# High-Squint FMCW SAR Imaging via Wavenumber Domain Algorithm

Authors: Hui Bi; Hao Li; Wei Liang
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

Nowadays, because the frequency modulation continuous-wave (FMCW) synthetic aperture radar (SAR) shows good potential in minimal transmission power and weight reduction of radar systems, it has been widely used and has become a common technique in modern short-range high-resolution earth observation. Different from the conventional pulsed mode, the stop-to-go approximation is not valid in FMCW SAR. Thus, the typical SAR imaging methods need to be modified to adapt the continuous-wave scheme in practical data processing. In this letter, a wavenumber domain algorithm (WDA) is derived and used for the FMCW SAR imaging under squint and high-squint cases. With the help of the exact scene recovery ability of the proposed WDA, we can achieve the accurate recovery of the considered scene even when using the collected high-squint data and hence obtain the well-focused high-resolution image of the considered scene. Experimental results with simulated and airborne data verify the effectiveness of the proposed method.


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## Body

## 1. Introduction

Synthetic aperture radar (SAR) is an active microwave imaging system usually mounted on a satellite or airplane to achieve high-resolution earth observation [1–3]. Compared to typical optical imaging, it has all-time and all-weather surveillance capability and hence has been widely used in early-warning, marine surveillance, and disaster monitoring applications [4]. Compared with a conventional pulse mode-based SAR system, frequency modulation continuous-wave (FMCW) SAR has shown good potential in minimal transmission power and weight reduction of radar systems and hence has been widely used and has become a common technique in modern short-range high-resolution earth observation [5–8]. Different from the typical pulsed imaging mode, FMCW SAR transmits the long sweeps in the data collection process, which will cause the variation of instantaneous slant range and hence makes the stop-to-go approximation not valid any more [5, 9]. To achieve the high-resolution FMCW SAR imaging, in recent years, several algorithms have been proposed and have shown good performance, e.g., range Doppler algorithm (RDA) [10, 11], nonlinear frequency scaling algorithm (FSA) [12], chirp-Z transform-based algorithm [13, 14], and back-projection algorithm (BPA) [15]. Compared with other FMCW SAR imaging methods, because BPA can use the recorded platform position, speed, and posture information effectively, it has shown better performance in SAR image focus from the data with serious trajectory errors. But the huge computational cost of BPA severely restricts its application in practical real-time scene monitoring and early warning. Although RDA and FSA do not need to consider the computational complexity, they have to face the problem caused by the slant range approximation, which will result in the imaging error especially under squint and high-squint cases. In SAR imaging, the two-dimensional (2-D) azimuth-range couple is increased as the squint angle increases, which becomes severe and cannot be ignored in the high-squint case. Therefore, novel algorithms are needed to solve this problem especially in the high-squint FMCW SAR imaging. In 1992, Bamler et al. proposed the wavenumber domain algorithm (WDA) and applied it for pulsed SAR imaging successfully [16, 17]. Because an accurate signal form is used for image focus, WDA has shown the excellent high-squint data-based high-resolution SAR imaging ability. In 2010, Wang et al. introduced WDA to the side-looking FMCW SAR data processing [9, 18]. With the help of an accurate focus function, WDA has also achieved the high quality scene recovery under the FMCW SAR imaging mode. In this letter, an accurate representation of the WDA algorithm is derived for the FMCW SAR image focusing in squint and high-squint cases. Experimental results show that the proposed method can achieve accurate recovery of the considered scene even when using the collected high-squint data, which means that it can be regarded as a general technique for FMCW SAR imaging in any squint angle. The rest of this letter is organized as follows. Section 2 provides an introduction of the FMCW SAR imaging model with the squint angle. In Section 3, we show the detailed processing procedures of the proposed WDA method for FMCW SAR imaging under squint and high-squint cases. In Section 4, the proposed WDA-based high-squint FMCW SAR method is validated by simulated data to show its advantages in high-resolution image focus. Experimental results via real airborne data as well as a performance analysis are shown in Section 5. Finally, conclusions are drawn in Section 6.


## 2. FMCW SAR Imaging

In FMCW SAR, letstr=expj2πf0tr·expjπKrtr2 denote the transmitted signal, where f0 is the carrier frequency, Kr is the chirp rate of frequency modulated signal, and tr is the range time. For a single point target Pτ0;r0 with zero Doppler time τ0 and closest slant range r0 between Pτ0;r0 and platform trajectory along the line of sight (LOS) direction, we can express the received backscattered energy as
(1)gτ,tr,τ0;r0=στ0;r0·expj2πf0tr−τd·expjπKrtr−τd2,where τd is the round-trip delay time of the wave propagation. Using the reference signal
(2)greftr=exp−j2πf0tr−τc·exp−jπKrtr−τc2,for the dechirp process in the FMCW SAR system, where rc is the reference slant range for the dechirp-on-receive operation, and τc=2αrc/c with the “Doppler factor” α=c2/c2−v2. Then, the received dechirp signal can be expressed as
(3)gdeτ,tr,τ0;r0=gτ,tr,τ0;r0·greftr=στ0;r0exp−j2πf0τd−τc·exp−j2πKrτd−τctr−τc·expjπKrτd−τc2.Let Rτ denote the instantaneous slant range at time τ=ta+tr with ta being the azimuth time and θ represent the squint angle. Then, the slant range at the receiver can be expressed as Rτ+τd, and
(4)τd=Rτ+Rτ+τdc,with
(5)Rτ=r02+v2τ−τ02−2r0vτ−τ0sinθ,Rτ+τd=r02+v2τ+τd−τ02−2r0vτ+τd−τ0sinθ,where v and c are the velocity of the platform and light, respectively. After applying (5) into (4), τd has an exact expression, i.e.,
(6)τd=2αRτc+v2c2τ−τ0−r0vsinθc2.The last term in (3) is the residual video phase (RVP). After removing the RVP term in the range frequency domain by phase multiplication operation, we have
(7)gdeτ,tr,τ0;r0=στ0;r0·exp−j2πf0τd−τc·exp−j2πKrτd−τctr−τc.Using the replacementfr=Krtr−τc, (7) can be rewritten as
(8)gdeτ,fr,τ0;r0=στ0;r0·exp−j2πf0+frτd−τc.Substitutingτ=ta+tr and introducing (6) into (8), we have
(9)gdefr,ta,tr,τ0;r0=στ0;r0·exp−j4παf0+fr·Rta+trc+v2c2ta+tr−τ0−rcc−r0vsinθc2.Then, the Fourier transformation of (9) with respect to ta is
(10)Gdefa,fr,tr,τ0;r0=∫gdefr,ta,tr,τ0;r0·exp−j2πfatadta=στ0;r0∫exp−jΦ1fa,fr,ta,tr,τ0;r0dta,with
(11)Φ1fa,fr,ta,tr,τ0;r0=4παf0+fr·Rta+trc+v2c2ta+tr−τ0−rcc−r0vsinθc2+2πfata.To obtain the desired reference spectrum of a point target, by using the principle of stationary phase (POSP), we set the first-order derivative ofΦ1 to be zero to obtain the stationary phase center tp as
(12)tp=τ0−tr+r0v·hfa,fr,where
(13)hfa,fr=1−cfa/2αvf0+fr+v/c2sinθ−cfa/2αvf0+fr+v/ccosθ1−cfa/2αvf0+fr+v/c2.After introducing (12) into (10), we have
(14)Gdefa,fr,τ0;r0=στ0;r0·exp−jΦfa,fr,τ0;r0,with
(15)Φfa,fr,τ0;r0=4παf0+frr0c·1+hfa,fr2−2hfa,frsinθ+cfa2αvf0+fr+vchfa,fr−2πfafrKr+2πfaτ0+2πfaτ0−4παf0+frrcc−4παfarcc−4παf0+frr0vsinθc2.


## 3. WDA-Based Squint and High-Squint Imaging

The procedures of WDA for squint and high-squint FMCW SAR imaging are summarized as follows.


### 3.1. RVP Removal

In the range frequency domain, a phase multiplication operation is used to remove the RVP term in (3).


### 3.2. Reference Function Multiplication (RFM)

Then, the reference functionGreffa,fr;r0,rref=expjΦreffa,fr;r0,rref with
(16)Φreffa,fr;r0,rref=4παf0+frrrefc·1+hfa,fr2−2hfa,frsinθ+cfa2αvf0+fr+vchfa,fr−2πfafrKr4παf0+frrcc−4παfarcc−4παf0+frr0vsinθc2,is used to remove the range-invariant phase in the 2-D frequency domain. After RFM, the signal becomes
(17)Gffa,fr,τ0;r0,rref=Gdefa,fr,τ0;r0·Greffa,fr;r0,rref=στ0;r0·exp−jΦrpfa,fr,τ0;r0,rref,with residual phase
(18)Φrpfa,fr,τ0;r0,rref=4παf0+frr0−rrefc·1+hfa,fr2−2hfa,frsinθ+cfa2αvf0+fr+vchfa,fr−2πfaτ0,where rref is the reference slant range. After RFM, the range walk continuous motion is compensated to obtain a coarsely focused image.


### 3.3. Stolt Interpolation

ForGffa,fr,τ0;r0,rref, the Stolt interpolation,
(19)f0+fr1+hfa,fr2−2hfa,frsinθ+cfa2αvf0+fr+vchfa,fr→f0+f1,can be achieved by replacing the range frequency fr with a new one f1. Then, the signal phase Φf in 2-D frequency domain becomes linearly depending on the new range frequency variable f1, i.e.,
(20)Φffa,f1,τ0;r0,rref=4παf0+f1r0−rrefc−2πfaτ0.


### 3.4. Image Focusing

Finally, we perform 2-D inverse Fourier transform on the signal after Stolt interpolation to obtain the focused image, i.e.,(21)gfτ,tr,τ0;r0,rref=waτ−τ0·wrtr−2αr0−rrefc,where wr· and wa· are the envelopes in the range and azimuth directions, respectively.


## 3.1. RVP Removal

In the range frequency domain, a phase multiplication operation is used to remove the RVP term in (3).


## 3.2. Reference Function Multiplication (RFM)

Then, the reference functionGreffa,fr;r0,rref=expjΦreffa,fr;r0,rref with
(16)Φreffa,fr;r0,rref=4παf0+frrrefc·1+hfa,fr2−2hfa,frsinθ+cfa2αvf0+fr+vchfa,fr−2πfafrKr4παf0+frrcc−4παfarcc−4παf0+frr0vsinθc2,is used to remove the range-invariant phase in the 2-D frequency domain. After RFM, the signal becomes
(17)Gffa,fr,τ0;r0,rref=Gdefa,fr,τ0;r0·Greffa,fr;r0,rref=στ0;r0·exp−jΦrpfa,fr,τ0;r0,rref,with residual phase
(18)Φrpfa,fr,τ0;r0,rref=4παf0+frr0−rrefc·1+hfa,fr2−2hfa,frsinθ+cfa2αvf0+fr+vchfa,fr−2πfaτ0,where rref is the reference slant range. After RFM, the range walk continuous motion is compensated to obtain a coarsely focused image.


## 3.3. Stolt Interpolation

ForGffa,fr,τ0;r0,rref, the Stolt interpolation,
(19)f0+fr1+hfa,fr2−2hfa,frsinθ+cfa2αvf0+fr+vchfa,fr→f0+f1,can be achieved by replacing the range frequency fr with a new one f1. Then, the signal phase Φf in 2-D frequency domain becomes linearly depending on the new range frequency variable f1, i.e.,
(20)Φffa,f1,τ0;r0,rref=4παf0+f1r0−rrefc−2πfaτ0.


## 3.4. Image Focusing

Finally, we perform 2-D inverse Fourier transform on the signal after Stolt interpolation to obtain the focused image, i.e.,(21)gfτ,tr,τ0;r0,rref=waτ−τ0·wrtr−2αr0−rrefc,where wr· and wa· are the envelopes in the range and azimuth directions, respectively.


## 4. Experiment via Simulated Data

In order to validate the derived WDA-based squint and high-squint FMCW SAR imaging method, we first perform the experiment based on the simulated data. The simulated scene includes three point targets, which are located on the near (T1, (Rc −25 m, −25 m)), center (T2, (Rc, 0)), and far ranges (T3, (Rc +25 m, 25 m)). Rc=1414m is the center slant range of the considered scene. Simulation parameters are shown in Table 1. Figure 1 shows the WDA-recovered images of three simulated point targets from the collected data with different squint angles. It is seen that when θ=0∘, i.e., the side-looking case, WDA can focus all targets well, which is consistent with the result in [9]. After increasing θ to 30∘, we can see that the proposed method can also achieve the accurate recovery of the considered scene. Further increasing the squint angle to 45∘, FMCW SAR can be regarded as in the high-squint case. The derived WDA-based method also reconstructs the scene successfully. While for the complicated simulated scene with multitargets, as shown in Figure 2, it is seen that the proposed method also can recover the point targets well even under the high-squint case with a 45∘ squint angle.Table 1
Simulation parameters.
ParameterValueRadar center frequency9.6G HzPulse repetition frequency3200 HzBandwidth500 MHzPlatform height1000 mDown-looking angle45°Figure 1
Recovered image from the collected data with squint angleθ=0∘ (left column), θ=30∘ (middle column), and θ=45∘ (right column). The targets from top to bottom rows are T1, T2, and T3, respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Figure 2
Recovered image from the collected data of multitargets with squint angle (a)θ=0∘, (b) θ=30∘, and (c) θ=45∘.
(a)(b)(c)The above experimental results agree with the theoretical analysis, i.e., the derived method achieves the accurate recovery of the considered scene even when using the collected squint and high-squint data. It means that the proposed method in this letter can be regarded as a general technique for both side-looking and squint and high-squint FMCW SAR imaging and can be used further for the practical data processing.


## 5. Experiment via Real Data

In this section, to further support our viewpoint, the derived method is introduced to the practical airborne FMCW SAR data processing. The WDA-focused images of different surveillance regions are shown in Figures3 and 4. From Figures 3 and 4, it is seen that WDA can reconstruct the considered scene accurately, such as buildings and the airport runway, which illustrates that the method presented in this letter is effectively used for real FMCW SAR imaging.Figure 3
Image of airport area recovered by WDA.Figure 4
Image of urban area recovered by WDA.


## 6. Conclusion

In this letter, we introduce WDA to FMCW SAR imaging for the cases of squint and high-squint. The exact theoretical derivation of the squint and high-squint FMCW SAR imaging helps us to construct a universal model for WDA-based image focusing, which can make our derived model be used for FMCW SAR imaging from the collected data with any squint angle. Experimental results based on simulated and practical data validate the derived method in image-exact focusing and its potential in real airborne data processing.



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