Wireless Sensor Network, 2009, 3, 152-162
doi:10.4236/wsn.2009.13021 ctober 2009 (http://www.SciRP.org/journal/wsn/).
Published Online O
High Resolution MIMO-HFSWR Radar Using Sparse
Frequency Waveforms
Guohua WANG, Yilong LU
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore
Received April 21, 2009; revised May 12, 2009; accepted May 15, 2009
Abstract
In high frequency surface wave radar (HFSWR) applications, range and azimuth resolutions are usually lim-
ited by the bandwidth of waveforms and the physical dimension of the radar aperture, respectively. In this
paper, we propose a concept of multiple-input multiple-output (MIMO) HFSWR system with widely sepa-
rated antennas transmitting and receiving sparse frequency waveforms. The proposed system can overcome
the conventional limitation on resolutions and obtain high resolution capability through this new configura-
tion. Ambiguity function (AF) is derived in detail to evaluate the basic resolution performance of this pro-
posed system. The advantages of the system of fine resolution and low peak sidelobe level (PSL) are demon-
strated by the AF analysis through numerical simulations. The impacts of Doppler effect and the geometry
configuration are also studied.
Keywords: MIMO, HFSWR, Radar, Sparse Frequency Waveform
1. Introduction
HIGH frequency surface wave radar (HFSWR) is a
low-cost radar system that adopts vertically polarized
high frequency electromagnetic signals which propagate
along the ocean surface. A preferable property of
HFSWR is that it can detect and track ship and aircraft
targets beyond the horizon. Due to this reason, HFSWR
has a wide range of applications in both civil and mili-
tary fields. For conventional HFSWR systems, its range
resolution is highly restricted by the bandwidth of avail-
able clear channels in a congested spectrum environment
[1,2], while the azimuth resolution is also constrained by
the physical dimension of the radar antenna aperture.
Multiple-input multiple-output (MIMO) radar is now
getting much intention for various applications such as
detection, estimation, and imaging etc. MIMO radar can
transmit at transmitters multiple waveforms that are di-
vidual at the receivers so that it can obtain more degrees
of freedom compared with conventional radars that
transmit single waveform [3–6]. With widely distributed
antennas, angular diversity can be fully achieved to
compete with the target scintillations [7,8]. Meanwhile,
MIMO radars with widely distributed antennas can gain
high resolution by coherent processing [8]. Like distrib-
uted MIMO radar, a single-input multiple-output (SIMO)
radar system with sparse coherent receiving aperture can
also achieve high resolution on the order of one wave-
length with limited bandwidth as reported in [9].
Sparse frequency waveform problem has been studied
in [10,11] and literatures therein. For HFSWR, sparse
frequency waveform can provide large flexibility to
choose clear channels and thereby reduce interferences
from assigned channels. Motivated by the potential
benefits from sparse frequency waveforms and high
resolution capacity of coherent multistatic radar and
MIMO radar systems, we in this paper propose a novel
MIMO-HFSWR using sparse frequency waveforms to
break down the limitation on range and azimuth resolu-
tions of conventional HFSWR. Ambiguity Function (AF)
is derived in detail and fully investigated in this paper to
analyze the performance of the proposed system. Unlike
that of paper [8], we take Doppler effect in the AF for
analysis. Through AF analysis it is demonstrated that this
system has high flexibility in operation and attractive
improvement on resolution in the restricted geographical
condition as well as the congested spectrum environment.
In particular, by using the widely separated antennas, it
abates the aperture limitation as well as the rigorous land
requirement successfully. In addition, by using sparse

G. H. WANG ET AL. 153
frequency waveform it not only takes more clear chan-
nels into use to compete with co-channel interference but
also reduces the peak sidelobe level (PSL).
The reminder of this paper is organized as follows.
The AF of the proposed MIMO-HFSWR using sparse
frequency waveforms is derived in Section 2. Based on
AF analysis and simulations the system is evaluated in
terms of resolution capacity and PSL performance, with
zero Doppler frequency in Section 3 and under the fac-
tors of geometric configurations and Doppler effects in
Section 4. Finally, conclusions and future work are out-
lined in Section 5.
2. Ambiguity Function of MIMO-HFSWR
System Using Sparse Frequency
Waveforms
Ambiguity function is an important tool in conventional
radar analysis as it shows radar’s inherent capacity of
discriminating targets associated with different time de-
lay and Doppler frequency. Thus, in this paper, we also
employ AF to evaluate the performance of the proposed
system.
The proposed MIMO-HFSWR system consists of M
transmitters transmitting M waveforms and N receivers.
Each transmitter is assigned a distinct channel with
starting frequency fm, m=1, 2,…, M. Thus, collectively,
the transmitting waveforms will have a sparse spectrum,
because which we call the transmitting waveforms sparse
frequency waveform. All antennas are arbitrarily located
with mutual separation distance larger than several
wavelengths in a 3-dimensional space. Figure 1 shows
the system configuration, where Rn and Tm refer to the
n-th receiver and the m-th transmitter, respectively.
Each transmitter and each receiver are located at a point
represented by a 3-dimensional vector in the Cartesian
coordinate system. For example, the m-th transmitter is
associated with a vector ct,m=[xm, y
m, z m], and the n-th
receiver cr,n=[xn, y
n, z n]. For simplification, this paper
considers only single point target case. And the target is
assumed to be located at a general point x = [x, y, z] with
constant velocity of v = [vx, vy, vz]. As the antennas are
widely distributed, each of them will view the target with
m

m

Figure 1. MIMO radar configuration.
a different angle. Angle variables θ and φ refer to the
true elevation and azimuth as illustrated in Figure 1. We
also assume that the phases and time at the transmitters
and receivers are synchronized in advance. Meanwhile,
the signal attenuation in different path is assumed to be
the same.
Let x
m(t) be the signal transmitted by the m-th trans-
mitter that meets the requirement of narrow band as-
sumption. It is expressed as




exp( 2)
mmm
x
tjfts  t

(1)
where sm(t) is the baseband waveform of the m-th trans-
mitter. After the signal impinged back from the target to
the n-th receiver, the echo is:

()
M
nm nmnm
m=1
et=x t-exp-j2π
f
dt 
 (2)
where γ is the complex reflection coefficient of the target,
τnm is the round-trip delay, and fdnm is the Doppler fre-
quency of the echo at the n-th receiver due to the m-th
transmitter. We take the assumption that the target stops
during the pulse transmission and reception. Then τnm is
the round-trip delay at the start of observation time, and
has the form






0
22
22
//
/
/
mn
nm TRt
mmm
nnn
rtcrtc
x
xyyzz
xxyyzzc



 
 
c (3)
where c is the velocity of light in the media that the
transmitters, receivers and targets are located in. In the
case that the transmitters or the receivers are mounted on
moving platform, the platform velocity can also be easily
included in (3). For different scatter, τnm is a function of
variables x, y, and z. Thus, through Taylor-series analysis
at a reference point x0=[x0, y0, z0], (3) can be changed to




 

 
 

000
[,,]
[,,]
0
0
0
0/ 0/
0/ 0/
coscoscos cos
cossincos sin
sin sin
mn
nm TRxyz
mn
TR
xyz
mm nn
mm nn
mn
0'
nm nm
rcrc
rcrc
xx
c
yy
c
zz
c

 
 















(4)
where




000
[,,]
0/ 0/
0m n
nm TR
x
yz
rcrc   (5)
and
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G. H. WANG ET AL. 154
 
 

0
0
0
coscoscos cos
cossincos sin
sin sin
mm nn '
nm
mm nn
mn
xx
c
yy
c
zz
c
 

 







 


(6)
fdnm has the form
  
1mn
nmT R
m
d

f
drtr
dt 
t
(7)
Again, through Taylor-series analysis (7) can be
changed to

 













0
222 22
cossincos sin coscoscos cos
sin sin
2
x
mymz mx nynz n
dnm
mmmmmnnn
ymm nn xmm nn
mm
zm n
m
vxxvyyvzzvxxvy yvzz
f
xxyyzzxxyyzz
v v
v
 

 
 


 

(8)
It can be easily proved that (8) is a tantamount expres-
sion of conventional Doppler frequency of bistatic radar.
As each transmitting waveform is assigned to a dis-
tinct channel, orthogonality holds for all the transmitting
signals. Thus, at each receiver, signals from M transmit-
ters can be firstly separated by down-converting into M
channels. Then, for each channel, a matched filter of
corresponding transmitting waveform is employed at the
interested range cell centered at x0= [x0, y0, z0]. Thus, the
m-th filter output at the n-th receiver can be expressed as


  



0
00 *
''
,,
exp2exp 2
exp 2,
nm nm
nm nm
nm
mnmmnm mnmnm
mmm nmnm
y
=jfst jfdtstdtn
=jf fd+nt
 



 


xx v
A

t

(9)
where Amm is the correlation between the m-th transmit-
ting waveform and its delay-Doppler shifted version, and
nnm (t) is the noise component of the output.
Collectively, there are NM outputs after matched fil-
tering. By coherently summing all these outputs we can
get:
 
2
00
11
,, ,,
NM
nm
nm
y

 Axx vxx v (10)
Besides, different waveforms may obtain different
Amm, thus waveforms also play a key role in the MIMO
radar ambiguity function. We take Linear Frequency
Modulation (LFM) waveforms as an example to illustrate
this point. A conventional LFM waveform defined by
u(t)=rect(t/T)exp(jπkt2) has an correlation function like
[12]:
  
2
A,exp1sin1
LFM
vjk cvTB
TT

 

 
 

 

 

Because both τ’nm and fdnm in (9) are affected by azi-
muth angles and elevation angels, the geometry configu-
ration represented by a matrix C consisting of all the
azimuth and elevation angels should be included in the
ambiguity function. Ignoring the noise-based component
and discarding the target reflection coefficient in (10),
we can define the normalized ambiguity function for the
proposed system as:
(12)
where v is the Doppler frequency, τ is the time-delay, k is
the chirp rate, T is the pulse width, B is the bandwidth.
From (12), it can be easily inferred that the bandwidth of
single waveform adopted will impact the performance of
the MIMO-HFSWR system.
3. Resolution Capacity and PSL


022
2
''
11
1
,,,
,
nm nm
NM
mmmnm
nm
MN
exp -j2πff





xx vC
A
Performance
d
(11)
In this section, the resolution capacity and PSL of the
MIMO-HFSWR system using sparse frequency wave-
form are assessed by setting the Doppler frequency to
zero in AF analysis. Sparse frequency waveforms con-
sisting of stepped frequency linear frequency modulation
signals are investigated. For simplification we just study
a simplified 2-dimensional configuration.
As τ’nm and fdnm are related to x−x0, y−y0, z−z0, vx, vy,
vz, azimuth angles and elevation angels, the range and
azimuth resolution as well as the effects of velocity com-
ponents and geometry configuration can be assessed
through the AF analysis.

G. H. WANG ET AL. 155
(a)
(b)
Figure 2. MIMO AF of (a) Sparse frequency waveforms (b) Single LFM signal, at zero Doppler frequency.
As we can see from above analysis, the ambiguity
function of the proposed MIMO-HFSWR system de-
pends on the system geometry configuration confined by
all azimuth and elevation angles. Thus, we can ignore the
true position of transmitters and receivers. We here take
nine transmitters and nine receivers located evenly over
spatial region of (−π/4, π/4) for φ. Each transmitter will
emit one LFM with an assigned start frequency. The
pulse width is 100 us for all transmitters. The bandwidth
of each LFM pulse is 500 kHz. The nine start frequencies
of LFM waveforms are defined as the sequence of {5, 6,
7, 8, 9, 8, 7, 6, 5} MHz. Orthogonality can be achieved
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G. H. WANG ET AL. 156
by sequentially transmitting at transmitters or by setting
the first five LFM waveforms to be up-chirps and the left
four down-chirps [12]. We in this paper utilize the first
mechanism. As a comparison, we also take an ambiguity
function from a single LFM waveform with the same
bandwidth and pulse width as well as the start frequency
of 9 MHz. We also suppose to transmit it sequentially in
time domain so that we can separate at each receiver the
returns from different transmitters. By central coherent
processing we can also get the results of AF as showed in
Figure 2(b), which seems the same as that in [8].
The mesh plots of the AF are showed in Figure 2 and
more details on resolutions and sidelobe characteristics
are given in Figure 3 and Table 1. As is obvious from
both Figure 3 and Table 1, the resolutions of MIMO-
HFSWR are at the level of one wavelength for both
(a)
(b)
Figure 3. Resolution and sidelobe performances of sparse frequency waveforms (solid line) and single LFM signal (dotted line)
along (a) x-axis and (b) y-axis.

G. H. WANG ET AL. 157
Table 1．Resolution and sidelobe characters of different waveforms.
Item Sparse frequency waveforms LFM
Resolution of x 0.31 wavelength 0.58 wavelength
PSL-x –14.9 dB –3.2 dB
Resolution of y 0.9 wavelength 1.6 wavelength
PSL-y –13.5 dB –16.5 dB
(a)
(b)
Figure 4. Resolution capacity in 3 point targets case (a) Sparse frequency waveforms (b) Single LFM waveform.
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G. H. WANG ET AL. 158
sparse frequency waveforms and single LFM waveform.
And the resolutions of sparse frequency waveforms are
even better than those of common LFM signals. This is a
great improvement for azimuth resolution and even for
range resolution from conventional several kilometers to
several tens meters. Meanwhile, as HFSWR always
works in a highly congested spectrum environment, the
sparse frequency waveform approach can provide better
flexibility on choosing available channels than wave-
forms confined in only one channel. The sidelobes of
x-axis and y-axis are well below –13 and –14 dB for
sparse frequency waveforms, respectively, which is a
significant improvement compared with the side lobe
level from the single LFM waveform within the same
channel. It demonstrates that the sidelobe levels can be
suppressed by frequency diversity in random arrays [13].
This is another advantage of sparse frequency waveform.
Multiple targets case are illustrated in Figure 4, where
(a)
(b)
Figure 5. Bandwidth effect on resolution and sidelobe performance along (a) x-axis and (b) y-axis. Solid line is associated with
bandwidth 50 KHz, while the dotted line is 500KHz.

G. H. WANG ET AL. 159
(a)
(b)
Figure 6. Ambiguity functions of (a) Configuration 1 and (b) Configuration 2 in case 1 with velocity [vx, vy] = [500, 500] m/s.
Configuration 1 (9×9), Configuration 2 (5×5), both evenly distributed in (−π/4, π/4).
three targets are located in [0, 0], [0, 10], and [–10, –10].
The coordinate system is expressed in multiples of
wavelength. As we can see, by using sparse frequency
waveform set, the system can better distinguish different
targets than by using waveform set in the same channel.
Bandwidth effect is illustrated in Figure 5. We take a set
of sparse waveforms like that mentioned above. The dif-
ference is that the bandwidth is 50 KHz for each wave-
form. From Figure 5 we can see that even with smaller
bandwidth, the resolution capacity is not much impacted.
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G. H. WANG ET AL. 160
(a)
(b)
Figure 7. Resolution and sidelobe performance along (a) x-axis and (b) y-axis in case 1 with velocity [vx, vy] = [500, 500] m/s.
Configuration 1 (9×9), Configuration 2 (5×5), both evenly distributed in (−π/4, π/4).
However, the PSL performance is deteriorated. The PSL
in x-axis is about –12.9 dB and is about –10 dB in y-axis
for this waveform set. Thus, we can see that the larger
the bandwidth adopted, the lower the sidelobes in both
x-axis and y-axis.
In this case study, the simulation results demonstrate
that MIMO-HFSWR with sparse frequency waveform
has superior resolution than conventional HFSWR in
both range and downrange domain. Sidelobe levels can
be suppressed by using sparse frequency waveforms.

G. H. WANG ET AL. 161
Further work will be focused on the suppression of
sidelobe levels by waveforms with effective frequency
diversity scheme.
4. Doppler and Geometry Factor
As HFSWR is always operated in Doppler circumstances,
the AF with Doppler effects should be further investi-
gated. Meanwhile, unlike monostatic radar the distrib-
uted MIMO radar is confined by the geometry configura-
tion. Thus the geometry factor should also be investi-
gated. Two cases are given below to investigate the
Doppler and configuration effect.
In Case 1, we study the configuration effect. Target of
this case is with velocity of [vx, vy] = [500, 500] m/s. This
is a high velocity target case corresponding to air targets.
There are two configurations. For Configuration 1, in the
region of (–π/4, π/4) there are nine transmitters and nine
receivers, evenly distributed. The transmitting wave-
forms are defined as those in Section 3 except that each
one has 10 kHz bandwidth. For practical HFSWR appli-
cation, only a limited number of continuous clear chan-
nels with bandwidth of a few kilo-Herz in the 3-30 MHz
high frequency band can be found and used at a time
when interference is considered [1,2]. Thus, 10 kHz
bandwidth is used for a much more similitude in real
condition of the HFSWR system. For Configuration 2, in
the same region of (–π/4, π/4) there are five transmitters
and five receivers, evenly distributed. The waveforms are
the first five used in Configuration 1 of Case 1. Thus, the
total spectra employed by these two configurations are
the same. As illustrated in Figure 6, Figure 7, both con-
figurations in this case show high resolution capabilities.
However, Configuration 1 with more transmit-receive
pairs shows better sidelobe performance in both x-axis
and y-axis. The PSL in x-axis is about –12 dB and is
about –10.5 dB in y-axis for Configuration 1. Based on
our numerous simulation experiments, it is found that as
more pairs of transmitter and receiver are set in a much
wider spatial region, the resolutions can be slightly im-
proved and the PSLs of y-axis and x-axis can be further
reduced. However, systematic study on the PSLs reduc-
tion through geometry optimization will be explored in
the future.
In Case 2: we have four velocity settings like [0, 0]
m/s, [100, –100] m/s, [–10, 5] m/s, and [500, 500] m/s.
The geometry configuration in Case 2 is the same as
Configuration 1 in Case 1. We also take the waveform
set of Configuration 1of Case 1 in this case study. Figure
8 shows the results of Case 2. We can see from Figure 8
that the proposed system shows similar characteristics in
different Doppler context, which means the resolution
and PSL performance are both insensitive to Doppler
frequency. Thus, for both high speed air targets and low
(a) v=[0,0] m/s (b) v=[100, −100] m/s
(c) v=[−10,5] m/s (d) v=[500,500] m/s
Figure 8. Ambiguity functions of different velocity with Configuration 1.
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G. H. WANG ET AL. 162
velocity surface targets, the proposed system can also
have high resolution performance.
5. Conclusions
In this paper, the concept of distributed MIMO-HFSWR
radar transmitting sparse frequency waveforms is pro-
posed. The AF of this proposed system is derived in de-
tail. Potential advantages of the proposed system on
resolution capacity and PSL performance are assessed
through AF analysis and simulations. The impacts of
Doppler effects and the geometry configuration factor
are also studied. It has been found that the system has
several distinguished characteristics. Firstly, the range
resolution and the azimuth resolution can be improved to
the level of one wavelength, namely, only tens meters
and the PSL is reduced to a much lower level with sparse
frequency waveforms. Meanwhile, the resolutions are
not restricted by individual bandwidth while the PSL can
benefit from large bandwidth. Secondly, the performance
of fine resolution and low PSL are insensitive to the
Doppler effects. Thus, for both high speed air and low
velocity surface targets, the proposed system also has
high performance. Thirdly, the resolution capacity and
PSL performance can be optimized through geometry
configuration optimization. In addition, multistatic con-
figuration provides large flexibility to find a proper place
to locate the radar transmitters and receivers; by using
sparse frequency waveforms, it is much easier to find
more available channels in different locations thus the
co-channel interference can be avoided and the perform-
ance can be further improved. Further studies will be
conducted on the surveillance strategy and high quality
waveforms with better AF performance. Meanwhile,
synchronization problem should also be paid special at-
tention to so that coherent processing can be conducted
perfectly.
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