An Efficient Imaging Method for Medium-Earth-Orbit Multichannel SAR-GMTI Systems

Abstract

Medium-Earth-orbit (MEO) synthetic aperture radar (SAR) has the advantages of short revisit time and wide coverage, and thus is a potential tool for implementing ground moving target indication (GMTI) tasks. In the paper, aiming at MEO SAR’s problems of low signal-to-noise ratio and limited computation resource, an efficient imaging method is proposed for MEO multichannel SAR-GMTI systems with relatively low resolution. The proposed imaging method is designed with the consideration of both static scenes and ground moving targets, and it can simultaneously correct the range cell migrations of static scenes and multiple moving targets of no Doppler ambiguity. It needs only four Fourier transforms and twice phase multiplications, and thus is computationally efficient. Moreover, moving targets’ signal characteristics, including the azimuth and range displacements and along-track interferometric phase, in the SAR image obtained by the proposed imaging method are figured out. Experimental results validate the proposed imaging method and the theoretical analyses.

1. Introduction

Synthetic aperture radar (SAR) is an active microwave sensor that is able to obtain a high-resolution image of a scene, regardless of the weather and daylight [1,2]. Moreover, modern SAR usually has the operation mode of ground moving target indication (GMTI). SAR-GMTI systems can image the illustrated area and indicate ground moving targets in the SAR image simultaneously, and have played an important role in applications such as traffic monitoring and military reconnaissance [3,4,5,6,7].

With the development of satellite technology, GMTI with spaceborne SAR has attracted a lot of attentions in recent years [8,9,10]. Low-Earth-orbit (LEO) spaceborne SAR-GMTI has been widely studied. However, since the altitude of the orbit is relatively low, LEO SARs suffer the problems of long revisit time and limited coverage, which severely limits their applications in GMTI. To address this issue, many attentions have been paid to SARs with higher orbital altitude [11,12,13,14], such as medium-Earth-orbit (MEO) and geosynchronous-orbit (GEO) SARs. Compared with GEO SAR, MEO SAR has shorter synthetic aperture time, lower transmission cost and lower system complexity, and thus is more suitable for GMTI applications.

Many studies on MEO SAR have been conducted in recent years [15,16,17,18,19,20]. The observation performance of MEO SAR and the signal characteristics of static targets had been studied in [15]. In [16], the author studied the orbit design, coverage, and revisit time of MEO SAR, and concluded that MEO SAR had great value due to its advantages of wide coverage and short revisit time. In [17,18,19], the imaging of the stationary scene was studied, and several imaging algorithms that could address the 2-D spatial of the signal and the curvature of the radar track were proposed. However, studies on MEO SAR-GMTI are still rare.

The focusing of ground moving targets can improve the signal-to-clutter-noise ratio (SCNR), thus improving the GMTI performance of the system [21,22]. Particularly for MEO SAR, the high orbital altitude makes it suffer the problems of low signal-to-noise ratio (SNR) and limited computation resource. Therefore, it is essential to propose an imaging method with high computational efficiency for MEO SAR-GMTI systems to improve their performance. However, the existing imaging methods of MEO SAR [17,18,19] are designed only for the stationary scene and thus may not be suitable for SAR-GMTI systems. For example, [19] proposed an imaging algorithm based on a novel coordinate system. Since the imaging coordinate used in this algorithm depends on the Doppler parameters of the stationary scene, it cannot accurately correct the range cell migrations (RCMs) of moving targets.

In this paper, we propose an efficient imaging method suitable for the MEO SAR-GMTI system. This method is designed with the consideration of both the stationary scene and ground moving targets. It utilizes the fact that, in a 2-D frequency domain, the couplings between the range and azimuth of static targets and the moving targets of no Doppler ambiguity are approximately identical, and thus the RCMs of these targets can be corrected simultaneously via a phase multiplication. The proposed imaging method is very efficient because it needs only four Fourier transforms (FTs) and twice phase multiplications. Thus, the proposed imaging method is very suitable for MEO SAR, which suffers the problem of limited energy and computation resources.

The analysis and understanding of moving targets’ signal characteristics in the image domain are very important for the development of the GMTI method [23], e.g., the clutter suppression and motion parameter estimation methods. Therefore, one other objective of this paper is to reveal moving targets’ signal characteristics in the SAR image obtained by the proposed imaging method. Specifically, in the paper, the analytical expressions for the azimuth and range displacements and the along-track interferometric (ATI) phase of the target are derived, and their variation with the target’s parameters are investigated.

Note that the MEO SAR-GMTI system in this paper is assumed to have a relatively low resolution. The considerations are as follows: (1) A relatively low resolution can improve the SNR and reduce the data volume. This is important for MEO SAR, which suffers limited energy and computation resources and relatively high system complexity because of the increased orbital altitude; (2) For GMTI, a relatively low resolution can reduce the probability of Doppler ambiguity of moving targets.

The remainder of this paper is organized as follows. In Section 2, the target’s range equation model and 2-D spectrum for MEO multichannel SAR are derived. The proposed method is introduced in Section 3. The signal characteristics of the target in SAR image are analyzed in Section 4. In Section 5, experimental results are presented to verify the work of this paper. In Section 6, some discussions are presented. Finally, the paper is concluded in Section 7.

2. Signal Modelling

2.1. Range Equation Modelling

The MEO SAR geometrical model containing one moving target is illustrated in Figure 1, where OXYZ refers to the Earth-centered rotating (ECR) coordinate system. The origin of the ECR coordinate system coincides with the earth’s core, the OX axis points from the core of Earth to the prime meridian in the equatorial plane, the OZ axis points to the north pole, and the OY axis is determined by the right-hand rule. The target’s longitude and latitude at t_a = 0 are assumed to be δ_lng and δ_lat, where t_a is the azimuth slow time. Besides, v_lng, a_lng, v_lat and a_lat respectively represent the velocity along the longitude, acceleration along the longitude, velocity along the latitude, and acceleration along the latitude of the moving target. In addition, the radar’s beam centerline is assumed to cross the target at t_a = 0.

Based on the kinematics, the coordinates of the satellite in the ECR coordinate system can be expressed as [20]

$$\begin{array}{l}{x}_s(t_a)=R_s(t_a)\cos \left(\omega +f(t_a)\right)\cos \left({\Omega }_G(t_a)-\Omega \right)+R_s(t_a)\sin \left(\omega +f(t_a)\right)\sin \left({\Omega }_G(t_a)-\Omega \right)\cos {\theta }_i\\ y_s(t_a)=-R_s(t_a)\cos \left(\omega +f(t_a)\right)\sin \left({\Omega }_G(t_a)-\Omega \right)+R_s(t_a)\sin \left(\omega +f(t_a)\right)\cos \left({\Omega }_G(t_a)-\Omega \right)\cos {\theta }_i\\ z_s(t_a)=R_s(t_a)\sin \left(\omega +f(t_a)\right)\sin {\theta }_i,\end{array}$$

(1)

with

$$R_s(t_a)=\frac{a(1-e^2)}{1+e\cdot \cos \left(f(t_a)\right)},$$

(2)

where ω means the argument of perigee, f (t_a) represents the satellite’s true anomaly, a indicates the orbit’s semi-major axis, and e represents the eccentricity, Ω is the right ascension of ascending node, θ_i defines the orbit’s inclination, and Ω_G(t_a) = Ω_G0 + ω_et_a presents the Greenwich Hour Angle (GHA), where ω_e refers to Earth’s rotational angular velocity, and Ω_G0 indicates the GHA at t_a = 0. The detailed derivation of (1) can be found in [20].

Based on Figure 1, the target’s coordinates in the ECR coordinate system are given by [20]

$$\begin{array}{l}{x}_t(t_a)=R_e\cos {\delta }_l{}_{at}\cos {\delta }_{lng}-\left(v_{lat}{t}_a+0.5a_{lat}{t}_a^2\right)\sin {\delta }_{lng}-\left(v_{lng}{t}_a+0.5a_{lng}{t}_a^2\right)\sin {\delta }_l{}_{at}\cos {\delta }_{lng}\\ y_t(t_a)=R_e\cos {\delta }_l{}_{at}\sin {\delta }_{lng}+\left(v_{lat}{t}_a+0.5a_{lat}{t}_a^2\right)\cos {\delta }_{lng}-\left(v_{lng}{t}_a+0.5a_{lng}{t}_a^2\right)\sin {\delta }_l{}_{at}\sin {\delta }_{lng}\\ z_t(t_a)=R_e\sin {\delta }_l{}_{at}+\left(v_{lng}{t}_a+0.5a_{lng}{t}_a^2\right)\cos {\delta }_l{}_{at},\end{array}$$

(3)

where R_e refers to Earth’s radius.

Assuming that the MEO multichannel SAR has N channels in azimuth direction, and the distance between two adjacent channels is d, as shown in Figure 2. The dotted line in the figure represents the motion track of the satellite, and the azimuth axis of the antenna is parallel to the satellite’s velocity vector v_s(t_a) because of the zero-Doppler steering. Therefore, the position vector of the effective phase center of the nth (n = 1, 2, …, N) channel can be represented by r_s,n(t_a) = r_s(t_a) − (n − 1)d·v_s(t_a)/v_s(t_a), and the range from the target to the nth channel can be expressed as

$$\begin{array}{cc}{R}_n(t_a)& =\left|{\mathit{r}}_{\mathit{t}}(t_a)-{\mathit{r}}_{\mathit{s},\mathit{n}}(t_a)\right|\hfill \\ & =\left|{\mathit{r}}_{\mathit{t}}(t_a)-{\mathit{r}}_{\mathit{s}}(t_a)+(n-1)d\frac{v_s(t_a)}{v_s(t_a)}\right|,\hfill \end{array}$$

(4)

where r_t(t_a) = [x_t(t_a) y_t(t_a) z_t(t_a)] and r_s(t_a) = [x_s(t_a) y_s(t_a) z_s(t_a)] are the position vectors of the moving target and the satellite, respectively, and |.| indicates modulo operation to the vector. In this paper, the bold letters represent vectors, and the letters of italic fonts represent the norms of vectors.

To facilitate the design of the GMTI method, the target’s range history is generally approximated by a Taylor series expansion around the time that the beam centerline crosses the target (i.e., t_a = 0). In the case of MEO SAR, because of the obvious curve of the trajectory, a third-order Taylor series expansion should be adopted. By exploiting the vector projection and matrix differential, the cubic-approximated range equation for the multichannel system is given by [20]

$$R_n(t_a)\approx R_0+\left(l_1+(n-1)d\cdot \alpha \right)t_a+l_2{t}_a^2+l_3{t}_a^3,$$

(5)

with

$$R_0\approx \left|r_t(0)-r_s(0)\right|$$

(6)

$$l_1\approx v_{tr}$$

(7)

$$l_2\approx \frac{v_{t0}^2-2v_{ta}{v}_{s0}+v_{s0}^2-v_{tr}^2}{2R_0}-\frac{a_{sr}}{2}+\frac{a_{tr}}{2}$$

(8)

$$l_3\approx \frac{3a_{sa}{v}_{s0}-j_{sr}{R}_0}{6R_0}-\frac{3a_{ta}{v}_{s0}}{6R_0}$$

(9)

$$\alpha =\frac{v_{ta}-v_{s0}}{R_0}+\frac{a_{sr}}{v_{s0}},$$

(10)

where v_s0 = |v_s(0)| and v_t0 = |v_t(0)| are, respectively, the velocities of the radar and target at t_a = 0. In addition, v_tr and a_tr represent the target’s radial velocity and acceleration; v_sr, a_sr and j_sr are, respectively, the radial projections of the radar’s velocity, acceleration and jerk; v_ta and a_ta define the target’s along-track velocity and acceleration; a_sa is the radar’s acceleration. They can be expressed as [20]

$$v_{tr}=v_t(0)\frac{{[r_t(0)-r_s(0)]}^T}{R_0},$$

(11)

$$v_{sr}=v_s(0)\frac{{[r_t(0)-r_s(0)]}^T}{R_0},$$

(12)

$$a_{tr}=\frac{a_t{[r_t(0)-r_s(0)]}^T}{R_0},$$

(13)

$$a_{sr}=\frac{a_s(0){[r_t(0)-r_s(0)]}^T}{R_0},$$

(14)

$$j_{sr}=\frac{j_s(0){[r_t(0)-r_s(0)]}^T}{R_0},$$

(15)

$$v_{ta}=\frac{v_t(0)v_s{(0)}^T}{\left|v_s(0)\right|},$$

(16)

$$a_{ta}=\frac{a_t{v}_s{(0)}^T}{\left|v_s(0)\right|},$$

(17)

$$a_{sa}=\frac{a_s(0)v_s{(0)}^T}{\left|v_s(0)\right|},$$

(18)

where $v_s(0)$, $a_s(0)$, and $j_s(0)$ are, respectively, the velocity, acceleration and jerk vectors of satellite at t_a = 0; $v_t(0)$ and $a_t(0)$ indicate the velocity and acceleration vectors of the target at t_a = 0.

2.2. Derivation of the Target’s 2-D Spectrum

Supposing that a chirp signal is transmitted, then the signal after demodulation and range compression can be expressed as [1]

$$s_n(t_r,t_a)=p_r\left(t_r-\frac{2R_n(t_a)}{c}\right)w_a(t_a)\exp \left\{-j\frac{4\pi }{\lambda }{R}_n(t_a)\right\},$$

(19)

where $p_r(\cdot )$ defines the function of range impulse response,$w_a(\cdot )$ indicates the azimuth window function, t_r is the range time, c represents the velocity of light, and λ defines the wavelength. In the paper, the channels are assumed to be well calibrated.

Applying a FT to (19) in the range direction, the target signal in the range frequency domain can be obtained:

$$S_n(f_r,t_a)=W_r\left(f_r\right)w_a(t_a)\exp \left\{-j4\pi \left(f_c+f_r\right)\frac{R_n(t_a)}{\lambda }\right\},$$

(20)

where f_r is the range frequency, $W_r(\cdot )$ refers to the envelope of range spectrum, f_c defines the carrier frequency.

The target’s 2-D spectrum can be obtained via taking a FT to (20) with respect to t_a. Since there are quadratic and cubic terms of t_a, the series reversion and the principle of stationary phase are used to solve the Fourier integral. The obtained expression for the target signal in the 2-D frequency domain is given by

$$S_n(f_r,f_a)=W_r(f_r)W_a(f_a)\exp \left\{j{\theta }_n(f_r,f_a)\right\},$$

(21)

with

$$\begin{array}{cc}{\theta }_n(f_r,f_a)& =-4\pi \frac{(f_c+f_r)}{c}\left(R_0-\frac{{\left(l_1+(n-1)d\alpha \right)}^2}{4l_2^{}}-\frac{{\left(l_1+(n-1)d\alpha \right)}^3{l}_3}{8l_2^3}\right)\hfill \\ & +2\pi (f_a+M\cdot PRF)\left(\frac{l_1+(n-1)d\alpha }{2l_2}+\frac{3{\left(l_1+(n-1)d\alpha \right)}^2{l}_3}{8l_2^3}\right)\hfill \\ & +\frac{\pi c}{f_c+f_r}{(f_a+M\cdot PRF)}^2\left(\frac{3\left(l_1+(n-1)d\alpha \right)l_3}{8l_2^3}+\frac{1}{4l_2}\right)-\frac{\pi c^2}{2{(f_c+f_r)}^2}{(f_a+M\cdot PRF)}^3(-\frac{l_3}{8l_2^3}),\hfill \end{array}$$

(22)

where $W_a(\cdot )$ indicates the envelope of azimuth spectrum, f_a defines the Doppler frequency, M represents the Doppler ambiguity number, PRF indicates the pulse repetition frequency.

To facilitate the design of the imaging method, θ_n (f_r, f_a) is decomposed via using power series expansions of ${(1+f_r/f_c)}^{-1}$ and ${(1+f_r/f_c)}^{-2}$ (keeping terms up to the third order), and also is rearranged according to the physical meaning. The decomposed and rearranged phase is given by

$$\begin{array}{cc}{\theta }_n(f_r,f_a)& =-\frac{4\pi R_0}{\lambda }+\frac{\pi {\left(l_1+(n-1)d\alpha \right)}^2}{\lambda l_2}+\frac{\pi {\left(l_1+(n-1)d\alpha \right)}^3{l}_3}{2\lambda l_2^3}+\frac{\pi \left(l_1+(n-1)d\alpha \right)}{l_2}(f_a+M\cdot PRF)\hfill \\ & +\frac{3\pi {\left(l_1+(n-1)d\alpha \right)}^2{l}_3}{4l_2^3}(f_a+M\cdot PRF)+\frac{\pi \lambda }{4l_2}{(f_a+M\cdot PRF)}^2+\frac{3\pi \lambda \left(l_1+(n-1)d\alpha \right)l_3}{8l_2^3}{(f_a+M\cdot PRF)}^2\hfill \\ & +\frac{\pi l_3{\lambda }^2}{16l_2^3}{(f_a+M\cdot PRF)}^3-2\pi f_r{\tau }_n(f_a)+\frac{\pi f_r^2}{F_n(f_a)}+\frac{\pi f_r^3}{M_n(f_a)},\hfill \end{array}$$

(23)

where $\tau (f_a)$ represents echo delay, $F_n(f_a)$ and $M_n(f_a)$ indicate the residual quadratic and cubic range frequency modulation rates, respectively. Their expressions are as follows

$$\begin{array}{cc}{\tau }_n(f_a)& =\frac{2R_0}{c}-\frac{{\left(l_1+(n-1)d\alpha \right)}^2}{2c{l}_2}-\frac{{\left(l_1+(n-1)d\alpha \right)}^3{l}_3}{4c{l}_2^3}\hfill \\ & +\frac{3c\left(l_1+(n-1)d\alpha \right)l_3}{16f_c{}^2{l}_2^3}{(f_a+M\cdot PRF)}^2+\frac{c}{8f_c{}^2{l}_2}{(f_a+M\cdot PRF)}^2+\frac{c^2{l}_3}{16f_c{}^3{l}_2^3}{(f_a+M\cdot PRF)}^3\hfill \end{array},$$

(24)

$$\frac{1}{F_n(f_a)}=\frac{3\lambda \left(l_1+(n-1)d\alpha \right)l_3}{8f_c{}^2{l}_2^3}{(f_a+M\cdot PRF)}^2+\frac{\lambda }{4f_c{}^2{l}_2}{(f_a+M\cdot PRF)}^2+\frac{3{\lambda }^2{l}_3}{16f_c{}^2{l}_2^3}{(f_a+M\cdot PRF)}^3,$$

(25)

$$\frac{1}{M_n(f_a)}=-\frac{3\lambda \left(l_1+(n-1)d\alpha \right)l_3}{8f_c{}^3{l}_2^3}{(f_a+M\cdot PRF)}^2-\frac{\lambda }{4f_c{}^3{l}_2}{(f_a+M\cdot PRF)}^2-\frac{{\lambda }^2{l}_3}{4f_c{}^3{l}_2^3}{(f_a+M\cdot PRF)}^3.$$

(26)

3. Proposed Imaging Method

In this section, an efficient imaging method is proposed. The secondary range compression (SRC) and RCM correction (RCMC) are accomplished efficiently in the 2-D frequency domain via phase multiplications. In addition, the azimuth compression is implemented in the range Doppler domain. The main steps of this imaging method are introduced below. Note that the signal of the reference channel (i.e., n = 1) is taken as an example to describe the proposed method in this section.

3.1. Secondary Range Compression

It can be seen from (23) that there are still quadratic and cubic terms of range frequency after range compression. If these terms are not compensated, defocus will occur in the range direction. Therefore, the operation of SRC is implemented, and the filter is constructed as follows

$$H_{SRC}(f_r,f_a)=\exp \{-\frac{j\pi f_r^2}{F_{ref}(f_a)}-\frac{j\pi f_r^3}{M_{ref}(f_a)}\},$$

(27)

with

$$\frac{1}{F_{ref}(f_a)}=\frac{3\lambda l_{10}{l}_{30}}{8f_c{}^2{l}_{20}^3}{f}_a{}^2+\frac{\lambda }{4f_c{}^2{l}_{20}}{f}_a{}^2+\frac{3{\lambda }^2{l}_{30}}{16f_c{}^2{l}_{20}^3}{f}_a{}^3,$$

(28)

$$\frac{1}{M_{ref}(f_a)}\text{=}-\frac{3\lambda l_{10}{l}_{30}}{8f_c{}^3{l}_{20}^3}{f}_a{}^2-\frac{\lambda }{4f_c{}^3{l}_{20}}{f}_a{}^2-\frac{{\lambda }^2{l}_{30}}{4f_c{}^3{l}_{20}^3}{f}_a{}^3,$$

(29)

where l₁₀, l₂₀ and l₃₀ define the coefficients of the linear, quadratic, and cubic terms of range history of the static target located at the center of the illuminated scene.

Comparing (25) with (28) and (26) with (29), respectively, one can see that there is a residual phase error after SRC by (27), and it can be expressed as

$$P{E}_r=\left|\frac{\pi f_r^2}{F_1(f_a)}+\frac{\pi f_r^3}{M_1(f_a)}-\frac{\pi f_r^2}{F_{ref}(f_a)}-\frac{\pi f_r^3}{M_{ref}(f_a)}\right|.$$

(30)

This phase error is very small (about 10 ⁻¹⁰) and much less than π/4, and thus it can be neglected. Therefore, the signal after SRC is given by

$$\begin{array}{cc}{S}_1(f_r,f_a)& =W_r(f_r)W_a(f_a)\exp \left\{j\left(-\frac{4\pi R_0}{\lambda }+\frac{\pi l_1^2}{\lambda l_2}+\frac{\pi l_1^3{l}_3}{2\lambda l_2^3}\right)\right\}\exp \left\{-j2\pi f_r{\tau }_1(f_a)\right\}\hfill \\ & \times \exp \left\{j\left(\frac{\pi l_1}{l_2}(f_a+M\cdot PRF)+\frac{3\pi l_1^2{l}_3}{4l_2^3}(f_a+M\cdot PRF)\right)\right\}\hfill \\ & \times \exp \left\{j\left(\frac{\pi \lambda }{4l_2}{(f_a+M\cdot PRF)}^2+\frac{3\pi \lambda l_1{l}_3}{8l_2^3}{(f_a+M\cdot PRF)}^2+\frac{\pi l_3{\lambda }^2}{16l_2^3}{(f_a+M\cdot PRF)}^3\right)\right\}.\hfill \end{array}$$

(31)

3.2. Range Cell Migration Correction

In order to improve the efficiency, a phase multiplication in 2-D frequency domain is used to accomplish RCMC. Based on (31), the RCM of target can be corrected by compensating the term containing f_r. Therefore, the RCMC filter can be constructed as follows

$$H_{RCMC}(f_r,f_a)=\exp \left\{\frac{j\lambda \pi f_r}{f_c}\left(\frac{1}{4l_{20}}{f}_a{}^2+\frac{3l_{10}{l}_{30}}{8l_{20}^3}{f}_a{}^2+\frac{\lambda l_{30}}{8l_{20}^3}{f}_a{}^3\right)\right\}.$$

(32)

By multiplying (32) into (31) and then performing inverse FT (IFT), one can obtain the range-Doppler domain signal as follows:

$$\begin{array}{cc}{S}_{1,rcmc}(t_r,f_a)& =W_a(f_a)p_r\left\{\begin{array}{c}{t}_r-\frac{2}{c}\left(R_0-\frac{l_1^2}{4l_2}-\frac{l_1^3{l}_3}{8l_2^3}\right)+\frac{c{f}_a{}^2}{f_c{}^2}\left(\frac{1}{8l_{20}}-\frac{1}{8l_2}\right)+\frac{c{f}_a{}^2}{f_c{}^2}\left(\frac{3l_{10}{l}_{30}}{16l_{20}^3}-\frac{3l_1{l}_3}{16l_2^3}\right)+\frac{c^2{f}_a^3}{f_c{}^3}\left(\frac{l_{30}}{16l_{20}^3}-\frac{l_3}{16l_2^3}\right)\hfill \\ -\left(\begin{array}{c}\frac{c}{f_c{}^2}\frac{1}{8l_2}(2f_a\cdot M\cdot PRF+M^2\cdot PR{F}^2)+\frac{c}{f_c{}^2}\frac{3l_1{l}_3}{16l_2^3}(2f_a\cdot M\cdot PRF+M^2\cdot PR{F}^2)\hfill \\ +\frac{c^2}{f_c{}^3}\frac{l_3}{16l_2^3}(M^3\cdot PR{F}^3+3f_a{}^{2}M\cdot PRF+3f_a\cdot M^2\cdot PR{F}^2)\hfill \end{array}\right)\hfill \end{array}\right\}\hfill \\ & \times \exp \left\{j\left(-\frac{4\pi R_0}{\lambda }+\frac{\pi l_1^2}{\lambda l_2}+\frac{\pi l_1^3{l}_3}{2\lambda l_2^3}\right)\right\}\exp \left\{j\left(\frac{\pi l_1}{l_2}(f_a+M\cdot PRF)+\frac{3\pi l_1^2{l}_3}{4l_2^3}(f_a+M\cdot PRF)\right)\right\}\hfill \\ & \times \exp \left\{j\left(\frac{\pi \lambda }{4l_2}{(f_a+M\cdot PRF)}^2+\frac{3\pi \lambda l_1{l}_3}{8l_2^3}{(f_a+M\cdot PRF)}^2+\frac{\pi l_3{\lambda }^2}{16l_2^3}{(f_a+M\cdot PRF)}^3\right)\right\}.\hfill \end{array}$$

(33)

From the range envelope of (33), one can see that there is still a residual RCM, which can be expressed as

$$\Delta RCM=\left|\begin{array}{c}\frac{{\lambda }^2{f}_a{}^2}{16}\left(\frac{1}{l_{20}}-\frac{1}{l_2}\right)+\frac{{\lambda }^2{f}_a{}^2}{16}\left(\frac{3l_{10}{l}_{30}}{2l_{20}^3}-\frac{3l_1{l}_3}{2l_2^3}\right)+\frac{{\lambda }^3{f}_a{}^3}{16}\left(\frac{l_{30}}{2l_{20}^3}-\frac{l_3}{2l_2^3}\right)\hfill \\ -\left(\begin{array}{c}\frac{{\lambda }^2}{16l_2}(2f_a\cdot M\cdot PRF+M^2\cdot PR{F}^2)+\frac{3{\lambda }^2{l}_1{l}_3}{32l_2^3}(2f_a\cdot M\cdot PRF+M^2\cdot PR{F}^2)\hfill \\ +\frac{{\lambda }^3{l}_3}{32l_2^3}(M^3\cdot PR{F}^3+3f_a{}^{2}M\cdot PRF+3f_a\cdot M^2\cdot PR{F}^2)\hfill \end{array}\right)\hfill \end{array}\right|.$$

(34)

Considering –PRF/2 < f_a < PRF/2, the maximum possible value of the residual RCM is given by

$$\Delta RC{M}_{\max }={\left|\Delta RCM\right|}_{f_a=PRF/2}.$$

(35)

For the targets moving with such a radial velocity that there is no Doppler ambiguity (i.e., M = 0), $\Delta RC{M}_{\max }$ will be smaller than half of a range resolution cell, and thus the residual RCM can be neglected. To validate this statement, numerical simulations are conducted. The MEO SAR parameters for simulations are presented in

Table 1. Parameters of MEO SAR.

Parameter	Value
Orbit’s height	6000 km
Range bandwidth	30 MHz
Sampling frequency	40 MHz
Azimuth resolution	6 m
Illumination time	3.3 s
Re	6371 km
R0	7548.7 km
PRF	1400 Hz
Ω	0
w	0
e	0
ΩG0	0
θi	90°
λ	0.03 m
d	2 m

. The result is illustrated in Figure 3, which shows the $\Delta RC{M}_{\max }$ of the targets with all possible velocities (note that, when there is no Doppler ambiguity, v_tr will be less than about 10 m/s). As can be seen from Figure 3, the maximum value of $\Delta RC{M}_{\max }$ is about 1.8 m. Therefore, $\Delta RC{M}_{\max }$ will not exceed half the range resolution cell in the cases that the range resolution is not high.

The RCMC filter shown in (32) is constructed without any parameter of the target. Therefore, this filter can simultaneously correct the RCMs of static scenes and the multiple moving targets of no Doppler ambiguity. Note that after the RCM is corrected there will be no defocusing in the range direction after azimuth compression.

3.3. Azimuth Compression

To finely focus the static scene, we perform the azimuth compression in the range Doppler domain. Based on (33), the azimuth compression filter is constructed as follows

$$H_{ac}(f_a)=\exp \left\{-j\left(\pi \lambda \frac{1}{4l_{20}}{f}_a{}^2+\pi \lambda \frac{3l_{10}{l}_{30}}{8l_{20}^3}{f}_a{}^2+\frac{\pi l_{30}{\lambda }^2}{16l_{20}^3}{f}_a{}^3\right)\right\}.$$

(36)

The target signal after azimuth compression is given by (the negligible residual RCM is omitted)

$$\begin{array}{cc}{S}_{1,ac}(t_r,f_a)& =W_a(f_a)p_r\left\{\begin{array}{l}{t}_r-\frac{2}{c}\left(R_0-\frac{l_1^2}{4l_2}-\frac{l_1^3{l}_3}{8l_2^3}\right)-\frac{c}{f_c{}^2}\frac{1}{8l_2}\left(2f_a\cdot M\cdot PRF+M^2\cdot PR{F}^2\right)\\ -\frac{c}{f_c{}^2}\frac{3l_1{l}_3}{16l_2^3}\left(2f_a\cdot M\cdot PRF+M^2\cdot PR{F}^2\right)-\frac{c^2}{f_c{}^3}\frac{l_3}{16l_2^3}\left(M^3\cdot PR{F}^3+3f_a{}^{2}M\cdot PRF+3f_a\cdot M^2\cdot PR{F}^2\right)\end{array}\right\}\hfill \\ & \times \exp \left\{j\pi \left(-\frac{4R_0}{\lambda }+\frac{l_1^2}{\lambda l_2}+\frac{l_1^3{l}_3}{2\lambda l_2^3}\right)\right\}\exp \left\{j\pi \cdot M\cdot PRF\cdot \left(\frac{l_1}{l_2}+\frac{3l_1^2{l}_3}{4l_2^3}+\frac{\lambda }{4l_2}\cdot M\cdot PRF+\lambda \frac{3l_1{l}_3}{8l_2^3}\cdot M\cdot PRF+\frac{l_3{\lambda }^2}{16l_2^3}\cdot M^2\cdot PR{F}^2\right)\right\}\hfill \\ & \times \exp \left\{j\pi \left(\frac{l_1}{l_2}+\frac{3l_1^2{l}_3}{4l_2^3}\right)f_a\right\}\exp \left\{j\pi \cdot M\cdot PRF\left(\frac{\lambda }{2l_2}+\lambda \frac{3l_1{l}_3}{4l_2^3}+\frac{3l_3{\lambda }^2}{16l_2^3}\cdot M\cdot PRF\right)f_a\right\}\hfill \\ & \times \exp \left\{j\pi \lambda \left[\left(\frac{1}{4l_2}-\frac{1}{4l_{20}}\right)f_a^2+\left(\frac{3l_1{l}_3}{8l_2^3}-\frac{3l_{10}{l}_{30}}{8l_{20}^3}\right)f_a^2+\left(\frac{\lambda l_3}{16l_2^3}-\frac{\lambda l_{30}}{16l_{20}^3}\right)f_a^3\right]\right\}\exp \left\{j\pi \frac{3l_3{\lambda }^2}{16l_2^3}\cdot M\cdot PRF\cdot f_a{}^2\right\}.\hfill \end{array}$$

(37)

The focused SAR image can be obtained via applying an azimuth IFT to (37). From (37) it can be seen that, for static targets, they can be finely focused. For moving targets, since there are residual azimuth modulations, they will suffer from azimuth defocusing. It should be noted that, since the RCMs of the moving targets without Doppler ambiguity have been corrected, the range defocusing will not occur after azimuth compression. Therefore, the SCNR of the target can be improved, which will improve the GMTI performance of the system.

4. Property Analysis of Moving Target in the Complex Image Domain

Analyzing and understanding the moving targets’ characteristics in the SAR image is of great significance for moving target indication. However, it is difficult to derive the analytical expression of the target signal in complex image domain, because there is residual azimuth modulation after azimuth compression. In this section, some properties of the target signal in the image obtained by the proposed imaging method, e.g., the target’s range displacement, azimuth displacement, and ATI phase, are quantitatively analyzed.

4.1. Range Displacement

The target’s location in range is displaced from R₀, which can be seen from the range envelope of (37). Thus, the target will suffer range displacement after focusing. Based on (37), the range displacement can be calculated as

$$r_{shift}=\frac{l_1^2}{4l_2}+\frac{l_1^3{l}_3}{8l_2^3}-\frac{c}{2}\left(\begin{array}{c}\frac{c}{f_c{}^2}\frac{1}{8l_2}(2f_a\cdot M\cdot PRF+M^2\cdot PR{F}^2)+\frac{c}{f_c{}^2}\frac{3l_1{l}_3}{16l_2^3}(2f_a\cdot M\cdot PRF+M^2\cdot PR{F}^2)\hfill \\ +\frac{c^2}{f_c{}^3}\frac{l_3}{16l_2^3}(M^3\cdot PR{F}^3+3f_a{}^{2}M\cdot PRF+3f_a\cdot M^2\cdot PR{F}^2)\hfill \end{array}\right).$$

(38)

Since the terms containing l₃ are smaller than a range cell, they can be neglected. Therefore, r_shift can be simplified as

$$\begin{array}{cc}{r}_{shift}& \approx \frac{l_1^2}{4l_2}-\frac{{\lambda }^2}{16l_2}\left(2f_a\cdot M\cdot PRF+M^2\cdot PR{F}^2\right)\hfill \\ & =\frac{{\lambda }^2}{16l_2}{f}_b^2,\hfill \end{array}$$

(39)

where f_b defines the target’s baseband Doppler center frequency, and it can be expressed as $f_b=-2v_{tr}/\lambda -M\cdot PRF$.

In order to reveal the relationship between the range displacement and the target’s parameters, the expressions of l₂ and f_b are substituted into (39), and r_shift can be rewritten as

$$\begin{array}{cc}{r}_{shift}& \approx \frac{{\lambda }^2{R}_0{\left(-2v_{tr}/\lambda -M\cdot PRF\right)}^2}{8(v_{t0}^2-v_{tr}^2-2v_{ta}{v}_{s0}+v_{s0}^2+a_{tr}{R}_0-a_{sr}{R}_0)}\hfill \\ & \approx \frac{{\lambda }^2{R}_0{\left(2v_{tr}/\lambda +M\cdot PRF\right)}^2}{8(-2v_{ta}{v}_{s0}+v_{s0}^2+a_{tr}{R}_0-a_{sr}{R}_0)}.\hfill \end{array}$$

(40)

Note that, since $v_{t0}^2-v_{tr}^2$ is far smaller than $-2v_{ta}{v}_{s0}+v_{s0}^2+a_{tr}{R}_0-a_{sr}{R}_0$, it is ignored.

From (39) one can see that the target’s range displacement is caused by its radial velocity, i.e., the range displacement will be equal to zero when the radial velocity is zero. Moreover, it is interesting that the range displacement is dependent on its baseband Doppler center frequency rather than the absolute Doppler center frequency. This property indicates that the range displacement is independent of Doppler ambiguity and will not be monotonically increasing with the target’s radial velocity. In addition, from (40) one can see that the range displacement is also related to the target’s radial acceleration, along-track velocity, and distance. However, it is very difficult to find out the variation of the range displacement with these parameters directly from (40). To address this issue, numerical simulations are conducted, with the results being presented in Figure 4.

Figure 4a illustrates the variation of the range displacement with the velocities of the target. It can be seen that the range displacement varies with v_tr periodically and gets the maximum value when the baseband Doppler center frequency equals to PRF/2. In addition, one can see that the range displacement is almost independent of v_ta. Figure 4b,c show the variation of the range displacement with the distance and radial acceleration of the target, respectively. It can be seen that the range displacement increases with R₀ approximately linearly, and is inversely proportional to a_tr.

4.2. Azimuth Displacement

Besides the range displacement, the motion of the target will also lead to an azimuth offset in SAR image. According to (37) and the properties of FT, the target’s azimuth displacement can be expressed as

$$\begin{array}{cc}{a}_{shift}& =v_g*\left(\frac{l_1}{2l_2}+\frac{3l_1^2{l}_3}{8l_2^3}+\lambda \frac{1}{4l_2}M\cdot PRF+\lambda \frac{3l_1{l}_3}{8l_2^3}M\cdot PRF+\frac{l_3{\lambda }^2}{32l_2^3}3M^2\cdot PR{F}^2\right)\hfill \\ & =v_g*\left(\frac{1}{2l_2}\left(l_1+\frac{\lambda }{2}M\cdot PRF\right)+\frac{3l_3}{8l_2^3}{\left(l_1+\frac{\lambda }{2}M\cdot PRF\right)}^2\right),\hfill \end{array}$$

(41)

where v_g is the velocity of the beam sweeping along the surface of earth.

Substituting the expressions of l₁, l₂ and l₃ into (41), a_shift can be rewritten as

$$\begin{array}{cc}{a}_{shift}& =\frac{v_g(3a_{sa}{v}_{s0}-j_{sr}{R}_0-3a_{ta}{v}_{s0})R_0^2}{2{(v_{t0}^2-2v_{ta}{v}_{s0}+v_{s0}^2-v_{tr}^2+a_{tr}{R}_0-a_{sr}{R}_0)}^3}{(v_{tr}+\frac{\lambda }{2}M\cdot PRF)}^2\hfill \\ & +\frac{v_g{R}_0}{(v_{t0}^2-2v_{ta}{v}_{s0}+v_{s0}^2-v_{tr}^2+a_{tr}{R}_0-a_{sr}{R}_0)}(v_{tr}+\frac{\lambda }{2}M\cdot PRF)\hfill \\ & \approx \frac{-v_{g}3a_{ta}{v}_{s0}{R}_0^2}{2{(v_{t0}^2-2v_{ta}{v}_{s0}+v_{s0}^2-v_{tr}^2+a_{tr}{R}_0-a_{sr}{R}_0)}^3}{(v_{tr}+\frac{\lambda }{2}M\cdot PRF)}^2\hfill \\ & +\frac{v_g{R}_0}{(v_{t0}^2-2v_{ta}{v}_{s0}+v_{s0}^2-v_{tr}^2+a_{tr}{R}_0-a_{sr}{R}_0)}(v_{tr}+\frac{\lambda }{2}M\cdot PRF)\hfill \\ & =\frac{-v_{g}3a_{ta}{v}_{s0}{R}_0^2}{2{(-2v_{ta}{v}_{s0}+v_{s0}^2+a_{tr}{R}_0-a_{sr}{R}_0)}^3}{\frac{\lambda }{4}}^2{f}_b^2-\frac{v_g{R}_0}{(-2v_{ta}{v}_{s0}+v_{s0}^2+a_{tr}{R}_0-a_{sr}{R}_0)}\frac{\lambda }{2}{f}_b^{}.\hfill \end{array}$$

(42)

Note that since the value of $v_g(3a_{sa}{v}_{s0}-j_{sr}{R}_0)R_0^2$ is much smaller than $3v_g{a}_{ta}{v}_{s0}{R}_0^2$, it can be ignored.

From (42), one can see that the azimuth displacement is also caused by the target’s radial velocity and is dependent on the baseband Doppler center frequency rather than the absolute Doppler center frequency. Moreover, the azimuth displacement is also dependent on the target’s along-track velocity, accelerations, and distance. In order to reveal the variation of the range displacement with these parameters, numerical experiments are carried out, with the results presented in Figure 5.

From Figure 5a it can be seen that the azimuth displacement varies with v_tr periodically and gets the maximum value when the baseband Doppler center frequency equals to PRF/2. In addition, one can see that the range displacement is almost independent of v_ta. The variation of the azimuth displacement with the target’s distance and accelerations are shown in Figure 5b,c. It can be seen that the range displacement increases with R₀ approximately linearly. From Figure 5c one can see that the azimuth displacement is inversely proportional to a_tr and almost independent of a_ta.

4.3. Along-Track Interferometry Phase

ATI phase is a very important parameter in the SAR-GMTI field. It can be utilized to estimate the radial velocity of the target. In addition, moving targets can be detected according to the interference phase diagram.

Based on (37), the signal of the second channel after registration is:

$$\begin{array}{cc}{S}_{2,ac,reg}(t_r,f_a)& =W_a(f_a)p_r\left\{\begin{array}{l}{t}_r-\frac{2}{c}\left(R_0-\frac{{(l_1+d\alpha )}^2}{4l_2}-\frac{{(l_1+d\alpha )}^3{l}_3}{8l_2^3}\right)-\frac{c}{f_c{}^2}\frac{1}{8l_2}\left(2f_a\cdot M\cdot PRF+M^2\cdot PR{F}^2\right)\\ -\frac{c}{f_c{}^2}\frac{3(l_1+d\alpha )l_3}{16l_2^3}\left(2f_a\cdot M\cdot PRF+M^2\cdot PR{F}^2\right)-\frac{c^2}{f_c{}^3}\frac{l_3}{16l_2^3}\left(M^3\cdot PR{F}^3+3f_a{}^{2}M\cdot PRF+3f_a\cdot M^2\cdot PR{F}^2\right)\end{array}\right\}\hfill \\ & \times \exp \left\{j\pi \left(-\frac{4R_0}{\lambda }+\frac{{(l_1+d\alpha )}^2}{\lambda l_2}+\frac{{(l_1+d\alpha )}^3{l}_3}{2\lambda l_2^3}\right)\right\}\exp \left\{j\pi \left(\frac{l_1+d\alpha }{l_2}+\frac{3{(l_1+d\alpha )}^2{l}_3}{4l_2^3}\right)f_a\right\}\hfill \\ & \times \exp \left\{j\pi \cdot M\cdot PRF\cdot \left(\frac{l_1+d\alpha }{l_2}+\frac{3{(l_1+d\alpha )}^2{l}_3}{4l_2^3}+\frac{\lambda }{4l_2}\cdot M\cdot PRF+\lambda \frac{3(l_1+d\alpha )l_3}{8l_2^3}\cdot M\cdot PRF+\frac{l_3{\lambda }^2}{16l_2^3}\cdot M^2\cdot PR{F}^2\right)\right\}\hfill \\ & \times \exp \left\{j\pi \cdot M\cdot PRF\left(\frac{\lambda }{2l_2}+\lambda \frac{3(l_1+d\alpha )l_3}{4l_2^3}+\frac{3l_3{\lambda }^2}{16l_2^3}\cdot M\cdot PRF\right)f_a\right\}\hfill \\ & \times \exp \left\{j\pi \lambda \left[\left(\frac{1}{4l_2}-\frac{1}{4l_{20}}\right)f_a^2+\left(\frac{3(l_1+d\alpha )l_3}{8l_2^3}-\frac{3l_{10}{l}_{30}}{8l_{20}^3}\right)f_a^2+\left(\frac{\lambda l_3}{16l_2^3}-\frac{\lambda l_{30}}{16l_{20}^3}\right)f_a^3\right]\right\}\exp \left\{j\pi \frac{3l_3{\lambda }^2}{16l_2^3}\cdot M\cdot PRF\cdot f_a{}^2\right\}.\hfill \end{array}$$

(43)

According to (37) and (43), the interference phase between channel 1 and channel 2 can be expressed as

$$\begin{array}{cc}{\varphi }_{AT{I}_{12}}& =\arg \left(S_{1,ac}(t_{r,}{f}_a)S_{2,ac,reg}{}^{\ast }(t_{r,}{f}_a)\right)\hfill \\ & =-\frac{2\pi l_{1}d\alpha }{\lambda l_2}-\frac{\pi d^2{\alpha }^2}{\lambda l_2}-\frac{3\pi l_1{}^{2}d\alpha l_3}{2\lambda l_2^3}-\frac{3\pi l_1{l}_3{d}^2{\alpha }^2}{2\lambda l_2^3}-\frac{\pi d^3{\alpha }^3{l}_3}{2\lambda l_2^3}-\frac{\pi d\alpha }{l_2}(f_a+M\cdot PRF)\hfill \\ & -\frac{6\pi l_1{l}_{3}d\alpha }{4l_2^3}(f_a+M\cdot PRF)-\frac{3\pi d^2{\alpha }^2{l}_3}{4l_2^3}(f_a+M\cdot PRF)\hfill \\ & -\pi \lambda \frac{3d\alpha l_3}{8l_2^3}{(f_a+M\cdot PRF)}^2-2\pi \frac{d}{v_{s0}}(f_a+M\cdot PRF),\hfill \end{array}$$

(44)

where * represents a conjugate operation on the signal.

Since the items containing d²α², d³α³ and l₃ are much smaller than other items, they can be ignored. Therefore, ${\varphi }_{AT{I}_{12}}$ can be simplified to

$${\varphi }_{AT{I}_{12}}\approx -\frac{\pi d\alpha }{l_2}M\cdot PRF+2\pi \frac{d}{v_{s0}}\frac{2}{\lambda }{v}_{tr}-2\pi \frac{d}{v_{s0}}M\cdot PRF.$$

(45)

In order to find out the variation of ATI phase with the target’s parameters, the expressions of α and l₂ are substituted into (45). Then, the ATI phase can be rewritten as:

$${\varphi }_{AT{I}_{12}}=\frac{4\pi d{v}_{tr}}{\lambda v_{s0}}-\frac{2\pi d}{v_{s0}}M\cdot PRF-\pi d\frac{2(v_{ta}-v_{s0})v_{s0}+2a_{sr}{R}_0}{(-2v_{ta}{v}_{s0}+v_{s0}^2+a_{tr}{R}_0-a_{sr}{R}_0)v_{s0}}M\cdot PRF.$$

(46)

It can be seen from (46) that the ATI phase is also caused by the target’s radial velocity and is only dependent on the radial velocity when there is no Doppler ambiguity. For the cases where Doppler ambiguity is happening, in addition to the radial velocity the ATI phase depends also on the along-track velocity, radial acceleration and distance. To find out the variation of the ATI phase with these parameters, numerical experiments are carried out, with the results being presented in Figure 6. Figure 6a shows the relationship between the ATI phase and velocities. One can find out that the ATI phase is proportional to v_tr, while the variation of the ATI phase with v_ta is very slight. In addition, the variation of ATI phase with the target’s distance and radial acceleration are shown in Figure 6b,c. One can see that the ATI phase is inversely proportional to R₀, and that the ATI phase is proportional to a_tr when there is Doppler ambiguity.

5. Experimental Results

In this section, numerical experiments are conducted to validate the proposed imaging method and the analysis presented in Section 4. The MEO SAR parameters used for simulation are given in Table 1. Five targets are simulated, including four moving targets and one stationary target. The parameters of these targets are given in

Table 2. Parameters of the targets.

Parameter	Target 1	Target 2	Target 3	Target 4	Target 5
vtr (m/s)	4	−3	4	0	5
vta (m/s)	6	8	12	0	15
atr (m/s2)	0.1	0.15	−0.1	0	0
ata (m/s2)	0.2	0.3	0.2	0	0.4
δlat (°)	10.016	10.008	10.02	10.012	10.002
δlng (°)	30	30	30	30	30

.

The results of RCMC are presented in Figure 7. Comparing Figure 7a with Figure 7b, it is evident that the five targets’ RCMs are corrected at the same time. In addition, the residual RCMs of five targets are also calculated by (34), which are 0.42 m, 0.85 m, 0.53 m, 0 m, and 0.16 m. They are so much smaller than a range cell that can be ignored.

The azimuth compression results are given in Figure 8, from which it can be found out that the targets are focused simultaneously. Furthermore, it can be seen that the static target (target 4) is finely focused. In addition, the four moving targets are also well focused in range direction with defocusing only in the azimuth direction.

In order to verify the target’s signal characteristic analysis presented in Section 4, the range and azimuth displacements and the ATI phases of the four moving targets are measured, with the results shown in

Table 3. Characteristic Measurement of Moving Targets.

	Range Displacement (m)		Azimuth Displacement (m)		ATI Phase (rad)
	Theoretical Value	Measured Value	Theoretical Value	Measured Value	Theoretical Value	Measured Value
Target 1	3.99	3.75	5904.7	5914	0.5555	0.5643
Target 2	2.19	3.75	4329.1	4325	−0.4375	−0.4676
Target 3	4.18	3.75	6592.2	6478	0.5833	0.5544
Target 5	6.61	7.5	7821.3	7808	0.7291	0.7229

. Note that the measured displacements are calculated based on target’s position in the SAR image, and thus they can only be the integer multiples of the size of a pixel. From Table 3 it can be seen that the measured values are very close to the theoretical ones. The differences between the theoretical range displacements and the measured ones are all less than half of the range length of a pixel (i.e., 3.75 m), which indicates the measured values being exactly consistent with the theoretical ones. The differences between the theoretical azimuth displacements and measured ones are slightly larger. This could be caused by the residual cubic azimuth modulation (see Equation (37)) and azimuth defocusing.

6. Discussion

In Section 5, the results show that the proposed imaging method can correct the RCMs of the five targets simultaneously, and thus the static target can be finely focused and the moving targets can be focused with defocusing only in the azimuth direction. Moreover, the results presented in Table 3 show the measured azimuth and range displacements in the SAR image and ATI phases. It is seen that the measured values are very close to the theoretical ones and thus verify the theoretical analysis of the target’s properties in the SAR image. In this section, we will further present discussions on the computational complexity of the proposed imaging method, moving target refocusing, and link-budget determination.

6.1. Computational Complexity Analysis

One of the main advantages of the proposed imaging method is its high computational efficiency. In this subsection, its computational complexity is discussed and compared with that of the method proposed in [19].

For the proposed method, since the range compression, SRC, and RCMC can all be implemented via phase multiplications in the 2-D frequency domain, one can utilize a phase multiplication to accomplish these operations. Therefore, based on the analysis presented in Section 3, one can see that the proposed method needs only two azimuth FT/IFTs, two range FT/IFTs, and twice phase multiplications. The computational complexity can then be given by

$$10N_a{N}_r{\log }_2(N_a)+10N_a{N}_r{\log }_2(N_r)+12N_a{N}_r,$$

(47)

where N_a and N_r are the number of azimuth samples and range samples of the data, respectively.

The imaging method proposed in [19] requires four range FT/IFTs, two azimuth FT/IFTs, four phase multiplications, and one azimuth interpolation. Supposing that the length of sinc interpolation kernel is M (M is set to be 16 in [19]), the overall computational complexity of this method is then

$$10N_a{N}_r{\log }_2(N_a)+20N_a{N}_r{\log }_2(N_r)+24N_a{N}_r+6M\cdot N_a{N}_r,$$

(48)

From (47) and (48) one can see that the proposed method is more efficient, because there are no interpolation and less range FTs/IFTs and phase multiplications.

6.2. Moving Target Refocusing

Although moving targets suffer azimuth defocusing in the SAR image obtained by the proposed imaging method, we can refocus them after target detection and data extraction. Figure 9 presents a flowchart of moving target detection and refocusing. First, the DPCA processing is applied to the registered SAR images to cancel the clutter. Then, the detection and data extraction are implemented. Finally, the operation of refocusing is applied to the extracted target data.

A refocusing method exploiting the maximum contrast criterion is proposed below. Assuming that the inverse operations of azimuth compression and RCMC have been applied to the extracted data, then we can utilize the following method to refocus the target

$$s_{refocused}\left(t_r,t_a;{\widehat{l}}_1,{\widehat{l}}_2,{\widehat{l}}_3\right)=IDF{T}_2\left\{S\left(f_r,f_a\right)\cdot H_{refocus}\left(f_r,f_a;{\widehat{l}}_1,{\widehat{l}}_2,{\widehat{l}}_3\right)\right\},$$

(49)

with

$$H_{refocus}\left(f_r,f_a;{\widehat{l}}_1,{\widehat{l}}_2,{\widehat{l}}_3\right)=\exp \left\{j2\pi f_r\left(\frac{3c{\widehat{l}}_1{\widehat{l}}_3}{16f_c{}^2{\widehat{l}}_2{}^3}{f}_a^2+\frac{c{f}_a^2}{8f_c{}^2{\widehat{l}}_2}+\frac{c^2{\widehat{l}}_3}{16f_c{}^3{\widehat{l}}_2{}^3}{f}_a^3\right)\right\}\exp \left\{-j\pi \lambda \left(\frac{1}{4{\widehat{l}}_2}{f}_a{}^2+\frac{3{\widehat{l}}_1{\widehat{l}}_3}{8{\widehat{l}}_2{}^3}{f}_a{}^2+\frac{{\widehat{l}}_3\lambda }{16{\widehat{l}}_2{}^3}{f}_a{}^3\right)\right\},$$

(50)

where S (f_r, f_a) is the signal after range compression and SRC, IDFT₂ (·) indicates 2-D IFT. In addition, ${\widehat{l}}_1$, ${\widehat{l}}_2$, and ${\widehat{l}}_3$ are the estimates of the coefficients of the range equation. They are estimated via the following maximum-contrast based way

$$({\widehat{l}}_1,{\widehat{l}}_2,{\widehat{l}}_3)=\arg \underset{l_{1,t},l_{2,t},l_{3,t}}{\max }\left\{Contrast\left[s_{refocused}\left(t_r,t_a;l_{1,t},l_{2,t},l_{3,t}\right)\right]\right\},$$

(51)

with

$$Contrast\left[s_{refocused}\left(t_r,t_a;l_{1,t},l_{2,t},l_{3,t}\right)\right]=\frac{\sqrt{E\left\{{\left[{\left|s_{refocused}\left(t_r,t_a;l_{1,t},l_{2,t},l_{3,t}\right)\right|}^2-E\left({\left|s_{refocused}\left(t_r,t_a;l_{1,t},l_{2,t},l_{3,t}\right)\right|}^2\right)\right]}^2\right\}}}{E\left\{{\left|s_{refocused}\left(t_r,t_a;l_{1,t},l_{2,t},l_{3,t}\right)\right|}^2\right\}},$$

(52)

where $E(\cdot )$ defines the spatial average, $Contrast(\cdot )$ indicates the image contrast.

Figure 10 and Figure 11 present the refocusing results of target 1 and target 2. One can see that these two targets are all well focused.

6.3. Link-Budget Determination

The high orbital altitude gives MEO SAR the advantages of short revisit time and wide coverage. However, it also requires radar having a high transmit power and large antenna area, because the signal energy decays rapidly with the distance. In the following, we will briefly discuss the relationships between the target’s SNR and radar’s transmit power and antenna area.

Since the MEO SAR considered in this paper is of relatively low resolution, the size of the ground moving target can be considered to be smaller than the area of a resolution cell. The SNR of the focused target can then be calculated as [1]

$$SNR=\frac{P_{ave}{G}_t{G}_r{\lambda }^3{\sigma }_{t}c}{256{\pi }^3{R}_0{}^{3}KT{B}_T{F}_n{L}_s{V}_e{\rho }_r{\rho }_a},$$

(53)

where P_ave is the average transmit power, G_t is the gain of the transmitting antenna, G_r is the gain of the receiving antenna, σ_t is the target’s radar cross section (RCS), R₀ is the distance from the target to the radar, K is the Boltzmann’s constant, T is the temperature of receiver, B_T is the bandwidth of the transmitted signal, F_n is the receiver noise figure, L_s is the system losses, V_e is the effective velocity of satellite, ρ_r and ρ_a are the range and azimuth resolution, respectively.

We assume that the multichannel system has three channels in azimuth, and the whole antenna is used for transmission. The SNR per channel can then be expressed as

$$\begin{array}{cc}SNR& =\frac{P_{ave}{\lambda }^3{\sigma }_{t}c}{256{\pi }^3{R}^{3}KT{B}_T{F}_n{L}_s{V}_e{\rho }_r{\rho }_a}\cdot 4\pi \eta \frac{A}{{\lambda }^2}\cdot 4\pi \eta \frac{A_r}{{\lambda }^2}\hfill \\ & =\frac{P_{ave}{\sigma }_{t}c\cdot {\eta }^2{A}^2}{48\pi R^{3}KT{B}_T{F}_n{L}_s{V}_e{\rho }_r{\rho }_a\lambda },\hfill \end{array}$$

(54)

where ƞ is the antenna efficiency, A is the antenna area, and A_r = A/3 is the area of the receiving antenna per channel.

Based on (54), numerical experiments are conducted to investigate the relationship between the average transmit power, antenna area, and SNR. A target with an RCS of 2 m² (≈small passenger car) is considered, and other parameters used in the experiments are given in Table 1 and

Table 4. Parameters for radar equation.

Parameter	Value
Ls	3 (dB)
Fn	5 (dB)
T	290 K
ƞ	0.85
K	1.38 × 10−23

. The experimental results are shown in Figure 12 and Figure 13. Figure 12 illustrates the dependence of SNR on the average transmit power and antenna area, with the color indicating the value of SNR (the unit is dB). In Figure 13, the required average transmit powers under various pairs of SNR and antenna area are plotted. From Figure 12 it can be seen that, in order to obtain satisfactory SNR, a relatively high transmit power and large antenna is required. In addition, as can be seen from Figure 13, when the antenna area is 180 m² [24], for desired SNR of 10 dB, 20 dB, 30 dB, and 40 dB, the required average transmit powers are 21.08 dBW, 31.08 dBW, 41.08 dBW, and 51.08 dBW, respectively.

7. Conclusions

This paper has proposed an efficient imaging method that can finely focus static scenes and coarsely focus the moving targets of no Doppler ambiguity at the same time. Based on the fact that, in the 2-D frequency domain, the couplings between the range and azimuth of static targets and the moving targets without Doppler ambiguity are approximately identical, the proposed method efficiently corrects the RCMs of these targets via a phase multiplication simultaneously. In addition, moving targets’ characteristics, including the range and azimuth displacements and ATI phases, in the SAR image focused by the proposed method have been analyzed. Furthermore, the dependence of these characteristics on the parameters of moving targets have been revealed. Experiments have been conducted to verify the proposed imaging method and theoretical analyses.

After moving targets have been efficiently and coarsely focused by the proposed method, their SCNRs will be improved. Therefore, the proposed method is very suitable for MEO SAR, which suffers from limited energy and computation resources due to the high orbital altitude. Moreover, the analysis of the target’s displacements and ATI phase in the SAR image will benefit the design of the GMTI method for MEO multichannel SAR.