568 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 10, NO. 3, MAY 2013
A Novel OFDM Chirp Waveform Scheme
for Use of Multiple Transmitters in SAR
Jung-Hyo Kim, Member, IEEE, Marwan Younis, Senior Member, IEEE,
Alberto Moreira, Fellow, IEEE, and Werner Wiesbeck, Fellow, IEEE
Abstract—In this letter, we present a new waveform technique
for the use of multiple transmitters in synthetic aperture radar
(SAR) data acquisition. This approach is based on the principle of
the orthogonal-frequency-division-multiplexing technique. Unlike
multiple subband approaches, the proposed scheme allows the
generation of multiple orthogonal waveforms on common spectral
support and thereby enables to exploit the full bandwidth for
each waveform. This letter introduces the modulation and the
demodulation processing in regard to typical spaceborne SAR
receive signals. The proposed processing techniques are verified
by a simulation for the case of pointlike targets.
Index Terms—Digital beamforming (DBF), multiple-input
multiple-output (MIMO) synthetic aperture radar (SAR), orthog-
onal frequency division multiplexing (OFDM), orthogonal wave-
form, SAR.
I. INTRODUCTION
T
HE USE of multiple transmitters is of a great interest in the
synthetic aperture radar (SAR) community. It provides an
increase of degrees of freedom and dramatically improves SAR
imaging performance when combined with multiple receivers, a
so-called multiple-input multiple-output (MIMO) constellation
[1]. However, the waveform design for multiple transmitters has
been the most important and challenging issue in realizing the
MIMO SAR concept. Particularly for spaceborne SAR, aside
from the orthogonality between waveforms for simultaneous
multiple pulse transmissions, a constant envelope of the wave-
form is desired, considering the high power amplifier (HPA) in
transmitters. The HPAs in spaceborne SAR systems operate in
saturation in order to generate the maximum output power and
to ensure a stable output power level for amplitude variations
in the HPA input signal [2]. A significant envelope variation
of waveform therefore yields a clipping effect on the output
waveform, and the orthogonality can be broken.
This letter presents a novel waveform scheme to meet the
aforementioned requirements, based on the principle of orthog-
onal frequency division multiplexing (OFDM) proposed in [3].
In OFDM systems, an available signal bandwidth is divided into
multiple subbands, which is specified to be narrower than the
Manuscript received April 20, 2012; revised June 29, 2012; accepted
July 16, 2012.
J.-H. Kim, M. Younis, and A. Moreira are with the Microwaves and Radar
Institute, German Aerospace Center (DLR), 82234 Oberpfaffenhofen, Germany
(e-mail: junghyo.kim@dlr.de).
W. Wiesbeck is with the Institute of High Frequency Technique and Elec-
tronics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/LGRS.2012.2213577
channel coherence bandwidth, in order to avoid the frequency-
selective channel effect [4]. The OFDM signaling is also paid
attention for SAR applications, and recently, SAR processing
techniques for OFDM waveforms were introduced in [5] and
[6]. A major problem of the typical OFDM signal (mainly in
spaceborne SAR) is the fast variation of the signal envelope.
We resolve the problem by combining the OFDM principle
with chirp waveforms. Therefore, the basic idea behind the
proposed waveform scheme is to exploit both the orthogonality
of subcarriers and intrinsic characteristics of traditional chirp
waveforms.
The importance of using chirp waveform in the proposed
scheme is emphasized in the following issues: First, one can
easily achieve the constant envelope of time domain wave-
forms aforementioned, which leads to a maximum efficiency
of the transmitter modules of a phased-array antenna. Second,
the chirp spectrum approaches a rectangular shape as the
time–bandwidth product increases so that its spectral efficiency
and signal-to-noise ratio can be maximized [2], [7]. Finally,
one can further exploit the linear frequency–time characteristics
of chirp in signal processing. It makes it easy to combine the
proposed scheme with existing SAR processing algorithms,
such as the dechirp-on-receive technique.
Focusing on a dual transmit antenna scenario, we develop
the novel waveform scheme consisting of the modulation and
demodulation algorithm, based on the conventional OFDM
processing, and validate its potential. Special attention has to be
paid to the demodulation since the classical OFDM demodula-
tion assumes that the delay length of a received signal is shorter
than the length of the cyclic prefix (CP), but a SAR echo signal
is generally much longer than the transmitted pulse length. In
this letter, we present a demodulation s trategy for such a long
SAR signal.
II. OFDM C
HIRP MODULATION
A. Principles
The basic idea behind the proposed technique is to exploit
the orthogonality of discrete frequency components, i.e., sub-
carriers. This means that the orthogonality of waveforms is
independent of the types of input sequences.
Assume the input sequence (spectrum) S[¯p] with N discrete
spectral components, which are separated by 2Δf as shown
in Fig. 1. First, the input sequence S[¯p] is interleaved by N
zeros, which is S
1
[p] in Fig. 1, and then, the interleaved input
sequence is shifted by Δf for the second data sequence S
2
[p].
These data sequences are transformed into the time domain by
the 2N -point inverse discrete Fourier transform (IDFT), which
1545-598X/$31.00 © 2012 IEEE
KIM et al.: NOVEL OFDM CHIRP WAVEFORM SCHEME FOR USE OF MULTIPLE TRANSMITTERS IN SAR 569
Fig. 1. Two orthogonal OFDM chirp waveforms are generated by zero
interleaving and shift of a single chirp spectrum as an input sequence.
is the OFDM modulation. As a result, we obtain two waveforms
modulated by two orthogonal subcarrier sets that are mutually
shifted by Δf . Thus, their demodulation must be performed by
2N-point DFT. It must be emphasized that both the sets contain
2N subcarriers but use only N subcarriers to carry the input
data, respectively.
B. Signal Model
As aforementioned, a chirp signal spectrum is used for the
OFDM modulation in this work. The chirp signal spectrum, as
input complex data, is obtained by
S[¯p]=F{s[n]} = F
exp
j · π · K
r
· (nT
s
)
2
(1)
where s[n] denotes the discrete time samples of a complex
chirp signal with the length of N, F{·}is the Fourier transform
operator, T
s
is the sampling interval, and K
r
is the chirp rate,
which is a ratio between the signal bandwidth B and the chirp
duration T
p
(K
r
= B/T
p
). Using (1), we generate two input
data sequences by the zero interleaving and shift as follows:
S
1
[p]=[S[0], 0,S[1], 0, ··· ,S[N − 1], 0] (2)
S
2
[p]=[0,S[0], 0,S[1], ··· , 0,S[N − 1]] (3)
where p =0, 1, 2,...,2N − 1. Both data sequences contain to-
tal 2N components, respectively. Therefore, they are modulated
by 2N-point IDFT. According to the Cooley–Tukey algorithm,
the DFT/IDFT can be performed by separate transforms with
respect to the odd and even components of the input. Since the
even components in S
1
are zeros, the inverse Fourier transform
of S
1
[p] is equivalent to that of S[¯p]. Substituting ¯p for p/2,the
modulated waveform s
1
is given by
s
1
[n]=
N−1
¯p=0
S[¯p] · exp
j
2π
N
¯pn
(4)
where n =0, 1, 2,...,2N − 1. Since the inverse Fourier trans-
form of S[¯p] is s[n] with the period of N , s
1
[n] can be expressed
by a repetition of s[n] over the length of 2N
s
1
[n]=s[n] · rect
n
N
+ s[n − N ] · rect
n − N
N
. (5)
In the same way, using (p +1)/2=¯p +(1/2), the other
modulated waveform s
2
is derived as
s
2
[n]=exp
j
π
N
n
N−1
¯p=0
S[¯p] · exp
j
2π
N
¯pn
(6)
and using s[n], it can be also described as
s
2
[n]=
s[n] · rect
n
N
+ s[n − N ] · rect
n − N
N
· exp
j
π
N
n
= s
1
[n] · exp
j
π
N
n
. (7)
Fig. 2. Real parts of OFDM chirp waveforms in time domain with 50-MHz
bandwidth and N = 4096. The phase change in s
2
(t) due to the subcarrier
offset must be remarked in comparison with s
1
(t).(a)s
1
(t).(b)s
2
(t).
Fig. 3. Generic schematic of the proposed OFDM demodulator followed
by the polyphase decomposition and parallel matched filters for the range
compression.
By introducing t = nT
s
and Δf =1/2NT
s
, it is noticed that
both waveforms are distinguishable by the subcarrier offset Δf .
These modulated OFDM waveforms are converted to analog
forms by a digital-to-analog converter. The OFDM waveforms
in the time domain are plotted in Fig. 2. Due to the zero inter-
leaving in the spectrum, the peak power level of the waveforms
will be reduced. However, according to Parseval’s theorem [8],
the total energy of the input sequence is conserved.
Using the aforementioned signal model, the OFDM wave-
forms can be also directly produced in the time domain.
The time domain generation is valuable since a band-limited
input sequence can cause an overshoot in time domain
waveforms [8].
III. OFDM SAR S
IGNAL DEMODULATION
This section is dedicated to the development of an OFDM
demodulation algorithm applicable to a received SAR signal,
which is typically much longer than the OFDM pulse length.
According to the classical OFDM demodulation scheme, the
orthogonality of subcarriers can be exploited only if the follow-
ing requirements are strictly fulfilled: synchronization between
the modulator and demodulator in length and no truncation of
the waveforms within IDFT/DFT windows. To meet the r equire-
ments, the demodulation algorithm developed in this section
consists of the segmentation of the echo signal by spatial
filtering (digital beamforming), the circular-shift addition, and
the conventional OFDM demodulation using DFT, as shown
in Fig. 3.
First of all, the spatial filters divide the echo signal into
multiple subsets satisfying Lemma 1, which will be derived in
the next section. Second, the circular-shift addition is carried
out so that 2N samples from each spatial filter are obtained. As
570 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 10, NO. 3, MAY 2013
Fig. 4. Spatial filter pattern parameters and geometric parameters. The wide
pattern indicates the transmit antenna beam illuminating the whole swath, and
the narrow pattern depicts the main lobe of a spatial filter beam.
a conventional OFDM demodulation, the 2N-point DFT block
demodulates the OFDM signals. Consequently, the polyphase
decomposition block easily separates the waveforms. The de-
composed signals are independently compressed by parallel
matched filters.
A. Segmentation: Spatial Filtering [Digital Beamforming
(DBF)]
A careful choice of weighting factors allows the steering of a
sharp and high-gain beam to the corresponding angular region.
This angular sectoring is called spatial filtering, which is an
important functionality of digital beamforming. This process
divides a whole swath into multiple subswaths so that the echo
length of each subswath is significantly reduced. The spatial
filter is time invariant, providing a constant gain over time, and
its pattern only varies with the elevation angle.
A signal model for the spatial filtering is developed on the
flat Earth model. Fig. 4 shows t he geometry used in the signal
model. The ground range from the nadir (x =0)to the swath
center is denoted by x
c
, and the whole swath width is 2x
o
.The
main lobe width of the spatial filter is denoted by Δθ
rx
, and
the boresight angle of the lth spatial filter beam is indicated by
the offset θ
l
from the incident angle θ
inc
. First, only the main
lobe signal r
l
(t) is considered
r
l
(t)=
M
T
j=1
x
f
x
n
C
2way
(θ) · a
j
(x) · s
j
t−
R
tx,j
+R
rx
c
o
dx (8)
where C
2way
(θ) is the two-way antenna pattern for the eleva-
tion angle θ, M
T
is the number of transmit antennas, a
j
(x)
denotes the complex scattering coefficient for the OFDM chirp
waveform s
j
(t), and x
n
and x
f
denote the nearest and the
furthest ground range covered by the main lobe of a spatial
filter at the height of h
o
. The subswath width is then given by
x
f
− x
n
. R
tx,j
and R
rx
are the ranges from the target to the
phase center of the jth transmit antenna and the receive array,
respectively.
In fact, a received signal by a single spatial filter is a
superposition of the main lobe and the sidelobe signals, i.e., the
signals from a desired subswath and the signals from adjacent
other subswaths, respectively. Therefore, it is required that the
spatial filtering sufficiently suppresses the sidelobe signals. Put
simply, the spatial filtered signal is desired to be equal to the
main lobe signal. The performance of the spatial filtering plays
a critical role in the demodulation. A further consideration is
made to the main lobe width since it determines the received
signal length. The main lobe width Δθ
rx
is defined as Δθ
rx
=
θ(x
f
) − θ(x
n
), and the subswath width S
sub
is given by [2]
S
sub
= x
f
− x
n
=Δθ
rx
·
h
o
cos
2
(θ
inc
− θ
l
)
. (9)
The spatial filtering leads to a selection of the subswath,
which corresponds to a specific range of delay times. The delay
length Δτ of a single subswath is proportional to the main
lobe width. Using (9), the relation between Δτ and Δθ
rx
is
formulated as follows:
Δτ =2· S
sub
·
sin(θ
inc
− θ
l
)
c
o
(10)
=2· Δθ
rx
·
h
o
· tan(θ
inc
− θ
l
)
c
o
· cos(θ
inc
− θ
l
)
.
In spite of the spatial filtering, the signal length within the
main lobe is still longer than the required demodulator (DFT)
length 2N . The following section presents a processing step to
convert the spatial filter signal to fit the length 2N.
B. Circular-Shift Addition
The approach presented in this section is to exploit the
periodicity of the discrete Fourier transform, which implies
that a finite segment of DFT is a single period of an infinitely
extended periodic signal [8]. In the case of periodic signals, the
time shift is represented by the phase rotation. If the time shift
reaches the signal period length, the phase rotation via the shift
becomes 2π, and the original and shifted signals are no longer
distinguishable. Therefore, on the premise that the maximum
delay length does not exceed a signal period, any delay of the
periodic signal can be represented within the single period.
Since the OFDM chirp pulse is a combination of two successive
chirps with the length of T
p
(see Fig. 2), the echo delay of
T
p
results in the 2π phase rotation. Therefore, the spatial filter
signal r
l
(t) must satisfy the following strict condition.
Lemma 1: The maximum delay length Δτ of a subswath
must be shorter than T
p
(Δτ<T
p
).
Assuming the maximum delay index of K within a subswath
(K<N), each delay is described as (n − k) · T
s
, where k =
0, 1, 2,...,K − 1. Using these discrete time indices n and k,
the received signal is given in a discrete signal form of
r
l
[n]=
⎛
⎝
M
T
j=1
K−1
k=0
C
2way
[k] · a
j
[k] · s
j
[n − k]
⎞
⎠
·rect
n
2N + K
(11)
where n =0, 1, 2,...,2N + K − 1. Note that the echo signal
length within the subswath is equal to (2N + K) · T
s
which
is the sum of the OFDM pulse length (2T
p
=2NT
s
) and the
delay length (KT
s
). If the first 2N samples are directly taken
by a DFT window, the 2N data samples contain only one entire
waveform, and the rest are truncated by the DFT window. In this
case, even though the IDFT/DFT is synchronized, its result will
be spoiled. To resolve this problem, we introduce an additional
KIM et al.: NOVEL OFDM CHIRP WAVEFORM SCHEME FOR USE OF MULTIPLE TRANSMITTERS IN SAR 571
Fig. 5. Received signals from the subswath with the length (a) before and
(b) after the circular-shift addition.
step, called circular-shift addition. This process brings the same
effect with the CP into the received signal. The signals before
and after the circular-shift addition are shown in Fig. 5(a) and
(b), respectively. Using two successive time windows, each with
lengths of 2N and K, respectively, the signal is divided into
two parts. The second term on the right-hand side is circularly
shifted and added to the first term in order to make the main lobe
signal periodic at each 2N sample. The circular-shift addition
process can be described by the modulo operation
r
l
[n]=
M
T
j=1
K
k=1
C
2way
[k] · a
j
[k] · s
j
[n − k
2N
] (12)
where n =0, 1, 2,...,2N − 1 and ·
2N
denotes the arith-
metic modulo 2N . Therefore, the circular-shift addition reduces
the signal length from 2N + K to 2N [see Fig. 5(b)]. Follow-
ing this, the first 2N samples contain all delayed waveforms so
that no truncation of the signal occurs. This procedure can be
replaced by the residual video phase correction technique [9].
C. Demodulation With 2N-DFT
The traditional demodulation by DFT with the length 2N is
now applicable to the OFDM chirp signals. Since the transmit-
ted signal s
j
[n] is the inverse Fourier transform of its spectrum
S
j
[p], the Fourier transform of (12) is described as
F{r
l
[n]}=
1
2N
2N−1
n=0
M
T
j=1
K−1
k=0
C
2way
[k] · a
j
[k]
·
2N−1
p=0
S
j
[p] · exp
j
2π
2N
(n − k)p
· exp
−j
2π
2N
np
. (13)
Rearranging (13) with respect to indices, the following formula
is obtained:
F{r
l
[n]}=
M
T
j=1
K−1
k=0
C
2way
[k] · a
j
[k] · exp
−j
2π
2N
kp
· S
j
[p]
(14)
where S
j
[p] for j =1, 2 is derived in (2) and (3). Finally, the
demodulated signal is given by
F{r
l
[n]} = H
1
· S
1
[p]+H
2
· S
2
[p] (15)
where H
1
and H
2
denote the channel transfer functions deliv-
ered by the OFDM waveforms s
1
and s
2
, respectively. The
demodulated signal is indeed separable with respect to each
waveform by the polyphase decomposition, which divides the
demodulated signal spectrum into odd and even components.
Hence, each spectrum length is reduced by N. This reduction
never leads to any loss of information since the waveforms s
1
and s
2
are actually carried by N subcarriers, and the other
N subcarrier frequencies are redundant zeros, so as to apply
2N-point DFT in demodulation. The channel transfer functions
H
1
and H
2
can be recovered by the conventional matched
filtering for each waveform
H
1
=
K−1
k=0
C
2way
[k] · a
1
[k] · exp
−j
2π
N
k
p
2
·|S
1
|
2
. (16)
In the same manner, the channel transfer function H
2
is
recovered
H
2
=
K−1
k=0
C
2way
[k] · a
2
[k] · exp
−j
2π
N
k
p
2
· exp
−j
π
N
k
·|S
2
|
2
(17)
whereby the second exponential term accounts for the linear
phase ramp depending on the target delay, which is caused by
the subcarrier spacing Δf . Therefore, there is a linear phase
difference between two images reconstructed by each OFDM
chirp waveform, which can be easily compensated in the time
domain.
IV. V
ERIFICATION
The OFDM chirp waveform scheme is verified by simula-
tions in this section. The simulation is carried out under the
following assumptions: use of the optimum spatial filter and the
simple delay channel (point scatterers). The optimum spatial
filter means that the antenna pattern is ideally covering the
angular range of interest and suppressing the sidelobe signals.
The channel is assumed only to cause delay and to include the
thermal noise. The simulation schematic is shown in Fig. 6.
The input sequence length N is equal to 1024 so that a total of
2048 subcarriers are used for the modulation and demodulation
within the bandwidth of 100 MHz. The modulated OFDM
chirp length 2T
p
is 17.067 μs. Hence, the maximum delay of
each channel never exceeds a T
p
of 8.53 μs. These OFDM
waveforms travel through two different channels, h
1
and h
2
,
and return to the receiver. Both of them are processed by a
single demodulator (receiver).
572 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 10, NO. 3, MAY 2013
Fig. 6. Simulation schematic for verification: Two independent channels and a single demodulator with DBF.
Fig. 7. OFDM demodulated discrete spectra of both waveforms within 3 and
4 MHz. The circular and triangular symbols indicate the spectral components
(subcarriers) of H
1
S
1
and H
2
S
2
, respectively. Δf =58.59 kHz.
Fig. 8. Impulse response functions (range profiles) of channel h
1
and h
2
:
(a) Original channel impulse response functions and (b) the reconstructed
impulse response functions from the OFDM chirp signals.
Fig. 7 presents the spectra of demodulated signals within
a 1-MHz range from 3 to 4 MHz for clarity. The circular
symbols denote the spectral components of H
1
S
1
, and the
triangular symbols indicate them for H
2
S
2
. It is clearly ob-
served that these two spectral components are out of joint, i.e.,
orthogonal.
Fig. 8(a) shows the original channel impulse response func-
tions h
1
and h
2
, defined for this simulation. These channels
contain several impulses, indicating the delay and scattering
coefficient of each target. The reconstructed channel impulse
responses are plotted in Fig. 8(b). Apart from a slight change
of the peak magnitude due to the Fourier transform of the
band-limited signal, both range profiles recovered by the single
demodulator and independent matched filters correspond to the
original channel impulse response functions.
V. C
ONCLUSION
In this letter, we proposed a novel waveform scheme, based
on the principle of OFDM technique combined with conven-
tional chirp waveforms. Regarding a typical spaceborne SAR
scenario, this letter dealt with the demodulation scheme com-
bined with DBF and verified it by simulation. The proposed
OFDM chirp waveforms preserve their orthogonality, as long as
they are processed by the presented demodulation scheme. As
a consequence, we intend to emphasize the following features:
First, the OFDM chirp waveform has the same characteristics of
the original input chirp signal. This aspect will be of a special
interest for the implementation of the MIMO SAR system
with the modern hardware and digital synthesis technologies.
Second, the novel waveform scheme has a high adaptabil-
ity, which implies that the novel waveform scheme can be
easily combined with other waveform schemes, such as the
space–frequency coding [10] or the multidimensional encoding
technique [11]. Finally, owing to the common spectral band, the
scheme shows high potential to use multiple SAR i mages pro-
duced using OFDM chirps for polarimetric and interferometric
applications. Future research will include the performance for
the proposed approach for distributed targets.
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