research article mimo-based forward-looking sar imaging algorithm and simulation · 2019. 7....

Research ArticleMIMO-Based Forward-Looking SARImaging Algorithm and Simulation

Ziqiang Meng, Yachao Li, Shengqi Zhu, Yinghui Quan, Mengdao Xing, and Zheng Bao

National Laboratory of Radar Signal Processing, Xidian University, Xi’an 710071, China

Correspondence should be addressed to Yachao Li; [email protected]

Received 25 March 2014; Revised 17 June 2014; Accepted 29 June 2014; Published 15 July 2014

Academic Editor: Wei Xu

Copyright © 2014 Ziqiang Meng et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Multiple-inputmultiple-output (MIMO) radar imaging can provide higher resolution and better sensitivity and thus can be appliedto targets detection, recognition, and tracking. Missile-borne forward-looking SAR (MFL-SAR) is a new and special MIMO radarmode. It has advantage of two-dimensional (2D) imaging capability in forward direction over monostatic missile-borne SAR andairborne SAR. However, it is difficult to obtain accurate 2D frequency spectrum of the target echo signal due to the high velocityand descending height of this platform, which brings a lot of obstacles to imaging algorithm design. Therefore, a new imagingalgorithm for MFL-SAR configuration based on the method of series reversion is proposed in this paper. This imaging methodcan 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 providea high resolution radar image of targets in long distance inde-pendent of weather conditions and sunlight illumination,and it is being more and more widely used in military andcivil fields [1–5]. Multiple-input multiple-output (MIMO)radar is an antenna system which transmits multiple probingsignals via its antennas and scattering signals are received bydistributed receiving antennas [6–9]. Application of MIMOradar to radar imaging could provide higher resolutionand 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) asshown in Figure 1, in which transmitting signals are transmit-ted simultaneously and the reflected signals at receivers canbe processed independently. This configuration can beapplied to missile precision terminal guidance because of itsadvantages of 2D imaging ability in forward direction overmonostatic SAR [13, 14]. Missile precision terminal guidancecould conduct good performance in targets detection, recog-nition, orientation tracking, and attacking [15, 16]. During themoving period, transmitter and receiver cooperate with each

other, and transmitter irradiates the imaging area with somesquint angle, while receiver receives the target echo signals inforward-lookingmode.Without loss of generality and for theconvenience of description, missile-borne bistatic forward-looking SAR (MBFL-SAR) configuration is considered in thispaper.

Effective and efficient imaging in this configuration is animportant technology for missile precision terminal guid-ance; therefore, imaging algorithm for MBFL-SAR is neces-sary and essential. Many investigations on SAR imaging havebeen published and some appealing approaches have beensuggested. Range-Doppler algorithm (RDA) [17–19] andchirp scaling algorithm (CSA) [20] are conventional mono-static algorithms which are applied to those configurationsthat have azimuth-invariant property. For bistatic SAR imag-ing, some other methods have been proposed. Soumekhderived the 2D spectrum expression of bistatic SAR andproposed a new RMA for bistatic configuration through twoapproximations in the derivation [21–23]. But this algorithmis limited to the bistatic parallel configuration because ofidentical velocity vectors of transmitter and receiver. Neoet al. and Davidson et al. proposed a nonlinear CS algorithmin [24, 25], but it neglected space-variance of Doppler fre-quency in range direction and influence of secondary range

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2014, Article ID 783949, 9 pageshttp://dx.doi.org/10.1155/2014/783949

2 International Journal of Antennas and Propagation

Target area

Transmitters

Receivers

Figure 1: An illustration of MFL-SAR configuration.

compression (SRC) on imaging. A simple operator named“dip move out” (DMO) is used to transform a bistatic surveyinto a monostatic one in [26, 27]. The initial dataset is usedto convolve with the so-called smile, a short operator, and theoutput is the equivalent monostatic data. Unfortunately, theso-called smile is space-variant both in range direction and inazimuth direction, and the changes of wave number in rangedirection are also neglected.

An imaging algorithm based on the method of seriesreversion [28, 29] for MBFL-SAR is suggested in this paper.By unfolding the Doppler frequency, the stationary phasepoint expansion coefficients are obtained,whichmakes it easyto get the 2D frequency spectrum of the target echo signal.Then imaging method can be obtained based on the 2D fre-quency spectrum. Range compression is implemented in therange frequency domain, and SRC can be finished in the 2Dfrequency domain. The RCM can be corrected in the range-Doppler domain and finally the imaging result can be gotthrough azimuth compression.

The rest of this paper is organized as follows. Geometricconfiguration and signalmodel ofMBFL-SAR are introducedin Section 2. Also, the difficulty in direct-using the principleof stationary phase (POSP) is discussed. In Section 3, thederivation of 2D frequency spectrum of the target echosignal is given based on the accurate approximation of theslant range history, in which the high-order terms cannot beignored owing to the presence of high velocities and accel-erations. The proposed imaging algorithm is described inSection 4 and numerical simulations are achieved to show theeffectiveness of our proposed method, and the resultsare given in Section 5. Finally, conclusions are drawn inSection 6.

2. Geometric Configuration and Signal Model

Figure 2 depicts geometric configuration of MBFL-SAR, inwhich both transmitter and receiver travel curvilinear

Point target

Z

R

HR

O

d

O

x

xpHT

→

�t

→

�r

yp

P(xp, yp, 0)

T

y

𝜓

Figure 2: Geometric configuration of MBFL-SAR.

descending motion. 𝑂 is the origin of coordinates, 𝑂 is theprojection of transmitter when the slow time 𝑡

𝑚= 0, and

|𝑂𝑂| = 𝑑. Transmitter is moving with the original velocity

vectors V⃗𝑡0= (V𝑡𝑥0, V𝑡𝑦0, V𝑡𝑧0) and acceleration vectors ⃗𝑎

𝑡=

(𝑎𝑡𝑥, 𝑎𝑡𝑦, 𝑎𝑡𝑧). And the ones of receiver are V⃗

𝑟0= (0, V

𝑟𝑦0, V𝑟𝑧0)

and ⃗𝑎𝑟= (0, 𝑎

𝑟𝑦, 𝑎𝑟𝑧), respectively. There exists an included

angle 𝜓 between the two planes in which the two platformstravel.The transmitter keeps illuminating the imaging area inside-looking mode, and the receiver looks in forward direc-tion. The receiver antenna gets the echo signal reflectedfrom the imaging area which the receiver moves towards.Assume that transmitter and receiver are at 𝑇

0(𝑑, 0,𝐻

𝑇) and

𝑅0(0, 0,𝐻

𝑅) when 𝑡

𝑚= 0, and velocity vectors of them at

the moment 𝑡𝑚(𝑡𝑚̸= 0) are V⃗

𝑡= (V𝑡𝑥, V𝑡𝑦, V𝑡𝑧) and V⃗

𝑟= (V𝑟𝑥,

V𝑟𝑦, V𝑟𝑧), respectively, where V

𝑡𝑥, V𝑡𝑦, and V

𝑡𝑧denote the veloc-

ity components of V⃗𝑡in the directions𝑥,𝑦, and 𝑧, respectively;

and V𝑟𝑥, V𝑟𝑦, and V

𝑟𝑧are the ones of V⃗

𝑟, respectively.

The bistatic slant range at time instant 𝑡𝑚can be obtained

as

𝑅𝑏𝑓(𝑡𝑚) = 𝑅𝑇(𝑡𝑚) + 𝑅𝑅(𝑡𝑚)

= √(𝑥𝑡− 𝑥𝑝)2

+ (𝑦𝑡− 𝑦𝑝)2

+ 𝑧2

𝑡

+ √(𝑥𝑟− 𝑥𝑝)2

+ (𝑦𝑟− 𝑦𝑝)2

+ 𝑧2𝑟,

(1)

with

𝑥𝑡= 𝑑 + V

𝑡𝑥0𝑡𝑚+ 0.5𝑎

𝑡𝑥𝑡2

𝑚, 𝑥

𝑟= 0,

𝑦𝑡= V𝑡𝑦0𝑡𝑚+ 0.5𝑎

𝑡𝑦𝑡2

𝑚,

𝑦𝑟= V𝑟𝑦0𝑡𝑚+ 0.5𝑎

𝑟𝑦𝑡2

𝑚,

𝑧𝑡= 𝐻𝑇+ V𝑡𝑧0𝑡𝑚+ 0.5𝑎

𝑡𝑧𝑡2

𝑚,

𝑧𝑟= 𝐻𝑅+ V𝑟𝑧0𝑡𝑚+ 0.5𝑎

𝑟𝑧𝑡2

𝑚,

(2)

where 𝑥𝑡, 𝑦𝑡, and 𝑧

𝑡represent the locations of transmitter in

directions 𝑥, 𝑦, and 𝑧, respectively; and 𝑥𝑟, 𝑦𝑟, and 𝑧

𝑟are the

ones of receiver, respectively.

International Journal of Antennas and Propagation 3

Suppose that the transmitted waveform is the linear fre-quency modulation (LFM), and scattering from 𝑃(𝑥

𝑝, 𝑦𝑝, 0)

to the receiver can be written as

𝑠 (�̂�, 𝑡𝑚) = 𝑤𝑟(�̂� −

𝑅𝑏𝑓(𝑡𝑚)

𝑐)𝑤𝑎(𝑡𝑚)

× exp[

[

𝑗𝜋𝛾(�̂� −𝑅𝑏𝑓(𝑡𝑚)

𝑐)

2

]

]

× exp[−𝑗2𝜋𝜆𝑅𝑏𝑓(𝑡𝑚)] ,

(3)

where �̂� is the range time, 𝜆 is the wavelength, 𝑐 is the speed oflight, and 𝛾 is the chirp rate.𝑤

𝑟(⋅) and𝑤

𝑎(⋅) are the range and

azimuth envelopes, respectively.By applying a range fast Fourier transform (FFT) to (3),

we can obtain

𝑠 (𝑓𝑟, 𝑡𝑚) = 𝑊

𝑟(𝑓𝑟) 𝑤𝑎(𝑡𝑚) ⋅ exp(−𝑗𝜋

𝑓2

𝑟

𝛾)

⋅ exp [−𝑗2𝜋𝑐(𝑓𝑟+ 𝑓𝑐) 𝑅𝑏𝑓(𝑡𝑚)] ,

(4)

where 𝑓𝑟represents the range frequency.

The 2D frequency spectrum of the echo signal can beobtained by an azimuth FFT, expressed as

𝑆 (𝑓𝑟, 𝑓𝑎) = ∫

+∞

−∞

𝑠 (𝑓𝑟, 𝑡𝑚) exp (−𝑗2𝜋𝑓

𝑎𝑡𝑚) 𝑑𝑡𝑚, (5)

where 𝑓𝑎represents the azimuth frequency, and the phase in

(5) can be written as

Θ(𝑡𝑚) = −𝜋

𝑓2

𝑟

𝛾−2𝜋

𝑐(𝑓𝑟+ 𝑓𝑐) 𝑅𝑏𝑓(𝑡𝑚) − 2𝜋𝑓

𝑎𝑡𝑚. (6)

By applying the POSP, we have

−2𝑐

𝑓𝑐+ 𝑓𝑟

𝑓𝑎=

𝛼𝑡1+ 2𝛼𝑡2𝑡𝑚+ 3𝛼𝑡3𝑡2

𝑚+ 4𝛼𝑡4𝑡3

𝑚

√𝑅2

𝑡0+ 𝛼𝑡1𝑡𝑚+ 𝛼𝑡2𝑡2𝑚+ 𝛼𝑡3𝑡3𝑚+ 𝛼𝑡4𝑡4𝑚

+𝛼𝑟1+ 2𝛼𝑟2𝑡𝑚+ 3𝛼𝑟3𝑡2

𝑚+ 4𝛼𝑟4𝑡3

𝑚

√𝑅2

𝑟0+ 𝛼𝑟1𝑡𝑚+ 𝛼𝑟2𝑡2𝑚+ 𝛼𝑟3𝑡3𝑚+ 𝛼𝑟4𝑡4𝑚

,

(7)

with

𝑅𝑡0= √(𝑑 − 𝑥

𝑝)2

+ 𝑦2𝑝+ 𝐻2

𝑇,

𝑅𝑟0= √𝑥2

𝑝+ 𝑦2𝑝+ 𝐻2

𝑅,

𝛼𝑡1= 2 [V

𝑡𝑥0(𝑑 − 𝑥

𝑝) − 𝑦𝑝V𝑡𝑦0+ V𝑡𝑧0𝐻𝑇] ,

𝛼𝑟1= 2 (−V

𝑟𝑦0𝑦𝑝+ V𝑟𝑧0𝐻𝑅) ,

𝛼𝑡2= V2𝑡𝑥0+ 𝑎𝑡𝑥(𝑑 − 𝑥

𝑝) + V2𝑡𝑦0− 𝑎𝑡𝑦𝑦𝑝+ V2𝑡𝑧0+ 𝑎𝑡𝑧𝐻𝑇,

𝛼𝑟2= V2𝑟𝑦0+ V2𝑟𝑧0− 𝑎𝑟𝑦𝑦𝑝+ 𝑎𝑟𝑧𝐻𝑅,

𝛼𝑡3= 𝑎𝑡𝑥V𝑡𝑥0+ 𝑎𝑡𝑦V𝑡𝑦0+ 𝑎𝑡𝑧V𝑡𝑧0,

𝛼𝑟3= 𝑎𝑟𝑦V𝑟𝑦0+ 𝑎𝑟𝑧V𝑟𝑧0,

𝛼𝑡4=1

4(𝑎2

𝑡𝑥+ 𝑎2

𝑡𝑦+ 𝑎2

𝑡𝑧) ,

𝛼𝑟4=1

4(𝑎2

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 stationarypoint 𝑡∗

𝑚from (7), so 2D frequency spectrum of the echo

signal cannot be obtained through direct POSP. Some otherapproaches should be taken to get 2D frequency spectrum ofthe echo signal.

3. Derivation of 2D Frequency SpectrumBased on Series Reversion

According to [28], it needs to take efficient approximation ofthe slant range 𝑅

𝑏𝑓(𝑡𝑚) before using the method of series

reversion. Because of the high velocity and acceleration in themissile platform, it needs to keep the terms up to the third-order

𝑅𝑏𝑓(𝑡𝑚) = 𝑅𝑏𝑓0+ 𝑘1𝑡𝑚+ 𝑘2𝑡2

𝑚+ 𝑘3𝑡3

𝑚+ ⋅ ⋅ ⋅ , (9)

with

𝑅𝑏𝑓0

= 𝑅𝑡0+ 𝑅𝑟0,

𝑘1=𝛼𝑡1

2𝑅𝑡0

+𝛼𝑟1

2𝑅𝑟0

,

𝑘2=𝛼𝑡2

2𝑅𝑡0

−𝛼2

𝑡1

8𝑅3

𝑡0

+𝛼𝑟2

2𝑅𝑟0

−𝛼2

𝑟1

8𝑅3

𝑟0

,

𝑘3=𝛼𝑡3

2𝑅𝑡0

−𝛼𝑡1𝛼𝑡2

4𝑅3

𝑡0

+𝛼3

𝑡1

16𝑅5

𝑡0

+𝛼𝑟3

2𝑅𝑟0

−𝛼𝑟1𝛼𝑟2

4𝑅3

𝑟0

+𝛼3

𝑟1

16𝑅5

𝑟0

.

(10)

To apply the method of series reversion, the linear rangecell migration (LRCM) should be removed firstly. LetΘ(𝑡𝑚) = 0, we have

−𝑐

𝑓𝑐+ 𝑓𝑟

𝑓𝑎= 2𝑘2𝑡𝑚+ 3𝑘3𝑡2

𝑚. (11)

Then through the method of series reversion, the expres-sion of the stationary phase point can be expressed byunfolded Doppler frequency as

𝑡𝑚(𝑓𝑎) = 𝐴

1(−

𝑐

𝑓𝑐+ 𝑓𝑟

𝑓𝑎) + 𝐴

2(−

𝑐

𝑓𝑐+ 𝑓𝑟

𝑓𝑎)

2

+ ⋅ ⋅ ⋅ ,

(12)

where 𝐴1= 1/2𝑘

2, 𝐴2= −3𝑘

3/8𝑘3

2.


Having obtained the stationary phase point, we shouldreintroduce the LRCM term to calculate 2D frequency spec-trum of the echo signal. By (5), (6), and (12), 2D frequencyspectrum of the echo signal is written as

𝑆MSR (𝑓𝑟, 𝑓𝑎)

= 𝑊𝑟(𝑓𝑟)𝑊𝑎(𝑓𝑎) exp[−𝑗𝜋

𝑓2

𝑟

𝛾] exp [𝑗Φ (𝑓

𝑟, 𝑓𝑎)] ,

(13)

where

Φ(𝑓𝑟, 𝑓𝑎) = −2𝜋

𝑓𝑐+ 𝑓𝑟

𝑐𝑅𝑏𝑓0− 𝜋

𝑓2

𝑟

𝛾

+ 2𝜋𝑐

4𝑘2(𝑓𝑐+ 𝑓𝑟)(𝑓𝑎+ (𝑓𝑐+ 𝑓𝑟)𝑘1

𝑐)

2

+ 2𝜋𝑘3𝑐2

8𝑘3

2(𝑓𝑐+ 𝑓𝑟)2(𝑓𝑎+ (𝑓𝑐+ 𝑓𝑟)𝑘1

𝑐)

3

.

(14)

It can be seen that the coefficient of the third-term inthe slant range is included in Φ(𝑓

𝑟, 𝑓𝑎), which indicates the

presence of the high-order terms introduced by high velocityand acceleration of this configuration into 2D frequency spec-trum. The imaging algorithm can be designed based on 2Dfrequency spectrum, which will be discussed in the nextsection.

4. Imaging Algorithm for MBFL-SAR

Todesign imaging algorithmefficiently, the phase term in (14)should be decomposed using Taylor series firstly because ofthe range/azimuth frequency coupling in 2D frequency spec-trum, and the series expansions are as follows:

Φ(𝑓𝑟, 𝑓𝑎) ≈ Φ

𝑟𝑔(𝑓𝑟) + Φ𝑎(𝑓𝑎) + Φrcm (𝑓𝑟, 𝑓𝑎)

+ Φsrc (𝑓𝑟, 𝑓𝑎) + Φres,(15)

with

Φ𝑟𝑔(𝑓𝑟) = −𝜋

𝑓2

𝑟

𝛾, (16)

Φ𝑎(𝑓𝑎)

= 2𝜋{1

4𝑘2

(𝑘2

1

𝑐+ 2𝑘1𝑓𝑎+𝑐

𝑓𝑐

𝑓2

𝑎)

+𝑘3

8𝑘3

2

[𝑘3

1

𝑐+ 3𝑘2

1𝑓𝑎+ 3𝑘1

𝑐

𝑓𝑐

𝑓2

𝑎+ (

𝑐

𝑓𝑐

)

2

𝑓3

𝑎]} ,

(17)

Φrcm (𝑓𝑟, 𝑓𝑎)

= 2𝜋𝑓𝑟{−𝑅𝑏𝑓0

𝑐+1

4𝑘2

[𝑘2

1

𝑐− 𝑐(

𝑓𝑎

𝑓𝑐

)

2

]

+𝑘3

8𝑘3

2

[𝑘3

1

𝑐− 3𝑘1𝑐(𝑓𝑎

𝑓𝑐

)

2

− 2𝑐2(𝑓𝑎

𝑓𝑐

)

3

]} ,

(18)

Table 1: Parameters used in the simulations.

Wavelength 0.02m

Bandwidth 50MHz

Sampling frequency 100MHz

Pulse duration 2 𝜇s

Scene center location (0, 4500, 0)

Fringe point location (300, 4800, 0)

𝜓 10∘

PRF 10KHz

𝐻𝑇

35 km

𝐻𝑅

30 km

V⃗𝑡0

(434.1, 2462.0, −2000) m/s

⃗𝑎𝑡

(−8.7, −49.2, 30) m/s2

V⃗𝑟0

(0, 2500, −2000) m/s

⃗𝑎𝑟

(0, −50, 30) m/s2

Φsrc (𝑓𝑟, 𝑓𝑎)

= 2𝜋{1

4𝑘2

𝑐

𝑓𝑐

((𝑓𝑟

𝑓𝑐

)

2

− (𝑓𝑟

𝑓𝑐

)

3

)𝑓2

𝑎

+𝑘3

8𝑘3

2

[3𝑘1

𝑐

𝑓𝑐

((𝑓𝑟

𝑓𝑐

)

2

− (𝑓𝑟

𝑓𝑐

)

3

)𝑓2

𝑎

+ (𝑐

𝑓𝑐

)

2

(3(𝑓𝑟

𝑓𝑐

)

2

− 4(𝑓𝑟

𝑓𝑐

)

3

)𝑓3

𝑎]} ,

(19)

Φres = −2𝜋𝑓𝑐𝑅𝑏𝑓0

𝑐. (20)

The term in (16) represents the range compression termwhich is independent of the azimuth frequency 𝑓

𝑎. Thus the

data can be range-compressed in the range frequency azim-uth time domain or in the 2D frequency domain alternatively.The term in (17) denotes the azimuth compression term,dependent only on 𝑓

𝑎and used in the azimuth compression.

It should be implemented after range compression andRCMC in the range time azimuth frequency domain as 𝑘-coe-fficients are range variant.The term in (18) indicates the RCMtermwhich is linearly dependent on the range frequency𝑓

𝑟. It

needs to be compensated because of the coupling betweenrange and azimuth. Similar to the azimuth compression term,this term should be implemented in the range time azimuthfrequency domain. Note that −2𝜋(𝑅

𝑏𝑓0/𝑐)𝑓𝑟is the linear

phase representing the location of target point; the energyof the target will be focused within the corresponding rangecell after RCMC. It should be paid attention to in the imagingprocess. The term in (19) is the SRC term which indicates the


Table 2: Image quality parameters using the proposed method.

Parameter Theoretical one Center point Fringe point

Azimuth Range Azimuth Range Azimuth Range

PSLR (dB) −13.26 −13.26 −13.27 −13.15 −13.25

ISLR (dB) −9.8 −10.07 −9.97 −10.04 −9.97

1

0.8

0.6

0.4

0.2

0

600600

400400

200 2000 0

Nor

mal

ized

ampl

itude

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th (samples)

(a) 2D impulse response

200 300 400 500 600 700 800

0

Azimuth (samples)

Nor

mal

ized

ampl

itude

(dB)

−40

−35

−30

−25

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200 300 400 500 600 700 800

0

Range (samples)

Nor

mal

ized

ampl

itude

(dB)

−40

−35

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(c) Range impulse response

Figure 3: Center point results using the proposed range model.

coupling between𝑓𝑟and𝑓𝑎. It may cause significant degrada-

tion in the imaging resolution if uncompensated and is com-pensated in the 2D frequency domain. The last residual termin (20) is independent of the range frequency or the azimuthfrequency. It has no effect on the imaging focus and thus canbe neglected in MBFL-SAR configuration.

Consequently, the main procedure of the proposed algo-rithm follows the following steps.

(1) Use 2D FFT to transform the echo signal to 2Dfrequency domain as shown in (13).

(2) Design the match filtering function in the 2D fre-quency domain to compensate the range compression

term, the SRC term, and the residual term. Thematch filtering function𝐻

1(𝑓𝑟, 𝑓𝑎) can be obtained as

follows:

𝐻1(𝑓𝑟, 𝑓𝑎) = exp [−𝑗 (Φ

𝑟𝑔(𝑓𝑟) + Φsrc (𝑓𝑟, 𝑓𝑎) + Φres)] .

(21)

(3) Apply range inverse fast Fourier transform (IFFT) toconvert the signal to the range time azimuth fre-quency domain and correct the RCM term.

(4) Finish azimuth compression bymultiplying the rangetime azimuth frequency domain signal with the


1

0.8

0.6

0.4

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600600400

400200

2000 0

Nor

mal

ized

ampl

itude

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th (samples)

(a) 2D impulse response

200 300 400 500 600 700 800

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Azimuth (samples)

Nor

mal

ized

ampl

itude

(dB)

−40

−35

−30

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(b) Azimuth impulse response

200 300 400 500 600 700 800

0

Range (samples)

Nor

mal

ized

ampl

itude

(dB)

−40

−35

−30

−25

−20

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−5

(c) Range impulse response

Figure 4: Fringe point results using the proposed range model.

Azimuth (samples)

Rang

e (sa

mpl

es)

200 400 600 800 1000

200

400

600

800

1000

(a) Quadratic range model

Azimuth (samples)

Rang

e (sa

mpl

es)

200 400 600 800 1000

200

400

600

800

1000

(b) Proposed range model

Figure 5: Contour plots comparison using different range models.


200 300 400 500 600 700 800

0

Azimuth (samples)

Nor

mal

ized

ampl

itude

(dB)

−40

−35

−30

−25

−20

−15

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(a) Quadratic range model

200 300 400 500 600 700 800

0

Azimuth (samples)

Nor

mal

ized

ampl

itude

(dB)

−40

−35

−30

−25

−20

−15

−10

−5

(b) Proposed range model

Figure 6: Azimuth impulse responses comparison using different range models.

azimuth match filtering function, which can beobtained as follows:

𝐻2(𝑡𝑚, 𝑓𝑎) = exp [−𝑗 (Φ

𝑎(𝑓𝑎))] . (22)

(5) Perform azimuth IFFT to obtain a well-focused SARimage.

5. Simulations and Results

In this section, several examples are provided to illustrate theperformance of the proposed method. Because of the com-plex configuration, the high-order terms in the slant rangecannot be ignored. If inappropriately approximated, theimaging focusing resolution may be affected significantly.Some simulations are conducted to compare the influences ofkeeping the terms up to the quadratic term (quadratic rangemodel) 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 theimaging area using the proposed range model are given inFigures 3 and 4. In order to confirm the necessity and validityusing the higher order range model in this special configu-ration, some simulations are also conducted to compare thedifferences using the quadratic rangemodel and the proposedrange model, and the results are shown in Figures 5 and 6(here, the center point is selected in comparison).

Figure 3 depicts the results of the center point usingthe proposed range model. Figure 3(a) is the 2D impulseresponse; Figures 3(b) and 3(c) are the azimuth impulseresponse and the range impulse response, respectively. The2D impulse response of the fringe target point using theproposed range model is shown in Figure 4(a), and theazimuth impulse response and the range impulse response aregiven in Figures 4(b) and 4(c), respectively. We can see fromFigures 3 and 4 that both center point and fringe point arewell focused in range direction and azimuth direction. For the

complexity of theMBFL-SAR configuration, if the slant rangeis not approximated accurately enough, the phase term in the2D frequency spectrum will not be compensated completelyin the azimuth direction. It is easy to find that the imageof the point target suffers from distortion and main-lobebroadening when the slant range keeps the terms up to thequadratic term in Figures 5(a) and 6(a). On the other hand, ifwe are using the proposed rangemodel, the imaging focusingperformance can be obviously improved, as given in Figures5(b) and 6(b). It is obvious that the image is well focusedin the azimuth direction, which confirms it necessary toapproximate appropriately in the MBFL-SAR configuration.

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

6. Conclusions

In this paper, we developed an imaging algorithm for MBFL-SAR configuration, a special MIMO radar mode.TheMBFL-SAR imaging geometric model is established and the signalmodel is analyzed firstly. The high-order term in the slantrange cannot be neglected because of the presence of highvelocity and acceleration and the addition of two square-root terms. Also, the difficulty in direct-using the POSP inthis configuration is discussed. Then 2D frequency spectrumis deduced using the method of series reversion based onthe appropriate approximation of the slant range. Based onthe derived spectrum, the corresponding imaging method is


developed. The high-order term introduced from this con-figuration is compensated, together with range compression,at the beginning of the algorithm. RCMC is finished in therange time azimuth frequency domain as the range-variant 𝑘-coefficients.The focused SAR image can be obtained throughthe azimuth compression. The feasibility and efficiency ofthe proposed approach are validated with the simulationexperiments.

Conflict of Interests

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

Acknowledgments

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

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