I. INTRODUCTION
Multistatic Radar Imaging of
Moving Targets
LING WANG, Member, IEEE
MARGARET CHENEY, Member, IEEE
Rensselaer Polytechnic Institute
BRETT BORDEN
Naval Postgraduate School
We develop a linearized imaging theory that combines the
spatial, temporal, and spectral aspects of scattered waves. We
consider the case of fixed sensors and a general distribution of
objects, each undergoing linear motion; thus the theory deals with
imaging distributions in phase space. We derive a model for the
data that is appropriate for narrowband waveforms in the case
when the targets are moving slowly relative to the speed of light.
From this model, we develop a phase-space imaging formula that
can be interpreted in terms of filtered backprojection or matched
filtering. For this imaging approach, we derive the corresponding
phase-space point-spread function (PSF). We show plots of the
phase-space point-spread function for various geometries. We also
show that in special cases, the theory reduces to: 1) range-Doppler
The use of radar for detection and imaging of
moving targets is a topic of great interest [1—29]. It
is well known that radar signals have two important
attributes, namely the time delay, which provides
information on the target range, and the Doppler shift,
which can be used to infer target down-range velocity.
Classical “radar ambiguity” theory [30—32] shows that
the transmitted waveform determines the accuracy to
which target range and velocity can be obtained from
a backscattered radar signal.
Another well-developed theory for radar imaging
is that of synthetic-aperture radar (SAR) imaging.
SAR combines high-range-resolution measurements
from a variety of locations to produce high-resolution
radar images [21, 33—35]. Such images, however,
suffer when there is unknown relative motion between
the target and radar platform; research addressing this
includes [3], [7]—[22]. This research generally does
not consider the possibility of using waveforms other
than high-range-resolution ones.
When high-Doppler-resolution waveforms
are transmitted from a moving platform, the
Doppler-shifted returns can also be used to form
images of a stationary scene [36].
These three theories are depicted on the coordinate
planes in Fig. 1. None of these theories address the
possibility of forming high-resolution moving-target
images in the case when different waveforms are
transmitted from different spatial locations and
received at spatially distributed receivers. Such a
theory is needed to address questions such as:
imaging, 2) inverse synthetic aperture radar (ISAR), 3) synthetic
aperture radar (SAR), 4) Doppler SAR, and 5) tomography of
moving targets.
Manuscript received November 25, 2008; revised October 15, 2009;
released for publication November 11, 2010.
IEEE Log No. T-AES/48/1/943610.
Refereeing of this contribution was handled by M. Rangaswamy.
This work was supported by the Air Force Office of
Scientific Research under Agreements FA9550-06-1-0017 and
FA9550-09-1-0013.
Ling Wang’s stay at Rensselaer was supported by the China
Scholarship Council.
The views and conclusions contained herein are those of the authors
and should not be interpreted as necessarily representing the official
policies or endorsements, either expressed or implied, of the Air
Force Research Laboratory or the U.S. Government.
Authors’ addresses: L. Wang, Dept. of Information and
Communication Engineering, Nanjing University of Aeronautics
and Astronautics, Nanjing 210016, China; M. Cheney, Dept.
of Mathematical Sciences, Rensselaer Polytechnic Institute,
Amos Eaton Hall, 110 Eighth St., Troy, NY 12180, E-mail:
(cheney@rpi.edu); B. Borden, Physics Dept., Naval Postgraduate
School, Monterey, CA 93943.
c 2012 IEEE
0018-9251/12/$26.00 °
230
Fig. 1. Notional diagram showing how our work (depicted by
question mark) relates to standard radar imaging methods.
In a multistatic system, which transmitters should
transmit which waveforms?
How many transmitters are needed, and where
should they be positioned?
How can data from such a system be used to form
an image of unknown moving targets?
What is the resolution (in position and in velocity)
of such a system?
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 48, NO. 1
JANUARY 2012
TABLE I
Table of Notation
Some work has been done to develop such a theory:
Ambiguity theory for bistatic systems has been
developed in [34], [35], [37], [38]; such systems allow
for estimation of only one component of the velocity
vector.
For multistatic systems, the work [23]—[29]
developed methods for moving-target detection.
Theory for use of a multistatic system for
imaging of a stationary scene is well known; see e.g.
[39]—[42].
Multistatic imaging of moving targets (phase-space
imaging) was developed in [43] for the case of fixed
transmitters and receivers (see Fig. 2). This theory
combines spatial, temporal, and spectral attributes of
radar data; in particular the theory exploits the actual
Doppler shift from moving targets. In appropriate
special cases, this theory reduces to the standard
imaging methods shown in Fig. 1.
The work [43] is extended in the present paper,
which sets forth the basic ideas in the special cases
that are of interest to radar-based imaging, and
undertakes an initial numerical exploration of the
properties of the imaging system. In particular, we
show that under some circumstances, this approach
allows for both (spatial) image formation and also
estimation of the full vector velocities of multiple
targets.
In Section II we develop a physics-based
mathematical model that incorporates not only the
waveform and wave propagation effects due to
moving scatterers, but also the effects of spatial
diversity of transmitters and receivers. In Section III
we show how a matched-filtering technique produces
images that depend on target velocity, and we
relate the resulting point-spread function (PSF) to
the classical radar ambiguity function. We show
four-dimensional plots of the PSF for three different
geometries. In Section IV we note that the general
theory of Section III reduces to familiar results,
including classical ambiguity theory, inverse SAR
(ISAR), SAR, and Doppler SAR. Finally, in Section V
we list some conclusions. Table I gives a list of
notations used throughout the paper.
II. MODEL FOR DATA
We model wave propagation and scattering by the
scalar wave equation for the wavefield Ã(t, x) due to a
source waveform s̃(t, x):
[r2 ¡ c¡2 (t, x)@t2 ]Ã(t, x) = s̃(t, x):
(1)
For example, a single isotropic source located at
y transmitting the waveform s(t) starting at time
t = ¡Ty could be modeled as s̃(t, x) = ±(x ¡ y)sy (t + Ty ),
where the subscript y reminds us that transmitters at
different locations could transmit different waveforms.
For simplicity, in this discussion we consider only
Symbol
Designation
x
t
y
sy (t)
Position (three-dimensional vector)
Time (scalar)
Transmitter position (three-dimensional vector)
Waveform transmitted by a transmitter located at y
¡Ty
Starting time of the waveform sy
!y
Carrier frequency of the transmitted waveform sy (t)
ky = !y =c
Wavenumber of the transmitted waveform sy
ay (t)
Slowly varying part of sy
v
Velocity (three-dimensional vector) of a moving
scatterer
Distribution of scatters with position x and velocity v
at time t = 0
Doppler scale factor for a scatter at position x
moving with velocity v.
Wavefield at time t and position x
Free-space field at x generated by a source at y
Receiver position (three-dimensional vector)
x ¡ z, three-dimensional vector from z to x
qv (x)
®x,v
Ã(t, x)
à in (t, x, y)
z
Rx,z
¹z,v
à sc (t, z, y)
1 + R̂x,z ¢ v=c
®x,v
Scattered field at the receiver location z due to the
field transmitted from position y
Doppler scale factor
!y ¯x,v
(Angular) Doppler shift
p
Position (three-dimensional vector) in the
reconstructed scene
Velocity (three-dimensional vector) of object in
reconstructed scene
Reconstruction of reflectivity of objects moving with
velocity u
Geometry-dependent weighting function for
improving the image quality
Point spread function (PSF)
Radar ambiguity function for the waveform used by
the transmitter located at y
Angle of reconstructed position vector p
Angle of reconstructed velocity u
u
Iu
J
K
Ay
μ
Á
localized isotropic sources; the work can easily be
extended to more realistic antenna models [44].
A single scatterer moving at velocity v corresponds
to an index-of-refraction distribution n2 (x ¡ vt):
c¡2 (t, x) = c0¡2 [1 + n2 (x ¡ vt)]
(2)
where c0 denotes the speed of light in vacuum. We
write qv (x ¡ vt) = c0¡2 n2 (x ¡ vt). To model multiple
moving scatterers, we let qv (x ¡ vt)d3 x d3 v be the
corresponding quantity for the scatterers in the volume
d3 x d3 v centered at (x, v). Thus qv is the distribution
in phase space, at time t = 0, of scatterers moving
with velocity v. Consequently, the scatterers at x give
rise to
Z
(3)
c¡2 (t, x) = c0¡2 + qv (x ¡ vt)d3 v:
We note that the physical interpretation of qv
involves a choice of time origin. The choice that is
particularly appropriate, in view of our assumption
about linear target velocities, is a time during which
WANG, ET AL.: MULTISTATIC RADAR IMAGING OF MOVING TARGETS
231
Fig. 2. Sensing geometry. Antennas are fixed and illuminate a scene in which there are multiple moving targets.
the wave is interacting with targets of interest. This
implies that the activation of the antenna at y takes
place at a negative time which we denote by ¡Ty ; as
mentioned above, we write s̃(t, x) = sy (t + Ty )±(x ¡ y).
The wave equation corresponding to (3) is then
¸
·
Z
2
¡2 2
3
2
r ¡ c0 @t ¡ qv (x ¡ vt)d v @t Ã(t, x)
= sy (t + Ty )±(x ¡ y):
(4)
From this point on, we drop the subscript on c0 , so
that c denotes the speed of light in vacuum.
In the absence of scatterers, the field from the
antenna is
sy (t + Ty ¡ jx ¡ yj=c)
à in (t, x, y) = ¡
:
(5)
4¼jx ¡ yj
(Details are given in the Appendix.)
We write à = à in + à sc and neglect multiple
scattering (i.e., use the weak-scatterer model) to obtain
the expression for the scattered field à sc :
Z
Z
±(t ¡ t0 ¡ jz ¡ xj=c)
à sc (t, z, y) =
qv (x ¡ vt0 )d3 v @t20
4¼jz ¡ xj
sy (t0 + Ty ¡ jx ¡ yj=c) 3
£
d x dt0 :
4¼jx ¡ yj
(6)
(For details of the derivation, see the Appendix.)
Many radar systems use a narrowband waveform,
which in our case is defined by
sy (t) = ay (t)e¡i!y t
(7)
where ay (t) is slowly varying, as a function of t, in
comparison with exp(¡i!y t), where !y is the carrier
frequency for the transmitter at position y. For such
waveforms, the second time derivative of sy obeys
s̈y (t) ¼ ¡!y2 sy (t). (For a justification of this and the
other approximations made here, see the Appendix;
for a derivation without this approximation, see [43].)
232
In (6), we make the change of variables x 7! x0 =
x ¡ vt0 , (i.e., change the frame of reference to one in
which the scatterer qv is fixed) to obtain
Z
Z
0
0
0
0
à sc (t, z, y) = ¡
±(t ¡ t ¡ jx + vt ¡ zj=c)
4¼jx0 + vt0 ¡ zj
qv (x )
4¼jx0 + vt0 ¡ yj
£ !y2 sy (t0 + Ty ¡ jx0 + vt0 ¡ yj=c)d 3 x0 d 3 v dt0 :
(8)
The physical interpretation of (8) is as follows: The
wave that emanates from y at time ¡Ty encounters a
target at time t0 ; this target, during the time interval
[0, t0 ], has moved from x0 to x0 + vt0 ; the wave scatters
with strength qv (x0 ), and then propagates from
position x0 + vt0 to z, arriving at time t. Henceforth we
drop the primes on x.
Next we assume that the target is slowly moving.
More specifically, we assume that jvjt and kjvj2 t2
are much less than jx ¡ zj and jx ¡ yj, where k =
!max =c, with !max the (effective) maximum angular
frequency of all the signals sy . In this case, expanding
jz ¡ (x + vt0 )j around t0 = 0 yields
jz ¡ (x + vt0 )j ¼ Rx,z + R̂x,z ¢ vt0
(9)
where Rx,z = x ¡ z, Rx,z = jRx,z j, and R̂x,z = Rx,z =Rx,z .
We can carry out the t0 integration in (8) as
follows: we make the change of variables
t00 = t ¡ t0 ¡ (Rx,z + R̂x,z ¢ vt0 )=c
(10)
which has the corresponding Jacobian
¯ 0¯
¯ @t ¯
1
1
¯
¯
:
¯ @t00 ¯ = j@t00 =@t0 j =
1 + R̂ ¢ v=c
x,z
We denote the denominator of this Jacobian as ¹z,v .
The argument of the delta function in (8) contributes
only when t00 = 0; (10) with t00 = 0 can be solved for t0
to yield
t ¡ Rx,z =c
:
(11)
t0 =
1 + R̂x,z ¢ v=c
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 48, NO. 1
JANUARY 2012
The argument of sy in (8) is then
jx + vt0 ¡ yj
c
Ã
!
Rx,y R̂x,y ¢ v 0
0
¼ t + Ty ¡
+
t
c
c
t0 + Ty ¡
=
μ
¶
Rx,y
Rx,z
t¡
¡
+ Ty (12)
c
c
¢ v=c
1 ¡ R̂x,y ¢ v=c
1 + R̂x,z
where we have used (9) and (11). The quantity
®x,v =
1 ¡ R̂x,y ¢ v=c
1 + R̂x,z ¢ v=c
¼ 1 ¡ (R̂x,y + R̂x,z ) ¢ v=c (13)
is the Doppler scale factor. If we write the Doppler
scale factor (13) as ®x,v ¼ 1 + ¯x,v , where
¯x,v = ¡(R̂x,y + R̂x,z ) ¢ v=c
(14)
then the quantity !y ¯x,v is the (angular) Doppler shift.
We note that the Doppler shift depends on the bistatic
bisector vector R̂x,y + R̂x,z (see Fig. 3).
With (12) and the notation of (13), (8) becomes
à sc (t, z, y)
=
¡!y2
Z
sy [®x,v (t ¡ Rx,z =c) ¡ Rx,y =c + Ty ]
(4¼)2 Rx,z Rx,y ¹z,v
qv (x)d3 x d 3 v:
(15)
We use the narrowband assumption (7) to write the
numerator of (15) as
sy [®x,v (t ¡ Rx,z =c) ¡ Rx,y=c + Ty ]
¼ ay (t ¡ Rx,z =c ¡ Rx,y =c + Ty )
£ expf¡i!y [(1 + ¯x,v )(t ¡ Rx,z =c) ¡ Rx,y =c + Ty ]g
(16)
where we have used the fact that ay is slowly varying
to replace ®x,v by 1. With (16), (15) becomes
Z
!y2 ei'x,v e¡i!y ®x,v t
à sc (t, z, y) ¼ ¡
(4¼)2 Rx,z Rx,y ¹z,v
£ ay (t + Ty ¡ (Rx,z + Rx,y )=c)qv (x)d3 x d3 v:
(17)
Here we have collected the time-independent terms in
the exponent into the quantity
'x,v = ky [Rx,y ¡ Ty + ®˜ x,v Rx,z ]
(18)
where ky = !y =c.
We recognize (17) as the superposition of
time-delayed and Doppler-shifted copies of the
transmitted waveform. The key feature of this model
is that it correctly incorporates the positions of the
transmitters and receivers, the transmitted waveform,
and the target position and velocity. In particular, the
Fig. 3. Bistatic range (sum of distances Rx,z and Rx,y ), and
bistatic bisector R̂x,y + R̂x,z .
time delay and Doppler shift depend correctly on the
target and sensor positions and on the target velocity.
III. IMAGING
We now address the question of extracting
information from the scattered waveform described
by (15).
Our image formation algorithm depends on
the data we have available. In general, we form a
(coherent) image as a filtered adjoint or weighted
matched filter [44], but the variables we integrate over
depend on whether we have multiple transmitters,
multiple receivers, and/or multiple frequencies.
The number of variables we integrate over, in turn,
determines how many image dimensions we can hope
to reconstruct. If our data depend on two variables,
for example, then we expect to be able to reconstruct
a two-dimensional scene; in this case the scene is
commonly assumed to lie on a known surface. If,
in addition, we want to reconstruct two-dimensional
velocities, then we are seeking a four-dimensional
phase-space image and thus we expect to need data
depending on at least four variables. More generally,
the problem of determining a distribution of positions
in space and the corresponding three-dimensional
velocities means that we seek to form an image in a
six-dimensional phase space.
A. Imaging Formula
We form an image Iu (p) of the objects with
velocity u that, at time t = 0, were located at position
p. Thus Iu (p) is constructed to be an approximation
to qu (p).
We form an image by matched filtering and
summing over all transmitters and receivers. From
(17) with qv (x) = ±(x ¡ p)±(v ¡ u), we see that the
predicted signal from a single point target at the
phase-space position (p, u) (i.e., a target at position
p traveling with velocity u) is
¡
WANG, ET AL.: MULTISTATIC RADAR IMAGING OF MOVING TARGETS
!y2 ei'p,u e¡i!y ®p,u t
a (t + Ty ¡ (Rp,z + Rp,y )=c):
(4¼)2 Rp,z Rp,y ¹z,u y
(19)
233
We denote the transmitters by y1 , y2 , : : : , yM , and the
receivers by z1 , z2 , : : : , zN . To obtain a phase-space
image Iu (p), we apply the (weighted) matched filter
corresponding to the phase-space location (p, u) to the
received signal à sc :
N X
M Z
X
In the second exponential of (22) we use (18) to
write
'x,v ¡ 'p,u = kym [Rx,ym ¡ Tym + (1 + ¯x,v )Rx,zn ]
¡ kym [Rp,ym ¡ Tym + (1 + ¯p,u )Rp,zn ]
= kym [¡c¢¿ym ,zn + ¯x,v Rx,zn ¡ ¯p,u Rp,zn ]
e¡i'p,u ei!ym ®p,u t Rp,zn Rp,ym ¹zn ,u
Iu (p) = (4¼)2
n=1 m=1
(23)
where
£ a¤ym (t + Tym ¡ (Rp,zn + Rp,ym )=c)Ã sc (t, zn , ym )Jn,m dt
c¢¿ym ,zn = [Rp,zn + Rp,ym ] ¡ [Rx,zn + Rx,ym ]
(24)
(20)
where Jn,m is a geometry-dependent weighting
function that can be inserted to improve the image
[44]. The weights Rp,z Rp,y ¹z,u are included so that they
will cancel the corresponding factors in the data when
p = x and u = v.
In allowing the imaging operation (20) to depend
on the transmitter ym , we are implicitly assuming
that, at the receiver located at zn , we can identify
the part of the signal à sc (t, zn , ym ) that is due to the
transmitter at ym . In the case of multiple transmitters,
this identification can be accomplished, for example,
by having different transmitters operate at different
frequencies, or possibly by quasi-orthogonal
pulse-coding schemes. We are also assuming a
coherent system, i.e., that a common time clock is
available to all sensors, so that we are able to form
the image (20) with the correct phase relationships.
The operation (20) amounts to geometry-corrected
and phase-corrected matched filtering with a
time-delayed, Doppler-shifted version of the
transmitted waveform, where the Doppler shift is
defined in terms of (14).
is the difference in bistatic ranges for the points x
and p. If in (22) we make the change of variables
t 7! t0 = t + Tym ¡ (Rx,zn + Rx,ym )=c and use the notation
(26), then we obtain
B. Image Analysis
We note that the bistatic ambiguity function of
(25) depends on the difference of bistatic ranges (24)
and on the difference between the velocity projections
onto the bistatic bisectors ¯p,u ¡ ¯x,v = (R̂x,ym + R̂x,zn ) ¢
v=c ¡ (R̂p,ym + R̂p,zn ) ¢ u=c.
For the case of a single transmitter and single
receiver (a case for which J1,1 = 1) and a point
target (which enables us to choose Tym so that the
exponential in the third line of (25) vanishes), and
leaving aside the magnitude and phase corrections,
(25) reduces to a coordinate-independent version of
the ambiguity function of [38].
In order to characterize the behavior of this
imaging system, we substitute (17) into (20),
obtaining
Z
K(p, u; x, v)qv (x)d3 x d3 v
Iu (p) =
(21)
where K, the PSF is
K(p, u; x, v) = ¡
N X
M Z
X
!y2m a¤ym
n=1 m=1
£ (t + Tym ¡ (Rp,zn + Rp,ym )=c)ei!ym [¯p,u ¡¯x,v ]t
£ aym (t + Tym ¡ (Rx,zn + Rx,ym )=c)ei['x,v ¡'p,u ]
£
Rp,zn Rp,ym ¹zn ,u
J dt:
Rx,zn Rx,ym ¹zn ,v n,m
(22)
The PSF characterizes the behavior of the imaging
system in the sense that it contains all the information
about how the true phase-space reflectivity distribution
qv (x) is related to the image Iu (p). If qv (x) is
a delta-like point target at (x, v), then the PSF
K(p, u; x, v) is the phase-space image of that target.
234
K(p, u; x, v) = ¡
M
N X
X
n=1 m=1
!y2m Aym (!ym [¯p,u ¡ ¯x,v ], ¢¿ym ,zn )
£ expf¡ikym ¯p,u [Rx,zn ¡ Rp,zn ]g
£ expfi!ym [¯p,u ¡ ¯x,v ][Rx,ym =c ¡ Tym ]g
£
Rp,zn Rp,ym ¹zn ,u
J
Rx,zn Rx,ym ¹zn ,v n,m
where
˜ ¿ ) = e¡i!y ¿
Ay (!,
Z
(25)
˜
dt
a¤y (t ¡ ¿ )ay (t)ei!t
(26)
is the radar ambiguity function for the waveform ay .
(Note that this definition includes a phase.) Equation
(25) says the following:
The multistatic phase-space PSF is a weighted
coherent sum of radar ambiguity functions evaluated
at appropriate bistatic ranges and bistatic velocities.
C.
Examples of the Point-Spread Function
The PSF contains all the information about the
performance of the imaging system. Unfortunately
it is difficult to visualize this PSF because it depends
on so many variables. In the case when the positions
and velocities are restricted to a known plane, the PSF
is a function of four variables.
We would like to know whether we can find
both the position and velocity of moving targets.
Ideally, the PSF is delta-like, and so we can obtain
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 48, NO. 1
JANUARY 2012
Fig. 4. This shows how Figs. 5—7 display the four-dimensional
PSF (27).
both position and velocity. If, however, the PSF is
ridge-like, then there will be uncertainty in some
directions or in some combination of positions and
velocities.
In order to look for possible ridge-like behavior,
we write the PSF as
K(p, u;x, v) = K(jpj(cos μ, sin μ), juj(cos Á, sin Á), x, v):
(27)
We plot the PSF for a fixed target position x and
target velocity v. We then sample μ and Á at intervals
of ¼=4, and for each choice of μ and Á, we plot
jpj versus juj. This process results in 9 £ 9 = 81
plots of jpj versus juj. Finally, to show the entire
four-dimensional space at a glance, we display all the
81 plots simultaneously on a grid, arranged as shown
in Fig. 4.
1) Simulation Parameters: Our strategy in the
simulations is to use a delta-like ambiguity function,
and investigate the effect of geometry on the overall
PSF. In all cases, we use a transmit time of Ty = 0.
a) Waveform: In the simulations, we use a
high-Doppler-resolution, linearly-frequency-modulated
(LFM) pulse train. The ambiguity function of this
pulse train has a “bed-of-nails” appearance [32],
with a delta-like central peak that has both velocity
resolution and range resolution that are good enough
for many applications. We assume that the extraneous
peaks of the ambiguity function occur in a region of
space and velocity where no targets are present.
In our simulations, we choose the duration of
each pulse to be Tp = 10 £ 10¡6 s, the pulse period
to be TR = 10¡4 s, and take N = 50 pulses in the pulse
train so that the duration of the entire pulse train is
T = NTR = 5 £ 10¡3 s. The bandwidth B of the pulse
train is chosen to be B = 3 £ 106 Hz. The FM rate
is B=Tp . We take the center frequency to be 30 GHz,
which corresponds to a wavelength of ¸ = 0:01 m.
The ambiguity function of this pulse train has the
following properties.
¡1
1) The Doppler resolution is 1=T = 200 s , which
in the ordinary monostatic case would correspond to
down-range velocity resolution of ¢V = ¸=(2T) =
1 m/s.
2) The first ambiguous Doppler value is 1=TR =
104 s¡1 , which would correspond to a monostatic
velocity ambiguity of ¢Vmax = ¸=(2TR ) = 50 m/s.
3) The delay resolution is 1=B ¼ :3 £ 10¡6 s,
which would correspond to a monostatic range
resolution of ¢r = c=(2B) = 50 m.
4) The first ambiguous delay is TR = 10¡4 s, which
corresponds to an ambiguous (monostatic) range
of ¢rmax = cTR =2 = 1:5 £ 104 m. This would be the
maximum measurable unique range.
b) Area of interest: The area of interest is a
circular region with radius of 1000 m; points within
this region differ by no more than 2000 m, which
is well within the unambiguous range of 15000 m.
We assume that there are no scatterers outside the
region of interest. The location of every point in the
region is denoted by the vector p = jpj(cos μ, sin μ).
The directions μ are sampled at intervals of ¼=4, while
the lengths r are sampled at intervals of 25 m.
c) Velocities of Interest: The velocities are written
u = juj(cos Á, sin Á). We consider velocity magnitudes
in the interval [0, 30] m/s. The magnitudes s are
sampled at intervals of 0.5 m/s and the directions Á
are sampled at intervals of ¼=4.
2) Examples:
a) Single transmitter, two receivers: The PSF for a
single transmitter and two receivers is shown in Fig. 5.
Here the transmitter is located at y = (0, ¡10000) m,
the receivers are located at z1 = (10000, 0) m and
z2 = (¡10000, 0) m, the scene of interest is a disk
centered at the origin with radius 1000 m, the target
location is 225(cos 45± , sin 45± ) m, and the target
velocity is 20(cos 0, sin 0) m/s.
We see that for this geometry, the PSF is indeed
ridge-like.
b) Circular geometry: Fig. 6 shows the PSF
for a circular arrangement of 8 transmitters and 10
receivers. The transmitters are equally spaced around
a circle of radius 10000 m; the receivers are equally
spaced around a circle of radius 9000 m. The scene
of interest has radius 1000 m. The true target location
is 225(cos 180± , sin 180± ) m and the true velocity is
20(cos 180± , sin 180± ) m/s. For this geometry, there
appear to be no ambiguities.
c) Linear array: Fig. 7 shows the PSF for a
linear array of 11 transmitters and a single receiver.
In this case, the transmitters are equally spaced along
the line ¡5000 m · x · 5000 m, y = 10000 m; the
receiver is located at (0, 10000) m; the true target
location is 225(cos 135± , sin 135± ) m and the target
velocity is 20(cos 45± , sin 45± ) m/s.
IV. REDUCTION TO FAMILIAR SPECIAL CASES
The formula (25) reduces correctly to a variety of
special cases. More details can be found in [43].
A. Range-Doppler Imaging
In classical range-Doppler imaging, the range
and velocity of a moving target are estimated from
measurements made by a single transmitter and
WANG, ET AL.: MULTISTATIC RADAR IMAGING OF MOVING TARGETS
235
Fig. 5. This shows geometry (not to scale) for one transmitter and two receivers, together with combined PSF.
coincident receiver. Thus we take z = y, J = 1, and
remove the integrals in (25) to show that the PSF
is simply proportional to the classical ambiguity
function.
1) Range-Doppler Imaging of a Rotating Target:
If the target is rotating about a point, then we can
use the connection between position and velocity to
form a range-Doppler target image. For the case of
turntable geometry, where for a point (x1 , x2 , 0), the
_ , with μ_
range is x1 and the down-range velocity is μx
2
the target rotation rate, (25) reduces to the classical
Ã
!
_
!0 μ(p
(p1 ¡ x1 )
2 ¡ x2 )
KRD (p, u; x, v) / Ay 2
,2
c
c
(28)
where !0 is the center frequency.
B. Inverse Synthetic-Aperture Radar
ISAR systems rely on the target’s rotational
motion to establish a pseudoarray of distributed
receivers, from which range measurements are
made. To show how ISAR follows from the results
of this paper, we fix the coordinate system to the
rotating target. Succeeding pulses of a pulse train
236
are transmitted from and received at positions z = y
that rotate around the target as ŷ = ŷ(μ). The pulses
typically used in ISAR are high range-resolution
(HRR) ones, so that each pulse yields a ridge-like
ambiguity function Ay (º, ¿ ) = Āy (¿ ), with Āy (¿ )
sharply peaked around ¿ = 0. Using this ambiguity
function in (25) shows that the imaging process
corresponds to backprojection [43].
C.
Synthetic-Aperture Radar
SAR also uses HRR pulses, transmitted and
received at locations along the flight path y =
y(³), where ³ denotes the flight path parameter.
For a stationary scene, u = v = 0, which implies
that ¯u = ¯v = 0. In this case, (25) corresponds to
backprojection over circles [45]. See Fig. 8.
D. Doppler SAR
Doppler SAR uses a high Doppler-resolution
waveform, such as an approximation to the ideal
single-frequency waveform s(t) = exp(¡i!0 t)
transmitted from locations along the flight path y =
y(³). If we transform to the sensor frame of reference,
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 48, NO. 1
JANUARY 2012
Fig. 6. This shows geometry (not to scale) for 8 transmitters and 10 receivers arranged in a circle and corresponding combined PSF.
then the entire scene is moving with velocity u = v =
_
¡y(³)
(see Fig. 9).
For a high Doppler-resolution waveform,
Ay (º, ¿ ) = Āy (º), where Āy (º) is sharply peaked around
º = 0. Using this waveform in (25) shows that the
imaging process corresponds to backprojection over
hyperbolas, as detailed in [36].
E. SAR for Other Ridge-like Ambiguity Functions
In the above scenario, SAR reconstruction can
be performed with a sensor moving along the path
y = y(³) while transmitting other waveforms. In
particular, waveforms (such as chirps) can be used
which produce a ridge-like ambiguity function along
any line º = ´¿ in the delay-Doppler plane [32]. Then
(25) corresponds to backprojection over the planar
curves Lp (³):
_
+ ´Rp,y(³) :
Lp (³) = ¡!0 R̂p,y(³) ¢ y(³)
(29)
The curve Lp (³) is the intersection of the imaging
plane with a surface generated by rotating a limaçon
of Pascal (see Fig. 10) about the flight velocity
vector. We note that for a slowly-moving sensor
and a side-looking beam pattern, the curve obtained
from the limaçon is almost indistinguishable from a
circle.
F. Moving Target Tomography
Moving target tomography has been proposed
in [23], which models the signal using a simplified
version of (17). For this simplified model, our
imaging formula (20) reduces to the approach
advocated in [23], namely matched-filter processing
with a different filter for each location p and for each
possible velocity u.
V.
CONCLUSIONS AND FUTURE WORK
We have developed a linearized imaging theory
that combines the spatial, temporal, and spectral
aspects of scattered waves.
This imaging theory is based on the general
(linearized) expression we derived for waves scattered
from moving objects, which we model in terms of a
distribution in phase space. We form a phase-space
image as a weighted matched filter.
WANG, ET AL.: MULTISTATIC RADAR IMAGING OF MOVING TARGETS
237
Fig. 7. This shows geometry (not to scale) for linear array of 11 transmitters and single receiver and corresponding combined PSF.
Fig. 8. SAR scenario: Stationary scene and sensor that (in start-stop approximation) does not move as it transmits and receives each
pulse. Between pulses, sensor moves to different location along flight path.
The theory allows for activation of multiple
transmitters at different times, but the theory is
simpler when they are all activated so that the waves
arrive at the target at roughly the same time.
238
The general theory in this paper reduces to familiar
results in a variety of special cases. We leave for the
future further analysis of the PSF. We also leave for
the future the case in which the sensors are moving
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 48, NO. 1
JANUARY 2012
Fig. 9. For Doppler SAR, we use frame of reference in which sensor is fixed and plane below is moving with uniform velocity.
Fig. 10. A limaçon of Pascal. Origin is located at transmitter position y.
independently, and the problem of combining these
ideas with tracking techniques.
APPENDIX
A. Details of the Derivation of (5) and (6)
1) The Transmitted Field: We consider the
transmitted field à in in the absence of any scatterers;
à in satisifies the version of (1) in which c is simply
the constant background speed c0 , namely
[r2 ¡ c0¡2 @t2 ]Ã in (t, x, y) = sy (t + Ty )±(x ¡ y):
(30)
Henceforth we drop the subscript on c.
We use the Green’s function g, which satisfies
WANG, ET AL.: MULTISTATIC RADAR IMAGING OF MOVING TARGETS
[r2 ¡ c¡2 @t2 ]g(t, x) = ¡±(t)±(x)
239
and is given by
g(t, x) =
±(t ¡ jxj=c)
:
4¼jxj
(31)
For the validity of (16), we use the Taylor
expansion of ay about the time (t ¡ Rx,z =c) ¡ tx , where
tx = Rx,y =c ¡ Ty :
ay [®x,v (t ¡ Rx,z =c) ¡ tx ]
We use (31) to solve (30), obtaining
Z
à in (t, x, y) = ¡ g(t ¡ t0 , jx ¡ yj)sy (t0 + Ty )±(x ¡ y)dt0 dy
sy (t + Ty ¡ jx ¡ yj=c)
=¡
:
4¼jx ¡ yj
(32)
2) The Scattered Field: We write à = à in + Ã̃ sc ,
which converts (1) into
Z
(33)
[r2 ¡ c¡2 @t2 ]Ã̃ sc = qv (x ¡ vt)d3 v @t2 Ã:
The map q 7! Ã̃ sc can be linearized by replacing the
full field à on the right side of (33) by à in . (This
is the Born or single-scattering approximation.) The
resulting differential equation we solve with the help
of the Green’s function; the result is
Z
à sc (t, z) = ¡ g(t ¡ t0 , jz ¡ xj)
Z
£ qv (x ¡ vt0 )d3 v @t20 Ã in (t0 , x)dt0 d3 x
(34)
which is (6).
B. Validity of Approximations used to Obtain the
Signal Model (17)
= ay [(t ¡ Rx,z =c) ¡ tx ] + a_ y [(t ¡ Rx,z =c) ¡ tx ]
£ (®x,v ¡ 1)(t ¡ Rx,z =c):
Since ®x,v ¡ 1 = O(v=c), we see that (16) is valid when
a_ y is bounded and jv=cj ¿ 1.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
1) Narrowband Waveforms: For a narrowband
waveform
sy (t, y) = ay (t, y)e¡i!y t
(35)
[7]
the second time derivative of sy is
s̈y (t) = ¡!y2 sy (t) ¡ 2i!y a_ y e¡i!y t + äy e¡i!y t
Ã
!
2ia_ y äy
2 ¡i!y t
= ¡!y e
+ 2 :
ay ¡
!y
!y
[8]
(36)
We see that the approximation s̈y (t) ¼ ¡!y2 sy (t) is
valid when a_ y =!y and äy =!y2 are negligible compared
with ay .
2) Slow-Mover Approximation: We expand R(t0 ) =
jz ¡ (x + vt0 )j around t0 = 0, yielding
0
_
+ O((t0 )2 ):
R(t ) = R(0) + R(0)t
0
[9]
[10]
(37)
Dividing by c, we obtain
0 ¯ ¯2 1
¯ v 0¯
t
v 0
jz ¡ (x + vt )j Rx,z
B¯c ¯ C
=
+ R̂x,z ¢ t + O @
A:
c
c
c
Rx,z =c
0
[11]
(38)
0
Thus we see that (9) is valid when jv=cjt ¿ 1; this is
also the same condition needed for validity of (12)
and (13).
240
(39)
[12]
Skolnik, M. I.
Radar Handbook (3rd ed.).
New York: McGraw-Hill Professional, 2008.
Brennan, L. E. and Reed, I. S.
Theory of adaptive radar.
IEEE Transactions on Aerospace and Electronic Systems,
AES-9 (Mar. 1973), 237—252.
Raney, R. K.
Synthetic aperture imaging radar and moving targets.
IEEE Transactions on Aerospace and Electronic Systems,
AES-7, 3 (May 1971), 499—505.
Ward, J.
Space-time adaptive processing for airborne radar.
MIT Lincoln Laboratory, Lexington, MA, Technical
Report 1015, 1994.
Klemm, R.
Principles of Space-Time Adaptive Processing.
London: IEE Publishing, 2002.
Cooper, J.
Scattering of electromagnetic fields by a moving
boundary: The one-dimensional case.
IEEE Transactions on Antennas and Propagation, AP-28, 6
(1980), 791—795.
Werness, S., et al.
Moving target imaging for SAR data.
IEEE Transactions on Aerospace and Electronic Systems,
26, 1 (Jan. 1990), 57—66.
Yang, H. and Soumekh, M.
Blind-velocity SAR/ISAR imaging of a moving target in
a stationary background.
IEEE Transactions on Image Processing, 2, 1 (Jan. 1993),
80—95.
Barbarossa, S.
Detection and imaging of moving objects with synthetic
aperture radar–Part 1: Optimal detection and parameter
estimation theory.
Proceedings of the IEE-F, Radar and Signal Processing,
139 (Feb. 1992), 79—88.
Barbarossa, S.
Detection and imaging of moving objects with synthetic
aperture radar–Part 2: Joint time-frequency analysis by
Wigner-Ville distribution.
Proceedings of the IEE-F, Radar and Signal Processing,
139 (Feb. 1992), 89—97.
Friedlander, B. and Porat, B.
VSAR: A high resolution radar system for detection of
moving targets.
IEE Proceedings on Radar, Sonar and Navigation, 144
(1997), 205—218.
Perry, R. P., Dipietro, R. C., and Fante, R. L.
SAR imaging of moving targets.
IEEE Transactions on Aerospace and Electronic Systems,
35, 1 (Jan. 1999), 188—200.
IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 48, NO. 1
JANUARY 2012
[13]
Jao, J. K.
Theory of synthetic aperture radar imaging of a moving
target.
IEEE Transactions on Geoscience and Remote Sensing, 39,
9 (Sept. 2001), 1984—1992.
[28]
Landi, L. and Adve, R. S.
Time-orthogonal-waveform-space-time adaptive
processing for distributed aperture radars.
In Proceedings of the 2007 International Waveform
Diversity and Design Conference, 2007, 13—17.
[14]
Fienup, J. R.
Detecting moving targets in SAR imagery by focusing.
IEEE Transactions on Aerospace and Electronic Systems,
37, 3 (July 2001), 794—809.
[29]
Capraro, G. T., et al.
Waveform diversity in multistatic radar.
In Proceedings of the International Waveform Diversity and
Design Conference, Lihue, HI, Jan. 2006.
[15]
Dias, J. M. B. and Marques, P. A. C.
Multiple moving target detection and trajectory estimation
using a single SAR sensor.
IEEE Transactions on Aerospace and Electronic Systems,
39, 2 (Apr. 2003), 604—624.
[30]
Cook, C. E. and Bernfeld, M.
Radar Signals.
New York: Academic, 1965.
[31]
Woodward, P. M.
Probability and Information Theory, with Applications to
Radar.
New York: McGraw-Hill, 1953.
[32]
Levanon, N.
Radar Principles.
Hoboken, NJ: Wiley, 1998.
[33]
Franceschetti, G. and Lanari, R.
Synthetic Aperture Radar Processing.
Boca Raton, FL: CRC Press, 1999.
[16]
Kircht, M.
Detection and imaging of arbitrarily moving targets with
single-channel SAR.
IEE Proceedings on Radar, Sonar and Navigation, 150, 1
(Feb. 2003), 7—11.
[17]
Pettersson, M. I.
Detection of moving targets in wideband SAR.
IEEE Transactions on Aerospace and Electronic Systems,
40 (2004), 780—796.
[34]
[18]
Minardi, M. J., Gorham, L. A., and Zelnio, E. G.
Ground moving target detection and tracking based on
generalized SAR processing and change detection.
Proceedings of SPIE, vol. 5808, 2005, 156—165.
Willis, N. J.
Bistatic Radar.
Norwood, MA: Artech House, 1991.
[35]
Barbarossa, S. and Farina, A.
Space-time-frequency processing of synthetic aperture
radar signals.
IEEE Transactions on Aerospace and Electronic Systems,
30, 2 (Apr. 1998), 341—358.
Willis, N. J.
Bistatic radar.
In M. I. Skolnik, Radar Handbook (2nd ed.), New York:
McGraw-Hill, 1990.
[36]
[20]
Soumekh, M.
Fourier Array Imaging.
Upper Saddle River, NJ: Prentice-Hall, 1994.
Borden, B. and Cheney, M.
Synthetic-aperture imaging from high-Doppler-resolution
measurements.
Inverse Problems, 21 (2005), 1—11.
[37]
[21]
Soumekh, M.
Synthetic Aperture Radar Signal Processing with MATLAB
Algorithms.
Hoboken, NJ: Wiley-Interscience, 1999.
Willis, N. J. and Griffiths, H. D.
Advances in Bistatic Radar.
Raleigh, NC: SciTech Publishing, 2007.
[38]
Tsao, T., et al.
Ambiguity function for a bistatic radar.
IEEE Transactions on Aerospace and Electronic Systems,
33 (1997), 1041—1051.
[22]
Stuff, M., et al.
Imaging moving objects in 3d from single aperture
synthetic aperture data.
In Proceedings of the IEEE Radar Conference, 2004,
94—98.
[39]
Colton, D. and Kress, R.
Inverse Acoustic and Electromagnetic Scattering Theory.
New York: Springer, 1992.
[40]
Devaney, A. J.
Inversion formula for inverse scattering within the Born
approximation.
Optics Letters, 7 (1982), 111—112.
[41]
Devaney, A. J.
A filtered backpropagation algorithm for diffraction
tomography.
Ultrasonic Imaging, 4 (1982), 336—350.
[42]
Varsolt, T., Yazici, B., and Cheney, M.
Wide-band pulse-echo imaging with distributed apertures
in multi-path environments.
Inverse Problems, 24, 4 (Aug. 2008).
[43]
Cheney, M. and Borden, B.
Imaging moving targets from scattered waves.
Inverse Problems, 24 (2008), 1—22.
[44]
Nolan, C. J. and Cheney, M.
Synthetic aperture inversion.
Inverse Problems, 18 (2002), 221—236.
[45]
Nolan, C. J. and Cheney, M.
Synthetic aperture inversion for arbitrary flight paths and
non-flat topography.
IEEE Transactions on Image Processing, 12 (2003),
1035—1043.
[19]
[23]
[24]
[25]
[26]
[27]
Himed, B., et al.
Tomography of moving targets (TMT).
Sensors, Systems, and Next-Generation Satellites,
Proceedings of SPIE, vol. 4540, 2001, 608—619.
Bradaric, I., et al.
Multistatic radar systems signal processing.
In Proceedings of the IEEE Radar Conference, 2006,
106—113.
Bradaric, I., Capraro, G. T., and Wicks, M. C.
Waveform diversity for different multistatic radar
configurations.
In Proceedings of the Asilomar Conference, 2007.
Adve, R. S., et al.
Adaptive processing for distributed aperture radars.
In Proceedings of 1st Annual IEE Waveform Diversity
Conference, Edinburgh, Nov. 2004.
Adve, R. S., et al.
Waveform-space-time adaptive processing for distributed
aperture radars.
In 2005 IEEE International Radar Conference Record,
2005, 93—97.
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241
Ling Wang received the B.S. degree in electrical engineering, and the M.S.
and Ph.D. degrees in information acquirement and processing from Nanjing
University of Aeronautics and Astronautics, Nanjing, China, in 2000, 2003, and
2006, respectively.
In 2003, she joined Nanjing University of Aeronautics and Astronautics,
where she is currently an associate professor in the Department of Information
and Communication Engineering. She was a postdoctoral research associate in the
Department of Mathematical Sciences and Department of Electrical, Computer,
and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY, from
February 2008 to May 2009. Her main research interest is radar imaging.
Margaret Cheney received the B.A. in mathematics and physics from Oberlin
College, Oberlin, OH, in 1976 and the Ph.D. degree in mathematics from Indiana
University, Bloomington, in 1982.
She held positions at Stanford University and Duke University before moving
to Rensselaer Polytechnic Institute, where she is a professor of mathematics.
Most of her work has been in inverse problems, including those that arise in
quantum mechanics, acoustics, and electromagnetic theory. She has been working
specifically in radar imaging since 2001.
Brett Borden received the B.S. degree in physics from the University of
Wisconsin—Madison in 1978 and the Ph.D. degree in physics from the University
of Texas at Austin in 1986.
From 1980 to 2002 he worked in the Research Department of the Naval Air
Warfare Center Weapons Division in China Lake, CA. In 2002, he moved to the
Physics Department at The Naval Postgraduate School.
242
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JANUARY 2012