# High-Resolution Squinted SAR Imaging Algorithm Applied in the Near-Field Environment

Authors: Shijia Wang; Shibo Wang; Wanli Liu
Journal: Complexity
 (2020)

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## Abstract

In the squinted synthetic aperture radar (SAR) imaging of the near-field environment, range-dependent characteristic of squint angle cannot be ignored, which causes azimuth-dependent range cell migration (RCM) after linear range walk correction (LRWC). In this study, an efficient SAR imaging algorithm applied in the near-field environment is proposed. In the processing of the range focusing, LRWC is firstly used to remove the linear RCM. Then, the residual LRCM is expanded into azimuth-invariant and azimuth-variant terms in consideration of the residual LRCM of azimuth-dependent. Range cell migration azimuth scaling (RCMAS) is designed to remove the azimuth-variant term before secondary range compression (SRC) and range compression (RC). In the azimuth focusing, azimuth distortion compensation (ADC) is performed to compensate the azimuth distortion, following which azimuth nonlinear chirp scaling (ANCS) is applied to equalize the frequency modulation (FM) rate for azimuth compression (AC). The simulated results show that more accurate and improved imaging result can be obtained with the proposed algorithm.


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## Body

## 1. Introduction

Due to the ability of high-resolution imaging for the observed scene with any time and any weather, synthetic aperture radar (SAR) has become one of the most attractive radar techniques [1, 2]. There are many imaging modes for SAR, such as stripmap [3, 4], spotlight [5, 6], and sliding spotlight [7, 8]. Particularly, the squinted SAR in the stripmap is widely applied in military and civilian applications. By adjusting the squint angle, SAR can flexibility choose the imaging scene to meet diverse needs. However, the squinted SAR also brings new difficulties and challenges. As the squinted angle increases, the signal property varies a lot and the accurate focusing becomes difficult [1, 9, 10]. Therefore, it is necessary to study imaging algorithm for high-resolution squinted SAR.A lot of algorithms have been proposed in the past years, for example, range-Doppler (RD) algorithm [11–13], chirp scaling (CS) algorithm [14–19], omega-K algorithm [20, 21], and backprojection (BP) algorithm [22, 23]. Theoretically, omega-K algorithm and BP algorithm can be applied in any imaging geometry and any squint angle without any approximation. However, the low efficiency and high computational load limit its practical application. As the squint angle increases, the range cell migration (RCM) and range-azimuth coupling become severe. This degrades the performance of RD algorithm. CS algorithm and its modifications have shown significant potential to process high-resolution squinted SAR data. Sun et al. [15] analyzed the spectrum of squinted data and used the squint minimization processing proposed in [24] to reduce the range-azimuth coupling. Then, an azimuth scaling function was created to equalize the Doppler rate of the targets in the same range cell. Liu et al. [16] first used linear range walk correction (LRWC), secondary range compression (SRC), and range compression (RC) to accomplish the range focusing. Then, in order to solve the problem of the azimuth-dependent frequency modulation (FM) rate caused by LRWC, a filter function and a scaling function, called the nonlinear chirp scaling (NCS) algorithm, were constructed. Chen et al. [17] built the accurate signal model of the missile-borne SAR which took into account not only the missile velocity but also the acceleration. In addition, the azimuth distortion of image was considered in the modified azimuth NCS algorithm. Li et al. [18] and An et al. [19] analyzed the characteristics of the azimuth phase terms in the case of bistatic and monostatic SAR, respectively. The modified scaling coefficients were derived by eliminating the higher-order phase approximation and azimuth-independent cubic phase term.All of the above proposed algorithms processed the squinted SAR data acquired in a far-field environment. In this case, the wavefront of the electromagnetic wave transmitted by a radar was usually considered as a plane because of the very large distance between the radar and target. However, in a near-field environment, the spherical wavefront of the electromagnetic wave cannot be ignored. This will result in the squint angle of target varying with the range. Therefore, the imaging algorithm of high-resolution squinted SAR in the near-field environment needs to be redesigned.The remainder of this paper is organized as follows. The radar signal model is described in Section2. Section 3 gives the proposed imaging algorithm including seven procedures. Simulation results are provided to verify the effectiveness of the proposed algorithm in Section 4. Finally, this study is concluded in Section 5.


## 2. Signal Model

The general geometry of squinted SAR imaging in the near-field environment is shown in Figure1. The radar moves along the direction parallel to the X-axis at a constant velocity v. Or is the synthetic aperture center where the azimuth time ta is equal to zero. θ is the radar squint angle. The imaging scene is located at OXZ plane. L is the distance between the radar trajectory and OXZ plane, and H is the height of the imaged scene center. r0 is the slant range from the target Ttn,R to the radar trajectory along the radar line-of-sight direction. R is the instantaneous slant range between the radar and the target T when the radar is at azimuth time ta. OrF is the half of the synthetic aperture length, whose value is vta. The target T is located at the azimuth time tn, which satisfies HT = vtn. The instantaneous squint angle of target T is denoted by θq, which is expressed as θq=sin−1Ltanθ/r0. The targets A∼E are used in the following simulation.Figure 1
General geometry of SAR imaging in the near-field environment.In Figure1, the instantaneous range R and its Taylor expansion at ta = tn are given by(1)Rta,r0=v2ta−tn2+r02−2vr0ta−tnsinθq≈r0+k1ta−tn+k2ta−tn2+k3ta−tn3+k4ta−tn4,where k1, k2, k3, and k4 are specifically expressed as(2)k1=−vsinθq,k2=v2cos2θq2r0,k3=v3cos2θqsinθq2r02,k4=v4cos2θq4sin2θq−cos2θq8r03.Assuming a Gaussian pulse is transmitted by the radar, the demodulated radar echo signalS0tr,ta received from target Ttn,R is given by(3)S0tr,ta=wrtr−2Rta,r0cwataptr−2Rta,r0cexp−j4πfccRta,r0,where tr is the fast time, ta is the slow time, wrtr is the range time envelope, wata is the azimuth time envelope, c is the speed of light, ptr is the Gaussian pulse, and fc is the carrier frequency.As shown in Figure1, the instantaneous squint angle θq of target T satisfies θq=sin−1Ltanθ/r0. It is obvious that θq is dependent on slant range r0 when L and θ are invariant. In order to clearly explain the influence of range-dependent squint angle on the RCM, simulation with the parameters listed in Table 1 was carried out. Supposed that there were two target points A (1.15 m, 0 m, 0.10 m) and C (1.15 m, 0 m, 1.50 m). As shown in Figure 1, the targets A and C were in the lowest and highest position of the imaging scene, respectively. They were located in the same azimuth cell. The squint angles of A and C were 29.97° and 24.79°, respectively. Figure 2 shows linear RCM (LRCM) and LRCM error between targets A and C. Figure 3 shows quadratic RCM (QRCM) and QRCM error between targets A and C. In Figure 2(b), LRCM error varied linearly with the azimuth time. The maximum LRCM error was approximately equal to the range resolution cell (0.0882 m). Therefore, in order to acquire the high-resolution image, the impact of range-dependent squint angle on the LRCM must be taken into account. In Figure 3(b), QRCM error was much smaller than the range resolution cell. Therefore, the variance of the QRCM and higher-order RCM caused by squint angle can be ignored.Table 1
Simulation parameters.
ParametersValuesSignal carrier frequency4.3 GHzSample frequency50 HzRadar moving velocity0.1 m/sScene center height0.8 mL2.0 mSignal bandwidth1.7 GHzRadar squint angle30°Synthetic aperture length2 mScene width1.4 mFigure 2
Diagram of LRCM and LRCM error between targets A and C: (a) LRCM; (b) LRCM error.
(a)(b)Figure 3
Diagram of QRCM and QRCM error between targets A and C: (a) QRCM; (b) QRCM error.
(a)(b)


## 3. Imaging Algorithm

The proposed algorithm includes the range focusing and the azimuth focusing, as shown in Figure4. The range focusing consists of four steps. First, LRWC is used to remove most of the LRCM. Second, azimuth-dependent residual LRCM is removed through range cell migration azimuth scaling (RCMAS) processing. Third, SRC is applied to remove the coupling of range and azimuth. Finally, the range focusing is accomplished by RC processing. The azimuth focusing is resolved by three steps. The azimuth distortion compensation (ADC) is firstly performed to compensate the azimuth distortion. Then, the azimuth-dependent FM rate is equalized by azimuth NCS (ANCS). Finally, AC is operated to finish the azimuth focusing. The detailed derivations and explanations are introduced in the following.Figure 4
Flowchart of the proposed imaging algorithm.


### 3.1. Linear Range Walk Correction (LRWC)

Substituting (1) into (3) and applying the range fast Fourier transform (FFT) yields(4)S1fr,ta=WrfrwataPfrexp−j4πfc+frcr0exp−j4πfc+frck1ta−tnexp−j4πfc+frck2ta−tn2exp−j4πfc+frck3ta−tn3+k4ta−tn4,where fr is the range frequency, Wrfr is the range frequency envelope, and Pfr is the frequency response of the Gaussian pulse. The second exponential term in (4) represents the LRCM term, which is the major component of RCM. The third and fourth exponential terms are the QRCM and higher-order RCMs. According to (4), LRWC function is given as follows:(5)HLRWCfr,ta=exp−j4πfc+frcvsinθsta,where θs is the squint angle of the reference point. Multiplying (4) with (5) yields(6)S2fr,ta=WrfrwataPfrexp−j4πfc+frcrLRCMexp−j4πfc+frck1resta−tn+k2ta−tn2exp−j4πfc+frck3ta−tn3+k4ta−tn4,where rLRCM = r0 + vsinθstn is the new slant range corresponding to r0 after LRWC processing. k1res = (−vsinθq) − (−vsinθs) is the coefficient of residual LRCM.In (6), the existence of coefficient k1res indicates that LRCM has not been completely removed after LRWC. Figure 5 shows the schematic diagram of LRWC processing. There are five targets P0∼P4, and target P0 is regarded as the reference point. Targets P1, P0, and P2 have the same beam center crossing time ta = 0, and targets P1 and P3 and P2 and P4 are located in the same range cell, respectively. In Figure 5(b), the black dotted lines represent the target LRCMs. LRCM trajectories of P1 and P3 have the same slope because these two targets have the same slant angle. This can also be explained for P2 and P4. Since the target squint angle varies with the slant range, targets P1, P0, and P2 have different LRCMs. As shown in Figure 5(c), the red dotted line denotes LRWC function. After LRWC processing, targets P3, P0, and P4 are located in the same range cell. However, targets P3 and P4 still have the residual LRCMs, and the slopes of their residual LRCM trajectories are different. This problem will be discussed in the next section.Figure 5
Diagram of LRWC processing: (a) distribution of five targets; (b) LRCM trajectory; (c) residual LRCM trajectory.
(a)(b)(c)


### 3.2. Range Cell Migration Azimuth Scaling (RCMAS)

From previous analysis, after LRWC processing, the residual LRCMs of targets P3, P0, and P4 located in the same range cell are azimuth-dependent. Not only the residual LRCM but also the QRCM and higher-order RCM vary with the target azimuth position. As introduced in Section 2, azimuth-dependent residual LRCM needs to be considered, and azimuth-dependent QRCM and higher-order RCM are too smaller which are ignored. Therefore, in order to equalize the azimuth-dependent residual LRCM, k1res is expanded in terms of tn as follows:(7)k1res=k1c+∂k1res∂tntn=k1c+k1atn,where k1c is the azimuth-invariant term and k1a is the coefficient of the azimuth-variant term. Based on (7), a perturbation function is designed to remove the azimuth-variant term of residual LRCM as follows:(8)HRCMASfr,ta=expj4πfc+frcp2ta2,where p2 is the coefficient in the perturbation function and will be determined later. Multiplying (6) with (8) yields(9)S3fr,ta=WrfrwataPfrexp−j4πfc+frcrLRCMexp−j4πfc+frck1resta−tn+k2ta−tn2exp−j4πfc+frck3ta−tn3+k4ta−tn4expj4πfc+frcp2ta2.Then, makingtη=ta−tn and substituting (7) into (9) yields(10)S4fr,ta=WrfrwataPfrexp−j4πfc+frcφtn,tη,where φtn,tη is expressed as(11)φtn,tη=rLRCM+k1ctη+k1atntη+k2tη2+k3tη3+k4tη4−p2tη2+2p2tntη+p2tη2.In order to remove the azimuth-dependent residual LRCM, the azimuth-variant coefficientsk1a should be eliminated. In other words, the azimuth-variant terms should be zero in (11), and it satisfies the following equation:(12)k1atntη−2p2tntη=0.Through solving (12), the coefficient p2 in the perturbation function is obtained as(13)p2=k1a2.Substitute (13) into (10) and rewrite (10) as(14)S5fr,ta=WrfrwataPfrexpj4πfc+frcp2tn2exp−j4πfc+frcrLRCM+k1nta−tnexp−j4πfc+frck2nta−tn2+k3nta−tn3exp−j4πfc+frck4nta−tn4,where the coefficients k1n, k2n, k3n, and k4n are expressed as(15)k1n=k1c,k2n=k2−p2,k3n=k3,k4n=k4.In (14), the first exponential term is a constant which depends on target position. This term has no effect on the focusing process. It can be ignored if a magnitude image is the final product [14, 25]. Therefore, this term is ignored in the following algorithm.


### 3.3. Secondary Range Compression (SRC)

With the principle of stationary phase (POSP) and the method of series reversion (MSR) [26, 27], (14) is transformed into the 2D frequency domain as follows:(16)S6fr,fa=WrfrWafaPfrexpjϕfr,faexp−j2πfatn,where Φfr,fa is expressed as (17), and the second exponential term represents the target azimuth position.(17)ϕfr,fa=−2π2fc+frcrLRCM+2π14k2nc2fc+frfa+2fc+frck1n2+2πk3n8k2n3c2fc+fr2fa+2fc+frck1n3+2π9k3n2−4k2nk4n64k2n5c2fc+fr3fa+2fc+frck1n4.Expanding (17) into a power series of fr yields(18)ϕfr,fa=ϕ0fa+ϕ1fafr+ϕ2fafr2+∑n=3∞ϕnfafrn,where Φ0fa, Φ1fa, Φ2fa, and Φnfa can be found in Appendix.In (18), Φ0fa is the azimuth modulated term, Φ1fa contains the information of residual RCM, Φ2fa represents the coupling of range and azimuth, also called secondary SRC term, and Φnfa (n ≥ 3) shows the higher-order cross coupling terms. In this paper, the range of reference point was used to realize the range compensation, and SRC function is given as follows:(19)HSRCfr,fa=exp−jϕ−ϕ0exp−j4πrLRCMcfr.Multiplying (19) with (16) yields(20)S7fr,fa=WrfrWafaPfrexpjϕ0faexp−j4πrLRCMcfrexp−j2πfatn.


### 3.4. Range Compression (RC)

RC is a procedure in which the echo signal is multiplied by the conjugate of the radar transmitting signal to improve the range resolution. In this study, RC function is given as follows:(21)HRCfr=Pfr¯,where Pfr¯ is the conjugate of the Gaussian pulse. Multiplying (21) with (20) and transforming the result into the range time domain by range inverse FFT (IFFT) yields(22)S8tr,fa=WafasincBrtr−2rLRCMcexpjϕ0faexp−j2πfatn,where Br is the signal bandwidth.


### 3.5. Azimuth Distortion Compensation (ADC)

RewritingΦ0fa as ψfa and expanding ψfa into Taylor’s series of fa at fa = 0 yields(23)ϕ0fa=ψfa≈πψ0+πψ1fa+πψ2fa2+πψ3fa3+πψ4fa4,where ψ0, ψ1, ψ2, ψ3, and ψ4 are expressed in Appendix. The first term is the inherent phase information, the second term represents the azimuth distortion, the third one is the azimuth modulation term, and the fourth and fifth show the cubic and quartic phase terms. The azimuth distortion term may cause the deviation of the target azimuth position. In order to eliminate the azimuth distortion term, the azimuth distortion compensation function is described as follows:(24)HADCfa=exp−jπψ1fa.Substituting (23) into (22) and then multiplying with (24) yields(25)S9tr,fa=WafasincBrtr−2rLRCMcexp−j2πfatnexpjπψ0−π1Kafa2+πψ3fa3+πψ4fa4,where Ka is the azimuth FM rate expressed as follows:(26)Ka=−32k2n5fc8k2n4c+12k1nk2n2k3nc+3k2n2c9k3n2−4k2nk4n.


### 3.6. Azimuth Nonlinear Chirp Scaling (ANCS)

It can be seen that the azimuth FM rateKa contains k1n, k2n, k3n, and k4n, which are expressed in (15). Except k1n, which is a constant, the other three are azimuth-dependent. Therefore, Ka also varies with the target azimuth position. It is necessary to model the azimuth-dependent Ka. First, k2n, k3n, and k4n are expanded in terms of tn as follows:(27)k2n=k2nc+k2natn,k3n=k3nc+k3natn,k4n=k4nc+k4natn,where k2nc, k3nc, and k4nc are the azimuth-invariant terms and k2na, k3na, and k4na are the coefficients of the azimuth-variant terms. Then, Ka is expanded in terms of tn and kept up to the second-order as follows:(28)Ka=Ka0+Ks1tn+Ks2tn2.It can be seen from (28) that the azimuth-dependent characteristic of the azimuth FM rate is very similar to the range-dependent characteristic of range FM rate. Thus, the range nonlinear chirp scaling algorithm can also be applied to azimuth signal for equalizing the azimuth FM rate, which has been described in [15–18]. The derivation of ANCS is expressed in the following.First, the cubic phase termψ3 is evaluated along the azimuth position as follows:(29)ψ3=ψ3c+ψ3atn,where ψ3c is the azimuth-invariant term and ψ3a is the coefficient of the azimuth-variant term. In order to eliminate the azimuth-independent cubic phase term and quartic phase term, the phase compensation factor is given as follows:(30)HPA=exp−jπψ3cfa3exp−jπψ4fa4.Multiplying (30) with (25) yields(31)S10tr,fa=WafasincBrtr−2rLRCMcexp−j2πfatnexpjπψ0−π1Kafa2+πψ3atnfa3.Then, a fourth-order azimuth filter function is designed to filter the phase term, and this function is shown as follows:(32)HIFfa=expjπY3fa3+jπY4fa4,where Y3 and Y4 are the coefficients which will be determined later. Multiplying (32) with (31) yields(33)S11tr,fa=WafasincBrtr−2rLRCMcexp−j2πfatnexpjπψ0−π1Kafa2+πY3+ψ3atnfa3+πY4fa4.Transforming (33) into the azimuth time domain yields(34)S12tr,ta=watasincBrtr−2rLRCMcexpjπKata−tn2expjπY3+ψ3atnKa3ta−tn3expjπY4Ka4ta−tn4.Then, a fourth-order chirp scaling factor is constructed to eliminate the azimuth-dependent azimuth FM rate, and this factor is given by(35)HANCSta=expjπq2ta2+jπq3ta3+jπq4ta4.Multiplying (35) with (34) yields(36)S13tr,ta=watasincBrtr−2rLRCMcexpjπKata−tn2expjπY3+ψ3atnKa3ta−tn3expjπY4Ka4ta−tn4expjπq2ta2+jπq3ta3+jπq4ta4.The POSP is applied to transform (36) into the azimuth frequency domain:(37)S14tr,fa=Wafa−q2tnKa+q2sincBrtr−2rLRCMcexpjΘfa,where(38)Θfa=−2πfa+KatnKa+q2fa+πKa+q22Kafa−q2tn2+q2fa+Katn2+πKa+q23Y3+ψ3atnKa3fa−q2tn3+q3fa+Katn3+πKa+q24Y4Ka4fa−q2tn4+q4fa+Katn4.Substitute (28) into (38); Θfa is approximately expressed as a power series of tn and fa as follows:(39)Θfa=C0q2,q3,q4,Y3,Y4,fa,fa2,fa3,fa4+C1q2,q3,q4,Y3,Y4tnfa+C2q2,q3,q4,Y3,Y4tn2fa+C3q2,q3,q4,Y3,Y4tnfa2+C4q2,q3,q4,Y3,Y4tn2fa2+C5q2,q3,q4,Y3,Y4tnfa3+C6q2,q3,q4,Y3,Y4,tn,tn2,tn2,tn2+Θresq2,q3,q4,Y3,Y4,tn,fa.In (39), the first term is the azimuth-independent phase modulation. The second term is the target real azimuth position. The third can cause geometric distortion along target azimuth direction. The fourth, fifth, and sixth are the azimuth-dependent phase modulation terms. The seventh is independent of the azimuth frequency. And the last one, which represents the summation of other expansion series terms, is so small that it can be neglected.To eliminate the azimuth-dependent modulation and azimuth distortion, the coefficientsC1, C2, C3, C4, and C5 are set as follows:(40)C1=−πα,C2=0,C3=0,C4=0,C5=0,where α is the scaling factor (α≠0.5).Through solving (39) and (40), the coefficients q2, q3, q4, Y3, and Y4 are obtained:(41)q2=2α−1Ka0,q3=2α−13Ks1,q4=10α−5Ks2+32α−1ψ3aKa0312Ka0,Y3=Ks14α−13Ka032α−1,Y4=16α−5Ks2+34α−1ψ3aKa03122α−1Ka04,Substituting (28) and (41) into (37) yields(42)S15tr,fa=Wafa−q2tnKa+q2sincBrtr−2rLRCMcexp−j2πfatn2αexp−jπ1Ka0+q2fa2expjπY3Ka03+q3Ka0+q23fa3+jπY4Ka04+q4Ka0+q24fa4.It can be seen from (42) that the azimuth-dependent modulation and azimuth distortion have been completely eliminated. In (42), the first exponential term contains the azimuth position of target. The residual exponential term is the azimuth compression factor.


### 3.7. Azimuth Compression (AC)

AC is performed after ANCS processing. AC function is given by(43)HACfa=expjπ1Ka0+q2fa2exp−jπY3Ka03+q3Ka0+q23fa3exp−jπY4Ka04+q4Ka0+q24fa4.Multiplying (43) with (42) and transforming the result into the azimuth time domain yields(44)S16tr,ta=sincBrtr−2rLRCMcsincBata−tn2α,where Ba is the Doppler bandwidth. From (44), it is seen that the target is focused at (tn/2α, rLRCM), which needs to be corrected by geometry correction to obtain the real position using(45)tnc=tn2α2α1,rLC=rLRCM−tn2αvsinθs.


## 3.1. Linear Range Walk Correction (LRWC)

Substituting (1) into (3) and applying the range fast Fourier transform (FFT) yields(4)S1fr,ta=WrfrwataPfrexp−j4πfc+frcr0exp−j4πfc+frck1ta−tnexp−j4πfc+frck2ta−tn2exp−j4πfc+frck3ta−tn3+k4ta−tn4,where fr is the range frequency, Wrfr is the range frequency envelope, and Pfr is the frequency response of the Gaussian pulse. The second exponential term in (4) represents the LRCM term, which is the major component of RCM. The third and fourth exponential terms are the QRCM and higher-order RCMs. According to (4), LRWC function is given as follows:(5)HLRWCfr,ta=exp−j4πfc+frcvsinθsta,where θs is the squint angle of the reference point. Multiplying (4) with (5) yields(6)S2fr,ta=WrfrwataPfrexp−j4πfc+frcrLRCMexp−j4πfc+frck1resta−tn+k2ta−tn2exp−j4πfc+frck3ta−tn3+k4ta−tn4,where rLRCM = r0 + vsinθstn is the new slant range corresponding to r0 after LRWC processing. k1res = (−vsinθq) − (−vsinθs) is the coefficient of residual LRCM.In (6), the existence of coefficient k1res indicates that LRCM has not been completely removed after LRWC. Figure 5 shows the schematic diagram of LRWC processing. There are five targets P0∼P4, and target P0 is regarded as the reference point. Targets P1, P0, and P2 have the same beam center crossing time ta = 0, and targets P1 and P3 and P2 and P4 are located in the same range cell, respectively. In Figure 5(b), the black dotted lines represent the target LRCMs. LRCM trajectories of P1 and P3 have the same slope because these two targets have the same slant angle. This can also be explained for P2 and P4. Since the target squint angle varies with the slant range, targets P1, P0, and P2 have different LRCMs. As shown in Figure 5(c), the red dotted line denotes LRWC function. After LRWC processing, targets P3, P0, and P4 are located in the same range cell. However, targets P3 and P4 still have the residual LRCMs, and the slopes of their residual LRCM trajectories are different. This problem will be discussed in the next section.Figure 5
Diagram of LRWC processing: (a) distribution of five targets; (b) LRCM trajectory; (c) residual LRCM trajectory.
(a)(b)(c)


## 3.2. Range Cell Migration Azimuth Scaling (RCMAS)

From previous analysis, after LRWC processing, the residual LRCMs of targets P3, P0, and P4 located in the same range cell are azimuth-dependent. Not only the residual LRCM but also the QRCM and higher-order RCM vary with the target azimuth position. As introduced in Section 2, azimuth-dependent residual LRCM needs to be considered, and azimuth-dependent QRCM and higher-order RCM are too smaller which are ignored. Therefore, in order to equalize the azimuth-dependent residual LRCM, k1res is expanded in terms of tn as follows:(7)k1res=k1c+∂k1res∂tntn=k1c+k1atn,where k1c is the azimuth-invariant term and k1a is the coefficient of the azimuth-variant term. Based on (7), a perturbation function is designed to remove the azimuth-variant term of residual LRCM as follows:(8)HRCMASfr,ta=expj4πfc+frcp2ta2,where p2 is the coefficient in the perturbation function and will be determined later. Multiplying (6) with (8) yields(9)S3fr,ta=WrfrwataPfrexp−j4πfc+frcrLRCMexp−j4πfc+frck1resta−tn+k2ta−tn2exp−j4πfc+frck3ta−tn3+k4ta−tn4expj4πfc+frcp2ta2.Then, makingtη=ta−tn and substituting (7) into (9) yields(10)S4fr,ta=WrfrwataPfrexp−j4πfc+frcφtn,tη,where φtn,tη is expressed as(11)φtn,tη=rLRCM+k1ctη+k1atntη+k2tη2+k3tη3+k4tη4−p2tη2+2p2tntη+p2tη2.In order to remove the azimuth-dependent residual LRCM, the azimuth-variant coefficientsk1a should be eliminated. In other words, the azimuth-variant terms should be zero in (11), and it satisfies the following equation:(12)k1atntη−2p2tntη=0.Through solving (12), the coefficient p2 in the perturbation function is obtained as(13)p2=k1a2.Substitute (13) into (10) and rewrite (10) as(14)S5fr,ta=WrfrwataPfrexpj4πfc+frcp2tn2exp−j4πfc+frcrLRCM+k1nta−tnexp−j4πfc+frck2nta−tn2+k3nta−tn3exp−j4πfc+frck4nta−tn4,where the coefficients k1n, k2n, k3n, and k4n are expressed as(15)k1n=k1c,k2n=k2−p2,k3n=k3,k4n=k4.In (14), the first exponential term is a constant which depends on target position. This term has no effect on the focusing process. It can be ignored if a magnitude image is the final product [14, 25]. Therefore, this term is ignored in the following algorithm.


## 3.3. Secondary Range Compression (SRC)

With the principle of stationary phase (POSP) and the method of series reversion (MSR) [26, 27], (14) is transformed into the 2D frequency domain as follows:(16)S6fr,fa=WrfrWafaPfrexpjϕfr,faexp−j2πfatn,where Φfr,fa is expressed as (17), and the second exponential term represents the target azimuth position.(17)ϕfr,fa=−2π2fc+frcrLRCM+2π14k2nc2fc+frfa+2fc+frck1n2+2πk3n8k2n3c2fc+fr2fa+2fc+frck1n3+2π9k3n2−4k2nk4n64k2n5c2fc+fr3fa+2fc+frck1n4.Expanding (17) into a power series of fr yields(18)ϕfr,fa=ϕ0fa+ϕ1fafr+ϕ2fafr2+∑n=3∞ϕnfafrn,where Φ0fa, Φ1fa, Φ2fa, and Φnfa can be found in Appendix.In (18), Φ0fa is the azimuth modulated term, Φ1fa contains the information of residual RCM, Φ2fa represents the coupling of range and azimuth, also called secondary SRC term, and Φnfa (n ≥ 3) shows the higher-order cross coupling terms. In this paper, the range of reference point was used to realize the range compensation, and SRC function is given as follows:(19)HSRCfr,fa=exp−jϕ−ϕ0exp−j4πrLRCMcfr.Multiplying (19) with (16) yields(20)S7fr,fa=WrfrWafaPfrexpjϕ0faexp−j4πrLRCMcfrexp−j2πfatn.


## 3.4. Range Compression (RC)

RC is a procedure in which the echo signal is multiplied by the conjugate of the radar transmitting signal to improve the range resolution. In this study, RC function is given as follows:(21)HRCfr=Pfr¯,where Pfr¯ is the conjugate of the Gaussian pulse. Multiplying (21) with (20) and transforming the result into the range time domain by range inverse FFT (IFFT) yields(22)S8tr,fa=WafasincBrtr−2rLRCMcexpjϕ0faexp−j2πfatn,where Br is the signal bandwidth.


## 3.5. Azimuth Distortion Compensation (ADC)

RewritingΦ0fa as ψfa and expanding ψfa into Taylor’s series of fa at fa = 0 yields(23)ϕ0fa=ψfa≈πψ0+πψ1fa+πψ2fa2+πψ3fa3+πψ4fa4,where ψ0, ψ1, ψ2, ψ3, and ψ4 are expressed in Appendix. The first term is the inherent phase information, the second term represents the azimuth distortion, the third one is the azimuth modulation term, and the fourth and fifth show the cubic and quartic phase terms. The azimuth distortion term may cause the deviation of the target azimuth position. In order to eliminate the azimuth distortion term, the azimuth distortion compensation function is described as follows:(24)HADCfa=exp−jπψ1fa.Substituting (23) into (22) and then multiplying with (24) yields(25)S9tr,fa=WafasincBrtr−2rLRCMcexp−j2πfatnexpjπψ0−π1Kafa2+πψ3fa3+πψ4fa4,where Ka is the azimuth FM rate expressed as follows:(26)Ka=−32k2n5fc8k2n4c+12k1nk2n2k3nc+3k2n2c9k3n2−4k2nk4n.


## 3.6. Azimuth Nonlinear Chirp Scaling (ANCS)

It can be seen that the azimuth FM rateKa contains k1n, k2n, k3n, and k4n, which are expressed in (15). Except k1n, which is a constant, the other three are azimuth-dependent. Therefore, Ka also varies with the target azimuth position. It is necessary to model the azimuth-dependent Ka. First, k2n, k3n, and k4n are expanded in terms of tn as follows:(27)k2n=k2nc+k2natn,k3n=k3nc+k3natn,k4n=k4nc+k4natn,where k2nc, k3nc, and k4nc are the azimuth-invariant terms and k2na, k3na, and k4na are the coefficients of the azimuth-variant terms. Then, Ka is expanded in terms of tn and kept up to the second-order as follows:(28)Ka=Ka0+Ks1tn+Ks2tn2.It can be seen from (28) that the azimuth-dependent characteristic of the azimuth FM rate is very similar to the range-dependent characteristic of range FM rate. Thus, the range nonlinear chirp scaling algorithm can also be applied to azimuth signal for equalizing the azimuth FM rate, which has been described in [15–18]. The derivation of ANCS is expressed in the following.First, the cubic phase termψ3 is evaluated along the azimuth position as follows:(29)ψ3=ψ3c+ψ3atn,where ψ3c is the azimuth-invariant term and ψ3a is the coefficient of the azimuth-variant term. In order to eliminate the azimuth-independent cubic phase term and quartic phase term, the phase compensation factor is given as follows:(30)HPA=exp−jπψ3cfa3exp−jπψ4fa4.Multiplying (30) with (25) yields(31)S10tr,fa=WafasincBrtr−2rLRCMcexp−j2πfatnexpjπψ0−π1Kafa2+πψ3atnfa3.Then, a fourth-order azimuth filter function is designed to filter the phase term, and this function is shown as follows:(32)HIFfa=expjπY3fa3+jπY4fa4,where Y3 and Y4 are the coefficients which will be determined later. Multiplying (32) with (31) yields(33)S11tr,fa=WafasincBrtr−2rLRCMcexp−j2πfatnexpjπψ0−π1Kafa2+πY3+ψ3atnfa3+πY4fa4.Transforming (33) into the azimuth time domain yields(34)S12tr,ta=watasincBrtr−2rLRCMcexpjπKata−tn2expjπY3+ψ3atnKa3ta−tn3expjπY4Ka4ta−tn4.Then, a fourth-order chirp scaling factor is constructed to eliminate the azimuth-dependent azimuth FM rate, and this factor is given by(35)HANCSta=expjπq2ta2+jπq3ta3+jπq4ta4.Multiplying (35) with (34) yields(36)S13tr,ta=watasincBrtr−2rLRCMcexpjπKata−tn2expjπY3+ψ3atnKa3ta−tn3expjπY4Ka4ta−tn4expjπq2ta2+jπq3ta3+jπq4ta4.The POSP is applied to transform (36) into the azimuth frequency domain:(37)S14tr,fa=Wafa−q2tnKa+q2sincBrtr−2rLRCMcexpjΘfa,where(38)Θfa=−2πfa+KatnKa+q2fa+πKa+q22Kafa−q2tn2+q2fa+Katn2+πKa+q23Y3+ψ3atnKa3fa−q2tn3+q3fa+Katn3+πKa+q24Y4Ka4fa−q2tn4+q4fa+Katn4.Substitute (28) into (38); Θfa is approximately expressed as a power series of tn and fa as follows:(39)Θfa=C0q2,q3,q4,Y3,Y4,fa,fa2,fa3,fa4+C1q2,q3,q4,Y3,Y4tnfa+C2q2,q3,q4,Y3,Y4tn2fa+C3q2,q3,q4,Y3,Y4tnfa2+C4q2,q3,q4,Y3,Y4tn2fa2+C5q2,q3,q4,Y3,Y4tnfa3+C6q2,q3,q4,Y3,Y4,tn,tn2,tn2,tn2+Θresq2,q3,q4,Y3,Y4,tn,fa.In (39), the first term is the azimuth-independent phase modulation. The second term is the target real azimuth position. The third can cause geometric distortion along target azimuth direction. The fourth, fifth, and sixth are the azimuth-dependent phase modulation terms. The seventh is independent of the azimuth frequency. And the last one, which represents the summation of other expansion series terms, is so small that it can be neglected.To eliminate the azimuth-dependent modulation and azimuth distortion, the coefficientsC1, C2, C3, C4, and C5 are set as follows:(40)C1=−πα,C2=0,C3=0,C4=0,C5=0,where α is the scaling factor (α≠0.5).Through solving (39) and (40), the coefficients q2, q3, q4, Y3, and Y4 are obtained:(41)q2=2α−1Ka0,q3=2α−13Ks1,q4=10α−5Ks2+32α−1ψ3aKa0312Ka0,Y3=Ks14α−13Ka032α−1,Y4=16α−5Ks2+34α−1ψ3aKa03122α−1Ka04,Substituting (28) and (41) into (37) yields(42)S15tr,fa=Wafa−q2tnKa+q2sincBrtr−2rLRCMcexp−j2πfatn2αexp−jπ1Ka0+q2fa2expjπY3Ka03+q3Ka0+q23fa3+jπY4Ka04+q4Ka0+q24fa4.It can be seen from (42) that the azimuth-dependent modulation and azimuth distortion have been completely eliminated. In (42), the first exponential term contains the azimuth position of target. The residual exponential term is the azimuth compression factor.


## 3.7. Azimuth Compression (AC)

AC is performed after ANCS processing. AC function is given by(43)HACfa=expjπ1Ka0+q2fa2exp−jπY3Ka03+q3Ka0+q23fa3exp−jπY4Ka04+q4Ka0+q24fa4.Multiplying (43) with (42) and transforming the result into the azimuth time domain yields(44)S16tr,ta=sincBrtr−2rLRCMcsincBata−tn2α,where Ba is the Doppler bandwidth. From (44), it is seen that the target is focused at (tn/2α, rLRCM), which needs to be corrected by geometry correction to obtain the real position using(45)tnc=tn2α2α1,rLC=rLRCM−tn2αvsinθs.


## 4. Simulation

To verify the effectiveness of the proposed algorithm, the simulation with parameters in Table1 was carried out. The Gaussian pulse was defined as (46), and its waveforms in the time and frequency domain are shown in Figure 6.(46)ptr=exp−tr−3×2.2×1092.2×1092cos2π×4.3×109×tr.Table 2
Abbreviations.
Full nameAbbreviationSynthetic aperture radarSARRange cell migrationRCMLinear range walk correctionLRWCRange cell migration azimuth scalingRCMASSecond range compressionSRCRange compressionRCAzimuth distortion compensationADCAzimuth nonlinear chirp scalingANCSFrequency modulationFMAzimuth compressionACRange-DopplerRDChirp scalingCSBackprojectionBPNonlinear chirp scalingNCSLinear range cell migrationLRCMQuadratic range cell migrationQRCMFourier transformFFTInverse Fourier transformIFFTPrinciple of stationary phasePOSPMethod of series reversionMSRExtended nonlinear chirp scalingENCSFigure 6
The Gaussian pulse waveform in the (a) time domain and (b) frequency domain.
(a)(b)There were five targets in the simulation, whose 3D coordinates were A (1.15 m, 0 m, 0.10 m), B (1.15 m, 0 m, 1.04 m), C (1.15 m, 0 m, 1.50 m), D (0.67 m, 0 m, 1.50 m), and E (1.64m, 0m, 0.10m), respectively. The distribution of A∼E in the azimuth and range direction is shown in Figure7. The targets A, B, and C are located in the same azimuth cell. The targets D, B, and E will be lay in the same range cell after LRWC. The target B was chosen as the reference point. The coordinates of these five targets are set so that the performance of the proposed algorithm can be easily shown. The scaling factor α is set to 0.505. In this study, the imaging results are obtained without any processing of the weighting function or side-lobe control.Figure 7
The distribution of targets A∼E.Figure8 shows the RCM trajectories of targets D, B, and E before and after RCMAS. In Figure 8(a), it was observed that RCM trajectory of target B, as well as targets D and E, was approximated into a straight line, which illustrated LRCM was the major component of RCM. After the reference range LRWC, targets D, B, and E were located in the same range cell. LRCM of the target B had been eliminated. However, because of the range-dependent characteristic of the squint angle, targets D and E still had different residual LRCM. As shown in Figure 8(c), after RCMAS, RCM trajectories of D and E were azimuth-invariant, which proved that the azimuth-dependent residual LRCM caused by range-dependent squint angle had been eliminated.Figure 8
RCM trajectories of targets D, B, and E: (a) raw RCM trajectories; (b) RCM trajectories after LRWC; (c) RCM trajectories after RCMAS.
(a)(b)(c)In order to evaluate the performance of the proposed algorithm, two methods, the extended NCS (ENCS) algorithm in [19] and the proposed algorithm, were utilized to process the simulated SAR data. Figure 9 shows the imaging results using these two algorithms. The color in Figure 9 represents the scattering intensity of target. In the simulation imaging, the deep red denoted the target point, and the dark blue showed there was no target. Because of the influence of side lobe, there also existed light blue. It can be easily seen that five targets were focused on the true position in Figure 9(b). However, in Figure 9(a), targets A, C, D, and E all had the azimuth shift compared with the true position. Targets A and E shifted along the azimuth positive direction, and C and D shifted along the azimuth negative direction. In the ENCS algorithm, because the range-dependent characteristic of squint angle was not considered, there was no the residual LRCM term after LRWC, that is, k1res in (6) was equal to zero. This could cause that ψ1 was equal to zero and azimuth distortion compensation processing was not operated. Therefore, the focused targets had the azimuth shift using ENCS algorithm. In addition, there was defocus in targets A, C, D, and E because their RCMs were not fully removed. The proposed algorithm improved the image quality as shown in Figure 9(b). The imaging result contained the range and azimuth information of targets. Table 3 shows the target position error in the simulated images using ENCS algorithm and the proposed algorithm. The range errors of D and E with the ENCS algorithm were much larger than those with the proposed algorithm because RCMs of D and E did not make a precise compensation. In the azimuth, because of the impact of the azimuth shift, the errors of targets A, C, D, and E with ENCS algorithm were much larger than those with the proposed algorithm. This proved that the proposed algorithm performed well in the near-field environment.Figure 9
The imaging results using different algorithms: (a) ENCS algorithm; (b) proposed algorithm.
(a)(b)Table 3
Target position error in the simulated images.
TargetENCS algorithmProposed algorithmAzimuth (m)Range (m)Azimuth (m)Range (m)A0.0930.0060.0160.007B0.0040.0010.0100.002C0.1270.0030.0120.001D0.1250.0540.0020.010E0.1030.0510.0120.008


## 5. Conclusion

In the near-field environment, the range-dependent squint angle of the target needs to be considered, which causes that there exists the residual LRCM of azimuth-dependent after reference range LRWC. In order to overcome the azimuth-dependent characteristic of the residual LRCM, the residual LRCM is expanded into azimuth-invariant and azimuth-variant terms. A novel perturbation function in RCMAS is designed to remove the azimuth-variant term. In the azimuth focusing, a filter function and a scaling function are applied to eliminate the high-order Doppler phases and equalize FM rate. The whole procedures only consist of FFT, IFFT, and multiplication operations without interpolation, which means higher efficiency and easier implementation. The simulation proves that more accurate and improved imaging result can be obtained with the proposed algorithm.



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*Source: 1647957-2020-05-22.xml*