MIMO-SAR waveform separation based on
inter-pulse phase modulation and range-
Doppler decouple filtering
C.-Z. Meng, J. Xu, X.-G Xia, T. Long, E.-K Mao, J. Yang
and Y.-N. Peng
The coupling among multiple coded orthogonal waveforms occupying
the same frequency band may seriously affect the distributed terrain
imaging of multiple-input and multiple-output synthetic aperture
radar (MIMO-SAR). Based on the inter-pulse phase modulation
among different transmitting waveforms, a range-Doppler decouple
filtering method is presented to separate waveforms effectively.
Finally, numerical experiments are provided to demonstrate the
effectiveness of the proposed method.
Introduction: Due to the multiple transmitters and receivers, a
multiple-input and multiple-output (MIMO) radar system can obtain
larger system degrees of freedom (DOF) [1, 2]. Therefore, the spatial
diversity gain and coherent integration gain can be optimised [2] to
improve the target detection performance. Also, a MIMO-based
sparse array can enlarge the aperture and suppress the grating sidelobes,
simultaneously [1]. Furthermore, by combining the MIMO technique
and synthetic aperture radar (SAR), it is pointed out that the
MIMO-SAR [3] can remarkably improve the SAR multifunctional
performance and effectively deal with the well-known SAR imaging
trade-off between the large swath and the high azimuth resolution [4].
Unfortunately, the completely orthogonal waveforms may not be
realistically found and used, and the cross-correlation noises among
different waveforms occupying the same frequency band may seriously
degrade the ultimate MIMO-SAR image quality. This problem is inves-
tigated well in [5], which shows that when the waveform number is
larger than two, the synthetic integrated sidelobe level rate (SISLR) is
certainly larger than zero for each waveform due to the cross-correlation
noises. Firstly, to address this problem, the SAR Doppler property is
investigated in this Letter. Then, the inter-pulse phase modulation is
proposed. Subsequently, a novel two-dimensional (2D) range-Doppler
decouple filtering (RDDF) method is presented to improve the SISLR.
Finally, numerical experiments are provided to demonstrate the
effectiveness of the proposed method.
Signal model and proposed RDDF method: In a side-looking
MIMO-SARsystem,itisassumedthatthereareM orthogonal waveforms
{s
m
(
t
), m = 1, 2, ···, M} in the same frequency band with the same
bandwidth B which are transmitted by M transmitters simultaneously,
and their returns are received by N receivers where
t
is the fast time.
The azimuth antenna size of each T/R element is D. Then, for a scatterer
located at (r
B
, 0)in the terrain, its 2D returns in the nth receiver can be rep-
resented as
s
n
(
t
; t) =
M
m=1
s
m
w
a
(t) s
m
(
t
− t
n,m
)exp( j2
p
f
c
(t − t
n,m
)),
n = 1, 2, ···, N
(1)
where t is the slow time, i.e. the pulse-by-pulse sampling time,
s
m
is the
backscattering coefficient for the mth waveform, w
a
(·) is the antenna
azimuth pa ttern modulation, f
c
is carrier frequency, t
n,m
is the propagation
time between the radar and the scatterer for the T/R channel of the mth
transmitter and the nth receiver, which is caused by the varied slant
range of scatter against t during the synthetic aperture. It is shown from
(1) that the time-varying t
n,m
will cause the envelope shift and the phase
modulation jointly. Normally, the former is called range migration and
the latter is called Doppler modulation. To reconstruct the scatterer, the
range compression and range migration correction (RMC) should be
implemented, which can be done via interpolation or a scaling operator
[6, 7]. After RMC, the range-varying peak of the scatte rer may be
assumed located with a fixed delay t
a
= 2r
B
/c,wherec is the light
speed. Then, (1) can be approximately converted into the range-Doppler
domain via FFT in terms of t [6] for the mth waveform as
s
n,m
(
t
; f
t
) ≃ A
n,m
sinc(B(
t
− t
a
))W
a
( f
t
) exp ( j
p
f
2
t
/
g
a
)
+
M
i=1,i=m
g
n.m,i
(
t
− t
a
)W
a
( f
t
) exp ( j
p
f
2
t
/
g
a
)
(2)
where A
n,m
is the amplitude of the mth waveform, W
a
(·) is azimuth modu-
lation in the Doppler domain, f
t
is the Doppler frequency,
g
a
=−(2v
s
)
2
/
l
r
B
is the Doppler rate,
l
is wavelength, v
s
is pla tform
speed. Notably, g
n,m,i
(·) is the cross-corr elation between the ith and the
mth waveforms in the nth receiver, which is a noise-like function
without an obvious peak [1] due to the orthogonal waveform design. For
the side-looking SAR, the Doppler frequency can be approximated as
linear frequency modulated and f
t
[ (−B
a
/2, B
a
/2) where Doppler band-
width B
a
≃ 2v
s
/D. Consequently, the filter function can be defined in the
Doppler domain as
H
m
( f
t
) = exp (−j
p
f
2
t
/
g
a
)rect( f
t
/B
a
)(3)
where rect( · ) is the rectangular function, which performs the role of band-
pass filtering. By multiplying (3) to (2) and taking the inverse FFT in terms
of f
t
, we can generate the ultimate imaging result as
s
n,m
(
t
, t ) ≃ B
n,m
sinc(B(
t
− t
a
)) sinc (B
a
t )
+
M
i=1,i=m
g
n.m,i
(
t
− t
a
)sinc(B
a
t )
(4)
where the co-registration has been done to compensate the azimuth shift
among different T/R channels, and B
n,m
is the matched amplitude of the
mth waveform. Normally, due to the orthogonal waveform design,
we have |B
n,m
|≫|g
n.m,i
(·)| where |·| is the amplitude operator.
Nevertheless, the second term of (4) cannot be omitted, especially for
the distributed terrain where there are a great number of neighbouring scat-
terers around the reconstructed scatterer. In this case, the second term of
the neighbouring scatterers, i.e. the coupled cross-correlation noise, may
totally submerge the first term of (4). Consequently, the image quality
may be seriously affected [4], especially with the increase of M.To
solve the couple effect, the RDDF algorithm is proposed in this Letter
and the two main steps of RDDF are described as follows:
1. Inter-pulse modulation technique. It is shown from (2) that the differ-
ent waveforms occupy the same range and Doppler frequency domain.
Although orthogonal waveform design is adopted, the cross-correlation
noises cannot be omitted from (4). Can we distinguish different
from waveforms in a new domain? The answer is yes. Let us design
2D time-varying waveforms as {
w
m
(
t
, t ) = s
m
(
t
) exp ( j2
p
f
ac,m
t ), m =
1, 2, ···, M}wheref
ac,m
≥ (m − 1)B
a
. Not only are the orthogonal
waveforms s
m
(
t
) used against fast time
t
but it is also modulated
pulse-by-pulse by frequency f
ac,m
against slow time t.Thatis,fora
fixed slow time t, s
m
(
t
) modulated with phase terms exp ( j
u
m
) =
exp ( j2
p
f
ac,m
t ) are transmitted by M transmitters against fast time τ.
Now, (2) should be rewritten as
s
n,m
(
t
; f
t
) ≃ A
n,m
sinc (B(
t
− t
a
))W
a
( f
t
− f
ac,m
)
exp ( j
p
(( f
t
− f
ac,m
)
2
/
g
a
)) +
M
i=1,i=m
g
m,i
(
t
− t
a
)
W
a
( f
t
− f
ac,i
) exp ( j
p
( f
t
− f
ac,i
)
2
/
g
a
)
(5)
2. Range-Doppler decouple filtering. It is shown that different wave-
forms now occupy different Doppler areas. Let us redefine (3) as
H
m
( f
t
) = exp (−j
p
( f
t
− f
ac,m
)
2
/
g
a
) rect (( f
t
− f
ac,m
)/B
a
)(6)
By multiplying (6) to (5) and taking the inverse FFT in terms of f
t
,we
can generate the ultimate images as
s
n,m
(
t
, t ) ≃ B
n,m
sinc (B(
t
− t
a
)) sinc (B
a
t ) (7)
It is shown that the cross-correlation noise can be well suppressed by the
Doppler bandpass filter as (6). Furthermore, there are two problems that
should be mentioned. One is that non-orthogonal multiple waveforms
can also be separated by Doppler-only decoupling as (6), but better
performance can be expected by the proposed RDDF because the
Doppler spectrum is not strictly band-limited. The other is that
the proposed method is more suitable for airborne SAR where the
pulse repetition frequency (PRF) can be far larger than B
a
.
Numerical experiments: In this Section, three T
X
and one R
X
, distribu-
ted along the track uniformly and linearly, are used for an airborne
MIMO-SAR. Three coded orthogonal waveforms are designed in a
similar way as in [5]. The SAR main parameters are given as follows:
ELECTRONICS LETTERS 14th March 2013 Vol. 49 No. 6
Te chset CompositionLtd,Salisbury Doc: //techsetserver2/journal/IEE/EL/ISSUE/49-6/pagination/EL20130016.3d
Radar, sonar andnavigation
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