Focusing Highly Squinted FMCW-SAR Data Using the Modified Wavenumber-Domain Algorithm

The three-dimensional (3-D) acceleration, azimuth dependence, and real-time processing are the main issues to be solved in highly squinted frequency-modulation continuous-wave (FMCW) synthetic aperture radar (SAR) with a curved trajectory. To overcome these issues, a modified wavenumber-domain algorithm is developed in this article. The proposed modified wavenumber-domain algorithm mainly includes the following six aspects. First, a modified range model considering 3-D acceleration for a curved trajectory is established. Then, the spectrum compression and rotation operations are performed to guarantee further processing. Afterward, residual phase terms introduced by acceleration are bulk compensated for. Moreover, the azimuth dependence is eliminated by the azimuth resampling to achieve uniform focusing. Subsequently, the range–azimuth coupling is removed by the modified Stolt interpolation. Finally, to avoid image aliasing, the azimuth wavenumber domain is selected to focus the final image. The experimental results of both numerical and raw-measured FMCW-SAR data demonstrate the superiority of the proposed method.

and mapping fields, the SAR system requires compact size, lightweight, and low cost [14], which is suitable for frequencymodulation continuous-wave (FMCW) SAR.
Different from the conventional pulsed SAR, the start-stop approximation will not be held in FMCW-SAR because there exists the variation of the range history caused by the continuous motion during the pulse time, which means that the leading edge and the trailing edge of any transmitted pulse will experience different delay times.Consequently, the conventional pulsed SAR algorithms cannot immediately be applied to FMCW SAR due to the additional range walk and range-coupling terms.In addition, the cases of high-squint angle, azimuth dependence of Doppler parameters, real-time processing, and curved trajectory must also be considered.
For instance, when FMCW-SAR works at high-squint mode, the range-azimuth coupling of the echo signal is generally severe [15], which leads to the difficulty of SAR focusing.To solve this problem, the modified range-Doppler algorithm (RDA) [16] uses the modified range migration correction method to compensate for the continuous motion within the sweep and utilizes improved secondary range compression to mitigate range coupling.However, RDA cannot remove the high-order coupling phase terms, which will degrade the final imaging quality.Therefore, several chirp scaling algorithms [17] are proposed and successfully improve the high-squint SAR imaging performance.Especially, Meta et al. [18] proposed a modified frequency-domain method to correct the Doppler shift in the FMCW-SAR.In addition, to solve the issue of the azimuth dependence, some nonlinear chirp scaling (NCSs) [19], [20], [21], [22], [23] are developed and effectively eliminated the azimuth dependence of the Doppler parameters.Unfortunately, these methods are only suitable for full-aperture data processing.In order to meet the requirement of real-time processing, the subaperture data processing approach is more attractive.Therefore, Liang et al. [24] proposed an extended NCS approach combined with spectrum analysis to achieve small-aperture data processing.However, the spatial-variation (SV) range cell migration (RCM) is ignored in this work.Consequently, the final image-focused performance needs to be further improved.
On the other hand, the Stolt interpolation [25], [26], [27], [28], [29] is an excellent choice to correct the SV RCM without approximation.However, in the high-squint FMCW-SAR mode, an extra range-azimuth coupling phase term exists due to the continuous motion.Therefore, Wang et al. [30] proposed the wavenumber-domain method to focus high-squint FMCW-SAR data.But its range model is established on a straight trajectory with a constant velocity, which means it will be invalid when the SAR platform moves along with a curved trajectory.Consequently, the variation of velocity should be considered in the range model.Although some previous range models, such as the hyperbolic range equation, equivalent squint range model, and their extensions [31], [32], [33], [34], considered the variation of velocity, these models exist residual phase error that cannot be ignored in the case of highly squinted FMCW-SAR with a curved trajectory.Therefore, some time-domain imaging algorithms [35], [36], [37], [38], [39] are proposed to realize well focusing based on the accurate coherent integral, which is theoretically applicable to arbitrary trajectory.However, the vast computation burden limits the practical application of these time-domain algorithms.
Motivated by the discussion above, this article develops a modified wavenumber-domain algorithm for highly squinted FMCW-SAR with a curved trajectory.First, a modified range model considering three-dimensional (3-D) acceleration is established and analyzed in detail.Then, spectrum compression (SC) and rotation processing are performed to mitigate acceleration impacts and realize the maximum 2-D WS utilization.Afterward, the residual phase terms introduced by the 3-D acceleration are bulk compensated for.After that, the azimuth dependence is eliminated by the azimuth resampling to achieve uniform focusing.Subsequently, the modified Stolt interpolation removes the range-azimuth coupling phase terms.Finally, the final image is focused on the azimuth wavenumber domain through the data aligning processing, which can avoid numerous zero-padding operations.
The rest of this article is organized as follows.In Section II, we deeply describe the echo signal model with 3-D acceleration.Section III introduces the proposed method.Section IV presents the computational load of the proposed method.The results of numerical simulation and raw-measured SAR data processing are shown in Section V. Finally, Section VI concludes this article.

II. ECHO SIGNAL MODEL
Fig. 1 shows the highly squinted imaging configuration of FMCW-SAR with a curved trajectory.The SAR platform moves along the curvilinear descending ABD during the synthetic aperture time t a with the velocity vector − → v = (v x , 0, v z ) and the acceleration vector − → a = (a x , a y , a z ) in the coordinates O − XY Z, respectively.At this time, B(0, 0, h) is the position corresponding to the azimuth time t a = 0, and point C is the arbitrary position within the ABD.
From Fig. 1, the instantaneous range of point Q can be expressed as follows: where t a denotes the azimuth slow time and t r represents the range fast time.θ s denotes the squint angle.R 0 is the slant range from position B to target P .t n denotes the zero-Doppler time.
Due to the complicated expression of (1), the 2-D WS is hard to obtain.Therefore, according to the Taylor series expansion, the instantaneous slant range can be expressed as follows: where represents the coefficient of the Taylor series.
However, the Taylor coefficients k i cannot directly display the impacts of 3-D acceleration.Consequently, according to the ideal of motion compensation, (2) can be divided into acceleration term and nonacceleration term, which yields where v = (v 2 x + v 2 z ) 1/2 and v sin θ e = v x sin θ s + v z cos ϕ. ϕ is the angle between the height and slant range.θ e denotes the equivalent squint angle.A ai are the high-order coefficients related to the 3-D acceleration.
From (3), the first component represents the traditional hyperbolic range model; the other parts denote the residual components caused by acceleration.To further analyze the characterization of the range model, based on the parameters in Table I, the numerical experimental simulation of the third part is performed, and the results are shown in Fig. 2. From Fig. 2, it is obvious that the envelope error is far less than a quarter of range unit and the phase error is less than its critical value π/4.It means that the third part in (3) can be neglected.Consequently, the range Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE I SYSTEM PARAMETERS
Suppose the received signal is given by where w r (t r ) and w a (t a ) denote the range envelope and the azimuth envelope, respectively; they will be omitted in the following derivation: Δt r = t r − 2R ref /c, f c represents the carrier frequency, c denotes the speed of light, γ is the range chirp rate, and R ref is the reference slant range.After RVP correcting [31], the echo signal becomes where

III. MODIFIED WAVENUMBER-DOMAIN ALGORITHM
There are three main aspects that should be considered before using the wavenumber-domain algorithm.The first is eliminating acceleration effects and maintaining higher 2-D WS utilization.The second is the spectrum rotation that will cause the azimuth dependence problem.The last is the small-aperture processing without padding zeros.After that, the wavenumberdomain algorithm can be solved to the highly squinted FMCW-SAR with curved trajectory imaging problems.Now, the solutions to these aspects will be further discussed in detail as follows.

A. SC and Rotation
In the highly squinted FMCW-SAR with a curved trajectory, 3-D acceleration will result in azimuth bandwidth exceeding its azimuth support domain, as shown in Fig. 3(a), which is unsuitable for further 2-D WS processing.Therefore, the SC  operation is adopted to eliminate the impact of acceleration, which yields Fig. 3(b) shows the result of SC.However, the skew wavenumber spectrum (SWS) characteristic remains.In order to achieve Stolt interpolation, a rectangular region needs to be selected in the WS.One way to obtain a rectangular WS is to select an inscribed rectangle region in SDSR.Another way is to choose a circumscribed rectangle region out of the SWS.But both two ways will degrade the final image-focusing performance One way to deal with this issue is to correct the SWS into a rectangle WS, which is equivalent to rotating the SWS.The maximum WS utilization rate is realized.However, it will introduce interpolation errors.Fortunately, the linear range walk correction can achieve the same effect.Consequently, the wavenumber spectrum rotation (WSR) factor is The result of WSR is given in Fig. 4, and the signal is renovated as Observing the phase terms of (10), the first part denotes the traditional phase term without acceleration.Spectrum rotation processing introduces the second part.The third part represents the residual phase components introduced by SC.The fourth part denotes the residual high-order phase term, and the other components will be eliminated in the further processing.

B. Residual Phase Components Compensation
After applying azimuth fast Fourier transform (FFT), the signal (10) is renovated as follows: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where with From ( 13), the first phase term denotes an analytical WS; the second phase term is introduced by the spectrum rotation operation; the third exponential term can be directly compensated for in the next derivation; the fourth phase term denotes the high-order residual term; and the fifth phase term is the residual term of distortion.
In order to eliminate the impacts of residual high-order phase terms introduced by the acceleration, a bulk compensation function is given as follows: Through the high-order phase compensation, the signal is renovated as For further analysis, ( 16) is rewritten as where ψ res (k r , k x ) denotes the disturbance component introduced by the 2D-WS approximation, and ξ(k r , x n ) denotes the image deformation, which has no impact on the image focusing and can be ignored in the following analysis.ϕ(Δk r , k x ) represents the residual envelope.For further analysis, the simulation is made, as shown in Fig. 5, and the simulation parameters are shown in Table I.
From the above simulation results, we can see that the maximum residual envelope error is far less than a quarter of the range cell, and the maximum residual phase error exceeds π/4.Therefore, the effect of the residual envelope error can be ignored, while the impact of residual phase error should be considered.Consequently, the residual phase error compensation function can be given by The echo signal is renovated as

C. Azimuth Resampling and 2-D Decoupling
After the spectrum rotation processing, the original range position R 0 will be renovated as R 0 = R 0 + x n sin θ e .Therefore, the signal is simultaneously updated to From ( 21), there exists the azimuth dependence in the second exponential term due to the coupling between the quadratic term of azimuth wavenumber and azimuth position.In order to solve this issue, an azimuth wavenumber resampling method is performed, which yields After azimuth resampling, the signal is rewritten as (23) Observing (23), it is evident that the azimuth dependence has been eliminated.Furthermore, only the first order of azimuth wavenumber k x is related to the x n .
In addition, the second exponential term contains rangeazimuth coupling, which can be well decoupled through Stolt mapping.Hence, a reference function multiplication is introduced before employing Stolt mapping to accomplish bulk Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.azimuth compression, which yields where Rs is the reference range corresponding to the scene center.Then, the signal is expressed as where ΔR = R 0 − R s , and the modified Stolt mapping is designed as After the modified Stolt interpolation, the signal is written as Obviously, the 2-D coupling is eliminated, and only the linear term of range wavenumber k y exists in the range phase term.

D. Data Aligning and Focusing Processing
In the condition of subaperture data processing, the interval of azimuth distance corresponds to short in highly squinted FMCW-SAR.Once the final image is focused on the 2-D time domain, the targets will be located at the wrong position, which will lead to the image aliasing.As shown in Fig. 6(a), the horizontal dimension represents the azimuth wavenumber axis.The vertical direction denotes the phase change rate axis corresponding to the azimuth wavenumber.Assuming that there exist three targets A, B, and C at one range unit.The time-frequency distribution lines (TFDLs) are presented, where the dashed TFDLs denote the real curves after aliasing, and the solid TFDLs represent the ideal curves before aliasing.
One method to eliminate image aliasing is to pad zeros in the azimuth time domain.However, it will extremely increase the computational burden in the high-squint SAR mode.Consequently, a more efficient method is needed.Therefore, ( 29) is multiplied by a uniform data aligning function, which yields The TFDLs are shown in Fig. 6(b).The dashed lines will be reflected in the opposite direction.Fig. 6(c) shows the final result of data aligning processing; it is evident that the azimuth aliasing Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
has been removed, which yields At this moment, applying the range IFFT and the azimuth IFFT, the signal is renovated as where A r is a constant value.γ a = 2π/(λR 0 ) is the slope rate of the azimuth phase.It should be mentioned that the parabolic approximation principle is used in the above pro- In order to select the azimuth wavenumber domain as the final focusing domain, the secondorder-deramp function is given by After azimuth deramp processing, the signal is expressed as • exp (jγ a x n cos θ e X a ) where the first exponential is only related to the linear term of the azimuth position, and the second exponential denotes the geometric deformation.Then, performing azimuth FFT, the signal becomes where A a is a constant related to small-aperture length.Fig. 7 shows the flowchart of the proposed algorithm.

IV. COMPUTATIONAL COST ANALYSIS
As mentioned above, the proposed algorithm mainly includes four procedures.
1) SC and rotation; 2) residual phase components compensation; 3) the azimuth resampling and 2-D decoupling; 4) the data aligning and focusing processing.Hence, in order to further analyze the computational complexity of the proposed algorithm, the computation cost is discussed here in detail.
Assuming that the processed echo data are a matrix of N r × N a size, where N r and N a denote the sample points in the range and azimuth direction, respectively.As shown in Fig. 7, the whole algorithm includes one interpolation, seven complex multiplications, and five FFTs.According to the authors in [38] and [39], one interpolation needs 2(2M ker − 1)N r N a operations, for each complex multiplication needs 6N r N a operations, and the floating-point process of each FFT can be derived   as 5N r N a log 2 [N r/a ].Consequently, the computation load of the whole procedure can be calculated as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.which is of order O(N 2 log 2 N ), where N is the 1-D size of the echo data.M ker denotes the interpolation kernel.

A. Numerical Simulation
In this section, the numerical simulations are performed to further demonstrate the effectiveness of the proposed algorithm.The numerical simulations use a highly squinted FMCW-SAR with a curved configuration, as shown in Fig. 8.The simulation parameters are listed in Table I.The uniform scattering of point targets on the ground is a 3 × 3 matrix.The whole scene is about 400 m in the range and azimuth direction.Fig. 9 shows the numerical simulation result achieved by the proposed algorithm.The range resolution and the azimuth resolution are both about 0.15 m.
In order to better evaluate the superiority of the proposed algorithm, a comparison of the proposed algorithm with the modified RD method [15] and the wavenumber method [27] by the numerical simulation is implemented, as shown in Fig. 10.The azimuth profiles of points T1-T3 achieved by the modified RD method are shown in Fig. 10(a); although the continuous motion has been removed, the high-order 2-D coupling phase terms introduced by high-squint angle are not considered.Hence, targets T1 and T3 are seriously defocused.Fig. 10(b) displays the azimuth profiles of targets T1-T3 achieved by the wavenumber-domain method.The range-dependent second-and higher order range-azimuth coupling components in the high-squint case have been well analyzed and considered.However, the variation of velocity is ignored in its range model.Therefore, targets T1 and T3 are defocused.The azimuth profiles of targets T1-T3 achieved by the proposed algorithm are given in Fig. 10(c).Because the 3-D acceleration is considered in the range model, the azimuth dependence is removed by the azimuth resampling approach, and the modified Stolt interpolation removes the range-azimuth coupling term.Consequently, targets T1-T3 are well focused.
Furthermore, for further quantitative analysis, the performance parameters of peak sidelobe ratio (PSLR) and integrated sidelobe (ISLR) of the three methods are given in Table II.The proposed method's performance parameters are close to the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.theoretical values of PSLR (−13.2 dB) and ISLR (−9.8 dB).In addition, to fully demonstrate the performance of the proposed method, the contour plots of targets T1-T3 achieved by three methods are compared in Fig. 11.It can be easily found that the results of the modified RD method [17] are worst.The targets T1 and T3 are completely defocused, as shown in Fig. 11(a).The wavenumber-domain method [30] results are given in Fig. 11(b).It is obvious that the main lobes and sidelobes of the targets T1 and T3 are coupled with each other.Fig. 11(c) shows the contour plots of the proposed algorithm.The main lobe and sidelobes of three targets, T1-T3, are well separated from each other and present an ideal "cross," which validates the excellent focusing performance.

B. Raw Data Processing
In the following, the raw-measured data processing is implemented on the highly squinted FMCW-SAR with a curved trajectory.The FMCW-SAR system works at strip-map mode with ka-band.
The system has a range bandwidth of 1.2 GHz.The squint angle is about 40 • .The center slant range is about 1.6 km.The velocity and acceleration vectors of transmitter are about (82.26, 0.36, −4.21) m/s and (0.61, 0.23, −1.82) m/s 2 , respectively.The image scene is about 600 m in the range direction and 420 m in the cross-range direction.The range resolution and cross-range resolution are both about 0.15 m.Then, a comparison of the modified RD method [18], the wavenumber-domain method [32], and the proposed method is made.
The result of real data achieved by the RD method is given in Fig. 12.Because the high-order 2-D coupling phase terms are neglected; thus, it cannot deal with highly squinted FMCW SAR data.Fig. 13 gives the result of real data processed by the wavenumber-domain method.Although the high-order rangeazimuth coupling components have been removed, its range model does not consider the velocity variation.Consequently, some residual phase errors introduced by the acceleration exist, which make the final image defocused, as shown in Fig. 15(b).Fig. 14 displays the results of real data processed by the proposed method.Because the velocity variation is considered in the range  model, the issue of azimuth dependence is solved by azimuth resampling, and the modified Stolt interpolation removes the range-azimuth coupling term.Consequently, the final image is well focused, and the image details have been well preserved, as shown in Fig. 15(c).
In addition, to further demonstrate the effectiveness of the proposed method, the comparison of the selected subdomain is shown in Fig. 15(c).The white dashed circle chooses an isolated point, and its azimuth impulse response via the three methods mentioned above is shown in Fig. 16.Furthermore, the performance parameters' of PSLR and ISLR of the three methods are given in Table III.It is obvious that the proposed method achieves the best image performance.

VI. CONCLUSION
This article develops a modified wavenumber-domain algorithm for highly squinted FMCW-SAR with a curved trajectory.First, the 3-D acceleration is considered in the modified range model.Then, the SC and rotation operations are performed to guarantee the 2-D WS.Afterward, the residual phase terms introduced by acceleration are bulk compensated for.Subsequently, the azimuth resampling and the modified Stolt interpolation methods are adopted to eliminate the azimuth dependence and the 2-D coupling, respectively.Moreover, to avoid padding zeros, the azimuth wavenumber domain is selected to focus the final image through data aligning processing.Finally, the numerical simulation and raw-measured FMCW-SAR data processing are performed to demonstrate the superiority of the proposed algorithm.Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 9 .
Fig. 9. Imaging results of point targets by the proposed method.

Fig. 12 .
Fig. 12. Result of raw data processed by the modified RD method.

Fig. 13 .
Fig. 13.Result of raw data processed by the wavenumber-domain method.

Fig. 14 .
Fig. 14. Results of real data processed by the proposed method.

Fig. 15 .
Fig. 15.Comparison of the selected subdomain.(a) Result processed by the modified RD method.(b) Result processed by the wavenumber-domain method.(c) Result processed by the proposed method.

Tinghao Zhang (
Graduate Student Member, IEEE) was born in Xianyang, China, in 1996.He received the B.S. degree in electronic and information engineering in 2018 from Xi'an University of Technology, Xi'an, China, where he is currently working toward the Ph.D. degree in signal processing with the National Laboratory of Radar Signal Processing.He is currently the Vice Chair of the IEEE GRSS Xidian University student branch chapter.His research interests include monostatic/bistatic SAR imaging and motion compensation.Yachao Li (Member, IEEE) was born in Jiangxi Province, China, in 1981.He received the M.S. and Ph.D. degrees in electrical engineering from Xidian University, Xi'an, China, in 2005 and 2008, respectively.He is currently a Professor with Xidian University.His current research interests include SAR/ISAR imaging, missileborne SAR imaging, ground-moving target indication, matching and orientation of SAR image, real-time signal processing based on FPGA and DSP technology, and distributed radar.