MIMO-Based Forward-Looking SAR Imaging Algorithm and Simulation

Abstract

Multiple-input multiple-output (MIMO) radar imaging can provide higher resolution and better sensitivity and thus can be applied to targets detection, recognition, and tracking. Missile-borne forward-looking SAR (MFL-SAR) is a new and special MIMO radar mode. It has advantage of two-dimensional (2D) imaging capability in forward direction over monostatic missile-borne SAR and airborne SAR. However, it is difficult to obtain accurate 2D frequency spectrum of the target echo signal due to the high velocity and descending height of this platform, which brings a lot of obstacles to imaging algorithm design. Therefore, a new imaging algorithm for MFL-SAR configuration based on the method of series reversion is proposed in this paper. This imaging method can implement range compression, secondary range compression (SRC), and range cell migration correction (RCMC) effectively. Finally, some simulations of point targets and comparison results confirm the efficiency of our proposed algorithm.

1. Introduction

Radar imaging is an emerging technology which can provide a high resolution radar image of targets in long distance independent of weather conditions and sunlight illumination, and it is being more and more widely used in military and civil fields [1–5]. Multiple-input multiple-output (MIMO) radar is an antenna system which transmits multiple probing signals via its antennas and scattering signals are received by distributed receiving antennas [6–9]. Application of MIMO radar to radar imaging could provide higher resolution and better sensitivity and thus can be applied to detection, recognition, and tracking of targets [10–12].

We consider a new MIMO radar system, that is, missile-borne multistatic synthetic aperture radar (MFL-SAR) as shown in Figure 1, in which transmitting signals are transmitted simultaneously and the reflected signals at receivers can be processed independently. This configuration can be applied to missile precision terminal guidance because of its advantages of 2D imaging ability in forward direction over monostatic SAR [13, 14]. Missile precision terminal guidance could conduct good performance in targets detection, recognition, orientation tracking, and attacking [15, 16]. During the moving period, transmitter and receiver cooperate with each other, and transmitter irradiates the imaging area with some squint angle, while receiver receives the target echo signals in forward-looking mode. Without loss of generality and for the convenience of description, missile-borne bistatic forward-looking SAR (MBFL-SAR) configuration is considered in this paper.

Figure 1 

An illustration of MFL-SAR configuration.

Effective and efficient imaging in this configuration is an important technology for missile precision terminal guidance; therefore, imaging algorithm for MBFL-SAR is necessary and essential. Many investigations on SAR imaging have been published and some appealing approaches have been suggested. Range-Doppler algorithm (RDA) [17–19] and chirp scaling algorithm (CSA) [20] are conventional monostatic algorithms which are applied to those configurations that have azimuth-invariant property. For bistatic SAR imaging, some other methods have been proposed. Soumekh derived the 2D spectrum expression of bistatic SAR and proposed a new RMA for bistatic configuration through two approximations in the derivation [21–23]. But this algorithm is limited to the bistatic parallel configuration because of identical velocity vectors of transmitter and receiver. Neo et al. and Davidson et al. proposed a nonlinear CS algorithm in [24, 25], but it neglected space-variance of Doppler frequency in range direction and influence of secondary range compression (SRC) on imaging. A simple operator named “dip move out” (DMO) is used to transform a bistatic survey into a monostatic one in [26, 27]. The initial dataset is used to convolve with the so-called smile, a short operator, and the output is the equivalent monostatic data. Unfortunately, the so-called smile is space-variant both in range direction and in azimuth direction, and the changes of wave number in range direction are also neglected.

An imaging algorithm based on the method of series reversion [28, 29] for MBFL-SAR is suggested in this paper. By unfolding the Doppler frequency, the stationary phase point expansion coefficients are obtained, which makes it easy to get the 2D frequency spectrum of the target echo signal. Then imaging method can be obtained based on the 2D frequency spectrum. Range compression is implemented in the range frequency domain, and SRC can be finished in the 2D frequency domain. The RCM can be corrected in the range-Doppler domain and finally the imaging result can be got through azimuth compression.

The rest of this paper is organized as follows. Geometric configuration and signal model of MBFL-SAR are introduced in Section 2. Also, the difficulty in direct-using the principle of stationary phase (POSP) is discussed. In Section 3, the derivation of 2D frequency spectrum of the target echo signal is given based on the accurate approximation of the slant range history, in which the high-order terms cannot be ignored owing to the presence of high velocities and accelerations. The proposed imaging algorithm is described in Section 4 and numerical simulations are achieved to show the effectiveness of our proposed method, and the results are given in Section 5. Finally, conclusions are drawn in Section 6.

2. Geometric Configuration and Signal Model

Figure 2 depicts geometric configuration of MBFL-SAR, in which both transmitter and receiver travel curvilinear descending motion. 𝑂 is the origin of coordinates, 𝑂^′ is the projection of transmitter when the slow time 𝑡_𝑚=0, and |𝑂𝑂^′|=𝑑. Transmitter is moving with the original velocity vectors 𝑣_𝑡0=(𝑣_𝑡𝑥0,𝑣_𝑡𝑦0,𝑣_𝑡𝑧0) and acceleration vectors 𝑎_𝑡=(𝑎_𝑡𝑥,𝑎_𝑡𝑦,𝑎_𝑡𝑧). And the ones of receiver are 𝑣_𝑟0=(0,𝑣_𝑟𝑦0,𝑣_𝑟𝑧0) and 𝑎_𝑟=(0,𝑎_𝑟𝑦,𝑎_𝑟𝑧), respectively. There exists an included angle 𝜓 between the two planes in which the two platforms travel. The transmitter keeps illuminating the imaging area in side-looking mode, and the receiver looks in forward direction. The receiver antenna gets the echo signal reflected from the imaging area which the receiver moves towards. Assume that transmitter and receiver are at 𝑇₀(𝑑,0,𝐻_𝑇) and 𝑅₀(0,0,𝐻_𝑅) when 𝑡_𝑚=0, and velocity vectors of them at the moment 𝑡_𝑚(𝑡_𝑚≠0) are 𝑣_𝑡=(𝑣_𝑡𝑥,𝑣_𝑡𝑦,𝑣_𝑡𝑧) and 𝑣_𝑟=(𝑣_𝑟𝑥,𝑣_𝑟𝑦,𝑣_𝑟𝑧), respectively, where 𝑣_𝑡𝑥, 𝑣_𝑡𝑦, and 𝑣_𝑡𝑧 denote the velocity components of 𝑣_𝑡 in the directions 𝑥, 𝑦, and 𝑧, respectively; and 𝑣_𝑟𝑥, 𝑣_𝑟𝑦, and 𝑣_𝑟𝑧 are the ones of 𝑣_𝑟, respectively.

Figure 2 

Geometric configuration of MBFL-SAR.

The bistatic slant range at time instant 𝑡_𝑚 can be obtained as 𝑅_𝑏𝑓𝑡_𝑚=𝑅_𝑇𝑡_𝑚+𝑅_𝑅𝑡_𝑚=𝑥_𝑡−𝑥_𝑝²+𝑦_𝑡−𝑦_𝑝²+𝑧²_𝑡+𝑥_𝑟−𝑥_𝑝²+𝑦_𝑟−𝑦_𝑝²+𝑧²_𝑟, (1) with 𝑥_𝑡=𝑑+𝑣_𝑡𝑥0𝑡_𝑚+0.5𝑎_𝑡𝑥𝑡²_𝑚,𝑥_𝑟𝑦=0,_𝑡=𝑣_𝑡𝑦0𝑡_𝑚+0.5𝑎_𝑡𝑦𝑡²_𝑚,𝑦_𝑟=𝑣_𝑟𝑦0𝑡_𝑚+0.5𝑎_𝑟𝑦𝑡²_𝑚,𝑧_𝑡=𝐻_𝑇+𝑣_𝑡𝑧0𝑡_𝑚+0.5𝑎_𝑡𝑧𝑡²_𝑚,𝑧_𝑟=𝐻_𝑅+𝑣_𝑟𝑧0𝑡_𝑚+0.5𝑎_𝑟𝑧𝑡²_𝑚, (2) where 𝑥_𝑡, 𝑦_𝑡, and 𝑧_𝑡 represent the locations of transmitter in directions 𝑥, 𝑦, and 𝑧, respectively; and 𝑥_𝑟, 𝑦_𝑟, and 𝑧_𝑟 are the ones of receiver, respectively.

Suppose that the transmitted waveform is the linear frequency modulation (LFM), and scattering from 𝑃(𝑥_𝑝,𝑦_𝑝,0) to the receiver can be written as 𝑠𝑡,̂𝑡_𝑚=𝑤_𝑟̂𝑅𝑡−^𝑏𝑓𝑡_𝑚𝑐𝑤_𝑎𝑡_𝑚̂𝑅×exp𝑗𝜋𝛾𝑡−^𝑏𝑓𝑡_𝑚𝑐²×exp−𝑗2𝜋𝜆𝑅_𝑏𝑓𝑡_𝑚, (3) where 𝑡̂ is the range time, 𝜆 is the wavelength, 𝑐 is the speed of light, and 𝛾 is the chirp rate. 𝑤_𝑟(⋅) and 𝑤_𝑎(⋅) are the range and azimuth envelopes, respectively.

By applying a range fast Fourier transform (FFT) to (3), we can obtain 𝑠𝑓_𝑟,𝑡_𝑚=𝑊_𝑟𝑓_𝑟𝑤_𝑎𝑡_𝑚𝑓⋅exp−𝑗𝜋^2𝑟𝛾⋅exp−𝑗2𝜋𝑐𝑓_𝑟+𝑓_𝑐𝑅_𝑏𝑓𝑡_𝑚, (4) where 𝑓_𝑟 represents the range frequency.

The 2D frequency spectrum of the echo signal can be obtained by an azimuth FFT, expressed as 𝑆𝑓_𝑟,𝑓_𝑎=^+∞_−∞𝑠𝑓_𝑟,𝑡_𝑚exp−𝑗2𝜋𝑓_𝑎𝑡_𝑚𝑑𝑡_𝑚, (5) where 𝑓_𝑎 represents the azimuth frequency, and the phase in (5) can be written as Θ𝑡_𝑚𝑓=−𝜋^2𝑟𝛾−2𝜋𝑐𝑓_𝑟+𝑓_𝑐𝑅_𝑏𝑓𝑡_𝑚−2𝜋𝑓_𝑎𝑡_𝑚. (6)

By applying the POSP, we have −2𝑐𝑓_𝑐+𝑓_𝑟𝑓_𝑎=𝛼_𝑡1+2𝛼_𝑡2𝑡_𝑚+3𝛼_𝑡3𝑡²_𝑚+4𝛼_𝑡4𝑡³_𝑚𝑅²_𝑡0+𝛼_𝑡1𝑡_𝑚+𝛼_𝑡2𝑡²_𝑚+𝛼_𝑡3𝑡³_𝑚+𝛼_𝑡4𝑡⁴_𝑚+𝛼_𝑟1+2𝛼_𝑟2𝑡_𝑚+3𝛼_𝑟3𝑡²_𝑚+4𝛼_𝑟4𝑡³_𝑚𝑅²_𝑟0+𝛼_𝑟1𝑡_𝑚+𝛼_𝑟2𝑡²_𝑚+𝛼_𝑟3𝑡³_𝑚+𝛼_𝑟4𝑡⁴_𝑚, (7) with 𝑅_𝑡0=𝑑−𝑥_𝑝²+𝑦²_𝑝+𝐻²_𝑇,𝑅_𝑟0=𝑥²_𝑝+𝑦²_𝑝+𝐻²_𝑅,𝛼_𝑡1=2𝑣_𝑡𝑥0𝑑−𝑥_𝑝−𝑦_𝑝𝑣_𝑡𝑦0+𝑣_𝑡𝑧0𝐻_𝑇,𝛼_𝑟1=2−𝑣_𝑟𝑦0𝑦_𝑝+𝑣_𝑟𝑧0𝐻_𝑅,𝛼_𝑡2=𝑣²_𝑡𝑥0+𝑎_𝑡𝑥𝑑−𝑥_𝑝+𝑣²_𝑡𝑦0−𝑎_𝑡𝑦𝑦_𝑝+𝑣²_𝑡𝑧0+𝑎_𝑡𝑧𝐻_𝑇,𝛼_𝑟2=𝑣²_𝑟𝑦0+𝑣²_𝑟𝑧0−𝑎_𝑟𝑦𝑦_𝑝+𝑎_𝑟𝑧𝐻_𝑅,𝛼_𝑡3=𝑎_𝑡𝑥𝑣_𝑡𝑥0+𝑎_𝑡𝑦𝑣_𝑡𝑦0+𝑎_𝑡𝑧𝑣_𝑡𝑧0,𝛼_𝑟3=𝑎_𝑟𝑦𝑣_𝑟𝑦0+𝑎_𝑟𝑧𝑣_𝑟𝑧0,𝛼_𝑡4=14𝑎²_𝑡𝑥+𝑎²_𝑡𝑦+𝑎²_𝑡𝑧,𝛼_𝑟4=14𝑎²_𝑟𝑦+𝑎²_𝑟𝑧. (8)

If Θ^′(𝑡^∗_𝑚)=0, we can get the stationary point 𝑡^∗_𝑚, but we can see that it is very difficult to determine the stationary point 𝑡^∗_𝑚 from (7), so 2D frequency spectrum of the echo signal cannot be obtained through direct POSP. Some other approaches should be taken to get 2D frequency spectrum of the echo signal.

3. Derivation of 2D Frequency Spectrum Based on Series Reversion

According to [28], it needs to take efficient approximation of the slant range 𝑅_𝑏𝑓(𝑡_𝑚) before using the method of series reversion. Because of the high velocity and acceleration in the missile platform, it needs to keep the terms up to the third-order 𝑅_𝑏𝑓𝑡_𝑚=𝑅_𝑏𝑓0+𝑘₁𝑡_𝑚+𝑘₂𝑡²_𝑚+𝑘₃𝑡³_𝑚+⋯, (9) with 𝑅_𝑏𝑓0=𝑅_𝑡0+𝑅_𝑟0,𝑘₁=𝛼_𝑡12𝑅_𝑡0+𝛼_𝑟12𝑅_𝑟0,𝑘₂=𝛼_𝑡22𝑅_𝑡0−𝛼²_𝑡18𝑅³_𝑡0+𝛼_𝑟22𝑅_𝑟0−𝛼²_𝑟18𝑅³_𝑟0,𝑘₃=𝛼_𝑡32𝑅_𝑡0−𝛼_𝑡1𝛼_𝑡24𝑅³_𝑡0+𝛼³_𝑡116𝑅⁵_𝑡0+𝛼_𝑟32𝑅_𝑟0−𝛼_𝑟1𝛼_𝑟24𝑅³_𝑟0+𝛼³_𝑟116𝑅⁵_𝑟0. (10)

To apply the method of series reversion, the linear range cell migration (LRCM) should be removed firstly. Let Θ^′(𝑡_𝑚)=0, we have −𝑐𝑓_𝑐+𝑓_𝑟𝑓_𝑎=2𝑘₂𝑡_𝑚+3𝑘₃𝑡²_𝑚. (11)

Then through the method of series reversion, the expression of the stationary phase point can be expressed by unfolded Doppler frequency as 𝑡_𝑚𝑓_𝑎=𝐴₁−𝑐𝑓_𝑐+𝑓_𝑟𝑓_𝑎+𝐴₂−𝑐𝑓_𝑐+𝑓_𝑟𝑓_𝑎²+⋯, (12) where 𝐴₁=1/2𝑘₂, 𝐴₂=−3𝑘₃/8𝑘³₂.

Having obtained the stationary phase point, we should reintroduce the LRCM term to calculate 2D frequency spectrum of the echo signal. By (5), (6), and (12), 2D frequency spectrum of the echo signal is written as 𝑆_MSR𝑓_𝑟,𝑓_𝑎=𝑊_𝑟𝑓_𝑟𝑊_𝑎𝑓_𝑎𝑓exp−𝑗𝜋^2𝑟𝛾exp𝑗Φ𝑓_𝑟,𝑓_𝑎, (13) where Φ𝑓_𝑟,𝑓_𝑎𝑓=−2𝜋^𝑐+𝑓_𝑟𝑐𝑅_𝑏𝑓0𝑓−𝜋^2𝑟𝛾𝑐+2𝜋4𝑘₂𝑓_𝑐+𝑓_𝑟𝑓_𝑎+𝑓_𝑐+𝑓_𝑟𝑘₁𝑐²𝑘+2𝜋³𝑐²8𝑘³₂𝑓_𝑐+𝑓_𝑟²𝑓_𝑎+𝑓_𝑐+𝑓_𝑟𝑘₁𝑐³. (14)

It can be seen that the coefficient of the third-term in the slant range is included in Φ(𝑓_𝑟,𝑓_𝑎), which indicates the presence of the high-order terms introduced by high velocity and acceleration of this configuration into 2D frequency spectrum. The imaging algorithm can be designed based on 2D frequency spectrum, which will be discussed in the next section.

4. Imaging Algorithm for MBFL-SAR

To design imaging algorithm efficiently, the phase term in (14) should be decomposed using Taylor series firstly because of the range/azimuth frequency coupling in 2D frequency spectrum, and the series expansions are as follows: Φ𝑓_𝑟,𝑓_𝑎≈Φ_𝑟𝑔𝑓_𝑟+Φ_𝑎𝑓_𝑎+Φ_rcm𝑓_𝑟,𝑓_𝑎+Φ_src𝑓_𝑟,𝑓_𝑎+Φ_res, (15) with Φ_𝑟𝑔𝑓_𝑟𝑓=−𝜋^2𝑟𝛾,Φ_𝑎𝑓_𝑎1=2𝜋4𝑘₂𝑘²₁𝑐+2𝑘₁𝑓_𝑎+𝑐𝑓_𝑐𝑓²_𝑎+𝑘₃8𝑘³₂𝑘³₁𝑐+3𝑘²₁𝑓_𝑎+3𝑘₁𝑐𝑓_𝑐𝑓²_𝑎+𝑐𝑓_𝑐²𝑓³_𝑎,Φ_rcm𝑓_𝑟,𝑓_𝑎=2𝜋𝑓_𝑟−𝑅_𝑏𝑓0𝑐+14𝑘₂𝑘²₁𝑐𝑓−𝑐^𝑎𝑓_𝑐²+𝑘₃8𝑘³₂𝑘³₁𝑐−3𝑘₁𝑐𝑓_𝑎𝑓_𝑐²−2𝑐²𝑓_𝑎𝑓_𝑐³,Φ_src𝑓_𝑟,𝑓_𝑎1=2𝜋4𝑘₂𝑐𝑓_𝑐𝑓_𝑟𝑓_𝑐²−𝑓_𝑟𝑓_𝑐³𝑓²_𝑎+𝑘₃8𝑘³₂3𝑘₁𝑐𝑓_𝑐𝑓_𝑟𝑓_𝑐²−𝑓_𝑟𝑓_𝑐³𝑓²_𝑎+𝑐𝑓_𝑐²3𝑓_𝑟𝑓_𝑐²𝑓−4^𝑟𝑓_𝑐³𝑓³_𝑎,Φ_res=−2𝜋𝑓_𝑐𝑅_𝑏𝑓0𝑐. (16)(17)(18)(19)(20)

The term in (16) represents the range compression term which is independent of the azimuth frequency 𝑓_𝑎. Thus the data can be range-compressed in the range frequency azimuth time domain or in the 2D frequency domain alternatively. The term in (17) denotes the azimuth compression term, dependent only on 𝑓_𝑎 and used in the azimuth compression. It should be implemented after range compression and RCMC in the range time azimuth frequency domain as 𝑘-coefficients are range variant. The term in (18) indicates the RCM term which is linearly dependent on the range frequency 𝑓_𝑟. It needs to be compensated because of the coupling between range and azimuth. Similar to the azimuth compression term, this term should be implemented in the range time azimuth frequency domain. Note that −2𝜋(𝑅_𝑏𝑓0/𝑐)𝑓_𝑟 is the linear phase representing the location of target point; the energy of the target will be focused within the corresponding range cell after RCMC. It should be paid attention to in the imaging process. The term in (19) is the SRC term which indicates the coupling between 𝑓_𝑟 and 𝑓_𝑎. It may cause significant degradation in the imaging resolution if uncompensated and is compensated in the 2D frequency domain. The last residual term in (20) is independent of the range frequency or the azimuth frequency. It has no effect on the imaging focus and thus can be neglected in MBFL-SAR configuration.

Consequently, the main procedure of the proposed algorithm follows the following steps.(1)Use 2D FFT to transform the echo signal to 2D frequency domain as shown in (13).(2)Design the match filtering function in the 2D frequency domain to compensate the range compression term, the SRC term, and the residual term. The match filtering function 𝐻₁(𝑓_𝑟,𝑓_𝑎) can be obtained as follows: 𝐻₁𝑓_𝑟,𝑓_𝑎=exp−𝑗Φ_𝑟𝑔𝑓_𝑟+Φ_src𝑓_𝑟,𝑓_𝑎+Φ_res. (21)(3)Apply range inverse fast Fourier transform (IFFT) to convert the signal to the range time azimuth frequency domain and correct the RCM term.(4)Finish azimuth compression by multiplying the range time azimuth frequency domain signal with the azimuth match filtering function, which can be obtained as follows: 𝐻₂𝑡_𝑚,𝑓_𝑎=exp−𝑗Φ_𝑎𝑓_𝑎. (22)(5)Perform azimuth IFFT to obtain a well-focused SAR image.

5. Simulations and Results

In this section, several examples are provided to illustrate the performance of the proposed method. Because of the complex configuration, the high-order terms in the slant range cannot be ignored. If inappropriately approximated, the imaging focusing resolution may be affected significantly. Some simulations are conducted to compare the influences of keeping the terms up to the quadratic term (quadratic range model) and the higher order term (proposed range model). Parameters used in the simulations are listed in Table 1. The imaging results of center point and fringe point in the imaging area using the proposed range model are given in Figures 3 and 4. In order to confirm the necessity and validity using the higher order range model in this special configuration, some simulations are also conducted to compare the differences using the quadratic range model and the proposed range model, and the results are shown in Figures 5 and 6 (here, the center point is selected in comparison).

Table 1 

Parameters used in the simulations.

(a) 2D impulse response

(b) Azimuth impulse response

(c) Range impulse response

Figure 3 

Center point results using the proposed range model.

(a) 2D impulse response

(b) Azimuth impulse response

(c) Range impulse response

Figure 4 

Fringe point results using the proposed range model.

(a) Quadratic range model

(b) Proposed range model

Figure 5 

Contour plots comparison using different range models.

(a) Quadratic range model

(b) Proposed range model

Figure 6 

Azimuth impulse responses comparison using different range models.

Figure 3 depicts the results of the center point using the proposed range model. Figure 3(a) is the 2D impulse response; Figures 3(b) and 3(c) are the azimuth impulse response and the range impulse response, respectively. The 2D impulse response of the fringe target point using the proposed range model is shown in Figure 4(a), and the azimuth impulse response and the range impulse response are given in Figures 4(b) and 4(c), respectively. We can see from Figures 3 and 4 that both center point and fringe point are well focused in range direction and azimuth direction. For the complexity of the MBFL-SAR configuration, if the slant range is not approximated accurately enough, the phase term in the 2D frequency spectrum will not be compensated completely in the azimuth direction. It is easy to find that the image of the point target suffers from distortion and main-lobe broadening when the slant range keeps the terms up to the quadratic term in Figures 5(a) and 6(a). On the other hand, if we are using the proposed range model, the imaging focusing performance can be obviously improved, as given in Figures 5(b) and 6(b). It is obvious that the image is well focused in the azimuth direction, which confirms it necessary to approximate appropriately in the MBFL-SAR configuration.

Table 2 lists the image quality parameters of the center point and fringe point using the proposed method to further illustrate the validity of the proposed algorithm. PSLR and ISLR represent peak side-lobe ratio and integrated side-lobe ratio, respectively. Besides, the theoretical PSLR and ISLR are −13.26 dB and −9.8 dB, respectively. We can see that the image quality parameters of both the center point and fringe point using our proposed method are all close to the theoretical ones, which indicate satisfactory imaging results and further validate the effectiveness and feasibility.

Table 2 

Image quality parameters using the proposed method.

6. Conclusions

In this paper, we developed an imaging algorithm for MBFL-SAR configuration, a special MIMO radar mode. The MBFL-SAR imaging geometric model is established and the signal model is analyzed firstly. The high-order term in the slant range cannot be neglected because of the presence of high velocity and acceleration and the addition of two square-root terms. Also, the difficulty in direct-using the POSP in this configuration is discussed. Then 2D frequency spectrum is deduced using the method of series reversion based on the appropriate approximation of the slant range. Based on the derived spectrum, the corresponding imaging method is developed. The high-order term introduced from this configuration is compensated, together with range compression, at the beginning of the algorithm. RCMC is finished in the range time azimuth frequency domain as the range-variant 𝑘-coefficients. The focused SAR image can be obtained through the azimuth compression. The feasibility and efficiency of the proposed approach are validated with the simulation experiments.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61001211 and 61303035, by the Fundamental Research Funds for the Central Universities (K5051202016), and by the Science Foundation for Navigation (20110181004).