Journal of Signal and Information Processing, 2013, 4, 14-18
doi:10.4236/jsip.2013.43B003 Published Online August 2013 (http://www.scirp.org/journal/jsip)
SAR Imaging of Moving Target Based on Quadratic Phase
Function
Chao Wang, Shaobin Li
School of Electronics information Engineering Harbin Institute of Technology Harbin, China.
Received March, 2013.
ABSTRACT
In this paper, a novel signal processing technique has been developed to refocus moving targets image from their
smeared responses in the Synthetic Aperture Radar (SAR) image according to the characteristics of the received signals
for moving targets. Quadratic Phase Function is introduced to the parameters estimation for moving target echo and
SAR imaging. Our method is available even under a low SNR environment and acquiring an exact SAR image of mov-
ing targets. The simulated results demonstrated the validity of the algorithm proposed.
Keywords: Synthetic Aperture Radar; Moving Targets; Quadratic Phase Function
1. Introduction
Airborne synthetic aperture radar (SAR) detection and
imaging of moving targets on the ground has been a sub-
ject of long-standing interest in both civil and military
applications such as battlefield reconnaissance, fire as-
sessment, traffic management and monitoring ocean cur-
rents. If there are moving targets in the observed scene,
SAR cannot simultaneously produce clear images of both
stationary and moving targets because they have different
Doppler frequency respectively, the target motion in-
duced phase errors, which interact with the matched filter
processing or cross-range compression, cause these im-
ages to be mislocated in the cross-range dimension and
smeared in both the cross-range and the range domains,
we can only focus on either the stationary target or the
moving target. How to detect moving targets in the
background of stationary objects called clutter is premise
for SAR imaging of moving targets, this problem is usu-
ally solved by applying the displaced phase center an-
tenna (DPCA) [1] or its modern extension, the space-
time adaptive processing (STAP) [2]. Radar returns from
terrain and stationary objects can be suppressed after
using DPCA or STAP, only the returns from moving
targets are used to reconstruct radar images.
Moving targets echo signal is generally characterized
by the linear frequency modulated (LFM) signal, before
obtain the focused images of moving targets, the pa-
rameters of LMF should be estimated firstly, and then
construct correct match filter of range and azimuth direc-
tions for imaging. In which, initial frequency and chirp
rate as the basic parameters of LMF, their estimation
problem has been an important research content of the
signal processing. In previous investigations, several
parameter estimation methods based on Maximum Like-
lihood (ML) [3] estimation cannot be implemented in
engineering because of a huge amount of computation.
Some signal processing methods such as Radon-Wigner
Transform (RWT) [4], Radon-Ambiguity Transform
(RAT) [5], Fractional Fourier Transform (FrFT) [6] have
been usually used in estimation of LMF parameters. It is
worth mentioning that WVD combine with Radon
Transform as a commonly used method of detection and
estimation for LMF can achieve an idea result. However,
there is cross-term interference associated with the WVD.
When the signal contains more than one component, the
WVD will generate cross-term interference between
components that occurs at spurious locations of the
time-frequency plane.
In this paper the Quadratic Phase Function (QPF) [7]
[8] is introduced to the SAR moving target imaging, and
a new SAR imaging algorithm based on the Quadratic
Phase Function is presented. This method can achieve an
idea result for moving targets imaging. The simulated
results demonstrate the validity of this algorithm.
2. Radar Returns of Moving Targets
In this section, we setablish the signal model for the
received signal in SAR imaging of moving targets, here,
take the ground coordinate system for the reference co-
ordinate system and without considering the rotation of

SAR Imaging of Moving Target Based on Quadratic Phase Function 15
the earth. The scene of a side-looking SAR and a moving
target is illustrated in Figure 1.
0
R
()
m
Rt
)0,,(
00
yx
)0,,(
tt
yx
),(
xx
av
),(
yy
av
Figure 1. Geometry of an airborne SAR and a moving tar-
get
In the case of a moving point target, we assume that at
a point target is located at , and the
radar platform is located at . Then at time t the
target moves to (00 yx
and the platform moves to. Where xand x
represent the azimuth velocity and acceleration of the
point in azimuth direction respectively, yand y rep-
resent the ground range velocity and acceleration of the
point in ground range direction respectively. The range
from the radar to the point target at time can be writ-
ten as
0t)0,,( 00yx
)
,atvy  v
v
t
,0,0( h
2/
2
ta x 
),0,(hvt )0,2/
2
ttvx y a
a
2
2
0
21
22
2
0
() [()
2
()
2
x
x
y
y
at
Rtvt vtx
at
yvt h
 
 ]
(1)
which can be seen by expanding and approximating [9]:
000
0
0
222
00 2
0
()
2
2
xy
xy xyx
xv yv xv
Rt Rt
R
vvvxaya vv
t
R


 

(2)
where the range component due to the radar motion is:
2
1
2
2
0
2
0])[()( hyxvttRradar  (3)
and the range component due to the target motion is:
21
2
22
2
arg 0
21
2
2
0
()[()()2( )]
22
()2()]
22
y
x
tet xy
y
x
xy
at
at
Rtvtvtvtx
at
at
vty vt
 
 
(4)
where 0 is the initial range at , because 0 is
much smaller than and is approximately equal
to , then the raw data can be described by:
R0tx
0
R0
y
0
R
00
2
22
0
0
2
()
exp{ 4}exp{ 2}
4
exp{[()] }
2
y
xy y
v
Rt
s
()tCjC jt
t
jv vvRa
R




 
 
(5)
In which
t
v
ty
shift


4)(  (6)
Equation (6) is a linear phase function due to the target
velocity in the radial direction, and
2
22
0
0
4
()( )2
defocusxyy t
tvvvRa
R





(7)
is a quadratic phase function determined by the relative
velocity between the radar and the moving target in the
x
-direction---the radial velocity, and acceleration of the
target.
From the above phase functions we can see that by
making use of a matched filter designed to match the
baseband returns from a stationary target, the linear
phase change due to the target’s radial velocity y in (6)
causes the image of the moving target to be shifted in the
cross-range direction, and the quadratic phase variation
in (7) causes the image of the moving target to be defo-
cused. So the present efforts in moving targets imaging
require that estimations be made of targets motion before
any imaging procedure can be performed.
v
3. Quadratic Phase Function
As to the LFM signal2
12
() exp()
s
tb atat
a a, where b
represents amplitude, 1and 2 represents initial fre-
quency and chirp rate respectively. The Quadratic Phase
Function Transform has been explained in detail by the
reference [7, 8]:


detstsutQP ju
 
 0
2
)()(),( (8)
Substitute into equation (8), we can obtain:
)(ts
 
0
)2()(2 22
2
2
21
),( 

deebutQP uajtataj (9)
The parameter 2 can be found through peak search
and the other parameters can be estimated by dechirp
technology.
a
From equation (8), it is demonstrated that the quadratic
phase function is bilinear transformation, therefore when
the signal contains more than one component, the QPF
will generate cross-term interference between compo-
nents that occurs at spurious locations of the time-fre-
quency plane. To reduce the cross-term interference,
multiple type and integral type of QPF had been defined
in the reference [10] and [11].

SAR Imaging of Moving Target Based on Quadratic Phase Function
16
In which multi type of QPF is defined as:


L
llutQPuMQP 1
),()( (10)
And integral type of QPF is defined as:


t
dtutQPuIQP),()( (11)
where l is of time sample points. From the equa-
tions (10) and (11), we can see that multi type or integral
type of QPF can reduce cross-term interference effec-
tively and then suitable for the analysis of LMF signal.
tL
4. Application of Quadratic Phase Function
for SAR Imaging
In the section3, we have presented the Quadratic Phase
Function algorithm to estimate the parameters of the
LFM signal. In this section, we will discuss the SAR
imaging techniques for moving targets based on Quad-
ratic Phase Function algorithm, and the new approach for
SAR imaging of moving targets can be illustrated explic-
itly as follows:
Step 1. Range compression. Definition of range com-
pression reference function is described by:
)exp()( 2 tKjtsref r   (12)
where rvKrrepresents chirp rate, while t repre-
sents time value in range direction. In the case of point
moving targets the acquired raw data after range com-
pression can be written as in

2
2
 )]([)]([)(
1tsrefFFTtsFFTIFFTts  (13)
where and represent Fourier Trans-
form operator and Inverse Fourier Transform operator
respectively.
][FFT][IFFT
Step 2. Azimuth compression function can be defined
as:
)exp()( 2
1mrm tKjtsref
  (14)
where mrepresents the time value in azimuth direction.
As to stationary targets, focused images could be
achieved after equation (13) multiplying with equation
(14), however the azimuth velocity of the moving targets
will cause smearing effect in azimuth direction when
processed by conventional image formation of stationary
points. At this moment, we multiply every range cell
with equation (14), the result can be written as in
t
)()()( 112 mmm tsreftsts  (15)
In order to counteract these smearing effects and get
focused image, the parameters of should be
taken into account during processing and for this purpose,
the parameters should be estimated firstly, this is
achieved by the following method named Quadratic
Phase Function in step 3.
)(
2m
ts
Step 3. Estimate the chirp rate

ˆ of , which
can be written as

m
ts2






,maxarg
ˆ
,F

(16)
where

  ,F is the Quadratic Phase Function of


m
ts2.
Step 4. According to the parameters which have al-
ready been estimated in step3, amended azimuth com-
pression function is contrasted as in
])
ˆ
(exp[)(2
2mrm tKjtsref
  (17)
Step 5. Using equation (17) to reconstruct azimuth
compression, we can obtain the focused SAR image of
the moving targets finally. The detailed method is as fol-
lows:





 ][][,22mmm tsrefFFTtsFFTIFFTttS 

(18)
Based on the procedure above, we can obtain the fo-
cused image of moving targets with the Quadratic Phase
Function technique. The flow chart of this algorithm is
illustrated in Figure 2.
Azimuth
IFFT
SAR imagingRange
compression
function
Azimuth
compression
raw date
azimuth
FFT Azimuth
compressi-
on
function
Parameter
estimation
Range
compression
New azimuth
compression
function QPF
Figure 2. The flow chart of SAR imaging based on Quad-
ratic Phase Function.
5. Computer Simulations
To demonstrate the effectiveness of the algorithm devel-
oped in the preceding section, we applied it to simulate
data samples. We assume that there are five point targets
in the same imaging plane, the radar operates at the fre-
quency 0
f
=10 GHz, the bandwidth of LFM signals is B =

SAR Imaging of Moving Target Based on Quadratic Phase Function 17
60 MHz, the pulse repetition frequency is PRF = 500, the
pulse width is w

= 10 μs, the sampling frequency is
Fs=70 MHz. The aircraft with the radar is moving along
the x-axis with velocity v = 120 m/s, while the radar’s
ground distance from the origin of the coordinate system
connected to the targets is 8 Km at t = 0. The motion
parameters of the targets are given in Table 1.
Table 1 motion parameters of the targets used in experi-
ment.
No. 1 2 3 4 5
x0(m) 0 30 5 -5 0
y0(m) 0 0 30 15 -30
Vx(m/s) 3 2.2 3.1 0 3
Vy(m/s) 0.7 1 0 0 1
ax(m/s2) 0 0 0 0 0
ay(m/s2) 0 0 0 0 0
Figure 3 is the SAR image of moving targets based on
Range-Doppler algorithm without parameter estimation,
we can see that the image is blurred in azimuth direction.
Azimuth
Range
50 100 150 200250 300350
50
100
150
200
250
300
Figure 3. SAR imaging without parameters estimation.
Figure 4 is the result of SAR imaging after parameter
estimation, compared with Figure 3, we can see that the
blurred of azimuth direction has been removed and the
moving targets has been focused.
Figure 5 and Figure 6 are the result of SAR imaging
when the SNR is -10 dB and -15 dB respectively, we can
see that this method still effective when noise exists.
Azimuth
Range
50 100 150 200 250 300 350
50
100
150
200
250
300
Figure 4. SAR imaging based on Quadratic Phase Function.
Azimuth
Ran ge
50100 150 200250 300 350
50
100
150
200
250
300
Figure 5. SAR imaging based on Quadratic Phase Function
with the SNR is -10 dB.
Azimuth
Range
50 100 150200 250 300350
50
100
150
200
250
300
Figure 6. SAR imaging based on Quadratic Phase Function
with the SNR is -15 dB.

SAR Imaging of Moving Target Based on Quadratic Phase Function
18
6. Conclusions
In this paper, we introduce Quadratic Phase Function
algorithm to estimate the parameters of the LFM signal
and propose a new algorithm for moving targets imaging,
which can acquire an exact SAR image even under a
SNR environment. The effectiveness and practicability of
this imaging approach are demonstrated by means of
numerical experiments using simulated data.
REFERENCES
[1] H. S. Wang and C. Mainlobe, “Clutter Cancellation by
DPCA for Space Based Radars,” IEEE Aero Space Ap-
plications Conference, Crested Butte, CO, USA: IEEE,
1991. pp. 1- 128.
[2] R. Klemm, “Application of Space-time Adaptive Proc-
essing,” The Institution of Electrical Engineers, London,
2004.
[3] T. J. Abatzoglou, “Fast Maximum Likelihood Joint Esti-
mation of Frequency and Frequency Rate,” IEEE Trans-
actions on Aerospace and Electronic System, Vol. 22, No.
6, 1986, pp. 708-715. doi:10.1109/TAES.1986.310805
[4] J. C. Wood and D. T. Barry, “Linear Signal Synthesis
Using the Radon-Wigner Transforms,” IEEE Transac-
tions On Signal Processing, Vol. 42, No. 8, 1994, pp.
2105-2111. doi:10.1109/78.301845
[5] M. S. Wang, A. K. Chan and C. K. Chui, “Linear Fre-
quency-Modulated Signal Detection Using Ra-
don-ambiguity Transform,” Vol. 46, No. 3, 1998, pp.
571-586.
[6] R. Tao, B. Deng and Y. Wang, “Research Progress of the
Fractional Fourier Transform in Signal Processing,” Sci-
ence in China (Ser. F, Information Science), Vol. 49, No.
1, 2006, pp. 1-25.
[7] P. O’Shea, “A New Technique for Estimating Instanta-
neous Frequency Rate,” IEEE Signal Processing Letters,
Vol. 9, No. 8, 2002, pp. 251-252.
doi:10.1109/LSP.2002.803003
[8] P. O’Shea, “A Fast Algorithm for Estimating the Pa-
rameters of a Quadratic FM Signal,” IEEE Transactions
On Signal Processing, Vol. 52, No. 2, 2004, pp. 385-393.
doi:10.1109/TSP.2003.821097
[9] V. C. Chen “Time-Frequency Transforms for Radar Im-
aging and Signal Analysis,” Artech House. 2002.
[10] L. Hong, “Analysis of Multi-component LFM Signals by
the Integrated Quadratic Phase Function,” Journal of
Electronics & Information Technology, Vol. 31, No. 6, pp.
2009, pp. 1363-1366.
[11] P. Wang and J. Y. Yang, “Multicomponent Chirp Signals
Analysis Using Product Cubic Phase Function,” Digital
Signal Processing, Vol. 16, No. 6, 2006, pp. 654-669.
doi:10.1016/j.dsp.2006.09.002