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An Improved RD Algorithm for Maneuvering Bistatic Forward-Looking SAR Imaging with a Fixed Transmitter Yue Yuan, Si Chen * and Huichang Zhao School of Electronic & Optical Engineering, Nanjing University of Science & Technology, Nanjing 210094, China; [email protected] (Y.Y.); [email protected] (H.Z.) * Correspondence: [email protected]; Tel./Fax: +86-25-8431-5843 Academic Editor: Jonathan Li Received: 13 April 2017; Accepted: 15 May 2017; Published: 19 May 2017

Abstract: In order to improve the azimuth resolution beyond what monostatic synthetic aperture radar (SAR) can achieve in the forward-looking area, an asymmetric configuration bistatic SAR system and its imaging algorithm are proposed in this paper. The transmitter is mounted on a fixed platform in side-looking mode while the receiver moves along a nonlinear trajectory in forward-looking mode. Due to the high velocity and acceleration of the maneuvering platform in both along-track and height direction, the traditional algorithms are no longer applicable. In this paper, a new algorithm based on the high precise 2-D frequency spectrum is proposed, which takes high-order Taylor series expansion terms of the slant range into consideration. The proposed algorithm compensates high-order range-azimuth coupling terms to guarantee the focus accuracy in SAR imaging. The simulation results and error analysis validate the effectiveness of the proposed algorithm and the correctness of our analysis. Keywords: 2-D frequency spectrum; method of series reversion (MSR); SAR imaging; bistatic forward-looking SAR (BFL-SAR)

1. Introduction Bistatic forward-looking synthetic aperture radar (BFL-SAR) has drawn much more attention because of its high resolution in both range and the azimuth direction, which fills in the blanks of conventional monostatic SAR. The BFL-SAR has many advantages, such as low probability interception, and immunity to physical attacks and electronic countermeasures, which is why it has significant potential in airplane navigation, missile guidance, and target detection. Many studies have been made on BFL-SAR [1–4]. The ground moving target (GMT) detection and imaging theories were discussed for BFL-SAR in [5,6]. Chen introduced the general situation and the method for BFL-SAR [7]. In addition, several experiments have been carried out and the results demonstrate the feasibility of BFL-SAR [8,9]. Differing from monostatic SAR, the slant range in bistatic SAR has two hyperbolic functions, which are defined as double square root (DSR) terms [10], thus, the principle of the stationary phase (POSP) could not be directly used in BFL-SAR. To derive the 2-D frequency spectrum, some researchers transformed the DSR term into a single square root regarded as the monostatic configuration [11]. Raw data simulations for bistatic SAR were proposed in [12,13], which are based on their 2-D frequency spectra. However, the platforms are moving horizontally without acceleration and vertical speed. Huang introduced an advanced hyperbolic approximation method to accurately fit the cubic term of the range history [14]. Some researchers adopted hyperbolic equivalent methods with compensating variation to obtain the 2-D frequency spectrum [15,16]. The equivalence error cannot be ignored, especially in a high-speed maneuvering Sensors 2017, 17, 1152; doi:10.3390/s17051152

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platform. Loffeld proposed a new method for 2-D frequency spectrum derivation [17]. The limitation high velocities and accelerations [18]. The method of series reversion (MSR) can obtain the In exact of Loffeld’s formula is the reduced accuracy with a difference between two stationary points. the spectrum, theoretically [19,20]. The accuracy mainly depends on the order of the expansion terms. application of missile terminal guidance and other maneuvering platforms, the platform moves with Zhang et al. used a high-order range equation and phase compensation to improve the accuracy of high velocities and accelerations [18]. The method of series reversion (MSR) can obtain the exact the range history and 2-D[19,20]. spectrum spectrum, theoretically The[21]. accuracy mainly depends on the order of the expansion terms. Based previous research, thisequation paper discusses asymmetric to bistatic configuration a Zhang et al.on used a high-order range and phasean compensation improve the accuracywith of the forward-looking and a [21]. stationary transmitter, in which the slant range is kept up to its range history andreceiver 2-D spectrum fourth-order Taylor series. Firstly, this linear range walk correction (LRWC) operation is performed Based on previous research, paper discusses an asymmetric bistatic configuration withina the range frequency azimuth domain. Secondly, thein method theslant series reversion forward-looking receiver andtime a stationary transmitter, which of the range is keptand upPOSP to its are adopted inTaylor the derivation of 2-D frequency spectrum. Then (LRWC) the rangeoperation compression, range cell fourth-order series. Firstly, linear range walk correction is performed in migration (RCM), secondary range compression (SRC), and high-order range-azimuth coupling the range frequency azimuth time domain. Secondly, the method of the series reversion and POSP terms are compensated in the 2-D frequency frequencyspectrum. domain. Then Finally, azimuth compression is are adopted in the derivation of 2-D the the range compression, range cell implemented by a secondary matched filter in compression the range Doppler migration (RCM), range (SRC),domain. and high-order range-azimuth coupling terms This paper is in organized as follows:domain. In Section 2, thethe geometry signal models are established. are compensated the 2-D frequency Finally, azimuthand compression is implemented by a In Sectionfilter 3, thein2-D spectrum is derived by MSR. Section 4 explores the proposed imaging matched thefrequency range Doppler domain. algorithm for BFL-SAR. Section 5 shows the simulation results and and signal error models analysisare ofestablished. proposed This paper is organized as follows: In Section 2, the geometry algorithm. The of this letter is is given in Section 6. Section 4 explores the proposed imaging In Section 3, theconclusion 2-D frequency spectrum derived by MSR. algorithm for BFL-SAR. Section 5 shows the simulation results and error analysis of proposed algorithm. 2. Geometry andofSignal Model The conclusion this letter is given in Section 6.

The geometry configuration of BFL-SAR with a stationary transmitter and a maneuvering 2. Geometry and Signal Model receiver is shown in Figure 1. The receiver works in forward-looking mode and travels in a The geometry configuration BFL-SAR with a stationary and ain maneuvering receiver descending curvilinear path alongofcurve AB in plane yoz as thetransmitter antenna points the same direction. is shown in Figure 1.is mounted The receiver in forward-looking mode and a descending The fixed transmitter on aworks high tower where it can illuminate the travels imaginginarea easily. ta curvilinear path along curve AB in plane yoz as the antenna points in the same direction. Theoffixed and tr denote the slow time (azimuth time) and the fast time (range time). The positions the transmitter is mounted on a high tower where it can illuminate the imaging area easily. t and a transmitter and receiver are T0  X t , Yt , H t  and R0  0,0, H r  at ta  0 . The initial velocity andtr denote the slow time (azimuth time) and the fast time (range time). The positions of the transmitter   0,initial ary , arz  velocity , respectively. Therefore,of the acceleration receiver are V   0,V H,V)rz0at  taand and receiver of arethe T0 ( X = 0.aThe and acceleration the t , Yt , Ht ) and R0 0 (0, 0,ry0 r

2 2 receiver are 0, Vry0 , V 0, aryta, ,aHrzr , Vrespectively. Therefore, the location of the receiver at the time ta , supposing the location of V the isrz0 Rand 0 =receiver  0,Vry0a t= a  0.5ary rz0ta  0.5arzta 
is R 0, Vry0 ta + 0.5ary t2a , Hr + Vrz0 ta + 0.5arz t2a at the time ta , supposing the projection of receiver in
O at t  0 . P  xp , yp ,0  is an arbitrary projection of receiver in xoy is the coordinates’ center xoy is the coordinates’ center O at ta = 0. P xp , yp , 0 is an arbitrary aground point target in the imaging ground point target in the imaging area. The instantaneous slant range from the area. The instantaneous slant range from the transmitter and receiver to the target P attransmitter slow time taand are receiver the target and P at slow time t are R R , respectively. Rt and Rto , respectively. a t r r

T0  X 0 , Y0 , H t 

Height

z

Rt

R0  0, 0, H r 

 Vr

A

y B

Rr O P

Along-track y

x

Figure Figure1.1.Geometry Geometryconfiguration configurationofofthe theMBFL-SAR. MBFL-SAR.

Supposing that the transmitted signal is the linear frequency modulation (LFM) signal [9], and Supposing that the transmitted signal is the linear frequency modulation (LFM) signal [9], and
 , then we have the instantaneous slant range of the is x x p, ,yy p,,0 the position the positionof ofthe thepoint pointtarget target PP is p p 0 , then we have the instantaneous slant range of the high-speed which can can be be described described by: by: high-speedmaneuvering maneuvering MBFL-SAR MBFL-SAR RR(ttaa ) ,, which

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q


2
2 Xt − xp + Yt − yp + (Ht )2 R ( ta ) = Rt + Rr ( ta ) = r 2  2  + xp2 + Vry0 ta + 12 ary t2a − yp + Hr + Vrz0 ta + 12 arz t2a

(1)

where Rt and Rr (ta ) are the transmitter range and receiver range, respectively. The signal model of the BFL-SAR is similar to the monostatic SAR, except for the range history. The transmitter range is only dependent on the position of P and independent of slow time ta . Thus, the azimuth resolution depends on the motion of the receiver alone. 3. Derivation of 2-D Point Target Frequency Spectrum Generally, we can obtain the 2-D frequency spectrum easily after the azimuth FFT. However, due to the existence of velocities and accelerations in both along-track and along-height directions, it is difficult to calculate the stationary point directly. Here, we expand the instantaneous slant range R(ta ) into a Taylor series at ta = 0 as:   R(ta ) = Rt + Rr0 + (k1c + k1 )ta + k2 t2a + k3 t3a + k4 t4a + o t4a where: Rr0 =

q

xp2 + y2p + Hr2

−Vry0 ypc + Vrz0 Hr Rr0
Vry0 ypc − yp k1 = Rr0

k1c =

k2 =

(2)

(3) (4) (5)

− ary yp + arz Hr + Vry0 2 + Vrz0 2 − (k + k1c )2 2Rr0

(6)

k3 =

Vry0 ary + Vrz0 arz − 2(k1 + k1c )k2 2Rr0

(7)

k4 =

ary 2 + arz 2 2(k1 + k1c )k3 + k22 − 8Rr0 2Rr0

(8)

where Rr0 is the instantaneous range at ta = 0; ypc is the y-axis of scene center point Pc ; k1c is the linear coefficient at Pc which represents the coefficient of the linear range walk (LRW); k1 is the residual LRW coefficient of the target, which is far from the scene center; and k2 , k3 , and k4 are the quadratic, cubic, and quartic coefficients, respectively. Here, the Taylor expansion series is kept up to fourth-order term. In this bistatic configuration, we compensate the LRW at the scene center point Pc , so the LRWC function can be written as:   ( fc + fr ) H1 ( f r , ta ) = exp j2π k1c ta (9) c After LRWC, the signal can be written as:   f2 sr3 ( f r , ta ) = Wr ( f r ) aa (ta ) · exp − jπ γr h
i × exp − j2π ( fc +c fr ) Rt + Rr0 + k1 ta + k2 t2a + k3 t3a + k4 t4a

(10)

Then, we can derive the 2-D frequency spectrum by the method of POSP and series reversion as: sr4 ( f r , f a ) = Wr ( f r )Wa ( f a ) · exp[ jψ( f r , f a )]

(11)

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where:

 2 k1 1 − 2π Rt +ARr0 + 2π AA f + a 2 A  3  4 k1 A3 A3 A2 A2 − 2π 3 f a + A + 2π 4 f a + kA1 f r2 γ

ψ( f r , f a ) = −π

and:

  A =

c fc + fr

  A = − 3k3 2 8k3

2

A1 =

1 2k2

A3 =

9k23 −4k2 k4 16k52

(12)

(13)

For further analysis, the phase term should be decomposed into its Taylor series of f r at f r = 0, then reorganized as: ψe ( f r , f a ) = ψ0 ( f a ) + ψ1 ( f a ) f r + ψ2 ( f a ) f r2 (14) + ψ3 ( f a ) f r3 + ψ4 ( f a ) f r4 + ψa ( f a ) + ψr ( f r ) where:

f

ψ0 ( f a ) = −2π cc( Rt + Rr0 )  + π fcc A1 k21 − 23 A2 k31 + 21 A3 k41
+ π fcc A1 − 2A2 k1 − 3A3 k21 f a2
2 3 + π cf 2 − 23 A2 + 2A3 k1 f a3 + π cf 3 12 A3 f a4 c

ψ1 ( f a ) =

c

c f c2

A1 − 2A2 k1 + 3A3 k21 f a2 π   3 2 + π cf 3 43 A2 − 4A3 k1 f a3 − π cf 4


c

ψ2 ( f a ) =

3 2 A3

c




f a4

−π γ1 + π fc3 A1 − 2A2 k1 + 3A3 k21 f a2 c 2 3 + π cf 4 (−2A2 + 6A3 k1 ) f a3 + π cf 5 3A3 f a4 c c

(16)





ψ3 ( f a ) = π fc4 − A1 + 2A2 k1 − 3A3 k21 f a2 c
2 3 + π cf 5 83 A2 − 8A3 k1 f a3 − π cf 6 5A3 f a4 c

ψ4 ( f a ) =

(15)

c f c5

(17)

(18)

c

π A1 − 2A2 k1 + 3A3 k21 f a2   2 3 + π cf 6 − 10 A2 + 10A3 k1 f a3 + π cf 7 15 A3 f a4 3 c 2 c


  ψa ( f a ) = 2π A1 k1 − A2 k21 + A3 k31 f a   2 1 π ψr ( f r ) = − 2( Rt + Rr0 ) + A1 k21 − A2 k31 + A3 k41 f r c 3 2

(19) (20) (21)

The azimuth compression term ψ0 ( f a ) only depends on f a ; thus, it can be compensated in the range time azimuth frequency domain. ψ1 ( f a ) represents the residual RCM after LRWC. The first term in ψ2 ( f a ) denotes the range modulation while the others are the SRC terms. ψ3 ( f a ) and ψ4 ( f a ) are high-order range-azimuth coupling terms. ψ0 ( f a ) and ψr ( f r ) represent the range and azimuth positions of target P after LRWC, respectively. 4. Imaging Algorithm for MBFL-SAR In the aforementioned derivation, Equation (16) represents the phase of the 2-D frequency spectrum after LRWC. Then, the residual RCM, SRC, and high-order coupling terms are compensated in 2-D frequency domain, the function can be expressed as: h  i H2 ( f r , f a ) = exp − j ψ1 f r + ψ2 f r2 + ψ3 f r3 + ψ4 f r4

(22)

In the aforementioned derivation, Equation (16) represents the phase of the 2-D frequency spectrum after LRWC. Then, the residual RCM, SRC, and high-order coupling terms are compensated in 2-D frequency domain, the function can be expressed as: H 2  f r , f a   exp   j  1 f r   2 f r 2   3 f r 3   4 f r 4  

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(22)

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Transforming the result by range IFFT, the azimuth compression is completed by multiplying the function given as:

Transforming the result by range IFFT, the azimuth compression is completed by multiplying the H 3  tr , f a   exp   j 0  (23) function given as: H3can f a )obtained = exp(− jψ0 )the azimuth IFFT operation, which can(23) (tr , be Finally, the 2-D focused SAR image after beFinally, writtenthe as: 2-D focused SAR image can be obtained after the azimuth IFFT operation, which can

be written as:

   ynew       Rnew   sr5  tr , ta   sinc  Br  tr Rnew  ynew   sinc  Ba  ta  sr5 (tr , ta ) = sinc Br  tr − c   sinc  Ba tvax − 

c

(24)

vx

(24)

 1 1  2 22 1 3 4 1 32 2 3 4 y and k A− A k and . A2 k 2 + A3 k 3 . where  Ay 1Ak31k1− A k + A k where RnewRnew = RRt t+ R Rrr0 0 +  A1kA 1 1 2 k1  3 1  1knew 1  A= 2 k1 A 1 1 3 22 1  2 3 1 new 2 2 31 TheThe compensation function sets the center point as the reference target point with acceptable compensation function sets the center point as the reference target point with an an acceptable error, which will be discussed in the next section. error, which will be discussed in the next section. To To summarize thetheoperations scheme ofofthe theproposed proposedBFL-SAR BFL-SAR imaging summarize operationsabove, above, the the block block scheme imaging algorithm is shown in in Figure 2.2. algorithm is shown Figure

H1  f r , t a 

H 2  fr , fa 

H 3  tr , f a 

Figure 2. Block diagram of proposed imaging algorithm.

Figure 2. Block diagram of proposed imaging algorithm.

5. Simulations Results and Error Analysis 5.1. Simulations and Results To demonstrate the efficiency of this algorithm, a simulation is presented in this section. The experiments are carried out in this section with the parameters shown in Table 1. Table 1. Simulation parameters. Parameters

Values

Parameters

Values

Velocity (m/s) Acceleration (m/s2 ) Transmitter position Scene center Initial receiver position Synthetic aperture length (m)

(0, 1000, −30) (0, −1, −5) (−20,000, 3000, 2000) (0, 3000, 0) (0, 0, 5000) 200

Carrier frequency (GHz) Pulse duration (µm) Bandwidth (MHz) Range samples Azimuth samples

10 1.5 150 2048 2048

The positions of these 3 × 3 scattering targets are shown in Figure 3. In the simulation, we set point target T5 as the reference center point. Point targets T4, T5, and T6 are located in the same

Scene center Initial receiver position Synthetic aperture length (m)

(0, 3000, 0) (0, 0, 5000) 200

Range samples Azimuth samples

2048 2048

of 10 of these 3  3 scattering targets are shown in Figure 3. In the simulation, 6we set point target T5 as the reference center point. Point targets T4, T5, and T6 are located in the same azimuth direction and in different range directions, which means they have the same instantaneous azimuth direction and in different range directions, which means they have the same instantaneous Doppler frequency and different range cells. Point targets T2, T5, and T8 are located indifferent Doppler frequency and different range cells. Point targets T2, T5, and T8 are located indifferent azimuth directions and in the same range direction, which means they have different instantaneous azimuth directions and in the same range direction, which means they have different instantaneous Doppler frequencies and the same range cell. Doppler frequencies and the same range cell.

Sensors 2017, 1152 The 17, positions

60 40

T1

T2

T3

T4

T5

T6

T7

T8

T9

Y(m)

20 0 -20 -40 -60 -60

-40

-20

0 X(m)

20

40

60

Figure3.3.Scattering Scatteringtarget targetpositions. positions. Figure

-80

-80

-60

-60

-40

-40 Azimuth(m)

Azimuth(m)

Figure 4 shows the point target spectrum and interim simulation results using the proposed Figure 4 shows the point target spectrum and interim simulation results using the proposed method. Figure 4a indicates the raw data from which we can see the skew due to the range azimuth method. Figure 4a indicates the raw data from which we can see the skew due to the range azimuth coupling. Figure 4b shows the data after LRWC. The data is flattened, thus, the following steps can coupling. Figure 4b shows the data after LRWC. The data is flattened, thus, the following steps can be be performed easier. Figure 4c depicts the results after range compression, RCMC, SRC, and performed easier. Figure 4c depicts the results after range compression, RCMC, SRC, and high-order high-order coupling compensation, where each point target response is focused in the same range coupling compensation, where each point target response is focused in the same range cell. Figure 4d cell. Figure 4d shows the results of these 3  3 scattering targets after azimuth compression, and we shows the results of these 3 × 3 scattering targets after azimuth compression, and we can find the can find the geometric distortion in the image due to the LRWC operation. The imaging result after geometric distortion in the image due to the LRWC operation. The imaging result after geometric geometric distortion Sensors 2017, 17, 1152 correction is shown in Figure 5. 7 of 11 distortion correction is shown in Figure 5.

-20 0 20

-20 0 20

40

40

60

60

80

80 2.54

2.56

2.58 2.6 Range(m)

2.62

2.64

2.54

2.56

4

x 10

-100

-100

-50

-50

0

50

100

2.62

2.64 4

x 10

(b)

Azimuth(m)

Azimuth(m)

(a)

2.58 2.6 Range(m)

0

50

2.58

2.585

2.59 2.595 Range(m)

(c)

2.6

100

2.605 4

x 10

2.58

2.585

2.59 2.595 Range(m)

2.6

2.605 4

x 10

(d)

Figure 4. Interim simulation results using the proposed method. (a) Raw data; (b) After LRWC; (c)

Figure 4. Interim simulation results using the proposed method. (a) Raw data; (b) After LRWC; After range compression, RCMC, SRC, and high-order coupling compensation; (d) After azimuth (c) After range compression, RCMC, SRC, and high-order coupling compensation; (d) After compression. azimuth compression. -100

(m)

-50

Range(m)

Range(m)

4

x 10

(c)

4

x 10

(d)

Figure 4. Interim simulation results using the proposed method. (a) Raw data; (b) After LRWC; (c)

After range compression, RCMC, SRC, and high-order coupling compensation; (d) After azimuth Sensors 2017, 17, 1152 compression.

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-100

Azimuth(m)

-50

0

50

100

2.58

2.585

2.59 2.595 Range(m)

2.6

2.605 4

x 10

Figure 5. Imaging result after geometry correction. Figure 5. Imaging result after geometry correction.

5.2. Error Analysis 5.2. Error Analysis In this paper, the main errors come from the following aspects: approximation of the slant range In this paper, the main errors come from the following aspects: approximation of the slant range R  ta  , approximation of the 2-D frequency spectrum derivation, the Taylor expansion of the 2-D R(ta ), approximation of the 2-D frequency spectrum derivation, the Taylor expansion of the 2-D frequency spectrum, and approximation the compensation function. Here, analyze the error frequency spectrum, and approximation ofof the compensation function. Here, wewe analyze the error toto prove effectiveness this algorithm. prove thethe effectiveness of of this algorithm. Firstly, the equivalence slant range is kept the fourth-order series expansion Firstly, the equivalence slant range is kept up toup thetofourth-order Taylor Taylor series expansion of R(ta ),of , where the higher-order terms are ignored. The ideal one is shown in Equation (2) and R t  a the higher-order terms are ignored. The ideal one is shown in Equation (2) and the error the where canerror be can be expressed as: expressed as: h i 2 3 4 ∆R(ta ) =RRrt(ta) R − t Rr0 + k + k t + k t (25) ( ) 2 a + k33 ta +4 k 4 ta a 2 1c 1  a  r  a    Rr0   k1c  k1  ta  k2ta  k3ta  k4ta  (25) Taking the scene center point as the reference point, the parameters are listed in Table 1, and the Taking the scene center point as the reference point, the parameters are listed in Table.1, and the results are shown Figure 6. Sensors 2017, 1152 in in 8 of 11 results are17,shown Figure 6. -7

4

x 10

3 2 error/m

1 0 -1 -2 -3 -4 -0.1

-0.05

0 ta/s

0.05

0.1

Figure6.6.Slant Slantrange rangeapproximation approximationerror errorof ofhigh highorders. orders. Figure

The maximum range approximation error is 3 107 m in Figure 6, which contributes to the The maximum range approximation error is 3 × 10−7 m in Figure 6, which contributes to the phase phase error in Equation (5) as: error in Equation (5) as: ( f cf + f r ) ∆ψ(fr , t af)r , t= −2π  c r  ∆R R (tta a) (26) a   2 cc Using the simulation parameters into Equation (1), the phase error absolute value of our Using the simulation parameters into Equation (1), the phase error absolute value of our approximation is about 6.3 × 10−5 rad. It meets the demand of ∆ψ ≤ π4 .   the 6.3 105 rad . It meets the demand of  by . method of POSP and series approximation is about Secondly, the approximation of the 2-D frequency spectrum 4 reversion introduces the error. This error partly depends on the range approximation, namely, it Secondly, the approximation of the 2-D frequency spectrum by the method of POSP and series is related to the first error. Another aspect is the remaining Taylor expansion orders of f r . In this reversion introduces the error. This error partly depends on the range approximation, namely, it is algorithm, the 2-D frequency spectrum expansion of f r ignores the terms higher than fourth-order. related to the first error. Another aspect is the remaining Taylor expansion orders of f r . In this algorithm, the 2-D frequency spectrum expansion of f r ignores the terms higher than fourth-order. To analyze this error, the method of numerical calculation is considered as a comparison, which can obtain the accurate 2-D frequency spectrum. Figure 7 shows the imaging results of point targets using the approximation of 2-D frequency spectrum derived in this paper and the method of numerical

approximation is about 6.3 105 rad . It meets the demand of  



. 4 Secondly, the approximation of the 2-D frequency spectrum by the method of POSP and series reversion introduces the error. This error partly depends on the range approximation, namely, it is related Sensors 2017,to 17,the 1152first error. Another aspect is the remaining Taylor expansion orders of f r . 8In of this 10 algorithm, the 2-D frequency spectrum expansion of f r ignores the terms higher than fourth-order. analyze this error, the method numerical calculation considered a comparison,which whichcan can ToTo analyze this error, the method ofof numerical calculation is is considered asas a comparison, obtain the accurate 2-D frequency spectrum. Figure 7 shows the imaging results of point targets using obtain the accurate 2-D frequency spectrum. Figure 7 shows the imaging results of point targets using theapproximation approximationofof2-D 2-Dfrequency frequencyspectrum spectrumderived derivedininthis thispaper paperand andthe themethod methodofofnumerical numerical the calculation,respectively. respectively. calculation, -50

Azimuth(m)

Azimuth(m)

-50

0

50 2.585

2.59

2.595 Range(m)

0

50 2.585

2.6 4

x 10

2.59

2.595 Range(m)

(a)

2.6 4

x 10

(b)

Figure 7. Focusing results of the point target using different 2-D spectra. (a) Series reversion; and (b) Figure 7. Focusing results of the point target using different 2-D spectra. (a) Series reversion; and Numerical calculation. (b) Numerical calculation.

The last one, also the greatest cause of algorithm accuracy, is the approximation of k-coefficients The last one, also the greatest cause of algorithm accuracy, is the approximation of k-coefficients which uses y pc instead of yp . Thus, we compare subimages of the center point T5 and edge points which uses ypc instead of yp . Thus, we compare subimages of the center point T5 and edge points T1 T1 and T9,17, which Sensors 2017, 17, 1152 can be shown in Figure 8. of 11 11 Sensors 1152 99 of and T9,2017, which can be shown in Figure 8. 15 15

-40 -40

10 10

55 55 50 50 45 45

55

Azimuth(m) Azimuth(m)

60 60

Azimuth(m) Azimuth(m)

Azimuth(m) Azimuth(m)

65 65

00 -5 -5 -10 -10

40 40 2.587 2.587

2.588 2.588 Range(m) Range(m)

2.589 2.589 4 104 xx 10

-45 -45 -50 -50 -55 -55 -60 -60 -65 -65

2.592 2.592

(a) (a)

2.593 2.593 Range(m) Range(m)

2.597 2.597

2.594 2.594 4 104 xx 10

(b) (b)

2.598 2.599 2.598 2.599 4 Range(m) Range(m) 104 xx 10

(c) (c)

Figure 8. 8. Focusing Focusing performance performance of of proposed proposed improved improved RDA: RDA: (a) (a) Target Target 1; 1; (b) (b) Target Target 5; 5; and and (c) (c) Target Target 9. 9. Figure Figure 8. Focusing performance of proposed improved RDA: (a) Target 1; (b) Target 5; and (c) Target 9.

From Figure Figure 8, 8, we we can can find find that that the the focusing focusing performance performance of of the the center center point point is is better better than than the the From From Figure 8, we can find that the focusing performance of the center point is better than the edge edge point. However, the edge point can still focus well with our algorithm. To make a clearer edge point. However, the edge point can still focus well with our algorithm. To make a clearer point. However, the edge point can still focus well with our algorithm. To make a clearer performance performance comparison comparison between between the the proposed proposed algorithm algorithm and and the the traditional traditional RDA, RDA, the the imaging imaging performance comparison between the proposed algorithm and the traditional RDA, the imaging result obtained by result obtained obtained by by traditional traditional RDA RDA is is shown shown Figure Figure 9. 9. result traditional RDA is shown Figure 9.

65 65 60 60 55 55 50 50 2.587 2.587

2.588 2.588 Range(m) Range(m)

(a) (a)

2.589 2.589 4 104 xx 10

-30 -30

20 20 15 15

Azimuth(m) Azimuth(m)

70 70

Azimuth(m) Azimuth(m)

Azimuth(m) Azimuth(m)

75 75

10 10 55 00 -5 -5

-35 -35 -40 -40 -45 -45 -50 -50 -55 -55

2.592 2.592

2.593 2.593 Range(m) Range(m)

(b) (b)

2.594 2.594 4 104 xx 10

2.597 2.598 2.597 2.598 Range(m) Range(m)

2.599 2.599 4 x 104 x 10

(c) (c)

Figure 9. 9. Focusing Focusing performance performance of of traditional traditional RDA: RDA: (a) (a) Target Target 1; 1; (b) (b) Target Target 5; 5; (c) (c) Target Target 9. 9. Figure Figure 9. Focusing performance of traditional RDA: (a) Target 1; (b) Target 5; (c) Target 9.

Comparing the the results, results, it it is is easy easy to to find find that that the the proposed proposed algorithm algorithm generally generally offers offers better better Comparing focusing capabilities capabilities than than the the traditional traditional RDA. RDA. Analyzing Analyzing the the data data in in Figures Figures 88 and and 9, 9, the the resolution resolution focusing , peak sidelobe ratio (PSLR) and integrated sidelobe ratio (ISLR) are used to evaluate the focusing   , peak sidelobe ratio (PSLR) and integrated sidelobe ratio (ISLR) are used to evaluate the focusing performance. T1 T1 and and T5 T5 are are chosen chosen for for analysis. analysis. The The results results are are listed listed in in Table Table 2. 2. We We can can find find that that performance. higher resolution and the theoretical PSLR and ISLR are obtained by the proposed algorithm. higher resolution and the theoretical PSLR and ISLR are obtained by the proposed algorithm.

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Comparing the results, it is easy to find that the proposed algorithm generally offers better focusing capabilities than the traditional RDA. Analyzing the data in Figures 8 and 9, the resolution ρ, peak sidelobe ratio (PSLR) and integrated sidelobe ratio (ISLR) are used to evaluate the focusing performance. T1 and T5 are chosen for analysis. The results are listed in Table 2. We can find that higher resolution and the theoretical PSLR and ISLR are obtained by the proposed algorithm. Table 2. Characteristic parameters of center point and edge point. Range ρ (m)

PSLR (dB)

Azimuth ISLR (dB)

ρ (m)

PSLR (dB)

ISLR (dB)

5.3 5.1

−13.5 −13.5

−10.2 −10.2

10.5 8.8

−22.7 −19.5

−20.3 −17.2

Proposed improved RDA Edge point T1 Center point T5

3.5 3.5

−13.5 −13.1

−10.8 −10.0

Traditional RDA Edge point T1 Center point T5

6.5 5.5

−20.8 −20.7

−22.2 −19.9

6. Conclusions In this paper, an imaging algorithm for a fixed-transmitter BFL-SAR is proposed, which takes high-order Taylor series expansion terms of the slant range into consideration. Firstly, the signal model of BFL-SAR with fixed transmitter is built. Secondly, the high-precision 2-D frequency spectrum is derived by the method of series reversion. This method uses the fourth-order Taylor expansion of the slant range after LRWC operation. Then, the residual RCM, SRC, and high-order coupling terms are compensated in the 2-D frequency domain. Finally, a SAR image can be obtained by 2-D compression. The simulation results and error analysis prove the feasibility and effectiveness of the proposed imaging algorithm and verify the correctness of the theory analysis. Acknowledgments: This work was supported by the Natural Science Foundation of Jiangsu Province (Jiangsu Provincial Natural Science Foundation) under grant BK20160848, the Fundamental Research Funds for the Central Universities under grant No.30917011315, and the Research and Innovation Plan for Graduate students of Jiangsu Higher Education institutions under grant KYLX16_0449. Author Contributions: Y. Yuan and S. Chen conceived the algorithm; Y. Yuan performed the experiments; Y. Yuan and S. Chen analyzed the data; H. Zhao contributed simulation and analysis tools; Y. Yuan wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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