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Extended Nonlinear Chirp Scaling Algorithm for High-Resolution Highly Squint SAR Data Focusing Daoxiang An, Member, IEEE, Xiaotao Huang, Member, IEEE, Tian Jin, Member, IEEE, and Zhimin Zhou, Member, IEEE
Abstract—In this paper, an extended nonlinear chirp scaling (ENLCS) algorithm for focusing synthetic aperture radar data acquired at high resolution and highly squint angle is proposed. The whole processing of the ENLCS consists of the following three steps. First, a linear range walk correction is used to remove the linear component of target range cell migration (RCM) and to mitigate the range–azimuth coupling of the 2-D spectrum. Second, a bulk second range compression (SRC) is performed in the 2-D frequency domain for compensating the residual RCM, SRC term, and higher order range–azimuth coupling terms. Third, a modified azimuth NLCS (ANLCS) operation is applied to equalize the azimuth frequency modulation rate for azimuth compression. By adopting higher order approximation processing and by properly selecting the scaling coefficients, the proposed modified ANLCS operation has better accuracy and little image misregistration. The overall focusing procedure of the ENLCS algorithm only involves fast Fourier transform and complex multiplication, which means easier implementation and higher efficiency. The experimental results with simulated data prove the effectiveness of the proposed algorithm. Index Terms—Highly squint angle, linear range walk correction (LRWC), nonlinear chirp scaling (NLCS), synthetic aperture radar (SAR).
I. I NTRODUCTION
N
OWDAYS, synthetic aperture radar (SAR) [1]–[3] has become one of the most attractive radar techniques because it can provide high-resolution images of an observed area, day and night and nearly independent of weather conditions, and can work on different types of platforms, including trucks, helicopters, unmanned aerial vehicles, satellites, etc. In conventional SAR imaging mode, the antenna is pointed at broadside, i.e., the pointing direction of the antenna is nearly perpendicular to the flight path. However, in some special SAR systems, there is a constant offset angle between the pointing direction of the antenna and the zero Doppler direction (i.e., the direction perpendicular to the flight path), and the offset angle is the
Manuscript received February 17, 2011; accepted December 23, 2011. Date of publication March 5, 2012; date of current version August 22, 2012. This work was supported in part by the National Natural Science Foundation of China under Grant 60972121, by the Programs for New Century Excellent Talents in University under grant NCET-07-0223 and NCET-10-0895, by the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (FANEDD) under grant 201046, and by the Hunan Provincial Innovation Foundation for Postgraduate under Grant CX2009B009. The authors are with the School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China (e-mail:
[email protected];
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2012.2183606
so-called squint angle. It should be noted that this definition of squint SAR is only suitable for the stripmap SAR, while a different definition is adopted in the squinted spotlight SAR [1], [2]. Compared to the broadside SAR, the squint SAR has some special advantages. For example, by adjusting the squint angle, the squint SAR can give repeat observation on the same region during its whole flight track. Besides the airborne monostatic SAR system, the squint imaging model is also widely adopted in the satellite SAR [3], [4] or some kinds of bistatic SAR data processing [5], [6]. In contrast to the broadside SAR, the squint SAR brings new complexities and challenges due to its different signal properties [4] induced by the special imaging geometry and makes accurate focusing more difficult. In the past years, a number of algorithms had been proposed for squint SAR imaging, such as the range Doppler algorithm (RDA) [7], [8], the chirp scaling algorithm (CSA) [9] and its modifications [10]–[14], the omega-K algorithm [3], [15], [16], the high squint subaperture (HSS) algorithm [17], [18], the nonlinear chirp scaling (NLCS) algorithm [5], [19]–[22], etc. These algorithms greatly improve the quality of squint SAR image. However, all of the aforementioned algorithms have their advantages and limitations. For example, the RDA and the CSA have relatively simple implementation and higher efficiency, but they are only suitable for processing the low-/moderate-resolution squint SAR data [3], [12]. The reason is that, in these algorithms, range cell migration (RCM) is corrected in the range Doppler domain or 2-D frequency domains. However, as the squint angle increases, a larger linear RCM will be introduced into the echo signal. Then, the range–azimuth coupling, involving the envelope and phase of the echo signal, becomes more serious, which will degrade the performance of the algorithms. Theoretically speaking, the omega-K algorithm can be applied to process the SAR data with any squint angle, but its practical application is limited by the complex and low-efficiency Stolt interpolation operation. Compared to the aforementioned algorithms, the HSS algorithm [18], proposed by Yeo et al., is a good choice for the highly squint SAR data focusing. In the HSS algorithm, a preprocessing of linear range walk correction (LRWC) is first applied to the echo signal to remove the linear component of the signal RCM and to reduce the range–azimuth coupling. Then, a subaperture processing strategy is implemented for resolving the depth of focus (DOF) problem induced by the LRWC operation, resulting in a large scene of well-focused highly squint data. However, the HSS algorithm requires meticulous bookkeeping of indexes when the subapertures are combined coherently, and subaperture processing increases the complexity of the overall imaging procedure.
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The NLCS algorithm is adopted by Wong and Yeo to process the stripmap mode monostatic or bistatic SAR data [5]. In the NLCS algorithm, the LRWC is also first applied to mitigate the range–azimuth coupling. Then, a simple phase multiplies with a cubic phase perturbation function for the monostatic SAR, or a quartic phase perturbation function for the bistatic SAR is performed to identify the azimuth frequency modulation (FM) rates of all targets at the same range gate. At last, the same azimuth matched filter is implemented in that range gate to achieve processing efficiency. In recent years, further developments [6], [20]–[22] of the NLCS algorithm are carried out for handling those kinds of SAR with the more complex imaging geometry. Based on the previous research work [5], this paper explores the application of the NLCS algorithm for processing SAR data acquired at high resolution and highly squint angle. Our research work can be grouped into two main areas: on one hand, in order to improve the imaging accuracy of traditional NLCS algorithm, a more accurate echo signal model is used. Based on the signal model, an LRWC operation is first performed, and then, a bulk second range compression (BSRC) method is introduced to compensate the residual RCM, the second range compression (SRC) term, and the higher order range–azimuth coupling phase. On the other hand, the impacts of higher order approximation terms are analyzed on the expansion of azimuth signal, and then, a modified azimuth NLCS (ANLCS) method is proposed with a higher accuracy imaging performance. By properly selecting the scaling and filtering coefficients, the modified ANLCS cannot only equalize the azimuth FM rate but also eliminate the azimuth-dependent geometric distortion of the focused image, and thus, the nonuniform interpolation for focused image misregistration correction used in the traditional NLCS algorithm is avoided. This paper is organized as follows. Section II gives the echo signal model of the monostatic squint SAR. Section III presents the echo signal LRWC processing, and the BSRC method as well as the range blocks processing method for large swath data focusing is also given. In Section IV, we first carry out a deep analysis on the expansion terms of azimuth signal. Then, a modified ANLCS algorithm for azimuth compression is proposed and detailed derived. Section V shows the flowchart of the proposed extended NLCS (ENLCS) algorithm, and point target simulation in highly squinted imaging is employed to validate the proposed algorithm in Section VI. Finally, we present the concluding remarks in Section VII.
Fig. 1.
Geometric model of squinted SAR imaging.
between the azimuth footprint center O and the target P . Line AA is parallel to line OO and intersects line X at A . Let us assume a linear FM (LFM) pulse is transmitted by the radar, and the demodulated radar signal ss(ta , τ ; r0 ) received from target P (xp ; r0 ) is given by ) ss(ta , τ ; r0 ) = wr [τ − 2r(ta ; r0 )/c] wa (ta 4πfc 2r(ta ; r0 ) r(ta ; r0 ) exp jπγ τ − × exp −j c c (1) where τ is the fast time, ta is the slow time, fc is the carrier frequency, γ is the range chirp rate, and c is the light speed. wr (·) and wa (·) are the range and azimuth envelopes, respectively. According to the relationships of the triangle AP A shown in Fig. 1, we can get the expression of the instantaneous slant range r(ta ; r0 ), which is given by r(ta ; r0 ) = r02 + v 2 (ta − tp )2 − 2r0 v(ta − tp ) sin θs . (2) Expanding (2) at ta = tp to its Taylor series and manipulating it, then we have [18] r(ta ; r0 ) ≈ r0 − v(ta − tp ) sin θs +
v 2 cos θs (ta − tp )2 2r0
v 3 sin θs cos2 θs (ta − tp )3 + · · · 2r02 λ λ λ = r0 − fdc ta − fdr t2a − f˙dr t3a + · · · 2 4 12 +
where
(3)
II. S QUINT SAR S IGNAL M ODEL
⎧ 2v sin θs ⎪ ⎨ fdc = λ 2v fdr = − λr0 cos2 θs ⎪ ⎩ f˙dr = − 6v3 sin θs2cos2 θs . λr
The geometric model in Fig. 1 provides the basis for a simple stripmap squint SAR imaging. The SAR sensor travels along the azimuth direction which is parallel to the X axis, and the move velocity is v. X is the line pass through the target P (xp ; r0 ) and parallel to the X axis. Point O is the platform position at the azimuth time zero, O is the azimuth center of the footprint, r0 = OO is the closest slant range between the target P and platform trajectory along the squint direction, and the instantaneous slant range at time ta is r(ta ; r0 ). The spatial squint angle is defined as θs . xp = vtp is the azimuth distance
In (3), λ = c/fc is the wavelength, and −(λ/2)fdc ta , −(λ/4)fdr t2a , and −(λ/12)f˙dr t3a are called the linear range walk (LRW), the range cell curvature, and the cubic range migration terms, respectively. In broadside SAR, the total RCM is dominated by the quadratic component. However, in squinted SAR, the linear component is usually much larger than the quadratic component. Therefore, in some squinted SAR imaging with moderate exposure time, only the expansion terms up to the secondorder are kept [5], and all of the remainder higher order terms
(4)
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are neglected due to their little impacts. However, the impacts of these higher order terms become serious when the squint angle gets larger, the resolution gets higher, and/or the center frequency gets lower. In this paper, we start the derivation by keeping the Taylor expansion series (3) up to the higher order terms, which gives the algorithm enough accuracy in focusing for high-performance, high-resolution, and highly squinted SAR.
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we assume that the signal has been compressed in the range dimension, and all of the following derivations are carried out based on the range compressed signal. The second exponential term in (6) represents the LRW term. From (6), the LRWC factor is 4π(fr + fc ) vta sin θs . (7) HLRWC (ta , fr ) = exp −j c Multiplying (6) with (7) gives the signal after the LRWC
III. R ANGE F OCUSING VIA THE BSRC Now, let us consider the range-focusing problem of the echo signal in high-resolution highly squint SAR. In this paper, the range-focusing problem is resolved by two steps: LRWC and BSRC. The LRWC is performed in the range frequency and azimuth time domain by using a simple phase factor multiplication, whose function is to remove the linear component of RCM to reduce the cross coupling of echo signal and to simplify the following imaging processing procedure. The BSRC operation is performed in the 2-D frequency domain by a bulk reference function multiply for a selected reference range, whose function is to compensate the residual RCM, SRC phase, and higher order coupling phases. A. LRWC The Taylor expansion series of the slant range r(ta ; r0 ) can be approximately rewritten as r(ta ; r0 ) ≈ r02 + v 2 cos2 θs (ta − tp )2 − v(ta − tp ) v 3 sin θs cos2 θs + (ta − tp )3 . 2r02
(5)
Comparing (3) to (5) is equal to ignoring the odd components of the higher order (≥5th) expansion terms. Because the more higher order terms are kept, (5) has higher accuracy than the traditional third-order approximation [18]. Substituting (5) into (1) and applying the range fast Fourier transform (FFT) yield sS1 (ta , fr ; r0 )
f2 = wa (ta )Wr (fr ) exp −jπ r γ 4π(fr +fc ) vta sin θs ×exp j c 4π(fr +fc ) vtp sin θs ×exp −j c 4π(fr +fc ) 2 2 2 2 r0 +v cos θs (ta −tp ) ×exp −j c 4π(fr +fc ) v 3 sin θs cos2 θs 3 ×exp −j (t −t ) (6) a p c 2r02
where Wr (·) represents the range frequency envelope. The first exponential term in (6) is the range FM term, which can be easily compensated by multiplying its complex conjugate in the range frequency domain. For simplicity of description,
sS2 (ta , fr ; r0 )
4π(fr + fc ) vtp sin θs = wa (ta )Wr (fr ) exp −j c 4π(fr + fc ) 2 2 2 2 r0 + v cos θs (ta − tp ) × exp −j c 4π(fr + fc ) v 3 sin θs cos2 θs 3 × exp −j (ta − tp ) . (8) c 2r02
Inspecting the third exponential term in (8), we can find that the function of LRWC is equal to transforming the squinted SAR signal to the broadside SAR signal, with a change of the moving velocity from v to v cos θs . Fig. 2 shows the trajectories and 2D spectra of simulated targets before and after the LRWC. It is easy to find that the skew phenomenon of the 2-D spectra [3], [4] is corrected after applying the LRWC, which means that the cross coupling of the 2-D spectrum is significantly mitigated. B. BSRC Applying the principle of stationary phase (POSP) to (8), we can transform the echo signal into the 2-D frequency domain as SS2 (fa , fr ; r0 ) = Wa (fa )Wr (fr ) exp(−j2πfa tp ) 4π(fr + fc ) vtp sin θs × exp −j c ⎧ ⎫ 2 2 1/2 ⎬ ⎨ 4πr λfa fr 0 1+ × exp −j − ⎩ ⎭ λ fc 2v cos θs × exp jr0
π(fr + fc ) λ3 fa3 sin θs c 4v 3 cos4 θs
×
1+
fr fc
2 −
λfa 2v cos θs
⎫ 2 −3/2 ⎬ ⎭
(9)
where Wa (·) represents the range frequency envelope. There are four exponential terms in (9). The first exponential term is target azimuth position; the second exponential term is range displacement, which is independent from target azimuth position and squint angle; and the third and fourth exponential terms are the cross-coupling term.
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Fig. 2. Simulated SAR data of a single target with a center frequency of 10 GHz, a squint angle of 50◦ , a spatial resolution of 1.0 m, and a velocity of 100 m/s. (Upper) Original signal. (Lower) After LRWC.
Let Φ(fr , fa ; r0 ) denote the summation of the phases of the third and fourth exponential terms in (9), and we have Φ(fa , fr ; r0 ) 4πr0 =− λ
fr 1+ fc
2 −
λfa 2v cos θs
Expanding the square root terms of (10) into a power series of fr yields Φ(fa , fr ; r0 ) ≈ φ0 (fa ; r0 ) + φ1 (fa ; r0 )fr + φ2 (fa ; r0 )fr2
2 1/2
+
∞
φm (fa ; r0 )frm
(12)
m=3
πr0 (fr + fc ) λ3 fa3 sin θs c 4v 3 cos4 θs 2 2 −3/2 λfa fr × 1+ − fc 2v cos θs 2 1/2 fr 4πr0 2fr 2 D (fa ) + =− + λ fc fc +
where ⎧ πr0 χ(fa ) 4πr0 ⎪ ⎪ φ0 (fa ; r0 ) = − λ D(fa )+ λD3 (fa ) ⎪ ⎪ ⎨ φ1 (fa ; r0 ) = − 4πr0 1 + πr03χ(fa ) − 3πr05χ(fa ) + 4πr0 c D(fa ) cD (fa ) cD (fa ) c 3πr0 χ(fa ) 3πr0 (5−D 2 ) 2πr0 1−D 2 (fa ) ⎪ φ2 (fa ; r0 ) = cfc D3 (fa ) − cfc D5 (fa ) + 2λf 2 D7 χ(fa ) ⎪ ⎪ c ⎪ ⎩ 1 ∂ m Φ(fa ,fr ;r0 ) φm (fa ; r0 ) = m! . m ∂fr (13)
πr0 (fr + fc ) λ3 fa3 sin θs c 4v 3 cos4 θs 2 −3/2 fr 2fr 2 × D (fa ) + + fc fc
Inspecting (12), φ0 (fa ; r0 ) is due to the azimuth modulation, all of the components in φ1 (fa ; r0 ), excluding the last component, are due to the residual RCM, and φ2 (fa ; r0 ) shows the range–azimuth coupling of the 2-D spectrum, which is also called the SRC term. The remainder terms denote the higher order cross-coupling terms. Due to the range-dependent characteristic of the aforementioned terms, it is impossible to perform a precise compensation for all of the targets at different ranges in the 2-D frequency domain. To overcome this problem, it is necessary to analyze the range-dependent characteristic of the residual RCM and the cross-coupling phases. Since the higher order (≥3rd) cross-
+
(10)
where D = 1 − (λfa /2v cos θs )2 denotes the range migration parameter. Let χ(fa ) = (λ3 sin θs /4v 3 cos4 θs )fa3 , and under the following assumption 2fr /fc + (fr /fc )2 1. D2 (fa )
(11)
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Multiplying (14) with (9) and transforming the result into the range Doppler domain (i.e., range time and azimuth frequency domain) by the range inverse FFT yield Ss3 (fa , τ ; r0 )
2rLRWC = Wa (fa )sinc Br τ − c 4π × exp −j vtp sin θs exp(−j2πfa tp ) λ 4πr0 πr0 χ(fa ) × exp −j D(fa ) exp j (15) λ λ D3 (fa )
Fig. 3. Diagram for the residual RCM and QP characteristic evaluation. (a) Diagram of the residual RCMs of targets T0–T2. (b) Diagram of the residual RCMEs between targets T1 and T2 and target T0, respectively. (c) Diagram of the QP values of targets T0–T2. (d) Diagram of the QPEs between targets T1 and T2 and target T0, respectively. From (b) and (d), it is easy to find that the range dependence of the residual RCM and QP is very weak in high-resolution highly squint SAR.
coupling phases are very small [22], we just consider the rangedependent characteristic of the residual RCM and the SRC term, i.e., the quadratic phase (QP). Considering a 0.75-m resolution X-band system with a squint angle of 50◦ , three targets labeled as T0, T1, and T2 are located in the imaged scene at ranges of 15, 17.5, and 20 km, respectively. Fig. 3 shows the diagrams for the residual RCM and QP characteristic evaluation. In Fig. 3(a), it is easy to find that the residual RCM and QP of all the targets are very large, which have serious impacts on target-focusing quality. In Fig. 3(b), it is easy to find that the residual RCM error (RCME) and QP error (QPE) are smaller than one range resolution cell and π/4, respectively, and this means that we can ignore the impacts of the range dependence of the residual RCM and the coupling phases. Based on the aforementioned analysis, we perform the phase compensation in the 2-D frequency domain by setting the reference range rref , and this phase compensation method is the BSRC. The BSRC factor is given by HBRC (fa , fr ; rref ) 4πrref fr = exp −j (Φ(fa , fr ; rref )−φ0 (fa ; rref ))− . c (14) The exponential phase term in (14) is equal to extracting the azimuth modulation term from (12) and keeping the remainder terms, including the residual RCM, SRC, and higher order cross-coupling phases.
where rLRWC = r0 + xp sin θs is the new slant range after the echo signal undergoes the LRWC procedure. The first exponential term in (15) carries the inherent phase information of the target. This term is important in applications such as interferometry and polarimetry, but it has no impacts on the target-focusing quality, so this term is neglected in the following discussion. The second exponential term in (15) represents target azimuth position. The third and fourth exponential terms are due to the azimuth modulation. Inspecting (15), we can find that the LRWC results in an azimuth-dependent displacement in the range envelope. In other words, the targets that have different initial slant ranges now may lie in the same range cell. This displacement brings two problems. First, when the azimuth size of the scene is larger, the displacement might provoke data wrap around in the range direction, which results in the false imaging results. Reference [22] carried out analysis on the data wrap around problem, and the range extension and azimuth block methods are proposed for overcoming the problem. Second, the azimuth-dependent displacement will hamper the direct use of the traditional azimuth filtering processing. This is the so-called DOF problem [17], [18]. The method for resolving this problem will be discussed in detail in Section IV. C. Range Block Processing In the aforementioned BSRC processing, the residual RCM and QP as well as the higher order range–azimuth coupling terms are assumed to be range independent, and the bulk compensation is implemented by multiplying a reference function with a selected range. However, the aforementioned assumption is only valid for the limited range swath size [3], [6]. When the range swath width is larger than the limited swath size, a range block processing technique should be used. Within a processing block, the residual RCM and the QP should be kept within a fraction of one range resolution cell and the assumed tolerance phase, respectively. Moreover, the processing block size can be determined by the imaging geometry and the specific SAR parameters (see the Appendix). In fact, in most squinted SAR, the range dependence of the residual RCM and QP is weak, so the block size can be set to very large in most cases, even in the high-resolution highly squint SAR. Usually, the whole range swath just needs to be divided into two or three blocks. The larger the block
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size is, the higher is the imaging efficiency. Moreover, the BSRC operation integrated with the range blocks processing can be performed before the LRWC operation to avoid the range extension problem induced by the LRWC. Remarks 1: Besides the BSRC method, there are other algorithms that can be used for signal range focusing, such as the Stolt interpolation [23], [24] or the new CSA [22]. Compared to the BSRC, the Stolt interpolation and new CSA can remove the range variance of the residual RCM when performing range focusing, so a relatively larger range block data can be processed. However, the Stolt interpolation and new CSA also have some limitations. First of all, from the analysis in the previous section, we know that the LRWC introduces a displacement in the signal range envelope, which makes the targets that have different initial slant ranges lie in the same range cell. This displacement makes the reference functions used in the Stolt interpolation or new CSA mismatch with the signals of the targets offset the azimuth reference position. Due to this mismatch, new phase errors are generated in range focusing, and these errors will limit the azimuth block size that the Stolt interpolation or the new CSA can deal with. Second, in the new CSA, a reference quadratic function for the selected range is used for the SRC. The residual range-dependent QPE also limits the range block size. At last, compared to the BSRC, the implementation of Stolt interpolation and new CSA is more complicated. In our opinion, in practical conditions, people should make a reasonable choice according to the specific SAR parameters and the imaging algorithm performance (such as the accuracy, efficiency, etc).
IV. A ZIMUTH C OMPRESSION VIA THE M ODIFIED ANLCS In [22], an ANCS algorithm is proposed for performing the azimuth compression. However, the ANCS does not take account of the influence of the higher order phases (HOPs; ≥4th), which may cause defocusing and smearing in the compressed pulse in high-resolution highly squint SAR imaging. Moreover, the ANCS algorithm adopts the first-order approximation of the azimuth FM rate. As the squint angle gets larger or the resolution gets higher, the accuracy of the aforementioned approximation gets lower, which will degrade its performance on processing high-resolution highly squint SAR data. To obtain the higher accuracy azimuth compression, a modified ANLCS algorithm is derived in this section.
Ψ(fa ; r0 ) ≈ − +
Let Ψ(fa ; r0 ) denote the phase of the range-focused signal (15), and rewrite it as 2 1/2 fa 4πr0 Ψ(fa ; r0 ) = −2πtp fa − 1− λ faM
4πr0 r0 − 2πtp fa + f2 λ 2vfaM cos θs a 2πr0 sin θs 3 3 cos θ fa + · · · λfaM s
= ψ0 + ψ1 fa + ψ2 fa2 + ψ3 fa3 +
∞
ψn fan .
(17)
n=4
The coefficients ψ0 , ψ1 , ψ2 , ψ3 , and ψn are given by ⎧ 4πr0 ψ1 = −2πtp ⎪ ⎨ ψ0 = −2πrλ ; 2πr0 sin θs ψ2 = λf 2 0 = − Kπa ; ψ3 = λf 3 cos θs aM aM ⎪ ⎩ 1 ∂ n Ψ(fa ;r0 ) ψn = n! ∂f n
(18)
a
where Ka = −(2v 2 cos2 θs /λr0 ) is the azimuth FM rate at range r0 . For simplicity of description, we called the summation of the terms above the third-order as the HOP. Based on (17), let us carry out an analysis on the influence of the cubic phase and the HOP, which are usually neglected in the low/moderate resolution squint SAR imaging [5], [17], [18]. Considering an X-band SAR system with a slant range of 20 km, squint angle of 50◦ , and moving velocity of 100 m/s, Fig. 4(a) and (b) shows the diagrams for the cubic phase and HOP evaluation for typical azimuth resolutions (ρa ). It is easy to find that the cubic phase is very large, while the HOP is small. However, the cubic phase and HOP increase when the center frequency gets lower, the squint angle gets larger, and/or the resolution gets higher. In general SAR imaging, when the HOP is larger than the tolerance phase (such as π/4 or π/2), its impact on target-focusing quality cannot be neglected anymore. B. Azimuth-Dependent Characteristic Evaluation Now, let us carry out a deep evaluation on the azimuthdependent characteristic of the azimuth FM rate and the cubic phase as well as the HOP. First of all, to model the azimuth dependence of the FM rate, we substitute rLRWC = r0 + vtp sin θs into Ka and expand it in terms of tp and keep up to the second-order as Ka = −
A. Azimuth Echo Signal Model
2 −3/2 fa 2πr0 sin θs fa3 + 1− 3 λ cos θs faM faM
where faM = 2v cos θs /λ. Under the assumption of |fa | |faM |, expanding into a power series in fa yields
≈− −
2v 2 cos2 θs λ(rLRWC − vtp sin θs ) 2v 2 cos2 θs 2v 2 cos2 θs v sin θs − tp λrLRWC λrLRWC rLRWC 2v 2 cos2 θs v 2 sin2 θs 2 tp 2 λrLRWC rLRWC
= KLRWC + KLRWC · Ks · tp (16)
+ KLRWC · Ks2 · t2p
(19)
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Fig. 4. Diagrams for the azimuth expansion phase evaluation for typical resolutions. (a) Cubic phase versus squint angle. (b) HOP versus squint angle. Compared to (a), the HOP is much smaller than the cubic phase, but both of them get larger as the squint angle gets larger, the resolution gets higher, and/or the wavelength gets longer.
where KLRWC and Ks are given by
2
2
cos θs KLRWC = − 2vλrLRWC v sin θs Ks = rLRWC .
(20)
From (19), it is easy to find that the azimuth-dependent characteristic of the azimuth FM rate is very similar to the range-dependent characteristic of the range FM rate. Therefore, we can apply the chirp scaling principle used in the NCS algorithm [12], [14] to azimuth signal for equalizing the azimuth FM rate, and this method is the so-called ANLCS algorithm [5]. Compared to the traditional first-order approximation [5], [6], [22], the approximation of (19) has higher accuracy. Simulated data are used to demonstrate the accuracy improvement by adopting different order approximation. Considering a 0.75m resolution X-band system with a slant range of 20 km, squint angle of 50◦ , and moving velocity of 100 m/s, Fig. 5(a) shows the relationship between the azimuth FM rate and azimuth position. It can be seen that the first-order approximation has larger fit error than the second-order approximation, and the difference gets larger as the azimuth position increases. Furthermore, Fig. 5(b) shows the relationship between the azimuth QPE induced by the different order approximation of the azimuth FM rate and azimuth position. It is easy to find that, for the same azimuth QPE, a much larger scene can be processed by adopting the second-order approximation than by adopting the first-order approximation. Similarly, to model the azimuth dependence of the cubic phase and HOP, substituting rLRWC = r0 + vtp sin θs into the coefficients ψn (n ≥ 3) yields ψn = ψn,LRWC + ψn,s · tp where ψn,LRWC and ψn,s are given by ψn,LRWC = ψr0n rLRWC ψn,s = − ψr0n v sin θs .
(21)
(22)
Substituting (18), (21), and (19) into (17) yields 2rLRWC Ss3 (fa , τ ; rLRWC ) =Wa (fa )sinc Br τ − c 4π(rLRWC −vtp sin θs ) ×exp −j λ ∞ fa2 +j (ψn,LRWC ×exp jψ1 fa −jπ Ka n=3 +ψn,s ·tp )fan .
(23)
The first exponential term in (23) is a constant, which has no impacts on target-focused quality, so we ignore it in the following derivation. Using (23), we can carry out an evaluation on the azimuthdependent characteristic of the cubic phase and HOP. Considering a 1-m resolution X-band system with a slant range of 20 km and a moving velocity of 100 m/s, Fig. 6(a) and (b) shows the diagrams of the azimuth-dependent cubic phase and the azimuth-dependent HOP versus azimuth position for typical squint angles. In Fig. 6, we can find that the azimuth-dependent cubic phase and azimuth-dependent HOP get larger when the squint angle increases and/or the azimuth position increases. In Fig. 6(a), when the azimuth position is larger than 2.4 km at a squint angle of 50◦ or when the azimuth position is larger than 0.7 km at a squint angle of 60◦ , the azimuth-dependent cubic phase is larger than π/4. In such cases, the azimuth-dependent cubic phase will degrade target-focused quality, and its impact must be considered in SAR imaging. Compared to the azimuthdependent cubic phase, the azimuth-dependent HOP is greatly small in a much larger scene size, as shown in Fig. 6(b). Therefore, it is reasonable to ignore the azimuth-dependent HOP in most squint SAR cases. However, it should be noted that the aforementioned phases get larger when the center frequency gets lower, the squint angle gets lager, and/or the resolution gets higher. Therefore, before the imaging, an evaluation for the impacts of the azimuth-dependent phases is suggested to carry out according to the specific SAR parameters.
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Fig. 5. Diagrams for the azimuth FM rate approximation evaluation. (a) Azimuth FM rates for different order approximation. (b) Relationships between the QPE induced by the approximated azimuth FM rate versus azimuth position. It is easy to find that the second-order approximation has higher accuracy than the first-order approximation. In other words, for the same accuracy, the second-order approximation is valid in the scene with larger azimuth size.
Fig. 6. Diagrams for the azimuth-dependent phase evaluation for typical squint angles. (a) Azimuth-dependent cubic phase versus azimuth position. (b) Azimuthdependent HOP versus azimuth position. Compared to (a), the azimuth-dependent HOP is much smaller than the azimuth-dependent cubic phase, but both of them get larger as the squint angle gets larger, the resolution gets higher, and/or the wavelength gets longer.
C. Derivation of the Modified ANLCS Now, let us start the derivation of the proposed modified ANLCS algorithm based on the analysis presented in the previous sections. For simplicity, in the following derivation, we just consider the azimuth dependence of the cubic phase and ignore the azimuth dependence of the HOP. Based on this consideration, the azimuth-dependent component of the cubic phase and the HOP in (23) can be simply compensated by multiplying with their conjugate. The compensation factor is given by ∞ 3 n ψn f a . HI (fa ; rLRWC ) = exp −jψ3,LRWC fa − j n=4
(24) Multiplying (24) with (23) yields 2rLRWC Ss4 (fa , τ ; rLRWC ) = Wa (fa )sinc Br τ − c 2 fa 3 + jψ3,s · tp · fa . (25) × exp jψ1 fa − jπ Ka Before performing the ANLCS operation, a fourth-order azimuth filter should be first applied, which is used for deriving
the expressions for the chirp scaling phase function coefficients. The fourth-order filter factor is given by (26) HIF (fa ) = exp jπ Y3 fa3 + Y4 fa4 . The azimuth filtering step is then represented by Ss5 (fa , τ ; rLRWC ) = Ss4 (fa , τ ; rLRWC ) × HIF (fa ) 2rLRWC = Wa (fa )sinc Br τ − c × exp(jψ1 fa ) f2 × exp −jπ a + jπ Y3 + ψ3,s tp fa3 Ka + jπY4 fa4 (27) = ψ3,s /π. Transforming (27) into the azimuth time where ψ3,s domain by the POSP yields 2rLRWC ss5 (ta , τ ; rLRWC ) = wa (ta − tp )sinc Br τ − c × exp jπKa (ta − tp )2 × exp jπ Y3 + ψ3,s tp Ka3 (ta − tp )3 × exp jπY4 Ka4 (ta − tp )4 . (28)
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In the aforementioned transformation, the cubic and quartic phases are assumed to be very small, and their impacts on the stationary point evaluation are ignored, i.e., the signal (27) is assumed to be an LFM dominantly, with the small non-LFM component. From (28), we can find that the expression of the azimuth modulation signal is similar to the range modulation, but they have quite different time scales. After performing the azimuth fourth-order filtering, a fourthorder chirp scaling factor is introduced for eliminating the azimuth dependence of the azimuth FM rate and the azimuthdependent cubic phase. The fourth-order chirp scaling factor is given by (29) HANLCS (ta ) = exp jπq2 t2a + jπq3 t3a + jπq4 t4a . Multiplying (28) with (29) yields ss6 (ta , τ ; rLRWC )
2rLRWC = wa (ta −tp )sinc Br τ − c ×exp jπq2 t2a +jπq3 t3a +jπq4 t4a ×exp jπKa (ta −tp )2 +jπ Y3 +ψ3,s tp Ka3 (ta −tp )3 (30) +jπY4 Ka4 (ta −tp )4 .
Transforming (30) into the azimuth frequency domain by using the POSP again yields sS6 (fa , τ ; rLRWC ) f a − q2 tp 2rLRWC = Wa sinc Br τ − Ka + q 2 c f a + Ka t p fa × exp −j2π Ka + q 2 π Ka (fa − q2 tp )2 × exp j 2 (Ka + q2 ) +q2 (fa + Ka tp )2 π Y3 + ψ3,s tp Ka3 (fa − q2 tp )3 × exp j 3 (Ka + q2 ) +q3 (fa + Ka tp )3 π Y4 Ka4 (fa − q2 tp )4 × exp j (Ka + q2 )4 4 +q4 (fa + Ka tp ) . (31)
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Let Θ(fa ) denote the phase term of (31), and we have f a + Ka t p fa Ka + q 2 π Ka (fa − q2 tp )2 + 2 (Ka + q2 ) +q2 (fa + Ka tp )2 π Y3 + ψ3,s + tp Ka3 (fa − q2 tp )3 3 (Ka + q2 ) +q3 (fa + Ka tp )3 π Y4 Ka4 (fa − q2 tp )4 + 4 (Ka + q2 ) (32) +q4 (fa + Ka tp )4 .
Θ(fa ) = − 2π
Using the approximation described in (33), shown at the bottom of the page, we can express as a power series of tp and fa and reorder it as follows: Θ(fa ) ≈ A(q2 , q3 , q4 , Y3 , Y4 , fa , fa2 , fa3 , fa4 ) + B(q2 , q3 , q4 , Y3 , Y4 )tp fa + C(q2 , q3 , q4 , Y3 , Y4 )t2p fa + D(q2 , q3 , q4 , Y3 , Y4 )tp fa2 + E(q2 , q3 , q4 , Y3 , Y4 )t2p fa2 + F (q2 , q3 , q4 , Y3 , Y4 )tp fa3 + G(q2 , q3 , q4 , Y3 , Y4 , tp , t2p , t3p , t4p ) + φres (q2 , q3 , q4 , Y3 , Y4 , fa , tp ).
(34)
There are seven terms in (34). The first term is due to the azimuth-independent phase modulation. The second term is the target real azimuth position. The third term is the main factor that causes the geometric distortion of nonuniform shift in the azimuth direction. Because of the term t2p , the shift is always to the right, and the targets furthest away from the scene azimuth center suffer the most shift. In [5], an interpolation method is used for removing this geometric distortion and moves the focused target to the correct position. The fourth and fifth terms as well as the sixth term are due to the azimuth-dependent phase modulation, which are the main factors that limit the size of the DOF. The seventh term is independent of the azimuth frequency, which has no impacts on target-focusing quality, and can be reasonably ignored if a magnitude image is the final product. However, in the interferometric SAR, the impact of this term should be considered. The last term denotes the summation of all of the other expansion series terms, which are very small, and their impacts are neglected. The coefficients of (34) are listed in Table I.
⎧ 1 KLRWC Ks2 KLRWC Ks 1 2 ⎪ Ka +q2 ≈ KLRWC +q2 − (KLRWC +q2 )2 tp − (KLRWC +q2 )2 tp ⎪ ⎪ ⎪ 2KLRWC Ks2 2KLRWC Ks 1 1 ⎪ ⎨ (K +q 2 ≈ (K 2 − (K 3 tp − (KLRWC +q2 )3 − a 2) LRWC +q2 ) LRWC +q2 ) 3KLRWC Ks2 3KLRWC Ks 1 1 ⎪ ⎪ (Ka +q2 )3 ≈ (KLRWC +q2 )3 − (KLRWC +q2 )4 tp − (KLRWC +q2 )4 − ⎪ ⎪ ⎪ ⎩ 4KLRWC Ks2 4KLRWC Ks 1 1 ≈ − t − 4 4 5 p (Ka +q2 ) (KLRWC +q2 ) (KLRWC +q2 ) (KLRWC +q2 )5 −
2 KLRWC Ks2 2 (KLRWC +q2 )4 tp 2 2 3KLRWC Ks 2 (KLRWC +q2 )5 tp 2 6KLRWC Ks2 2 (KLRWC +q2 )6 tp
(33)
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TABLE I C OEFFICIENTS OF (34)
TABLE II C OEFFICIENTS VALUES OF (37)
To eliminate the azimuth-dependent geometric distortion and the azimuth dependence of the azimuth filtering, we set the coefficient B(q2 , q3 , q4 , Y3 , Y4 ) to −π/α, where α = 0.5 is an introduced constant factor. Moreover, all of the other coefficients of the terms, which contain tkp fal , k = 0, l = 0 in (34) are set to zero. Then, we have five equations for five unknowns as ⎧ B(q2 , q3 , q4 , Y3 , Y4 ) = −π/α ⎪ ⎪ ⎪ ⎨ C(q2 , q3 , q4 , Y3 , Y4 ) = 0 (35) D(q2 , q3 , q4 , Y3 , Y4 ) = 0 ⎪ ⎪ , q , q , Y , Y ) = 0 E(q ⎪ 2 3 4 3 4 ⎩ F (q2 , q3 , q4 , Y3 , Y4 ) = 0. Solving (35), the parameters are given by ⎧ q2 = KLRWC (2α − 1) ⎪ ⎪ KLRWC Ks (2α−1) ⎪ ⎪ q 3 = ⎪ 3 ⎪ 2 ⎨ K 2 (10α−5)+3ψ3,s KLRWC (2α−1) q4 = s 12 ⎪ s (4α−1) ⎪ Y3 = 3KK ⎪ 3 (2α−1) ⎪ LRWC ⎪ ⎪ 2 2 Ks (16α−5)+3ψ3,s KLRWC (4α−1) ⎩ Y4 = . 12K 3 (2α−1)
× exp −jπ
(36)
LRWC
Similarly, to obtain the expression of A(q2 , q3 , q4 , Y3 , Y4 , fa , fa2 , fa3 , fa4 ), we rewrite it as follows: A(q2 , q3 , q4 , Y3 , Y4 , fa , fa2 , fa3 , fa4 ) = A1 (q2 , q3 , q4 , Y3 , Y4 )fa + A2 (q2 , q3 , q4 , Y3 , Y4 )fa2 + A3 (q2 , q3 , q4 , Y3 , Y4 )fa3 + A4 (q2 , q3 , q4 , Y3 , Y4 )fa4 .
(37)
The coefficient values of (37) are listed in Table II. Combining (36), Tables I, and II, we can obtain the range Doppler domain echo signal after implementing the fourthorder NLCS operation, which is given by sS6 (fa , τ ; rLRWC ) f a − q2 tp = Wa 1(KLRWC + q2 ) − Ks tp (KLRWC + q2 )2 2rLRWC tp × sinc τ − exp −j2π fa c 2α
1 KLRWC + q2
fa2 + jπ
3 + q3 3 Y3 KLRWC f (KLRWC + q2 )3 a
4 + q4 4 Y4 KLRWC +jπ f . (KLRWC + q2 )4 a
(38)
Inspecting (38), it is evident that the azimuth-dependent characteristic of the azimuth modulation terms has been completely removed, and the azimuth-dependent geometric distortion is also corrected. From the envelope Wa [·] in (38), we can find that there is a shift in the azimuth spectrum. In most cases, this spectrum shift has no impacts on the imaging results because the azimuth signal is usually oversampled. Therefore, as long as the shift in the spectrum envelope stays within this constraint, no aliasing of the spectrum will occur. However, if the spectrum shift exceeds the constraint, the spectrum extension method should be applied to resolve the spectrum aliasing problem [22]. From (38), the azimuth compression factor is given by
1
fa2
HAC (fa ; rLRWC ) = exp jπ K +q LRWC 3 2 Y3 KLRWC +q3 3 ×exp −jπ f (KLRWC +q2 )3 a 4 Y4 KLRWC +q4 4 × exp −jπ f . (KLRWC +q2 )4 a
(39)
Multiplying (39) with (38) and transforming the resulting signal into the azimuth time domain, the final focused SAR image is
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obtained as
2(r0 + vtp sin θs ) ss7 (ta , τ ; r0 ) = sinc τ − c tp q2 tp ×sinc ta − ta . exp j 2α Ka + q 2
(40)
From (40), we can find that the target is focused at position (r0 + xp sin θs , (xp /2α), which is offset from the correct position. The method to resolve this problem is simple. First, multiplying the focused target signal with exp(−j2πxp sin θs fr ) in the range frequency and azimuth time domain, the displacement error in range direction can be eliminated. Second, adjusting the azimuth sampling interval Δx by multiplying with 2α puts the target at a position matching the real azimuth coordinates. D. Selection of the Scaling Factor Another point, the selection of the scaling factor α should be noted, which must be determined carefully and strictly. According to (36), it is easy to find that the factor α determines the values of the introduced parameters q2 , q3 , q4 , Y3 , and Y4 . In practical application, the selection of the scaling factor α should simultaneously meet the following two conditions. Condition 1: The value of α cannot be too close to 0.5. As mentioned in Section IV-B, in the Fourier transformation of the echo signal from (27) to (28) by using POSP, we assume that the cubic and quartic phases are very small, and we ignore their impacts. However, from (36), we can find that parameters Y3 and Y4 get larger as factor α gets closer to 0.5. If the values of Y3 and Y4 get too large, the assumption of ignoring the cubic and quartic phases in the POSP is not valid anymore. In such case, all of the following derivations are also incorrect. Condition 2: The value of α cannot be too far from 0.5. From (38), it is easy to find that the magnitude of the azimuth shift of the Doppler spectrum is equal to q2 tp . Obviously, the farther the scaling factor α from 0.5, the larger is the absolute value of q2 , and the larger is the azimuth shift of the Doppler spectrum. Therefore, the value of factor α should not be too far from 0.5 to avoid the spectrum aliasing problem. From the aforementioned analysis, we can find that the value of α should be set around of 0.5, not too close and not too far. In the SAR imaging processing, people can make decision on the selection of scaling factor α by experience and a small amount of trial-and-error testing according to the specific SAR parameters. Remarks 2: Of note is the relationship of the proposed modified ANLCS algorithm with those previously developed. On one hand, when we ignore the higher order (≥3rd) terms in (17) and the second-order term in (19) and let q2 = q4 = Y3 = Y4 = 0, the algorithm simplifies to the traditional NLCS algorithm [5]. On the other hand, when we ignore the higher order (≥4th) terms in (17) and the second-order term in (19) as well as the azimuth-independent cubic phase in (25) and let q4 = Y4 = 0, the algorithm simplifies to the ANCS algorithm [22]. The traditional NLCS and ANCS were all specifically developed for squint-mode processing, but the approximation error has been dealt with more generally in this paper.
Fig. 7. Flowchart of the proposed algorithm. TABLE III S IMULATION PARAMETERS
Fig. 8. Flight geometry and target distribution in the slant plane for the simulation.
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Fig. 9. Simulated SAR data processed using different imaging algorithms. The subimages from left to right in every row correspond to target 5, target 7, and target 9, respectively. (a) Original RD algorithm. (b) NLCS algorithm. (c) Proposed algorithm.
Referring to the derivation of the generalized frequencydomain algorithm [14], it is easy to obtain a more generalized derivation of the proposed algorithm by considering the azimuth dependence impacts of higher order (≥4th) phase terms in (17). Moreover, the higher order azimuth filter and higher order NLCS factor need to be introduced. However, it should be noted that the derivation gets more complex when the higher order azimuth-dependent phases are considered, so the precision is obtained at the cost of increased complexity. In practice, the required approximation order of the azimuth-dependent phase for image processing should be properly determined according to the given set of the specific SAR parameters. V. F LOWCHART OF THE ENLCS A LGORITHM Fig. 7 shows the flowchart of the proposed ENLCS algorithm. We can find that the whole processing procedure of the ENLCS algorithm only contains FFT transformation and
complex multiplication, so it has easier implementation and higher efficiency. VI. S IMULATION E XPERIMENTS To prove the effectiveness of the proposed algorithm on processing the high-resolution highly squint SAR data, the experiments with simulated data using airborne SAR parameters shown in Table III are carried out. The simulation uses an array of 15 targets, which are located in a 2 km × 1.8 km grid in the slant range plane in azimuth/range, as shown in Fig. 8. Targets 1–9 have the same range position and a distance of 250 m in the azimuth position. Targets 10–15 as well as target 5 have the same azimuth position and a distance of 300 in the range direction. Based on the parameters listed in Table III, it is easy to compute that the azimuth DOF is 71.9 m, which is much smaller than the scene azimuth width.
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Fig. 10. Simulated SAR data processed using different imaging algorithms. The subimages from left to right in every row correspond to target 10, target 11, and target 12, respectively. (a) Original RD algorithm. (b) NLCS algorithm. (c) Proposed algorithm.
In the imaging, target 5 located at the scene center is selected as the reference target. Three algorithms are applied to process the simulated data. To observe the compressed target properties more clearly and to make fair comparison, no weighting function or sidelobe control approach is used. Moreover, the geometric correction is also unused so that the major sidelobes of the targets are aligned with the horizontal and vertical axes, which is helpful for computing the measured parameters of the targets. Fig. 9 shows the subimages of target 5, target 7, and target 9, which are extracted from the entire focused SAR images. All of the extracted subimages are interpolated by zeros padding the spectrum in the energy gaps and inverse transforming. In Fig. 9, the subimages listed at the top row are obtained by traditional RD algorithm in the presence of ignoring the impacts of the residual RCM and the DOF problem. It is found that due to the lack the compensation of residual RCM, the azimuth sidelobes of the targets take a shape of curve and cross several range
bins. Moreover, as the distance from the target and the reference position increases, the target-focusing quality degrades quickly due to the influence of the mismatched azimuth FM rate. The middle row shows the imaging result obtained by the traditional NLCS algorithm [5]. Since the mismatched azimuth FM rate impact is removed, the focusing quality of targets is greatly improved. However, the azimuth sidelobes of the targets still curve due to the uncorrected residual RCM, and the azimuth sidelobes are unsymmetrical due to the uncompensated cubic phase and HOP in (17). The imaging results obtained by the proposed algorithm are shown at the bottom rows. In the imaging, the chirp scaling factor α is set to 0.55. Observing the results, we can find that all of the targets are well focused. In addition, Fig. 10 shows the extracted subimages of target 10, target 11, and target 12, which have the same azimuth position and different range positions. Observing Fig. 10, it is found that, due to lacking compensation of residual RCM, a worse subimage quality is obtained by the RD algorithm and the
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TABLE IV M EASURED PARAMETERS OF THE S ELECTED TARGETS
NLCS algorithm than by the proposed algorithm. In Fig. 10(c), we can find that all of the targets are well focused. The simulated results shown in Figs. 9 and 10 prove that the azimuth dependence of the azimuth FM rate and higher order (≥3rd) phases has no impacts on the focused quality of the targets at different range positions but has serious impacts on the focused quality of the targets at different azimuth positions. To further evaluate the performance of the proposed algorithm, the measured parameters peak sidelobe ratio (PSLR), integrated sidelobe ratio (ISLR), and spatial resolution (3 dB width) of targets 5–9 are calculated, and the results are listed in Table IV. Observing Table IV, it is found that nearly theoretical PSLR, ISLR, and spatial resolutions are obtained by the proposed algorithm. In contrast, some of the measured parameters of the targets in the images obtained by the other two algorithms are relatively worse, especially in the azimuth direction, which are much worse than the theoretical values. At last, Fig. 11 shows the azimuth position error of the targets in the SAR images obtained by the traditional NLCS algorithm and our proposed algorithm, respectively. Observing Fig. 11, we can find that, as the target apart from the reference azimuth position, the target position error of the NLCS algorithm increases. To remove this error, an extra interpolation operation [5] is necessary for moving targets to their real azimuth position. In contrast, in the image obtained by our proposed algorithm, the azimuth position errors of all of the targets are smaller than one azimuth resolution cell (0.5 m), and this level position error can be neglected in most cases. VII. D ISCUSSION AND C ONCLUSION In this paper, an ENLCS for the monostatic high-resolution highly squint SAR data processing has been discussed. In the
Fig. 11.
Azimuth position errors of targets.
algorithm, an LRWC is first performed to mitigate the echo signal cross coupling. Then, a BSRC operation is applied for compensating the residual RCM, SRC term, and higher order cross-coupling terms. At last, a modified ANLCS operation is used for the azimuth compression, which not only resolves the DOF problem with higher accuracy but also corrects the image misregistration without interpolation. From the viewpoint of computational load, the whole imaging procedure of the proposed algorithm only contains the FFT operation and complex multiplication, so it has high efficiency and is suitable for the real-time squint SAR system. The simulation experimental results prove the effectiveness of the proposed algorithm. In the derivation of the proposed algorithm, the Taylor expansion approximations were used several times. All of the approximations are valid based on the special precondition. Therefore, when applying the ENLCS algorithm for the real SAR data imaging, especially in the long wavelength or superresolution highly squint SAR, an evaluation on the validity of the preconditions should be first carried out according to
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specific parameters. The evaluation can help users to determine the reasonable approximations in algorithm derivation and can make sure that all of the approximations are valid and accurate. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their very competent comments and helpful suggestions to improve this paper.
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[21] X. L. Qiu, D. H. Hu, and C. B. Ding, “An improved NLCS algorithm with capability analysis for one-stationary BiSAR,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 10, pp. 3179–3186, Oct. 2008. [22] G. C. Sun, X. W. Jiang, M. D. Xing, Z. J. Qiao, Y. R. Wu, and Z. Bao, “Focus improvement of highly squinted data based on azimuth nonlinear scaling,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 6, pp. 2308– 2322, Jun. 2011. [23] A. Reigber, E. Alivizatos, A. Potsis, and A. Moreira, “Extended wavenumber-domain synthetic aperture radar focusing with integrated motion compensation,” Proc. Inst. Elect. Eng.—Radar Sonar Navig., vol. 153, no. 3, pp. 301–310, Jun. 2006. [24] M. Vandewal, R. Speck, and H. süß, “Efficient and precise processing for squinted spotlight SAR through a modified Stolt mapping,” EURASIP J. Adv. Sig. Proc., vol. 2007, no. 1, pp. 1–7, Jan. 2007.
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Daoxiang An (S’10–M’11) received the B.S., M.S., and Ph.D. degrees in information and communication engineering from the National University of Defense Technology, Changsha, China, in 2004, 2006, and 2011, respectively. He is currently a Lecturer with the National University of Defense Technology. His research interests include ultrawideband SAR image formation, ultrawideband SAR motion compensation, and highresolution SAR image formation.
Xiaotao Huang (M’02) received the B.S. and Ph.D. degrees in information and communication engineering from the National University of Defense Technology, Changsha, China, in 1990 and 1999, respectively. He is currently a Professor with the National University of Defense Technology. His fields of interest include radar theory, signal processing, and radio frequency signal suppression.
Tian Jin (S’07–M’08) received the B.S., M.S., and Ph.D. degrees in information and communication engineering from the National University of Defense Technology, Changsha, China, in 2002, 2003, and 2007, respectively. He is currently an Associate Professor with the National University of Defense Technology. His fields of interest include radar imaging, automatic target detection, and machine learning. Dr. Jin’s Ph.D. dissertation was awarded as the National Excellent Doctoral dissertation of China in 2009.
Zhimin Zhou (M’08) received the B.S. degree in aeronautical radio measurement and control and the M.S. and Ph.D. degrees in information and communication engineering from the National University of Defense Technology, Changsha, China, in 1982, 1989, and 2002, respectively. He is currently a Professor with the National University of Defense Technology. His fields of interest include ultrawideband radar system and real-time signal processing. Dr. Zhou is a Senior Member of the Chinese Institute of Electronics.
