing of multiple moving targets, respectively. Finally, numerical simulation is performed to verify the validity of the proposed method.

Focus Improvement of Airborne High-Squint Bistatic SAR Data Using Modified Azimuth NLCS Algorithm Based on Lagrange Inversion Theorem

In this paper, a modified azimuth nonlinear chirp scaling (NLCS) algorithm is derived for high-squint bistatic synthetic aperture radar (BiSAR) imaging to solve its inherent difficult issues, including the large range cell migration (RCM), azimuth-dependent Doppler parameters, and the sensibility of the higher order terms. First, using the Lagrange inversion theorem, an accurate spectrum suitable for processing airborne high-squint BiSAR data is introduced. Different from the spectrum that is based on the method of series reversion (MSR), it is allowed to derive the bistatic stationary phase point while retaining the double square root (DSR) of the slant range history. Based the spectrum, a linear RCM correction is used to remove the most of the linear RCM components and mitigate the range-azimuth coupling, and, then, bulk secondary range compression is implemented to compensate the residual RCM and cross-coupling terms. Following this, a modified azimuth NLCS operation is applied to eliminate the azimuth-dependence of Doppler parameters and equalize the azimuth frequency modulation for azimuth compression. The experimental results, with better focusing performance, prove the high accuracy and effectiveness of the proposed algorithm.

Focusing Translational-Variant Bistatic Forward- Looking SAR Data Using the Modified Omega-K Algorithm

Accurate 2-D frequency spectrum (2-D FS) with acceleration, two-way range coupling terms, and spatial-variant Doppler parameters are the main problems to be solved in translational-variant bistatic forward-looking synthetic aperture radar (SAR) (TV BFSAR) with curved trajectory. For these issues, a modified omega-K imaging algorithm is derived in this article. The maximum usage of 2-D FS based on the method of series reversion (MSR) is achieved by linear range cell migration correction, and 2-D FS is linearized in bistatic range by using high-order polynomial fitting. Then, a method of azimuth resampling is introduced to implement compensation of spatial-variant Doppler parameters. Different from other bistatic omega-K methods, our newly proposed method focuses on the small-aperture data in the azimuth frequency domain to avoid azimuth aliasing without padding zeros and uses the frequency focusing position to study the model of spatial-variant phase. Simulation results and real data verify the effectiveness of the proposed method.

A novel AFNCS algorithm for super-resolution SAR in curve trajectory

With the improvement of SAR resolution, super-resolution SAR imaging is more and more widely used in the all-time and all-weather video surveillance and remote sensing imaging. One implementation of super-resolution SAR is that radar works in the spotlight mode. In the case with highly squinted angle and acceleration, the azimuth space variance and the coupling between range and azimuth will become serious in super-resolution imaging. Thus, an azimuth frequency nonlinear chirp scaling algorithm is proposed to solve this problem. Based on the acceleration model, the accurate 2-D spectrum is performed by adopting the method of series reversion. The space-variance of missile-borne SAR in curved flight path is analyzed and an azimuth polynomial phase filter is constructed to make the coefficients of the perturbation function has sufficient flexibility to eliminate the spatial-variant couple terms between range and azimuth. In addition, the gradient operation is used to expand the space-variant coefficients of azimuth modulation term, and the perturbation function is applied to eliminate the space-variant in azimuth direction. The proposed focusing method can process the missile-borne SAR data obtained in spotlight mode in highly squinted angle with high efficiency. The simulation results verify the effectiveness of the proposed imaging approach. The integration of the research in this paper and the deep learning will further pave the way of super-resolution SAR imaging applications in disaster monitoring, security and surveillance.

Bistatic-Range-Doppler-Aperture Wavenumber Algorithm for Forward-Looking Spotlight SAR With Stationary Transmitter and Maneuvering Receiver

Bistatic forward-looking spotlight synthetic aperture radar with stationary transmitter and maneuvering receiver (STMR-BFSSAR) is a promising sensor for various applications, such as the automatic navigation and landing of maneuvering vehicles. Because of the bistatic forward-looking configuration and the receiver’s maneuvers, conventional image formation algorithms suffer from high computational complexity or small size of a well-focused scene if applied to STMR-BFSSAR. In this article, we propose a wavenumber-domain algorithm for STMR-BFSSAR image formation, which is termed the bistatic-range-Doppler-aperture wavenumber algorithm (BDWA). First, a novel range model in bistatic-range and Doppler-aperture coordinate space instead of conventional Cartesian coordinate space is established by employing the elliptic polar coordinate system and the method of series reversion. The novel range model not only makes the echo’s samples to be regular along the direction of the bistatic-range wavenumber axis but also constructs a curved wavefront close to the true wavefront. Second, an operation termed wavenumber-domain gridding is conceived to regularize the echo’s samples along the Doppler-aperture wavenumber axis, which can be implemented by 1-D interpolation. The proposed algorithm significantly outperforms the conventional algorithms in terms of computational complexity and scene size limits. Both point and distributed targets are simulated for two STMR-BFSSAR systems with different parameters. The simulation results verify the validity and superiority of the proposed BDWA.

A Range Doppler Imaging Algorithm Based on the MSR Spectrum for the Multi-Aperutre Syntactic Sonar

For ignoring the azimuth variation of the “non-hop-go-hop” time, the existing imaging algorithm for multi-aperture have a poor imaging in moderate squint, so this paper proposes a multi-aperture range Doppler imaging algorithm based on the spectrum of method of series reversion. The exact range history is approximated once, and the error can be adjusted by the order of Taylor series expansion. Then, the analytical expression of the two-dimensional spectrum is obtained by the method of series reversion. In signal processing, the signal of the single aperture is handle with the squint range Doppler imaging algorithm, and the azimuth spectrum is superposed to each aperture, to obtain the image. Finally, the effectiveness and correctness of the algorithm is provided by the simulation experiments.

A Generalized Wavefront-Curvature-Corrected Polar Format Algorithm to Focus Bistatic SAR Under Complicated Flight Paths

Bistatic synthetic aperture radar (BiSAR) imaging is faced with two major challenges: large scene imaging and adaptability to unideal platform motion in practice. In order to deal with these two problems, a generalized wavefront-curvature-corrected polar format algorithm (PFA) is proposed in this article. The traditional PFA is little restricted on geometry configuration and platform motion, but its application to large scene imaging is limited by the far-field planar wavefront assumption. To solve this limitation, this article derives the phase error caused by wavefront curvature and analyzes its influence on both geometric distortion and defocusing effect in detail. Based on the analysis, we present a wavefront curvature completely correcting method through space-variant phase compensation using the analytical wavefront curvature phase in wavenumber-domain, which is derived through method of series reversion. What's more, an efficient realization of the space-variant phase compensation based on two-stage image division is given to avoid high overlap rate in the traditional image division method. The proposed method can obtain well focused and geometric undistorted image for BiSAR under complicated flight paths, and it also keeps the logarithmic complexity of traditional PFA. The effectiveness of the proposed method is verified by numerical simulations.

A Fast High-Resolution Imaging Algorithm for Helicopter-Borne Rotating Array SAR Based on 2-D Chirp-Z Transform

In view of the azimuth resolution of the helicopter-borne rotating array synthetic aperture radar (RoASAR) depending on the azimuth reconstruction angle and the sector distortion caused by the azimuth Deramp processing, this paper proposes an efficient helicopter-borne RoASAR high-resolution imaging algorithm based on two-dimensional (2-D) chirp-z transform (CZT). First, the high-order Taylor series expansion is performed on the slant range, and the accurate 2-D spectral expression of the point target is obtained by using the method of series reversion (MSR). Based on that, the space-variant characteristics of the range cell migration (RCM) terms are analyzed. After that, the space-variant RCM and the sector distortion effect caused by the azimuth Deramp processing are removed by using efficient 2-D CZT, thereby increasing the azimuth reconstruction angle and improving the azimuth resolution. The proposed algorithm is efficient without the interpolation operation, and it is easy to implement in real-time. Finally, the simulations are provided to verify the effectiveness of the proposed algorithm.

A CA-NCS algorithm in curve trajectory for smart global village

The development of embedded system makes it convenient to process SAR data in curve trajectory which caused by maneuvers with acceleration. To eliminate the effect of acceleration on imaging, a constant acceleration nonlinear chirp scaling (CA-NCS) algorithm in curve trajectory based on motion compensation is proposed. Through dividing the acceleration into forward-looking and cross-track acceleration, which could be divided into the acceleration vertical to the imaging plane and the other one in imaging plane further, we use vectorial methods to compensate the phase errors caused by acceleration. Moreover, for range migration we compensate it through NCS approach based on the accurate 2-D spectrum acquired by the method of series reversion (MSR). The system integrated with this algorithm can process SAR data in curve trajectory and reduce computation burden. In addition, the integration of the research in this paper and the machine learning will further identify the motion curve of the aircraft with cross-track acceleration quickly, and real-time SAR imaging can be realized which helps to achieve fast target localization in smart global village.

Azimuth Signal Multichannel Reconstruction and Channel Configuration Design for Geosynchronous Spaceborne–Airborne Bistatic SAR

In geosynchronous spaceborne-airborne bistatic synthetic aperture radar (GEO-BiSAR) system, the airborne platform achieves high-resolution imaging by passively receiving the signal from the interested scenario. In this paper, the Doppler characteristics of GEO-BiSAR and the individual contribution of the transmitter and the receiver are first analyzed. The airborne receiver is found to be the dominant contributor for the total Doppler bandwidth, which will lead to Doppler spectrum aliasing regarding the low pulse repetition frequency (PRF) adopted by the GEO-SAR. In order to suppress the Doppler ambiguity without adjusting the PRF of GEO-SAR, azimuth multichannel receiving technique is introduced to the airborne receiver. The multichannel transfer function is derived based on the method of series reversion and the spectrum reconstruction algorithm is then modified for multichannel GEO-BiSAR. Moreover, the reconstruction performance is closely related to the corresponding spacing between each channel (i.e., channel configuration). Therefore, the channel configuration design for GEO-BiSAR aims at optimizing the azimuth ambiguity-to-signal ratio with a satisfactory level of signal-to-noise ratio scaling factor by adjusting the channel configuration. The channel configuration design is modeled as a constrained single objective optimization problem (CSOP). Then, a channel configuration design method based on differential evolution and feasibility rule is proposed to solve the CSOP and obtain the channel configuration for the receiver with the optimal reconstruction performance. Finally, simulations results are presented to verify the effectiveness of the proposed method, and characteristics of channel configuration are analyzed in detail, which can be a practical guide for the implementation of multichannel GEO-BiSAR systems.

A Comparison of Point Target Spectrum Derived for Multi-Aperture SAS

Synthetic aperture sonar has an important application prospect in ocean resources exploration. The SAS receiver antennas generally have multi-aperture configuration, which lead that it is more difficult to obtain the analytical solution of the two-dimensional spectrum. In this paper, the analytical solutions of the multi-aperture SAS with two methods of displaced phase centre antenna (DPCA) and four-order the method of series reversion (MSR) are given respectively, and the phase errors and the imaging results of the two methods under the same conditions are compared by computer simulation. The results of the computer simulation show that MSR has a smaller phase error than the DPCA at the same squint angle, and the DPCA and MSR methods have the same image quality in the small squint angle, but the quality of MSR is much better than that of DPCA in the medium squint angle.

Two‐dimensional frequency decoupling method for curved trajectory synthetic aperture radar imaging

Synthetic aperture radar (SAR) imaging with curved trajectory is difficult due to its severe two-dimensional (2D) space variance. When processing the echo with azimuth-variance, the range cell migration and Doppler modulation of the targets with different azimuth locations are different, which brings a big challenge to image formation. Here, a 2D frequency decoupling method for this case is proposed. Firstly, the range history of the curved trajectory is accurately modelled based on fourth-order Taylor series expansion. Then, the 2D point target reference spectrum of the echo signal is obtained by the method of series reversion. The 2D space variance characteristics of two typical cases with curved trajectory, i.e. geosynchronous SAR and missile-borne SAR, are analysed. Based on the space variance characteristics, a 2D frequency decoupling step is implemented to handle the space-variant phase. Simulations are presented to demonstrate the effectiveness of the proposed algorithm.

Modified Chirp Scaling Algorithm for Circular Trace Scanning Synthetic Aperture Radar

For circular trace scanning synthetic aperture radar (CTSSAR) with a circular track, the conventional hyperbolic equation becomes inadequate to express the range history of a point target accurately, and when it comes to the wide swath observation and imaging, the range variance makes it even harder to focus the target on the edge of the scene. Thus, an expression with high-order terms is needed to approximate the range history and the range variance should also be considered in the imaging algorithm. In this paper, based on the method of series reversion, a fourth-order approximated range model is established for the CTSSAR processing and the 2-D spectrum is derived for the echo signal in CTSSAR with circular trajectory. At the same time, in order to deal with the range-variant range cell migration problem in large-area CTSSAR imaging, a modified chirp scaling algorithm is proposed to realize precise wide swath CTSSAR focusing. Experiments and analyses are performed to validate the effectiveness of the proposed algorithm.

Precise aperture‐dependent motion compensation for high‐resolution synthetic aperture radar imaging

Aperture-dependent motion compensation (MOCO) is crucial for high-resolution airborne synthetic aperture radar (SAR) imaging, which can suppress image degradation especially for SAR systems working at a high-frequency waveband, such as millimetre-wave (MMW) SAR. The current precise topography and aperture (PTA) dependent MOCO algorithm aims to compensate the azimuth-dependent motion error by using a modified azimuth matched filtering. However, the matched filter is deduced with the principle of stationary phase without considering the effects from the azimuth-dependent motion errors, which limits its precision in focusing MMW SAR data involving severe motion errors. This study proposes a refined PTA algorithm with a precise post matched filter established by the method of series reversion. The range-variant characteristic of residual motion errors is also investigated for an improved precision. Simulated and real MMW SAR data sets are used to validate the effectiveness of the proposal.