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56 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1,
NO. 1, JUNE 2007
Adaptive Waveform Design for Improved Detectionof Low-RCS
Targets in Heavy Sea Clutter
Sandeep P. Sira, Douglas Cochran, Antonia Papandreou-Suppappola,
Senior Member, IEEE, Darryl Morrell,William Moran, Member, IEEE,
Stephen D. Howard, and Robert Calderbank, Fellow, IEEE
Abstract—The dynamic adaptation of waveforms for trans-mission
by active radar has been facilitated by the availability
ofwaveform-agile sensors. In this paper, we propose a method to
em-ploy waveform agility to improve the detection of low
radar-crosssection (RCS) targets on the ocean surface that present
lowsignal-to-clutter ratios due to high sea states and low
grazingangles. Employing the expectation-maximization algorithm
toestimate the time-varying parameters for compound-Gaussiansea
clutter, we develop a generalized likelihood ratio test
(GLRT)detector and identify a range bin of interest. The clutter
estimatesare then used to dynamically design a phase-modulated
waveformthat minimizes the out-of-bin clutter contributions to this
rangebin. A simulation based on parameters derived from real
seaclutter data demonstrates that our approach provides around 10dB
improvement in detection performance over a nonadaptivesystem.
Index Terms—Detection, sea clutter, waveform design,
wave-form-agile sensing.
I. INTRODUCTION
THE detection of small targets on the ocean surface byactive
radar is particularly challenging due to the lowsignal-to-clutter
ratio (SCR) that can result from low grazingangles and high sea
states. Advances in radar technology thatpermit pulse-to-pulse
waveform agility provide many oppor-tunities for improved
performance. For example, waveformscan be adapted to match the
target characteristics and the envi-ronment or a desirable level of
estimation accuracy of specific
Manuscript received September 1, 2006; revised February 14,
2007. Thiswork was supported in part by the U.S. Department of
Defense under MultiUniversity Research Initiative Grant AFOSR
FA9550-05-1-0443 administeredby the Air Force Office of Scientific
Research; by DARPA, Waveforms for Ac-tive Sensing Program under NRL
Grant N00173-06-1-G006; and the Interna-tional Science Linkages
established under the Australian Government’s innova-tion statement
“Backing Australia’s Ability.” The associate editor coordinatingthe
review of this manuscript and approving it for publication was Dr.
MariaSabrina Greco.
S. P. Sira, D. Cochran, and A. Papandreou-Suppappola are with
the Depart-ment of Electrical Engineering, Arizona State
University, Tempe, AZ 85287USA (e-mail: [email protected];
[email protected]; [email protected]).
D. Morrell is with the Department of Engineering, Arizona State
UniversityPolytechnic Campus, Mesa, AZ 85212 USA (e-mail:
[email protected]).
W. Moran is with the Department of Electrical and Electronic
Engineering,University of Melbourne, Melbourne, Australia (e-mail:
[email protected]).
S. D. Howard is with the Defence Science and Technology
Organisation, Ed-inburgh, Australia (e-mail:
[email protected]).
R. Calderbank is with the Department of Electrical Engineering,
PrincetonUniversity, Princeton, NJ 08540 USA (e-mail:
[email protected]).
Color versions of one or more of the figures in this paper are
available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JSTSP.2007.897048
target parameters. This paper proposes an approach to
thesuppression of sea clutter using waveforms that are
designedon-the-fly to minimize the effect of clutter in areas of
interest,thereby improving target detection performance.
Recent work on dynamic waveform adaptation, to improvetracking
performance, for example, has often assumed perfectdetection [1],
[2] or simplistic clutter models [3], [4]. While thismay be
appropriate when the signal-to-noise ratio is high, nei-ther
assumption is justified in scenarios that involve heavy seaclutter.
Early investigations of detection in the presence of in-terference
in space-time adaptive processing [5], and waveformdesign for
clutter rejection [6], assumed that the clutter returnswere
independent, and identically Gaussian distributed. How-ever, when a
radar has a spatial resolution high enough to re-solve structure on
the sea surface, the Gaussian model fails topredict the observed
increased occurrence of higher amplitudesor spikes. This lead
researchers to use two-parameter distribu-tions to empirically fit
these longer tails [7]. As a result, the com-pound-Gaussian (CG)
model for sea clutter has now gained wideacceptance [8], [9] and
has been tested both theoretically [10]as well as empirically
[11].
Both coherent detection [9] and waveform optimization [12]in
non-Gaussian backgrounds require knowledge of the statis-tics of
the clutter echoes and, in [13] for example, the targetimpulse
response. However, the dynamic nature of the oceansurface
necessitates a reliance on estimates of the statistics,thus
precluding optimal solutions. Moreover, these approachestypically
seek optimal but fixed, or dynamically nonadaptive,waveform
designs, that fail to exploit the potential of waveformagility.
The main contribution of this paper is a methodology to adaptthe
transmitted waveform on-the-fly, based on online estima-tion of sea
clutter statistics for improved target detection. It ismotivated by
the fact that, in a radar, the signal obtained aftermatched
filtering at the receiver is a convolution of the ambi-guity
function of the transmitted signal with the radar scene [14],which
smears energy from one range-Doppler cell to another.Therefore, the
incorporation of information about the clutterinto the design of a
waveform whose ambiguity function min-imizes this smearing in the
region of interest, can improve theSCR and detection performance.
In our proposed method, thestatistics of the clutter at different
ranges are estimated using theexpectation-maximization (EM)
algorithm and are then used todesign a phase-modulated (PM)
waveform for the next trans-mission that improves the SCR. Using
clutter covariance esti-mates derived from real data, we provide
numerical simulationexamples to demonstrate around 10 dB
improvement in detec-tion performance for a single target. Although
the development
1932-4553/$25.00 © 2007 IEEE
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SIRA et al.: ADAPTIVE WAVEFORM DESIGN 57
is focused purely on range estimation, it is believed to be
appli-cable to range-Doppler estimation; this is the subject of
ongoingresearch.
The use of principal component analysis (PCA) for the
miti-gation of colored interference is well known [15]. This
approachcan be used to exploit the spatial and temporal
correlations in seaclutter returns [11] to mitigate them as was
demonstrated in [16].The primary difficulty with this approach
revolves around theestimation of the interference covariance matrix
[17], [18] andthe subsequent determination of a low-rank
interference sub-space. Assuming a high pulse-repetition frequency
(PRF), weuse the CG model to demonstrate that we can form a
wave-form-independent estimate of the clutter subspace. The
orthog-onal projection of the received signal into this subspace
providesappreciable clutter suppression.
The paper is organized as follows. In Section II, we describethe
CG model for sea clutter and the processing of the receivedsignal.
Section III presents the estimation of the clutter statistics,while
Section IV describes the generalized likelihood ratio test(GLRT)
detector. The design of a PM waveform is described inSection V, and
simulation examples are presented in Section VI.
II. SIGNAL AND SEA CLUTTER MODELING
We consider a medium-PRF radar that transmits pulsesin a dwell
on a region of interest before switching to othertasks. Each dwell
consists of two sub-dwells, Sub-dwells 1and 2, during which
identical pulses of the waveforms
and are transmitted, respectively. The impliedwaveform agility
is thus between sub-dwells rather than ona pulse-to-pulse basis.
Each pulse returns a view of the radarscene which consists of
clutter and the target, if present. Inthis section, we describe the
CG model for sea clutter and theprocessing of the received
signal.
A. Sea Clutter Modeling
According to the CG model, sea clutter returns are believed tobe
the result of two components: a speckle-like return that arisesdue
to a large number of independent scattering centers reflectingthe
incident beam, anda texture caused by large-scale swell struc-tures
that modulates the local mean power of the speckle
return[7],[19].ThespecklegivesrisetolocallyGaussianstatistics,char-acterized
by short correlation time ( ms), while the texturedecorrelates much
less rapidly ( s) [7], [8], [11]. The texturecomponent also
exhibits spatial correlation that depends on therange resolution,
sea state, and wind speed [20]. Its probabilitydistribution has
been the subject of much investigation. From var-ious studies, sea
clutter has been modeled to have K, log-normal,or Weibull
distributions [7]. In this paper, we will not assume anyparticular
distribution for the texture, as it is not needed for ourwaveform
design.
The radar scene is defined to consist of a number of
clutterscatterers and at most one point target, distributed in
range andDoppler. In each range-Doppler cell, the number of
scatterers isassumed to follow a Poisson distribution with a rate
determinedby the cell volume and a clutter density . The complex
reflec-tivity
(1)
of the th scatterer over pulses, conditioned on the texture, and
speckle covariance matrix with
denoting the identity matrix, is a circular complexGaussian
random vector with zero mean and covariance matrix
, or [9]. Thus, the samples in (1) arecorrelated and is
nonwhite. Given the texture and the specklecovariance matrix, the
reflectivities of two scatterers andare independent so that
(2)
where denotes the probability density function of given. Due to
the high PRF, the duration of each sub-dwell can be
made much smaller than the decorrelation time of the speckle,and
we assume that the radar scene is practically stationaryduring this
period. Thus, we assume that the number of scat-terers in each
cell, and the scatterers’ delays and Doppler shifts,are constant
during a sub-dwell. However, the scatterer am-plitudes may
fluctuate randomly because small changes inrange, on the order of
the radar wavelength, may cause signif-icant changes in the phase
of the received signal [21]. We willalso assume that the value of
the texture is identical for all clutterscatterers within a cell
and is fully correlated across a dwell [22].
B. Received Signal Processing
The processing of the received signal is identical in Sub-dwells
1 and 2. Accordingly, we will use to denote the trans-mitted signal
(instead of and ), and only differentiatebetween the sub-dwells
where necessary, by means of a sub-script. The received signal at
the th pulse, ,is given by
(3)
where and are the complex reflectivity, delay andDoppler shift,
respectively, of the target (if present), and
are the delay and Doppler shift of the th scatterer,
respec-tively, and is additive noise. We will henceforth assumea
high clutter-to-noise ratio so that the clutter is the
dominantcomponent and additive noise is negligible. Since we
onlyconsider transmitted signals of very short duration, the
Dopplerresolution is very poor. Therefore, we completely
ignoreDoppler processing and restrict our attention to delay or
rangeestimation alone.
The received signal in (3) is sampled at a rate to yield
asequence . Note that is a scalar becausea single sensing element
is being used to obtain it in contrast to[17], where an array of
sensors is used. The sampled signal ismatched filtered at each
sampling instant to yield the sequence
. We defineto be the vector of matched-filtered outputs at the
th delay orrange bin. Then,
(4)
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58 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1,
NO. 1, JUNE 2007
where defines the set of scatterers that lie in the th rangebin,
is the index of the range bin that contains the target, ifpresent,
and is the length of the transmitted signal
. Analogous to (1), in(4) is the target reflectivity, which,
assuming a Swerling I pointtarget [14], is distributed according to
. Thefunction in (4) is the autocorrelation function of atlag and
is given by
The concept of individual scatterers presented in this
sectioncan be thought of as a limiting case of incremental
scatteringcenters on the ocean surface [23], the combined responses
ofwhich give rise to the clutter return. It is more convenient
tothink of a single, aggregate scatterer in each bin with
complexamplitude
(5)
for the th range bin, so that (4) is replaced by
(6)
We henceforth take the view that the sea clutter return is
theresult of reflections from these aggregate scatterers. From
(2)and (5), we note that . Due to the indepen-dence in the
contributions of individual scatterers in (2), andthe assumption of
the texture invariance within a range-Dopplercell, the texture
associated with can be seen to be theproduct of the number of
scatterers in the cell and the commontexture value associated with
each one of them.
III. TEXTURE ESTIMATION
The GLRT detector and the waveform design algorithm, bothrequire
estimates of the clutter statistics. In this section, we de-scribe
the application of the expectation-maximization (EM) al-gorithm to
the estimation of the speckle covariance and the tex-ture values in
each range bin.
Define as theset of parameters upon which the probability
densities of
depend. Note that, although thetexture is a random process, we
consider its values in each binto be deterministic but unknown
variables because they are as-sumed to be constant across a dwell.
This approach is in con-trast to most sea clutter research, where
the parameters of theprobability distribution that is assumed to
model the texture areestimated, rather than its actual values (see
[24] for example).The speckle however, is treated as a random
variable and we es-timate its covariance matrix .
From (6), we note that there is a many-to-one mappingbetween the
scatterer reflectivities and the matched-filteredvector. This
mapping is noninvertible and precludes an
exact solution for . Therefore, we attempt to find the
max-imum-likelihood estimate of given the observed data. With
, we then seek
(7)
where is the probability density function of that de-pends on .
Since the maximization in (7) requires a compli-cated
multidimensional search, we instead attempt to find anestimate of
that maximizes , where the unobservedor complete data upon which
the observed or incomplete data
depends is and, where is the Kronecker delta. Re-
call from (5) and (6) that is the amplitude of the
aggregatescatterer in the th bin, is the target reflectivity, and
is theindex of the range bin in which the target is located. Thus,
weseek
(8)
which can be accomplished by the application of the EM
algo-rithm as follows [25].
Starting with an initial guess , let be the estimateof after
iterations. Then, the EM algorithm is given by theiteration of
(9)
(10)
where in (9) is the expectation operator. The details of
thecomputations in (9) and (10) are described next.
The mapping in (6) between the unobserved and observeddata is
linear, which indicates that the probability density func-tion of ,
given and , isa mixture of complex Gaussian densities. We thus
havethe linear transformation
(11)
where the first rows of the matrixare given by
where indicates the Kronecker product. Each successiveblock of
the matrix is obtained by circularly shifting
the previous block columns to the right.Using the conditional
independence of and in (5),
the probability density of , given , is complex
multivariateGaussian with zero mean and covariance
(12)
where is such that.
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SIRA et al.: ADAPTIVE WAVEFORM DESIGN 59
The log-likelihood function in (9) is therefore given by
(13)
where is a constant that does not depend on and de-notes the
matrix trace. Since is a linear function of , the con-ditional
expectation of (13), given , at is given by [26]
(14)
(15)
(16)
The maximization of (14) is equivalent to the minimizationof
(17)
Since in (12) is a block diagonal matrix, (17) may bewritten
as
(18)
where is the covariance matrix ofand is the th matrix on the
diagonal of .Taking the partial derivatives of (18) with respect to
andand equating them to zero, the maximum-likelihood estimate
of
is obtained from the simultaneous solution of
(19)
(20)
(21)
This solution must be found numerically and provides thevalue of
that maximizes (14). The algorithm defined by(9) and (10) is
iterated until successive changes in the parametervalues in drop
below a set threshold.
The form of (19) and (21) makes it computationally expen-sive to
solve because simplifications afforded by the Kroneckerproduct
cannot be used in (15) and (16) to obtain . A simplerapproach is to
make the approximation , which islikely in low-SCR scenarios. Then,
(19) reduces to
which, upon substituting for from (20) gives
(22)
the solution of which, together with (20), provides the value
ofin (10). Clearly, this approximation causes an inconsis-
tency in modeling the covariance of the return in the bin in
whichthe target is present, where it results in an overestimate of
thevalue of the texture. However, as described in Sections IV-B
andV, this inaccurate estimate is never used. On the other hand, in
abin where the target is not present, the weak contribution of
thetarget return is further reduced by the weighting of the
autocor-relation function. This can be neglected as long as the
autocor-relation function sidelobes are generally much smaller than
thepeak at the zeroth lag. In practice this is easily
achieved—theautocorrelation function of the LFM chirp we use in
Sub-dwell1 has a highest sidelobe level that is 16 dB below the
peak.
The use of numerical methods to solve (22) results in themajor
computational burden of this estimation procedure. Thetemporal
variations of the speckle covariance do not appear tohave been
reported upon. If it is considered to be slowly varyingover time,
there may not be a requirement to estimate in (22)at each dwell.
This can lead to significant savings in computa-tional effort.
IV. CLUTTER SUPPRESSION AND DETECTION
The dynamic design of is motivated by the need to im-prove the
SCR in a range bin that is to be interrogated for thepresence of a
target. At the end of Sub-dwell 1 therefore, wewish to identify a
putative target location. In this section, wedevelop a GLRT
detector that uses the estimates obtained inSection III. The same
detector is used at the end of Sub-dwell 2to provide the final
detection from this dwell.
A. Clutter Suppression
Prior to forming the detector, we seek to suppress the
seaclutter. This is motivated by the differences in the
correlationproperties of the target and clutter returns, as well as
the inac-curacy in the estimates of the texture. The latter, which
arisesdue to the lack of averaging in (20), deleteriously affects
detec-tion performance. A key requirement for this procedure is
theknowledge of the covariance matrix of the clutter returns
fromwhich the clutter subspace can be obtained. This is not
availableand must be estimated online. Next, we demonstrate that
the
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60 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1,
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clutter subspace can be obtained from the eigendecompositionof
the speckle covariance matrix , which we have estimatedin Section
III. The covariance matrix of the matched-filtered re-turn in (6)
is given by
From the CG model however, given the texture and covari-ance
matrix and are independent if . Also, thehave zero mean and
hence
(23)
where the scalar is a functionof the waveform due to the
dependence on . From (23), wesee that the eigenvectors of are
identical to those of , andare waveform-independent. The space
occupied by the clutter inthe matched-filtered return is thus
identical to that of , thespeckle covariance matrix. This obviates
the need to estimate
in (23).Let be the matrix, whose columns are the eigen-
vectors, obtained by eigendecomposition, of in (22),
corre-sponding to the smallest eigenvalues of that
togethercontribute 0.01% of its total energy. In our simulation
study, wehave observed that this typically corresponds to the 3–4
smallesteigenvalues out of . The projection
(24)
provides the component of the received signal that lies in
thespace where the clutter has the least energy and represents
aclutter-suppressed signal. We have assumed that the target
re-flectivity is circularly symmetric. Thus, some component of
theuseful echo does get canceled out while some fraction remainsin
the subspace orthogonal to the clutter, and thus in . Sincethe
clutter component in is very weak, the projection opera-tion in
(24) leads to improved SCR.
The determination of the interference subspace in
principalcomponent methods for clutter suppression has been
extensivelyresearched [17], [18], [27]. The method we have used is
rela-tively straightforward but still provides around 5 dB
improve-ment in detection performance at the end of Sub-dwell 1,
asshown in Section VI. This is significant because it helps to
limitthe number of range bins that have to be interrogated in
Sub-dwell 2 and thereby reduces system usage.
B. GLRT Detection
With respect to a given range bin, we define the hypothesesand
to refer to the presence of only clutter or clutter and
target, respectively. We seek to form a detector based on
thelog-likelihood ratio test, which for range bin is
(25)
where , and is a thresholdthat is set to obtain a specified
probability of false alarm, .Since we only have estimates for the
parameters in (25), weinstead form the GLRT so that
(26)
From (6) and (23), both probability density functions in(25) are
complex Gaussian. The detection problem in (26), istherefore a test
of Gaussian distributions with different vari-ances [28]. With the
estimates and obtained as describedin Section III, the
maximum-likelihood estimate can beobtained for an observed value of
. Then, following [28], itcan be shown that the test in (26)
reduces to
(27)
where are the eigenvalues of the covari-ance matrix
is a pre-whitened version of, where the columns of the unitary
matrix are the
eigenvectors of , and is a diagonal matrix of the corre-sponding
eigenvalues.
The GLRT detector requires the clutter statistics for eachrange
bin. As discussed in Section III, the value of the tex-ture in the
range bin where the hypothesis is true, is typ-ically
overestimated. This is detrimental to detection perfor-mance. One
way of avoiding this problem is to consider a targetthat is moving
fast enough so that it is located in a differentrange bin in
successive dwells. Then, if the GLRT detector isbeing formed for
the th bin at dwell , we can use the esti-mate obtained during
Sub-dwell 1 of the th dwell asthe clutter statistics for the
current dwell. Clearly, this requiresthe assumption that the
texture does not change significantly be-tween dwells, which is
reasonable if the time between dwells isabout 0.5–1 s.
It is important to select a suitable threshold in (27) topermit
efficient evaluation of detection performance via MonteCarlo
simulation. Assuming that all the values of in (27)are distinct, we
compute the threshold as the solution of
, where
This computation assumes an exponential distribution for thetest
statistic in (27) [28], which is an approximation because
theweights are data dependent. However, we do not use thisthreshold
to analytically compute detection performance and
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SIRA et al.: ADAPTIVE WAVEFORM DESIGN 61
make no claims to its optimality. Further, all the values of
canbe derived from data obtained in a previous dwell, thus
makingthem practically independent of in (27).
It is possible, of course, to perform the detection in (26)
onthe original matched-filtered signal , rather than on the
clutter-suppressed signal . We have found that this approach leads
toaround 5 dB loss in detection performance. The reason is that,
inthis case, all eigenvalues of the matrix would be used toform the
test statistic in (27). The texture estimation proceduredescribed
in Section III is not very accurate due to a lack ofaveraging,
leading to poorly estimated eigenvalues. The cluttersuppression
described in Section IV-A reduces the number ofeigenvalues used in
the detector from to , thus limiting theeffect of these
inaccuracies.
V. WAVEFORM DESIGN
At the end of Sub-dwell 1, suppose that we have decided to
in-terrogate range bin . Excluding bin itself, all bins in the
range
contribute out-of-bin clutter to thematched-filtered return and
thus to . The objective of ourwaveform design in Sub-dwell 2 is to
minimize this out-of-bincontribution. To achieve this, we design
the waveform inSub-dwell 2 such that its autocorrelation function
takesvalues close to zero at those lags where the clutter is
estimatedto have the highest energy.
The synthesis of waveforms whose ambiguity function bestmatches
a specified function was first investigated in [29]. Theapproach
involves the selection of an orthonormal set of basisfunctions
whose cross-ambiguity functions were shown toinduce another
orthonormal basis in the time-frequency plane.The coefficients of
the expansion of the specified or desiredambiguity function on this
induced basis were then used tosynthesize the waveform. This
approach was further developedusing least-squares optimization in
[30]. However, this methodcannot be easily used to synthesize
waveforms where only adesired autocorrelation function instead of
an ambiguity func-tion has been specified. In this section, we
develop a methodthat designs a phase-modulated waveform using
mean-squareoptimization techniques so as to achieve low
autocorrelationmagnitude values at specified lags [31].
A. Synthesis of Phase-Modulated Waveforms
Let represent a unimodular phase-modulated (PM)waveform given
by
(28)
where the phase modulation is expanded in terms of an
orthog-onal set of basis functions as
(29)
with the total waveform duration and ,where is the number of
samples in the designed signal. Asdescribed in Section V-B, we will
require the waveform dura-tion in (28) to be identical to that of .
Consequently, thenumber of samples in and is . The advantage of
theformulation in (28) is that PM waveforms can be easily
gener-ated, and the absence of any amplitude modulation permits
themaximum power of the radar to be employed.
Defining the autocorrelation function of as
we want to determine the coefficients in (29) that minimize
(30)
where , and represents the (possiblydisconnected) set of range
values for which the texture valuesare large. The minimization
problem posed in (30) can be seenas a special case of a larger
problem for which the PM signalwhose autocorrelation function best
approximates a specifiedfunction in a mean-square sense is
synthesized.
It is relatively straightforward to show that the
autocorrelationfunction for the PM waveform in (28) is
(31)
where, and . Using the squared magnitude
of , the gradient and Hessian of can be easily com-puted and the
minimization of (30) can be accomplished by theNewton-Raphson
method. Note that it is also possible to usethis approach to design
two waveforms whose cross-correlationfunction is small where the
clutter is strong, which may then beused in a mismatched filter for
Sub-dwell 2. This will howeverentail added computational
complexity.
B. Implementation
The texture estimation procedure described in Section III
pro-vides an estimate of the th texture
. We then aim to design a waveform whose autocorrelationfunction
is negligibly small where is large. For simplicity,we assume that
the signal duration of and are iden-tical. We choose the bins
corresponding to thelargest estimated texture values to form the
set , excluding
, which corresponds to the putative target cell. Since
maycontain positive and negative lags, we form the set in (30)
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62 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1,
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Fig. 1. Generation of K-distributed clutter. The in-phase (I)
and quadrature (Q)components of the complex speckle variables are
filtered to yield the temporalcorrelation specified by the speckle
covariance matrix�. The correlated textureis generated using the
memoryless nonlinear transformation (MNLT).
with positive lags only, that, together with their
correspondingnegative lags, account for all the elements of . Since
this setis discrete, the integral in (30) reduces to the
summation
where the elements of are all integral multiples of the
sam-pling interval .
VI. SIMULATIONS
Our simulation model consists of a single moving target thatis
observed by a single sensor in the presence of simulatedsea
clutter. Individual clutter scatterers that are randomlydistributed
in range and velocity are generated, and their am-plitudes are
sampled from a K-distribution. The received signalis generated
according to (3) and is processed as described inSection II-B.
Using a GLRT detector as in Section IV-B, MonteCarlo simulations
were used to obtain the receiver operatingcharacteristic (ROC)
curves presented in this section. We firstdescribe the generation
of the synthetic sea clutter.
A. Generation of Synthetic Sea Clutter
The simulation of sea clutter with meaningful correlations
hasbeen the subject of much research (see [32] for a review). We
usethe method described in [8] and shown in Fig. 1. The
scattereramplitudes in (3) are generated as
(32)
where is the texture and represents the speckle componentthat
follows a complex Gaussian distribution with zero meanand
covariance . For the purpose of the simulations, the texturefollows
a gamma distribution
where is a scale parameter and is a shape parameter. In
oursimulations, we use and which results in highlynon-Gaussian
clutter.
We aim to generate and in (32) with appropriatecorrelations.
While it is straightforward to generate correlatedspeckle variables
by independently sampling the real andimaginary parts from a zero
mean Gaussian distribution withcovariance matrix , gamma variates
with arbitrary correla-tions cannot be easily generated and a
number of alternative
Fig. 2. Correlation coefficient of the (a) speckle and (b)
texture for theNov7starea4 dataset, range cell 1, VV polarization,
of the OHGR database.
techniques have been proposed [33]–[35]. We use a
memorylessnonlinear transformation (MNLT) to generate the
correlatedgamma-distributed texture variables [36].
B. Speckle and Texture Temporal Correlations
In order to simulate sea clutter with appropriate
temporalcorrelations, we derived correlation estimates from
experi-mental clutter data, collected at the Osborne Head
GunneryRange (OHGR) with the McMaster University IPIX radar[37].
Specifically, we used the procedure in [11] to analyze theclutter
data in range cell 1 of the Nov7starea4 dataset with VV(vertical
transmit and vertical receive) polarization, and theresulting
speckle and texture temporal correlations are shown inFig. 2(a) and
(b), respectively. The variation of the correlationproperties of
sea clutter with time does not appear to havebeen reported upon. In
this simulation, we will assume thatthe speckle and texture are
stationary processes. In Section IIIhowever, the speckle covariance
and texture values are dynam-ically estimated and therefore the
stationarity assumption is notnecessary.
C. Simulation Setup
At the start of each simulation consisting of 25 dwells,
thetarget is located at a distance of 10 km from the sensor
andmoves away from it at a near-constant velocity of 5 m/s. At a
car-rier frequency of GHz, this results in a Doppler shiftof
approximately Hz. The clutter density is adjusted sothat we obtain
an average of 20 scatterers per bin, each of whichis distributed
uniformly in range over the extent of the range binand uniformly in
Doppler over Hz.
The waveform transmitted in Sub-dwell 1 of each dwellwas chosen
to be a linear frequency-modulated (LFM) chirp ofduration 1.5 s and
a frequency sweep of 100 MHz. Note thatthe LFM chirp was chosen
because it is widely used in radarand provides a useful benchmark
for performance comparison.In order to ensure a fair comparison,
the LFM chirp was chosen
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SIRA et al.: ADAPTIVE WAVEFORM DESIGN 63
Fig. 3. L norm of (a) the matched-filtered output kr k and (b)
the clutter-suppressed output kr k , for a typical dwell with SCR =
�35 dB. The truetarget range bin is marked with an asterisk.
to have the same time-bandwidth product as the designed
wave-form, . Each transmitted pulse has unit energy and a totalof
pulses were transmitted in each sub-dwell. The pulserepetition
interval (PRI) was 100 s so that the duration of eachsub-dwell was
1 ms, which is well within the decorrelation timeof the speckle
component, as seen in Fig. 2(a). The samplingfrequency was 100 MHz
so that the number of samples inthe signal was . The amplitude of
the target return wassampled from a zero mean complex Gaussian
process with co-variance matrix , where was chosen to satisfy
specifiedvalues of SCR. We define the SCR to be the ratio of the
targetsignal power to the total power of the clutter in the range
bincontaining the target. It is thus the SCR at the input to the
re-ceiver. For the waveform design, we use phase func-tions in
(29).
Example 1: Sea Clutter Suppression The first exampledemonstrates
the advantage of using the subspace-based ap-proach for the
suppression of sea clutter. In this example, onlythe Sub-dwell 1
signal, is transmitted. A plot of typicalvalues of the norms, and ,
is shown in Fig. 3for dB. The range bin that contains the targetis
marked with an asterisk. The ROC curves for this case areshown in
Fig. 4 for different SCR. For comparison, the ROCcurves obtained by
GLRT detection on the raw matched filtereddata in (6) are also
shown. It is apparent that a performanceimprovement exceeding 5 dB
SCR is obtained. The range binsthat are investigated in Sub-dwell 2
with a designed waveformare determined by the detections in
Sub-dwell 1. Thus, im-proved detection in Sub-dwell 1, as evidenced
by Fig. 4, resultsin fewer false alarms to be investigated and
leads to reducedsystem usage and improved efficiency.
Example 2: Waveform Design for improved Detection Inthe second
example, we examine the performance at the end ofSub-dwell 2. For
the waveform design, we used largestvalues of the texture estimates
to position the zeros of the auto-correlation function of . A
typical result of the waveformdesign algorithm is shown in Fig.
5(a), where the magnitude of
Fig. 4. ROC curves for a GLRT detector operating on unsuppressed
(dottedlines) and clutter-suppressed data (dashed lines). The
numbers on the curvesindicate SCR values in decibels.
Fig. 5. (a) Comparison of the magnitude of the autocorrelation
function of thedesigned PM waveform s [n] (dashed lines) with that
of the LFM chirp used ass [n] (solid lines). (b) A zoomed view of
the former with asterisks marking therange bins with large texture
values.
the autocorrelation function of is shown. From the zoomedview of
the nulls in Fig. 5(b), it can be observed that the value of
is indeed small in the range bins where the clutter wasestimated
to be strong (which are marked by asterisks). How-ever, it is also
evident that some sidelobes of the designed wave-form are much
higher than the corresponding sidelobes of theLFM chirp. Since
these sidelobes occur where weak or negli-gible clutter has been
estimated, their effect on the detectionprocess is negligible.
The ROC curves for different SCR values at the end of Sub-dwell
2 are shown in Fig. 6. Note that these are conditioned onthe actual
target range bin being interrogated. We also show theROC curves at
the end of Sub-dwell 1 for comparison. We canobserve around 10 dB
improvement in detection performancewhen dynamic waveform design is
used. For example, the ROC
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64 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1,
NO. 1, JUNE 2007
Fig. 6. ROC curves at the end of Sub-dwell 1 (dashed) and using
adaptive wave-form design in Sub-dwell 2 (solid). The numbers on
the curves indicate SCRvalues in dB.
curve for dB SCR at the end of Sub-dwell 2 is comparableto the
ROC curve for dB SCR at the end of Sub-dwell 1.Also, at a
probability of false alarm , the probabilityof detection improves
by 76% and 30% at SCR of dBand dB, respectively, when the designed
waveform is used.These gains may be attributed to the fact that the
false alarms dueto clutter are significantly reduced when the
designed waveformis used.
In order to further investigate the benefits of dynamic
wave-form adaptation, we consider a scenario where a dwell is not
di-vided into sub-dwells and all the pulses transmitted are
identical.Here, pulses identical to the LFM chirp described
inSection VI-C are transmitted and the resulting ROC curves
arecompared in Fig. 7 to the performance of the dynamic
waveformadaptation algorithm. As it may be expected, the
performancegains due to dynamic waveform adaptation reduce from
those inFig. 6 due to the increased pulse integration because the
returnsfrom 20 fixed LFM pulses are now processed together
ratherthan from 10 pulses. However, the gains are still around
6–7dB. It is also important to note that the processing of each
burstof identical pulses assumes that the radar scene has not
changedduring the transmission of the burst. Thus, as the pulse
durationis increased, it becomes less likely that this assumption
remainssatisfied.
VII. CONCLUSION
The problem of detecting small targets in sea clutter has
beenthe subject of much research and has led to the developmentof
clutter rejecting waveforms and improvements in
detectionperformance. The knowledge about the statistical
properties ofsea clutter gained in the last few decades was used to
further im-prove on these advancements. The application of dynamic
wave-form design to radar operations is a relatively new
developmentfollowing the availability of flexible waveform
generators andwaveform-agile sensors.
Fig. 7. ROC curves usingK = 20 LFM chirp pulses (dashed lines),
andK =10 LFM chirp followed by K = 10 dynamically designed
waveforms (solidlines). The numbers indicate SCR values in dB.
In this paper, we proposed an algorithm that utilizes the
ben-efit provided by waveform agility to improve detection
perfor-mance at low SCRs. Using a two-stage procedure, we first
gatherinformation about the clutter statistics and identify a
putativetarget location. The knowledge of the correlation
properties ofthe clutter over a short time period is exploited by a
simple, sub-space-based clutter suppression scheme that enhances
detectionperformance. The waveform for the next transmission is
dynam-ically designed using a mean square optimization technique
ap-plied to phase modulated waveforms so that its
autocorrelationfunction is small where the clutter is estimated to
be strong.This design of the waveform minimizes the smearing of
energyfrom out-of-bin clutter into the range bin under
investigation.A simulation study was presented to demonstrate the
perfor-mance of the algorithm, and reasonable gains in the
detectionperformance over the nonadaptive case were observed. The
ad-vantage of the waveform design can be more readily appreciatedif
the radar scene contains strong reflectors or emitters, such
asother targets or jammers, for example, whose range sidelobescan
mask a weak reflection from another target. The ability toposition
the sidelobes of the designed waveform may have sig-nificant
payoffs in such scenarios.
The waveform design development presented does not in-clude
Doppler processing. Although most elements of the al-gorithm can be
immediately extended to range-Doppler estima-tion, the requirement
to transmit longer waveforms to obtainmeaningful Doppler resolution
may invalidate the assumptionof complete correlation of the texture
across a dwell.
ACKNOWLEDGMENT
The authors thank Prof. M. Greco for providing them accessto the
OHGR sea clutter database (with permission from Prof. S.Haykin).
They also thank Prof. A. Nehorai and Prof. L. Scharffor insightful
discussions. They acknowledge the comments andsuggestions of the
reviewers which have resulted in substantialimprovements in this
paper.
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SIRA et al.: ADAPTIVE WAVEFORM DESIGN 65
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http://soma.crl.mcmaster.ca/ipix/dartmouth/index.html
Sandeep Sira received the M.Tech. degree fromthe Indian
Institute of Technology, Kanpur, India, in1999, and the Ph.D.
degree in electrical engineeringin 2007 from Arizona State
University (ASU),Tempe.
He was a Commissioned Officer in the Corpsof Signals, Indian
Army, from 1988 to 2003. Heis currently a Postdoctoral Research
Associate atASU. His research interests include
waveform-agilesensing, target tracking, and detection and
estimationtheory.
Douglas Cochran received the S.M. and Ph.D.degrees in applied
mathematics from Harvard Uni-versity, Cambridge, MA, and degrees in
mathematicsfrom the Massachusetts Institute of
Technology,Cambridge, and the University of California at SanDiego,
La Jolla.
He has been on the faculty of the Department ofElectrical
Engineering at Arizona State University(ASU), Tempe, since 1989 and
is also affiliated withthe Department of Mathematics and
Statistics. Since2005, he has served as Assistant Dean for
Research
in the Ira A. Fulton School of Engineering at ASU. Between 2000
and 2005,he was Program Manager for Mathematics at the Defense
Advanced ResearchProjects Agency (DARPA). Prior to joining the ASU
faculty, he was SeniorScientist at BBN Systems and Technologies,
Inc., during which time he servedas Resident Scientist at the DARPA
Acoustic Research Center and the NavalOcean Systems Center. He has
been a Visiting Scientist at the AustralianDefence Science and
Technology Organisation and served as a consultant toseveral
technology companies. His research is in applied harmonic
analysisand statistical signal processing.
Dr. Cochran was General Co-Chair of the 1999 IEEE International
Confer-ence on Acoustics, Speech, and Signal Processing (ICASSP-99)
and Co-Chairof the 1997 U.S.-Australia Workshop on Defense Signal
Processing. He hasalso served as Associate Editor for book series
and journals, including the IEEETRANSACTIONS ON SIGNAL
PROCESSING.
-
66 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1,
NO. 1, JUNE 2007
Antonia Papandreou-Suppappola (M’95–SM’03)received the Ph.D.
degree in electrical engineeringin 1995 from the University of
Rhode Island (URI),Kingston.
Upon graduation, she held a research faculty posi-tion at URI
with Navy funding. In 1999, she joinedArizona State University,
Tempe, and was promotedto Associate Professor in 2004. Her research
inter-ests are in the areas of waveform design for agilesensing,
integrated sensing and processing, time-fre-quency signal
processing, and signal processing for
wireless communications. She is the editor of the book
Applications in Time-Frequency Signal Processing (CRC, 2002).
Dr. Papandreou-Suppappola is the recipient of the 2002 NSF
CAREER awardand 2003 IEEE Phoenix Section Outstanding Faculty for
Research award. Sheserved as the Treasurer of the Conference Board
of the IEEE Signal ProcessingSociety from 2004 to 2006. She is
currently serving as an Associate Editor forthe IEEE TRANSACTIONS
ON SIGNAL PROCESSING and as a Technical CommitteeMember of the IEEE
Signal Processing Society on Signal Processing Theoryand Methods
(2003-2008).
Darryl Morrell received the Ph.D. degree in elec-trical
engineering in 1988 from Brigham Young Uni-versity, Provo, UT.
He is currently an Associate Professor in theDepartment of
Engineering, Arizona State Univer-sity at the Polytechnic Campus,
Mesa, where he isparticipating in the design and implementation ofa
multidisciplinary undergraduate engineering pro-gram using
innovative research-based pedagogicaland curricular approaches. His
research interestsinclude stochastic decision theory applied to
sensor
scheduling and information fusion. He has received funding from
the ArmyResearch Office, the Air Force Office of Scientific
Research, and DARPA toinvestigate different aspects of Bayesian
decision theory with applications totarget tracking, target
identification, and sensor configuration and schedulingproblems in
the context of complex sensor systems and sensor networks.His
publications include over 50 refereed journal articles, book
chapters, andconference papers.
William Moran (M’95) is Research Director ofMelbourne Systems
Laboratory and a Professo-rial Fellow with the Department of
Electrical andElectronic Engineering, University of
Melbourne,Melbourne, Australia. He has participated in nu-merous
signal processing research projects for U.S.and Australian
government agencies and industrialsponsors. He has published
extensively in bothpure and applied mathematics. He has authored
orco-authored well over 100 published mathematicalresearch
articles. His main areas of interest are
in signal processing, particularly with radar applications,
waveform designand radar theory, and sensor management. He also
works in various areas ofmathematics, including harmonic analysis
and number theory.
Dr. Moran is a Fellow of the Australian Academy of Science and a
Memberof Australian Research Council College of Experts. He also
serves as a Consul-tant to the Australian Department of Defence,
through the Defence Science andTechnology Organization.
Stephen D. Howard received the B.S, M.S., andPh.D. degrees from
La Trobe University, Melbourne,Australia, in 1982, 1984, and 1990,
respectively.
He joined the Australian Defence Science andTechnology
Organisation (DSTO) in 1991, wherehe has been involved in the area
of electronicsurveillance and radar for the past 14 years. He
hasled the DSTO research effort into the developmentof algorithms
in all areas of electronic surveillance,including radar pulse train
deinterleaving, precisionradar parameter estimation and tracking,
estimation
of radar intrapulse modulation, and advanced geolocation
techniques. Since2003, he has led the DSTO long-range research
program in radar resourcemanagement and waveform design.
Robert Calderbank (M’89–SM’97–F’98) receivedthe B.Sc. degree in
1975 from Warwick University,Warwick, U.K., the M.Sc. degree in
1976 from Ox-ford University, Oxford, U.K., and the Ph.D. degreein
1980 from the California Institute of Technology,Pasadena, all in
mathematics.
He is currently Professor of electrical engineeringand
mathematics at Princeton University, Princeton,NJ, where he directs
the Program in Applied andComputational Mathematics. He joined Bell
Tele-phone Laboratories as a Member of Technical Staff
in 1980, and retired from AT&T in 2003 as Vice President of
Research. Hehas research interests that range from algebraic coding
theory and quantumcomputing to the design of wireless and radar
systems.
Dr. Calderbank served as Editor in Chief of the IEEE
TRANSACTIONS ONINFORMATION THEORY from 1995 to 1998, and as
Associate Editor for CodingTechniques from 1986 to 1989. He was a
member of the Board of Governors ofthe IEEE Information Theory
Society from 1991 to 1996 and began a secondterm in 2006. He was
honored by the IEEE Information Theory Prize PaperAward in 1995 for
his work on the Z linearity of Kerdock and Preparata Codes(joint
with A. R. Hammons, Jr., P. V. Kumar, N. J. A. Sloane, and P.
Sole), andagain in 1999 for the invention of space-time codes
(joint with V. Tarokh andN. Seshadri). He received the 2006 IEEE
Donald G. Fink Prize Paper Awardand the IEEE Millennium Medal, and
was elected to the National Academy ofEngineering in 2005.