56 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL...

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

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)


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.


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


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


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)


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


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


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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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.


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.

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