260 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 1, NO. 4, OCTOBER 2004
Unambiguous SAR Signal Reconstruction From
Nonuniform Displaced Phase Center Sampling
Gerhard Krieger, Member, IEEE, Nicolas Gebert, and Alberto Moreira, Senior Member, IEEE
Abstract—The displaced phase center (DPC) technique will
enable a wide-swath synthetic aperture radar (SAR) with high
azimuth resolution. In a classic DPC system, the pulse repetition
frequency (PRF) has to be chosen such that the SAR carrier moves
just one half of its antenna length between subsequent radar
pulses. Any deviation from this PRF will result in a nonuniform
sampling of the synthetic aperture. This letter derives an inno-
vative reconstruction algorithm and shows that an unambiguous
reconstruction of a SAR signal is possible for nonuniform sam-
pling of the synthetic aperture. This algorithm will also have great
potential for multistatic satellite constellations as well as the dual
receive antenna mode in Radarsat 2 and TerraSAR-X.
Index Terms—Azimuth ambiguities, digital beamforming, dis-
placed phase center antenna (DPCA), high-resolution wide-swath
SAR, interferometry, synthetic aperture radar (SAR).
I. INTRODUCTION
W
IDE-SWATH imaging and high azimuth resolution pose
contradicting requirements on synthetic aperture radar
(SAR) system design. To overcome this fundamental limitation,
several innovative techniques have been suggested that use mul-
tiple receiver apertures to acquire additional samples along the
synthetic aperture. The apertures may be either on a single plat-
form like in the classical displaced phase center (DPC) tech-
nique [1]–[6] or on different platforms [7]–[9] leading to a mul-
tistatic SAR where the size of each individual receiver is re-
duced. Fig. 1 shows one example for each scenario, where a
single transmitter illuminates a wide swath and
subapertures
record simultaneously the scattered signal from the illuminated
footprint. Under ideal conditions, this will allow for a reduction
of the pulse repetition frequency (PRF) by a factor of
without
rising azimuth ambiguities. This reduction of the azimuth sam-
pling rate becomes possible by a coherent combination of the
individual receiver signals where the ambiguous parts of the
Doppler spectra cancel each other. Note that such an ambiguity
suppression can also be regarded as digital beamforming on re-
ceive where nulls in the joint antenna pattern are steered to the
ambiguous zones. For optimum performance, the along-track
displacement of the subapertures
relative to the
first receiver
should be chosen as
PRF
(1)
which will result in a uniform sampling of the received SAR
signal. In this equation, PRF is the pulse repetition frequency
of the transmitter and
is the velocity of the SAR carrier. Note
that for a multistatic constellation the
may be different for
Manuscript received March 24, 2004; revised May 13, 2004.
The authors are with the Microwaves and Radar Institute, German Aerospace
Centre (DLR), D-82234 Wessling, Germany (e-mail: Gerhard.Krieger@dlr.de).
Digital Object Identifier 10.1109/LGRS.2004.832700
Fig. 1. Multiple aperture sampling for a single-platform system (top) and a
distributed satellite array (bottom). The effective (bistatic) phase centers are
shown as squares. Solid squares correspond to samples of the synthetic aperture
for the illustrated transmitter (Tx) and receiver (Rx) positions. The dotted
squares are for previous and subsequent positions assuming an appropriately
chosen PRF.
each receiver, which enables a great flexibility in choosing the
along-track distance between the satellites. In a single-platform
system, all
will be zero. Since the subaperture distance and
the platform velocity are fixed in this case, a specific PRF will
be required
PRF
(2)
where we assume an antenna with
subaperture elements sep-
arated by
. The PRF in a single-platform DPC
system has thus to be chosen such that the SAR platform moves
just one half of the total antenna length between subsequent
radar pulses. However, such a rigid selection of the PRF may
be in conflict with the timing diagram for some incident angles.
It will, furthermore, exclude the opportunity to use an increased
PRF for improved azimuth ambiguity suppression.
II. G
ENERAL APPROACH
In the following, an innovative reconstruction algorithm
will be derived, which allows for an unambiguous recovery
of the Doppler spectrum even for a nonuniform sampling of
the SAR signal. The only requirement is that the samples do
1545-598X/04$20.00 © 2004 IEEE
KRIEGER et al.: UNAMBIGUOUS SAR SIGNAL RECONSTRUCTION 261
Fig. 2. (Left) Reconstruction for multichannel subsampling in case of three channels. (Right) Each reconstruction filter
P
consists of
n
bandpass filters
P
.
not coincide. Such an algorithm has a great potential for any
multiaperture system, be it multistatic with sparsely distributed
receiver satellites or be it a classic DPC antenna (DPCA) system
like Radarsat 2 [5] or TerraSAR-X [6]. The data acquisition in
a multiaperture SAR can be considered as a linear system with
multiple receiver channels, each described by a linear filter
with transfer function (cf. Fig. 2). For such a system,
there exist many powerful theorems in linear systems theory. Of
special importance for the present context is a generalization of
the sampling theorem according to which a bandlimited signal
is uniquely determined in terms of the samples of
the responses
of linear systems with input sampled
at
of the Nyquist frequency [10]. To be valid, the transfer
functions of the linear filters may be selected in a quite general
sense, but not arbitrarily (for details see [10]). A block diagram
for the reconstruction from the subsampled signals is shown in
Fig. 2. The reconstruction consists essentially of
linear filters
that are individually applied to the subsampled signals of
the receiver channels and then superimposed. Each of the recon-
struction filters
can again be regarded as a composition
of
bandpass filters , where . As shown in
[10], the reconstruction filters can be derived from a matrix
consisting of the transfer functions , which have to be
shifted by integer multiples of the PRF in the frequency domain
PRF PRF
.
.
.
.
.
.
.
.
.
PRF PRF
(3)
The reconstruction filters
are then derived from an
inversion of the matrix
as
PRF PRF
PRF PRF
.
.
.
.
.
.
.
.
.
.
.
.
PRF PRF
(4)
III. Q
UADRATIC PHASE APPROXIMATION
In principle, it is possible to use the multichannel SAR signal
model for a complete reconstruction of the scene reflectivity.
However, in order to concentrate on the essential steps, we will
only consider the azimuth modulation in the following deriva-
tion. The azimuth signal of a point target at azimuth time
and slant range for each individual receiver channel may be
described as
(5)
where
and define the envelope of the azimuth signal
arising from the projection of the transmit and receive antenna
patterns on the ground. The phase of the azimuth signal is pro-
portional to the sum of the transmit and receive paths that are
here approximated by the two square roots assuming a straight
flight path with velocity
. A further simplification arises if we
expand (5) in a Taylor series up to the second order that will lead
to the quadratic approximation
(6)
Here, the first exponential describes a constant phase offset
for a given slant range
that is equal for all receivers, while the
second exponential accounts for an additional constant phase
offset that is due to the different along-track displacements be-
tween each individual receiver and the transmitter. The time-
varying azimuth modulation of the bistatic SAR is then given
by the third exponential. By comparing (6) with the point target
response of a monostatic SAR
(7)
it becomes clear that the bistatic azimuth response evolves from
its monostatic counterpart by a time delay
and
a phase shift
if we substitute the antenna
footprints
and by and
. The bistatic antenna footprint is
then given by
(8)
262 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 1, NO. 4, OCTOBER 2004
Fig. 3. Quadratic phase approximation.
Therefore, the multiple aperture system can be regarded as a
monostatic SAR that is followed by additional time and phase
shifts for each receiver channel. This is illustrated in Fig. 3.
Since the joint antenna footprint may be different for each re-
ceiver, we have introduced a separate amplitude weighting of
the azimuth spectrum in each channel, but it is also possible
to separate the joint antenna pattern in two parts, where a first
weighting function in the monostatic SAR response accounts for
a weighting of the Doppler spectrum, which is common to all
receivers, and the second transfer function in each channel ac-
counts for residual deviations between the individual receivers.
These deviations may for example result from different Doppler
centroids due to the different along-track displacements, inaccu-
racies in the antenna pointing, or a mutual coupling between the
antenna subarrays in case of a single-platform DPC system. It
may be expected that such a separation in two transfer functions
will have an impact on the stability of the signal reconstruction,
but this topic deserves further investigation.
IV. A
NALYTIC EXAMPLE
As a simple example for an analytic derivation of the re-
construction filters we consider a bistatic SAR with two re-
ceiver channels having the same azimuth antenna pattern. This
two-channel system may serve as an example for the unam-
biguous recovery of the Doppler spectrum resulting from a SAR
signal recorded by the split antenna technique in TerraSAR-X
and Radarsat 2. Since it is possible to incorporate the transfer
function due to the antenna pattern into the monostatic SAR re-
sponse, the remaining difference between the two channels is
given by the two impulse responses
(9)
TABLE I
S
IMULATION PARAMETERS FOR
DUAL RECEIVE
ANTENNA
(10)
In the frequency domain, these impulse responses correspond to
(11)
(12)
The matrix
is then given by (13), shown at the bottom
of the page, and the inverse matrix
can be derived as in
(14), shown at the bottom of the page. The transfer functions of
the reconstruction filters
and are then
PRF
PRF
PRF
PRF
(15)
PRF PRF
(13)
PRF PRF
PRF
PRF
(14)
KRIEGER et al.: UNAMBIGUOUS SAR SIGNAL RECONSTRUCTION 263
Fig. 4. (Left) Azimuth response for one subaperture and (right) after nonuniform DPCA reconstruction.
Fig. 5. Reconstruction for different intersample offsets. (Left) 100% (uniform sampling with 2.4m/2.4m). (Middle) 33% (1.2m/3.6m). (Right) 5% (0.24m/4.56m).
PRF
PRF.
(16)
The numerator of the reconstruction filters can be regarded as
compensating the different time delays and phase shifts within
each branch, while the conjugate mirror structure in the denomi-
nator is responsible for a cancelation of the ambiguous frequen-
cies within each branch. It becomes also clear that the denomi-
nator will vanish for
and a reconstruction becomes
impossible due to the coinciding samples in azimuth.
V. S
IMULATION RESULTS
In this section, some simulation results for the nonuniform
DPCA reconstruction will be given. As an example, the dual re-
ceive antenna in TerraSAR-X [6] has been chosen. The relevant
system parameters are summarized in Table I.
Fig. 4 shows the results for a reconstruction from nonuniform
DPCA sampling. For better illustration, an extended scatterer
has been chosen in this simulation. Furthermore, independent
white noise has been added to each subaperture signal to account
for differences in the two receiver channels. It becomes clear
that the signal from each channel is strongly ambiguous (left),
while the coherent reconstruction of the two subsampled SAR
signals will provide a efficient ambiguity suppression (right).
Further investigations were carried out to evaluate the sensitivity
of the reconstruction result on the relative sample offset. Fig. 5
shows reconstruction examples for different degrees of nonuni-
formity in the azimuth sampling process. The intersample offset
is given in percent, where 100% corresponds to the optimum
offset with uniform sampling and 0% corresponds to coinciding
samples. The results in Fig. 5 indicate an increasing noise floor
in the reconstructed image for decreasing sampling offsets. Note
that there are no ambiguities even for a strong deviation from
the uniform sampling distance and the noise floor will only be-
come significant in case of a rather strong nonuniform sampling.
Further simulation results of the nonuniform DPC reconstruc-
tion have been presented in [11], where a multistatic satellite
array consisting of three receivers has been assumed (cf. Fig. 1,
bottom). This simulation shows that an unambiguous recon-
struction of the SAR signal is also possible in case of large in-
tersatellite separations and different Doppler centroids for each
receiver.
The reconstruction algorithm has also been tested with real
SAR data acquired by the DLR E-SAR system. Since there is
currently no displaced phase center mode in the E-SAR system,
monostatic data were used in this example. The data, which
have an original sampling frequency
, were filtered with an
ideal lowpass of bandwidth
to get SAR data
oversampled by a factor of 10. Then, the ambiguous inputs to
the two DPCA channels have been formed by taking every 20th
sample for each channel. Selecting adjacent samples for the two
channels will result in a maximum nonuniform sampling, while
a distance of ten samples will yield a uniform sampling. After
filtering with
and coherent summation, the monostatic SAR
processing follows. Fig. 6 compares the reconstruction result for
a maximum nonuniform sampling with the SAR image of one
ambiguous channel. It is clear, that the ambiguities are well sup-
pressed in the image after nonuniform DPCA reconstruction.
VI. D
ISCUSSION
An innovative algorithm for the suppression of azimuth ambi-
guities in case of nonuniform DPCA sampling has been derived.
This reconstruction algorithm can be regarded as a time-variant
digital beamforming on receive, which combines the individual
receiver signals in a linear space–time processing. The algo-
rithm is directly applicable to systems relying on the displaced
264 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 1, NO. 4, OCTOBER 2004
Fig. 6. (Left) Ambiguous image for one channel and (right) reconstruction after nonuniform DPC sampling.
phase center technique, like the Quad Array [2] and HRWS [3]
SAR or the dual receive antenna approaches in TerraSAR-X [6]
and Radarsat 2 [5], thereby avoiding any stringent PRF restric-
tion. The algorithm will also be useful for ambiguity suppres-
sion in a distributed SAR with multiple receiver satellites. How-
ever, any cross-track separation of the receivers will introduce
an additional phase in the received signals, which has to be com-
pensated, e.g., via the simultaneous acquisition of a digital ele-
vation model in case of multiple satellites [12], [13]. The joint
processing of these data will then lead to a nonlinear combina-
tion of cross-track interferometry and digital beamforming on
receive [7]. This combination will enable a distributed aperture
SAR with small antennas to serve a broad spectrum of different
applications [14], [15]. It is clear that such a multifunctional
SAR system will require further and more detailed investiga-
tions. This short letter has to be regarded as a first step in this
direction.
A
CKNOWLEDGMENT
The authors thank the anonymous reviewers for their valuable
comments and suggestions to improve the paper.
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