An efficient NLCS algorithm for maneuvering forward-looking linear array SAR with constant acceleration

Accepted Manuscript

An Efficient NLCS Algorithm for Maneuvering Forward-Looking Linear Array SAR with Constant Acceleration Si Chen , Yue Yuan , Shuning Zhang , Huichang Zhao PII: DOI: Reference:

S0165-1684(17)30374-2 10.1016/j.sigpro.2017.10.015 SIGPRO 6636

To appear in:

Signal Processing

Received date: Revised date: Accepted date:

16 June 2017 19 August 2017 11 October 2017

Please cite this article as: Si Chen , Yue Yuan , Shuning Zhang , Huichang Zhao , An Efficient NLCS Algorithm for Maneuvering Forward-Looking Linear Array SAR with Constant Acceleration, Signal Processing (2017), doi: 10.1016/j.sigpro.2017.10.015

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Highlights 

A realistic geometry configuration of the forward-looking linear array SAR is built.



Efficient range nonlinear chirp scaling algorithm is proposed for the forward looking linear-array SAR imaging. Thorough algorithm derivation and approximation error analysis are presented.



Numerical simulation illustrates easy implementation and high efficiency of

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

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the method.

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An Efficient NLCS Algorithm for Maneuvering Forward-Looking Linear Array SAR with Constant Acceleration Si Chen, Yue Yuan, Shuning Zhang, Huichang Zhao

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School of Electronic & Optical Engineering, Nanjing University of Science & Technology, Nanjing 210094, China

Abstract: Forward-looking linear-array synthetic aperture radar (FLLA-SAR), which can overcome the deficiency

of the traditional SAR on forward-looking imaging, continues to gain in significance due to the various

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applications, such as unmanned aerial vehicle (UAV) reconnaissance. To compensate the space-variant

characteristic of along-track frequency modulation (FM) rate caused by the high speed maneuvering UAV, an

efficient nonlinear chirp scaling (NLCS) algorithm is proposed in this paper. Based on the analysis of novel

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geometric configuration and the introduction of NLCS operation, the linear-array UAV SAR can reconstruct the

2-D image of the scout area. Finally, the validation of the proposed algorithm is done by exploiting the simulated

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data based on 3  3 array point targets and an airplane scattering model.

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Keywords: Forward-looking linear-array synthetic aperture radar (FLLA-SAR); forward-looking; nonlinear chirp scaling

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(NLCS); acceleration

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1. Introduction

Traditional synthetic aperture radar (SAR) [1]-[4] can reconstruct the 2-D image of the observed area day

and night with weather independent, which has been widely used in both civil and military applications. Nevertheless, azimuth imaging performance degrades seriously when the reconnaissance UAV SAR works in the forward-looking mode [5]-[6]. Forward-looking linear array SAR (FLLA-SAR) [6]-[8] can solve the

 Corresponding author. Tel.: +86 25 84315843; fax: +86 25 84315843. E-mail addresses: [email protected]. (S. Chen)

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special imaging problem effectively, which utilizes the linear-array antennas working in time division multiplexing access mode to achieve azimuth high-resolution. Owing to this advantage, FLLA-SAR has attracted more and more attentions in last decade, such as innovative imaging mode [9], and various imaging algorithms are studied in [8], [10]-[12].

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In the existing imaging algorithms for the FLLA-SAR, range-Doppler algorithm (RDA) [13], chirp scaling algorithm (CSA) [14], frequency scaling algorithm (FSA) [15], and compressive sensing method [16] are only fit for the traditional geometric configuration of FLLA-SAR, which neglects the high speed and

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acceleration of the platform. Unfortunately, for the practical UAV reconnaissance application, it is almost impossible to avoid accelerated movement in data acquisition, which will lead to the serious degradation of imaging performance. Theoretically speaking, the back projection algorithm (BPA) [17]-[18] can deal with

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this special geometric configuration, but it suffers from severe computational complexity. Recently, an extended FSA and an efficient 2-D chirp-z algorithm based on the classical SIREV system [9] were

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proposed in [15] and [19] respectively. However, both of the two methods took the high speed of the UAV

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into consideration, but neglected the acceleration in the imaging procedure, which would restrict their widespread applications. Moreover, both of the two algorithms were exploited for the frequency modulated

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continues wave (FMCW) SAR, not suitable for the FM pulsed SAR.

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Based on the above research results [16]-[19], this paper develops the application of an efficient NLCS algorithm for the FLLA-SAR. Firstly, the novel quasi geometric configuration and signal model of the FLLA-SAR are built. Then, an efficient NLCS algorithm is proposed to equalize the range-dependent frequency modulation (FM) rate and remove the geometric distortion. Finally, the azimuth focusing and residual video phase (RVP) compression are implemented to achieve the final image by multiplying their complex conjugate in range-Doppler domain.

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The remaining paper is organized as follows. Section 2 presents the geometry and signal model of the FLLA-SAR. Linear range walk correction (LRWC), 2-D spectrum derivation, NLCS operation, residual range cell migration correction (RCMC) and secondary range compression (SRC) are present in Section 3. Section 4 displays azimuth focusing and residual video phase (RVP) compression. The performance of the

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proposed method is investigated by simulated data in Section 5. Finally, Section 6 summarizes this paper.

2. FLLA-SAR Geometry and Signal Model

The FLLA-SAR configuration is shown in Fig. 1. x , y and z axis correspond to range, azimuth and

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elevation directions, respectively. The platform UAV flies along x-axis with high velocity v and constant acceleration a at height H . The forward-looking linear-array antennas are equably arranged along the UAV wings with spacing d, which work in the time division multiplexing access mode with a fixed switch

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velocity vs . Thus, the cross-track resolution is substantially increased by the virtual synthetic aperture,

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which is formed by the linear array antennas. Supporting that Os ( X s , Ys , 0) is the scene center, and

P( X P , YP , 0) is an arbitrary point target in the imaging area. The instantaneous slant range from the n-th

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antenna to the point target P at slow time ta is R  ta  .

z  

ideal position

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Height

y ( N 1)/ 2

y1

0

y1

actual position

v a

 

y( N 1)/ 2

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d

H

O

Os  X s , Ys , 0 

y Azimuth

P  X p , Yp , 0 

x Range

Fig. 1. Geometric configuration of the FLLA-SAR.

Assuming that the transmitted signal is linear frequency modulated (LFM) pulse signal, and the base-band

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signal reflected from point scatterer P can be written as 2  2 R  ta   2 R  ta       4   sr  tr , ta   ar tr  a t exp  j R t exp j  t      r  ,  a a a   c   c        

(1)

N 3 N 3 N 1   N 1 where ta denotes the slow time with a range ta    Tr ,  Tr , ,  Tr , 0, Tr , , Tr , Tr  , 2 2 2  2 

t r is the fast time, N denotes the number of antennas, Tr responds to the pulse repetition time,  is the

azimuth envelopes respectively.

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wavelength,  is the chirp rate, and c is the light speed. ar   and R  ta  represent the range and

According to the geometric configuration shown in Fig. 1, the range history R  ta  between the point

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scatterer P and n-th antenna element can be described as 2

1 2   R  ta    vta  ata2  X P    yn  YP  +H 2 , 2  

(2)

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where yn  vsta denotes the cross-track position of the n-th antenna. Observing (2), unlike the traditional FLLA-SAR signal model [9, 15, 19], both the high-speed and acceleration of platform are taken into

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consideration in our signal model, which maybe more useful in practical application.

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Besides, the relationship between the spacing d of adjacent antenna elements, switch speed vs and

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pulse repetition time Tr can be easily obtained as

vs 

d . Tr

(3)

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For further analysis, (2) can be expanded to its Taylor’s series at ta = 0, then we have

 vX  R  ta   R0   B1  P  ta  B2ta2  B3ta3  B4ta4  O  ta  , R0  

A1 vX P   B1  R  R 0 0   1  A2 A12   B2    3  2  R0 R0   where  , 3  B  1  A3  A1 A2  A1    3 2  R0 R03 R05    1  A4 A22  4 A1 A3 6 A12 A2 5 A14    7   B4    8  R0 R03 R05 R0  

R  X 2  Y 2  H 2 P P  0  A1    vX P  vsYP   2 2  A2  v  vs  aX P .  A  va  3  A4  a 2 

(4)

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In conventional broadside SAR, the total RCM is usually dominated by quadratic component, all the remainder high order terms are neglected due to their little impacts. While in FLLA-SAR, due to high speed and acceleration of the UAV platform, the actual transmitting and receiving position of each array antenna can’t keep in a straight line any more as shown in Fig. 1, which will change the instantaneous range and

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bring a great influence on the echo phase. Therefore, we start the derivation by keeping the Taylor expansion series up to their quartic term in this paper.

3. Range Compression by NLCS Algorithm

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As we pointed out earlier, the virtual synthetic aperture is a parabola-like curve. Thus, the high-order phase in range dimension is so large that it can’t be neglected any more in the image formation processing. Besides, the defocusing phenomenon caused by spatial variance of range FM rate also should be

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compensated.

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3.1. LRWC

Substituting (2) and (3) into (1), and applying the range Fourier transform (FT) yields

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 4  f c  f r  vX P   f2 sr1  f r , ta   Ar  f r  aa  ta  exp   j r  exp  j ta    c R0   

,

(5)

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 4  f c  f r    exp   j  R0  B1ta  B2ta2  B3ta3  B4ta4  c  

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where Ar   represents the range frequency envelope, f c is the carrier frequency, f r is range frequency variable. The second term denotes the first-order coupling term, i.e., line range walk term, which can be easily eliminated by using

 4  f c  f r  vX s  H1  f r , ta   exp   j ta  . c Rs   where Rs is the reference slant range. Multiplying (5) and (6) yields

(6)

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 4  fc  f r    f2 sr2  f r , ta   Ar  f r  exp   j r  aa  ta  exp   j  R0  B1ta  B2ta2  B3ta3  B4ta4  .  c    

(7)

As we know, the LRWC function is based on the reference point scatterer position, which will result in the residual phase error to affect the imaging quality in this procedure. Fig. 2 exhibits the residual phase error of reference point scatterer T1 with marginal point scatterers T5 and T6 shown in Fig. 5 respectively.

impacts can be ignored. 600

600

0.32767

0.32767

0.32767

-0.41922 400

0.1966

0.1966

azimuth time (samples)

azimuth time (samples)

400

0.1966

200

0.065534

0.065534

-0.065534

-0.065534

-0.065534

-200 -0.1966

-0.1966

-0.1966

-400

-0.083845 0

-0.41922

-0.41922

-0.25153

-0.25153

-0.083845

-0.083845

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0.065534 0

-0.25153 200

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From the Fig. 2, it can be easily found that the residual phase errors are smaller than  / 4 , which means the

0.083845

0.083845

0.083845

0.25153

0.25153

0.25153

0.41922

0.41922

0.41922

-200 -400

-0.32767

-0.32767

-600 -500 -0.45874

-0.32767

-0.45874 -0.458740 range frequency (samples)

500

(a)

-600 -500 0.58691

0.58691 0.586910 range frequency (samples)

500

(b)

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Fig. 2. Residual phase error of the reference scatterer T1 with marginal scatterers T5 and T6. (a) T1 with T5. (b) T1 with T6.

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Moreover, to evaluate the performance of LRWC operation, a simulation on the 2-D spectrum of a point scatterer before and after LRWC is carried out, which is shown in Fig. 3. Observing the result, we can find

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that the skew and excursion phenomenon of 2-D spectrum is disappeared, which means the cross coupling is

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mitigated significantly and the following procedure will be much easier. -150

Range frequency (samples)

-100

300

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Range frequency (samples)

250

350 400 450

-400

-200 0 200 Azimuth frequency (samples)

(a) Fig. 3. LRWC.

0 50 100

500 -600

-50

400

600

150 -600

-400

-200 0 200 Azimuth frequency (samples)

400

600

(b)

Illustration of LRWC performance in the proposed algorithm. (a) 2D spectrum before LRWC. (b) 2D spectrum after

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3.2. 2-D Spectrum of FLLA-SAR Data Next, utilizing the principle of stationary phase (POSP) and the method of series reversion [20], (7) can be transformed into 2-D frequency domain as

sr2  f r , fa   Ar  f r  Aa  fa  exp  j  f r , fa  ,

(8)

f r2

  f r , f a   



 2

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where

2  fc  fr  2  fc  fr    c R0  2 B1   fa  c 8B2  f c  f r   c 

2

9B32  4B2 B4  c3  f  2  fc  f r  B  2  fc  fr     2 f  B  2    a 1 1 2 3  a c c 32 B23  f c  f r   512 B25  f c  f r     3

B3c 2

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Expanding (9) into its Taylor series of f r at f r  0 yields

4

.

(9)



  f r , fa   0  fa   1  f a  f r  2  f a  f r2  3  f a  f r3  m  f a  f rm ,

(10)

m 3

where

2 4 2 3  2f  B B2 f B 3 B f  9 B3  4 B2 B4  B1 f c  3B 2 B  9 B3  4 B2 B4  B1  c   2  1  1 3 3   fa R0  1 c  1 33 c  5 5  c   2 B2  2 B2c 4 B2 c 32 B2 c 8B2 16 B2     2 2 2 2 2  c    9B3  4B2 B4  B1c  f 3 3B B c  9 B3  4 B2 B4  3B1 c 2 Bc  f a  2  3 3 2  ,  2   1 3   8B2 f c 16 B23 f c   32 B2 f c  a 64 B25 f c 64 B25 f c2    

9B

2 3

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 2

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0  f a   2  

 4 B2 B4  c 3

512 B25 f c3

f a4

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2 4 2 2  2R  c B2 B 3 B  9 B3  4 B2 B4  B1  3B1B3c  9 B3  4 B2 B4  3B1 c  2 0    fa  1  1 33   2     c   8B2 f c2 16 B23 f c2  2 B2c 4 B2 c 32 B25c 64 B25 f c2     , 2 2 2 3 2  Bc 9 B3  4 B2 B4  B1c  3 9 B3  4 B2 B4  3c 4   3  f a  2  2   fa  16 B23 f c3  32B25 f c3 512 B25 f c4  

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1  f a   2  

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2  f a   



2 2  c  3B c 2 9B32  4B2 B4  3B1c2  f 3 3B1B3c  9 B3  4 B2 B4  3B1 c  2 3       f     4 B2 f c3 8B23 f c3  a  16 B23 f c4  a  32 B25 f c3 32B25 f c4     ,

1

9B

 4 B2 B4  3c3

2 3

128B25 f c5

f a4

2 2  c  B c2 9 B32  4 B2 B4  B1c 2  3  3B1B3c  9 B3  4 B2 B4  3B1 c  2 3     fa 3  f a   2   f  2   8B2 f c4 16 B23 f c4  a  8B23 f c5  64 B25 f c4 16B25 f c5     ,

 2

9B

2 3

 4 B2 B4  5c3

256B25 f c6

f a4

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m  f a  

m 1    fa , f r  m!  f r m

. f r =0

Inspecting (10), 0  fa  is independent of range variable, which represents the azimuth modulation phase. Excluding the first component in 1  f a  , other components are the residual RCM after LRWC

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operation. The last component in 2  fa  is the range FM term, the residual components represent the range-azimuth coupling, which are also called the SRC terms. All the other terms represent the high-order cross-coupling terms.

For further derivation, the equation (8) can be rewritten as

where  

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  f r2    sr3  f r , f a   Ar  f r  Aa  f a  exp  j  2 f r    3  f a  f r3   exp  j0  f a  , K f     r a     

1  f a   , Kr  fa    . 2  f a  2

(11)

Expanding  into its Taylor series of X P at X P  0 and keeping up to the first-order term yields

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  X P    0   d   0   X P ,

P

0

d  X P  dX P

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where  0    X P  X

, 

(12)

.

X P 0

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From (12), we can find that the range image distortion is composed of  0 and  X P , where  0

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denotes the whole range displacement. The range displacement correction (RDC) can be implemented easily

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by multiplying its conjugate

H 2  f r , fa   exp   j 2 0 f r  .

(13)

After correcting the whole range displacement, the 2-D spectrum can be given as   f r2    sr4  f r , f a   Ar  f r  Aa  f a  exp  j  2 d f r    3  f a  f r3   exp  j0  f a   . K f   r a      

(14)

Moreover,  d   X P represents the range stretch of SAR image, which will be compensated in the following procedure.

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3.3. Efficient Range NLCS Algorithm For precision focusing, the space-variant characteristic of range FM rate must be taken into consideration in the following imaging procedure. Here, we take the linear component of range FM rate into consideration, and expand K r  fa  in terms of

Kr  Ks0  Ks1 d ,

where Ks0  Kr

 d =0

, Ks1 

dK r d d

  d =0

dK r dX P

XP

dX P 1 dK r  d    dX P d =0

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 d , i.e. (15)

. X P =0

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By convention, a cubic phase filtering (CPF) operation in range dimension should be first carried out, which can eliminate the cubic component of f r . The cubic phase filtering function is given as H3  f r   exp  j 2 Yf r3  ,

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Multiplying (16) with (14) yields

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where Y is an undetermined coefficient.

  f r2    sr5  f r , f a   Ar  f r  Aa  f a  exp  j  2 d f r    2 Ym f r3   exp  j0  f a   , K f     r a     

3  f a  . 2

(17)

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where Ym  Y 

(16)

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Ignoring the cubic phase's contribution to stationary phase point evaluation, the signal in range-Doppler

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domain can be obtained by using POSP, which can be written as 2 3 sr6  tr , fa   ar  tr  Aa  f a  exp  j Kr  tr   d   j 2 Ym K r3  tr   d   exp  j0  f a  .  

(18)

After the range cubic phase filtering operation, a cubic chirp scaling factor is introduced to eliminate the

space-variant of range FM rate, which can be expressed as H 4  tr   exp  j q2tr2  j 2 q3tr3  ,

where q2 and q3 are the undetermined parameters as Y.

(19)

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Multiplying (18) with (19), and transforming the results into 2-D frequency domain by utilizing POSP yields

sr7  f r , fa   Ar  f r  Aa  fa  exp  j0  fa  exp  j  f r , fa  , where the stationary phase point tr* 

f r  K r  d , K r  q2

 K  f  q2  d 2 q2  f r  K r  d 2  f r  K r  fr    r r   2 2 K r  q2  K r  q2     K r  q2 

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  f r , f a   2

(20)

2

.

Y K 3 f  q2  d 3  q3  f r  K r  d 3   3  m r  r   K r  q2 

Substituting equation (15) into the expressions 1/（Kr  q2）, 1/（Kr  q2）2 , 1/（Kr  q2）3 and

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expanding them to second order Taylor series at td  0 , then we can obtain

 1 1 K s1 K s12       d2  d 2 3  Ks0  q2   K r  q2 K s0  q2  K a0  q2   1 1 2 K s1 3Ks12     d   d2 .  2 2 3 4  Ks0  q2   Ks0  q2   Ks0  q2    K r  q2   1 1 3K s1 6 K s12     d   d2 3 3 4 5   K r  q2   K s0  q2   K s0  q2  K  q  s0 2  

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(21)

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Combining (15) and (21),   f r , fa  can be expressed by a power series of  d and f r , which can be written as

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  f r , f a   A  q2 , q3 , Ym , f a , f r2 , f r3   B  q2 , f a   d f r  C  q2 , q3 , f a , Ym   d2 f r

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 D  q2 , q3 , f a , Ym   d f r2  E  q2 , q3 , Ym , f a ,  d2 ,  d3 

,

where

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   1 Y K 3  q3 3  2 3 f r2  2 m s0 f  A  q2 , q3 , Ym , f a , f r , f r      3 r   Ks0  q2    K s0  q2    B  q2 , f a   2 K s0  K s0  q2   3K s02 Ym K s0 q22  q3    K s1 K s0 K s1  .   C  q2 , q3 , Ym , f a   2   2 3  K s0  q2  K s0  q2    Ks0  q2    K s0 Ym K s02 q2  q3   K s1  6  D  q2 , q3 , Ym , f a    2 3  Ks0  q2   Ks0  q2     q2K 2 K s03  q3  Ym q23   3 q2 K s0  2 3 2 2 s1    d E q , q , Y , f ,   ,            2 3 m a d d d 2 3  K s0  q2   K s0  q2   Ks0  q2   

(22)

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Inspecting (21), the first term denotes the range-independent modulation term. The second and third terms represent the linear and nonlinear range-dependent RCM respectively, which are the main factor that cause the geometric distoration in range direction and should be eliminated. The fourth one represents the range-dependent SRC term. The last term represents residual RCM. The other unlisted terms are too small to

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affect the imaging quality, which can be ignored reasonably. To eliminate the range-dependent RCM and SRC terms, the parameter B is set to 2 /  and all of other terms including f rn  dm（n  0, m  0） are set to 0. Thus, three equations can be achieved as

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 B  q2 , f a   2 /   C  q2 , q3 , f a , Ym   0 .   D  q2 , q3 , f a , Ym   0

(23)

Solving equation (23), the three undetermined parameters above can be achieved as

(24)

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K s1  2  1  Ym  6 K s03   1    q2  K s0   1 ,   q  K s1   1  3 6 

where   vs represents the distortion correction coefficient.

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be achieved as

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Substituting (22), (23) and (24) into (20), the signal in 2-D frequency domain after NLCS operation can

(25)

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   1   sr8  f r , f a   Ar  f r  Aa  f a  exp  j0  f a    exp  2 d f r  exp   j f r2   Ks0  q2     .   q2K  2 Ks03  q3  Ym q23   3  Ym Ks03  q3 3  q2 Ks0   2 2 s1   d   exp  j 2 f exp  j  d  j   3 r  2 3   Ks0  q2   Ks0  q2    Ks0  q2  Ks0  q2    

From (25), we can find that the space-variant characteristic of range FM rate has been compensated

completely. Therefore, the match-filter (MF) function in range dimension can be easily designed as

  Y K3  q   1  H 5  f r , f a   exp  j  f r2  2 m s0 33 f r3   . K  q  Ks0  q2    2    s0

(26)

Multiplying (25) with (26) and transforming the result into range-Doppler domain, then we can obtain the

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range focused image as        qK sr9  tr , f a   sinc  Br  tr  d   Aa  f a  exp  j0  f a   exp  j 2 s0  d2   K  q    s0 2   2  2 Ks03  q3  Ym q23   3    q2 Ks1    d   exp  j   2 3 K  q K  q         s0  2 s0 2   

.

(27)

4. Azimuth Compression and RVP Compensation

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where Br is the bandwidth of transmitted signal.

Inspecting (27), the azimuth modulation phase 0  fa  is space-dependent, which will lead to that it is

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impossible to make a precise compensation for all targets at different azimuth position. Nevertheless, limited by the aperture of linear-array antenna, the azimuth focusing depth is usually not very large, which can provide enough support to neglect the space-variant in azimuth focusing.

be expressed by

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According to the aforementioned analysis, the azimuth compression and RVP compensation funciton can

  q2K 2 Ks03  q3  Ym q23   3    q2 Ks0  2 s1   d  . (28) H 6  f a   exp j  0  f a   2 YP' f a  exp  j  d2  j   2 3 K  q K  q K  q    s0 2    s0 2  2   s0  2 3 B1 3B12 B3  9 B3  4B2 B4  B1 denotes the new target azimuth position.   2 B2 8B23 16 B25

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where YP' 



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

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Multiplying (27) with (28) and transforming the result into 2-D time domain, then we can achieve the final

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focused SAR image as

     sr10  tr , ta   sinc  Br  tr  d   sinc  Ba  ta  YP'  .    

(29)

Summarizing the above processing procedures, the flowchart of the proposed efficient NLCS algorithm

for FLLA-SAR can be shown as Fig. 4.

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Raw Data

SAR Image

Range FFT

Azimuth IFFT LRWC

Azimuth MF

H 1  f r , ta 

H 6  fa 

Azimuth FFT

Range IFFT Range MF

H 5  fr , fa 

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RDC

H 2  fr , fa  CPF

H3  fr 

Range FFT NLCS

H 4  tr 

Range IFFT

Flowchart of the proposed efficient range NLCS algorithm.

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Fig. 4.

5. Simulation Results

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To verify the effectiveness of proposed NLCS algorithm on processing the FLLA-SAR data, the experiments with simulated data using the parameters shown in Table 1 are carried out in this section.

Parameters

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UAV velocity v

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Table 1 Simulation Parameters Values

Parameters

Values

200m/s

Carrier frequency

40GHz

30m/s

Pulse duration

2.5s

UAV height H

500m

Signal bandwidth

50MHz

Antenna spacing d

0.06m

Switch velocity vs

1200m/s

320

PRF

20kHz

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UAV acceleration a

Antenna number N

2

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An 3  3 array target shown in Fig. 5 is chosen as the imaging target, the intervals of adjacent point

targets are 100m and 35m in range and azimuth directions, respectively. Selecting the scatterer T1 located at the scene center (3100m, 0m) as the reference target. Here, to evaluate the performance of the proposed method, we employ traditional NLCS algorithm [1, 3, 21] to make a comparison with the proposed algorithm, which only takes the fixed switch velocity of the antenna array element into consideration and neglects the high-speed and acceleration of UAV platform in

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the algorithm derivation and correction function design. In addition, no weighting function or side-lobe control is used in the two algorithms.

y T4

T5

 3100, 0  m 35m

T3

T1

T6

T2

x

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O

100m

Fig. 5. Point scatterers distribution for the simulation.

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Fig. 6 exhibits the complete focused image of the array point scatterer. Fig. 6(a) is obtained by using the traditional NLCS algorithm, while Fig. 6(b) displays the imaging result of the proposed efficient NLCS algorithm. Comparing Fig. 6(a) and (b), we can easily find that the proposed efficient NLCS algorithm can

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not only achieve higher precision image than the traditional method, but also have no geometric distortion.

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-50 2950

PT

0

3050

3100 3150 range(m)

(a)

3200

3250

0

-50 2950

3000

3050

3100 3150 range(m)

3200

3250

(b)

Images of the simulated FLLA-SAR data achieved by different imaging algorithms. (a) Traditional NLCS algorithm. (b)

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Fig. 6.

3000

50

az imut h(m)

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azimuth(m)

50

Proposed efficient NLCS algorithm.

Fig. 7 shows the sub-images of point targets T1, T2 and T4, where T1 and T2 locate in the same azimuth

and different range bin, on the contrary, T1 and T4 locate in the same range but different azimuth bin. Both of the two methods considered the space-variant of range FM rate during the imaging procedures, therefore, the point targets lie in different range position can focused as well as the scene center respectively.

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38

T1

6

T4

2 0

T2

4 azimuth(m)

36 azimuth(m)

azimuth(m)

4

34 32

2 0 -2

-2

30

-4 3080

3090

3100 3110 range(m)

-4 3080

3120

3090 3100 range(m)

3110

3180 3190 3200 3210 3220 range(m)

(a) 38

0 -2

T4

36 34 32

-4 3080

3090

3100 3110 range(m)

3120

3080

3090

3100 3110 range(m)

azimuth(m)

0 -2 -4 3180

3120

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4 2

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T1

2

azimuth(m)

azimuth(m)

4

T2

3190

3200 3210 range(m)

3220

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(b)

Fig. 7. Sub-images of the point scatterers T1, T4 and T2 achieved by using different imaging algorithms. (a) Traditional NLCS

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algorithm. (b) Proposed efficient NLCS algorithm.

Nevertheless, due to the neglect of UAV platform’s high-speed and acceleration, the sub-image quality

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obtained by the traditional NLCS algorithm is much worse than the proposed method, which can be easily

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found by Fig. 7. Besides, comparing sub-images of T1 and T4 with T2, we can obviously find that as the point scatterer is far away from the scene center, azimuth focusing quality degrades gradually because the

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space variant characteristic has been neglected in azimuth processing.

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IRW=6m PSLR=-14.90dB ISLR=-12.50dB

-20 -30

T1

-40 -50

3090

-20 -30

T2

-40

3100 3110 range(m)

-50

3120

3190

3200 3210 range(m)

-10 -20 -30

T4

-40 -50

3080

3090 3100 range(m)

3110

(a) IRW=2.43m PSLR=-14.14dB ISLR=-13.10dB

-20 -30

T1

-40 -50

-2

Magnitude(dB)

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-10

0 2 azimuth(m)

Magnitude(dB)

-2

0 2 azimuth(m)

4

-20

ED PT

-30

T4

30

32 34 azimuth(m)

IRW=6m PSLR=-13.27dB ISLR=-10.35dB

36

IRW=6m PSLR=-13.17dB ISLR=-10.34dB

0

Magnitude(dB)

-10 -20 -30

T1

-10 -20 -30

T2

-40

3100 3110 range(m)

-50 3180

3120

3190

3200 3210 range(m)

IRW=6m PSLR=-13.12dB ISLR=-10.30dB

0

Magnitude(dB)

Magnitude(dB)

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T2

(b)

0

AC

-30

IRW=2.46m PSLR=-14.65dB ISLR=-13.46dB

-10

-50

3090

-20

-50

4

-40

-50 3080

-10

-40

0

-40

IRW=3.6m PSLR=-16.34dB ISLR=-11.28dB

0

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Magnitude(dB)

0

3220

IRW=6m PSLR=-14.72dB ISLR=-12.59dB

0 Magnitude(dB)

-10

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-10

IRW=6m PSLR=-14.13dB ISLR=-11.46dB

0

Magnitude(dB)

Magnitude(dB)

0

-10 -20 -30

T4

-40 -50 3080

3090

3100 3110 range(m)

(c)

3120

3220

ACCEPTED MANUSCRIPT 18

-20 -30

T1

-40 -50 -4

-2

0 azimuth(m)

-10 -20 -30

T2

-40 2

4

-50 -4

-2

0 azimuth(m)

IRW=1.19m PSLR=-12.80dB ISLR=-10.01dB

0

Magnitude(dB)

Magnitude(dB)

-10

IRW=1.2m PSLR=-10.37dB ISLR=-9.74dB

0

Magnitude(dB)

IRW=1.2m PSLR=-12.91dB ISLR=-10.07dB

0

-10 -20 -30

T4

-40 2

4

-50

32

34 36 azimuth(m)

38

(d) Fig. 8. Two-dimensional profiles of T1, T2 and T4 by different methods. (a) Range profiles by traditional NLCS method. (b)

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Azimuth profiles by traditional NLCS method. (c) Range profiles by proposed method. (d) Azimuth profiles by proposed method.

In addition, to quantitatively compare the focusing quality of the two methods, the profiles of the two-dimensional impulse response are illustrated in Fig. 8. According to these profiles, the parameters

are evaluated, which are labeled in the graphs.

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including impulse response width (IRW), peak side-lobe ratio (PSLR) and integrated side-lobe ratio (ISLR)

Observing the graphs, we can find that nearly theoretical two dimensional IRW, PSLR and ISLR can be achieved by the proposed algorithm, while the performance parameters of the traditional NLCS algorithm

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have much degradation in both dimensions, especially in azimuth dimension, the parameter IRW is degraded

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by nearly two times of the theoretical value. This is mainly due to the neglect of the UAV platform’s high

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speed motion, just as the illustrated above.

Furthermore, to make a further evaluation on the proposed algorithm, Fig. 9 show the imaging

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results of an airplane model by using the proposed and traditional NLCS algorithm respectively. From

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the results, we can find that there is no shadow existing in the well-focused image by the proposed method. However, the traditional method cannot work as well as the proposed one, especially for the region that is far away form the scene center.

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750

750

700

700

650

650

600

600

550 500 450

550 500 450

400

400

350

350

300

300

250 500

1000 1500 azimuth (samples)

250

2000

(a)

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range (samples)

range (samples)

19

500

1000 1500 azimuth (samples)

2000

(b)

Fig. 9. Images of a simulated airplane by different imaging algorithms. (a) Traditional NLCS algorithm. (b) Proposed efficient

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NLCS algorithm.

6. Conclusion

In this paper, an efficient NLCS algorithm for the FLLA-SAR data has been presented and analyzed.

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Meanwhile, the complete mathematical theory of the algorithm has been derived. The keys of the proposed algorithm are taking the platform’s motion into consideration and introducing the NLCS operation, which

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can remove the space-variant characteristic of along-track FM rate. Furthermore, the method only needs FFT

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and complex multiplication operations, making it appropriate for actual application. Simulated experiments indicate that the proposed algorithm can achieve nearly theoretical two dimensional IRW, PSLR and ISLR,

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which are significantly better than the traditional one achieved, especially in azimuth dimension.

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Acknowledgment

This research was partially supported by Natural Science Foundation of Jiangsu Province (no.

BK20160848), Fundamental Research Funds for the Central Universities (no. 30917011315), National Science Foundation Research of China (NSFC) (no. 61571294). The authors would like to thank anonymous reviewers for their valuable comments and suggestions which lead to substantial improvements of this paper.

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