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LIANG et al.: HIGH-ORDER PHASE CORRECTION APPROACH FOR FOCUSING HS-SAR SMALL-APERTURE DATA 3
where W
r
(·) represents the range envelope in frequency
domain, w
a
(t
m
) denotes the azimuth envelope in time domain,
γ is the range chirp rate, c is the speed of light, f
r
and f
c
are the range frequency and the carrier frequency, respectively.
Observing (2), the first exponential term is the range frequency-
modulation (FM) term, which can be easily compensated for
by multiplying its complex conjugate. The second exponential
term comes from the azimuth modulation.
III. R
ANGE SIGNAL PROCESSING
The range processing of HS-SAR involves two steps: 1) the
range cell migration correction (RCMC) and 2) the range
focusing, which includes the range compression (RC) and the
secondary range compression (SRC). The RCMC contains the
linear range walk correction (LRWC) and the range curvature
correction (RCC). In general, the LRWC is often performed
in the range frequency and azimuth time domain as the first
step by using a linear phase multiplication, and the RCC oper-
ation is performed in the 2-D frequency domain by multiplying
a selected reference range compensating factor.
A. LRWC
It is a challenging task for HS-SAR data focusing due to the
strong coupling between the r ange and azimuth. The LRWC
operation can reduce the cross coupling and simplify the fol-
lowing processing operation. According to [13]–[19], the value
of the linear range walk is v sin θ
0
t
m
, so the compensating
factor is given by
H
LRW C
(f
r
,t
m
)=exp
−j
4πv sin θ
0
c
(f
c
+ f
r
)t
m
. (3)
B. RCC, RC, and SRC
Transforming the echo signal after the LRWC into the 2-D
frequency domain yields
SS(f
r
,f
a
; R
0
)=W
r
(f
r
) W
a
(f
a
)exp
−jπ
f
r
2
γ
· exp
−j
4πx
n
sin θ
0
c
f
r
· exp(−j4πR
0
cos θ
0
Φ
0
)
· exp
−j4πR
0
cos θ
0
Φ
1
+
sin
2
θ
0
cos θ
0
· c
f
r
· exp
−j4πR
0
cos θ
0
Φ
2
f
2
r
· exp
−j
2π
v
(f
a
+ f
dc
)(R
0
sin θ
0
+ x
n
)
.
(4)
The detailed derivation of (4) is shown in Appendix A,
including the expressions of Φ
0
, Φ
1
, and Φ
2
.In(4),f
a
denotes
the azimuth frequency, W
a
(·) represents the azimuth enve-
lope in frequency domain, and f
dc
is the Doppler center
frequency, with the value f
dc
=2v sin θ
0
/λ (λ is the carrier
wavelength). Furthermore, the second exponential term denotes
the deformation of the range position. The third exponential
term shows the azimuth frequency modulated item. The fourth
exponential term is the range curvature and the fifth exponen-
tial term is called the SRC. The last exponential term reflects
the real azimuth position, the azimuth deformation, and the
constant phase.
From (4), it is easy to get the RCC, the range compression
and the SRC factors, which have the complex conjugate forms
of the first, the fourth, and the fifth exponential terms. It should
be noted that, the RCC and SRC factors are both range depen-
dent, and we use the compensating factors at the reference range
as an acceptable approximation, where the reference range is
the slant range at the center line of the scene. Usually, when
the range swath is so large that the range dependence cannot
be neglected, a convincible method is used to divide the range
into several small blocks and each block has its corresponding
reference range [18].
After the range multiplication processing and transforming
the result into the range time domain by the inverse FFT, we
have
sS(
ˆ
t, f
a
; R
0
)=Sinc
B
r
ˆ
t −
2(R
0
+ x
n
sin θ
0
)
c
· W
a
(f
a
)exp(−j4πR
0
cos θ
0
Φ
0
)
· exp
−j
2π
v
(f
a
+ f
dc
)(R
0
sin θ
0
+ x
n
)
(5)
where
ˆ
t represents the range fast time. Observing the range
envelope Sinc(·) in (5) (the rectangular form of W
r
(·) has
been assumed), B
r
is the transmitted signal bandwidth, and
R
0
+ x
n
sin θ
0
is the new slant range, which varies with the
azimuth position x
n
. It means that the points lying in the same
range cell after range processing have different original slant
ranges. This dependence can induce the variation of the azimuth
FM rate and phase coefficients of high-order terms, and give
rise to the azimuth filtering mismatch for different targets in the
same range cell.
IV. A
ZIMUTH COMPRESSION BASED ON HIGH-ORDER
PHASE CORRECTION TECHNIQUE
According to Section III, we know that the value of Doppler
FM rate and the coefficients of high-order phases vary with
the azimuth position after the range processing, resulting in the
failure of uniform azimuth processing. In order to solve this
problem, a high-order phase correction approach (HPCA) for
small-aperture data is presented. Its main idea is to eliminate
the azimuth dependence by introducing a correction phase in
the azimuth frequency domain, to realize t he azimuth focusing
via a common reference function. In the following, the selec-
tion of the image focused domain and the azimuth dependent
characteristic are analyzed in detail first.
A. Selection of Focused Domain
It is convenient for HS-SAR on high-speed moving platforms
to adopt the small-aperture data processing. When the matched