78 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 1, JANUARY 2011

Adaptive OFDM Radar for Target Detection inMultipath Scenarios

Satyabrata Sen, Student Member, IEEE, and Arye Nehorai, Fellow, IEEE

Abstract—We develop methods for detecting a moving target inthe presence of multipath reflections, which exist, for example, inurban environments. We take advantage of the multipath prop-agation that increases the spatial diversity of the radar systemand provides different Doppler shifts over different paths. Weemploy a broadband orthogonal frequency division multiplexing(OFDM) signal to increase the frequency diversity of the systemas different scattering centers of a target resonate variably atdifferent frequencies. To overcome the peak-to-average powerratio (PAPR) problem of the conventional OFDM, we also useconstant-envelope OFDM (CE-OFDM) signaling scheme. First,we consider a simple scenario in which the radar receives onlya finite number of specularly reflected multipath signals. Wedevelop parametric measurement models, for both the OFDMand CE-OFDM signaling methods, under the generalized multi-variate analysis of variance (GMANOVA) framework and employthe generalized likelihood ratio (GLR) tests to decide about thepresence of a target in a particular range cell. Then, we proposean algorithm to optimally design the parameters of the OFDMtransmitting waveform for the next coherent processing interval.In addition, we extend our models to study the aspects of temporalcorrelations in the measurement noise. We provide a few numer-ical examples to illustrate the performance characteristics of theproposed detectors and demonstrate the achieved performanceimprovement due to adaptive OFDM waveform design.

Index Terms—Adaptive waveform design, asymptotic perfor-mance analysis, multipath, OFDM radar, target detection, urbanscenarios.

I. INTRODUCTION

T HE problem of detection and tracking targets in the pres-ence of multipath, particularly in urban environments,

are becoming increasingly relevant and challenging to radartechnologies. In [1], we have shown that the target detectioncapability can be significantly improved by exploiting multipleDoppler shifts corresponding to the projections of the targetvelocity on each of the multipath components. Furthermore, themultipath propagations increase the spatial diversity of the radarsystem by providing extra “looks” at the target and thus enablingtarget detection and tracking evenbeyond the line-of-sight (LOS)

Manuscript received January 04, 2010; accepted October 04, 2010. Date ofpublication October 11, 2010; date of current version December 17, 2010. Thiswork was supported by the Department of Defense under the Air Force Of-fice of Scientific Research MURI Grant FA9550-05-1-0443 and ONR GrantN000140810849. The associate editor coordinating the review of this manu-script and approving it for publication was Dr. Deniz Erdogmus.

The authors are with the Department of Electrical and Systems Engineering,Washington University in St. Louis, St. Louis, MO 63130 USA (e-mail:ssen3@ese.wustl.edu; nehorai@ese.wustl.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSP.2010.2086448

[2], [3]. Other areas of application in which multipath effectsare of primary interest are in low-angle tracking (sea-skimmers)[4]–[7], height finding [8], [9], and radar-aided navigation andlanding systems [10]. Similar problems have been addressed insonar literature due to bottom bounce in shallow waters [11],[12]. Note that in [13] we have demonstrated that the direction-finding capability of a radar system can be improved also byexploiting multipath reflections close to the sensors.

To resolve and exploit the multipath components it is gener-ally common to use short pulse, multi-carrier wideband radarsignals. We consider the orthogonal frequency division multi-plexing (OFDM) signaling scheme [14], [15], which is one ofthe ways to accomplish simultaneous use of several subcarriers.The use of OFDM signal mitigates the possible fading, resolvesthe multipath reflections, and provides additional frequency di-versity as different scattering centers of a target resonate at dif-ferent frequencies.

Although OFDM has been elaborately studied and com-mercialized in the digital communication field [16], it has notso widely been studied by the radar community apart froma few recent efforts [17]–[19]. One of major reasons of suchunpopularity is that OFDM has a time-varying envelope andthat originates a potentially high peak-to-average power ratio(PAPR) [20], [21]. A high value of PAPR demands for systemcomponents (e.g., transmitter’s power amplifier) with a largelinear region of operation. However, practical power amplifiersoperate over limited linear region, beyond which they saturatecausing nonlinear distortion to the signal [22].

Over the years, a number of approaches have been proposedto deal with the PAPR problem. A comprehensive survey ofPAPR reduction techniques can be found in [20] and [23, Ch.6]. One of such methods is to apply the phase modulationtransform that achieves the lowest possible PAPR (0 dB). Inthis work, besides considering conventional OFDM, we alsoinclude the constant-envelope OFDM (CE-OFDM) signalingscheme [21], [24]–[27], which is based on using a real-valuedbaseband OFDM signal to phase modulate the carrier.

First, we discuss a detection problem in which the radar hasthe complete knowledge of the first-order (or single bounce)specularly reflected multipath signals. We also assume thatthe clutter and measurement noise are temporally white. InSection II, we develop the measurement models, for boththe OFDM and CE-OFDM signaling schemes, under thegeneralized multivariate analysis of variance (GMANOVA)framework [28], [29]. Based on these models, in Section III, weformulate the detection problem as a hypothesis test to decideabout the presence of a target in a particular range cell. Dueto the lack of knowledge of all the parameters in our models,

1053-587X/$26.00 © 2010 IEEE

SEN AND NEHORAI: ADAPTIVE OFDM RADAR FOR TARGET DETECTION IN MULTIPATH SCENARIOS 79

we employ the generalized likelihood ratio (GLR) test [30, Ch.6]. We present numerical results to evaluate the performanceof these proposed detectors, as we do not have any analyticalexpressions to evaluate their performances.

Then, in Section IV, we propose a criterion to adaptivelycompute the parameters of the next transmitting waveform. Toconstruct such a criterion we first look into the performancecharacteristics of the GLR test statistics for both OFDM andCE-OFDM models assuming that the target velocity is known.However, this analysis does not characterize the detection per-formance of our detectors, in which the target velocity is un-known. The analysis with known target velocity shows that theGLR test results in constant false alarm rate (CFAR) detectorsfor both OFDM (with large number of temporal samples) andCE-OFDM (with finite number of temporal sample) models, andthe detection performances depend on the system parametersthrough the corresponding noncentrality parameters of the dis-tributions under alternate hypothesis. This implies that it is pos-sible to improve the detection performance by maximizing thesenoncentrality parameters. We apply this idea to our problem andformulate the optimization problem to select the parameters ofthe next transmitting waveform that maximizes the same ex-pression of the noncentrality parameter subject to a fixed trans-mission-energy constraint. For the OFDM model, we show thatthe solution of this optimization problem results in an eigen-vector corresponding to the largest eigenvalue of a matrix thatdepends on the target, clutter, and noise parameters. However,for the CE-OFDM model we cannot improve the detection per-formance in this way because the noncentrality parameter doesnot depend on the transmitting waveform.

Later in the paper, in Section V, we relax the assumptionof temporal whiteness to study the effects of temporally cor-related measurement noise process on our models. Temporalcorrelations exist in certain radar applications, in particular athigh pulse repetition frequencies (PRF) [31], [32]. To model thetemporal correlation matrix, we look into a branch of statisticsknown as the nearest neighbor analysis [33], [34], and presentthe consequent detection tests.

To illustrate the potential of our proposed detectors, wepresent numerical examples in Section VI. We find that thewideband OFDM model performs better than the narrowbandCE-OFDM model in exploiting the multipath reflections. Inaddition, we achieve significant performance improvement dueto adaptive OFDM waveform design. However, the CE-OFDMsignal lacks such an adaptive design as the detection per-formance does not depend on the transmitting coefficients.Finally, we give concluding remarks and some thoughts on afew unaddressed issues in Section VII.

Notations: We list here some notational convention that willbe used throughout this paper. We use math italic for scalers,lowercase bold for vectors, and uppercase bold for matrices. Fora matrix , , , , , , , ,and denote the transpose, conjugate-transpose, deter-minant, th entry, generalized inverse (such that

), trace, vec-operation, and block-diagonal vec-operation (de-fined in [35, eq. (7)]) of , respectively. represents an iden-tity matrix of dimension . forms a square matrix withnonzero entries only on the main diagonal. Additionally, ,

Fig. 1. Schematic representation of the multipath scenario.

, and are the inner-product, Kronecker product, and ele-ment-wise Hadamard product operators, respectively.

II. PROBLEM DESCRIPTION AND MODELING

We consider a far-field point target moving with a constantrelative velocity , with respect to the radar, in a multipath-richenvironment, as shown in Fig. 1. At the operating frequency,we assume that the reflecting surfaces produce only specular re-flections of the radar signal. We assume that the radar has thecomplete knowledge of the environment that is under surveil-lance. Hence, for every range cell the radar knows the numberof possible multipath between the radar and target and thedirection-of-arrival (DOA) unit-vectors ( ,

) along each such path. Under this scenario, we first intro-duce the parametric measurement models for both OFDM andCE-OFDM signaling techniques. Then, we discuss our statis-tical assumptions on the clutter and noise.

A. OFDM Measurement Model

We consider an OFDM signaling system [15] with ac-tive subcarriers, a bandwidth of Hz, and pulse durationof seconds. Let represents thecomplex weights transmitted over the subcarriers, satisfying

. Then, the complex envelope of the transmittedsignal can be represented as

(1)

where denotes the subcarrier spacing.Let be the carrier frequency of operation, then the transmittedsignal is given by

(2)

where represents the th subcarrier frequency.Interchanging the real and summation operators, we can alsorewrite (2) as

(3)

80 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 1, JANUARY 2011

where

(4)

represents the transmitted signal due to the th subcarrier only.Then, the received signal along the th path (represented by

the DOA vector ) due to only the th subcarrier can be writtenas

(5)

where is a complex quantity representing the scattering coef-ficient of the target along the th subchannel and th path;

where is the relative Doppler shiftalong the th path and is the speed of propagation; is theroundtrip delay between the radar and target along the th path;

represents the clutter and measurement noise along the thsubchannel and th path. Therefore, the received signal over all

available paths due to an -carrier OFDM signal is given by

(6)

and hence the corresponding complex envelope is given as

(7)

Let us assume at this point that the relative time gaps betweenany two multipath signals are very small in comparison to theactual roundtrip delays, i.e., for .These assumptions can be justified in systems where the pathlengths of multipath arrivals differ little (e.g., narrow urbancanyon where the range is much greater than the width). Fur-ther, let us denote as the roundtrip delay corresponding tothe range cell under consideration. Then, the information ofthe roundtrip delays can be automatically incorporated into themodel by choosing , ,where is the pulse repetition interval (PRI) and is thenumber of temporal measurements within a given coherentprocessing interval (CPI). Hence, corresponding to a specificrange cell containing the target, the complex envelope of thereceived signal at the output of the th subchannel is

(8)

where

(9)

Stacking the measurements of all subchannels into onecolumn vector of dimension , we get

(10)where

•• is an complex diagonal matrix that

contains the transmitted weights ;• is an complex

rectangular block-diagonal matrix where each nonzeroblock , ,represents the scattering coefficients of the target at the thsubchannel over all multipath;

•is an complex vector where

, ,contains the Doppler information of the target at the thsubchannel over all multipath;

• is a column vector containing the unknown target-ve-locity components;

• is an vectorof clutter returns, measurement noise, and co-channel in-terference.

Then, concatenating all the temporal data columnwise into anmatrix we obtain the OFDM measurement model as

follows:

(11)

where• ;• is an

matrix containing the Doppler information of the targetthrough the parameter ;

• is an matrix com-prising clutter returns, noise, and interference.

B. CE-OFDM Measurement Model

A CE-OFDM signal is realized by using a real-valued base-band OFDM signal to phase modulate the carrier. The complexenvelope of a CE-OFDM transmitted signal is represented as[21]

(12)

where is the modulation index in radians and messagesignal bears an OFDM signal structure

(13)

where are real-valued weights at dif-ferent subcarriers.

Assuming a narrowband signal model (which can be achievedwith small modulation index [21]), the complex envelope of the

SEN AND NEHORAI: ADAPTIVE OFDM RADAR FOR TARGET DETECTION IN MULTIPATH SCENARIOS 81

received signal corresponding to a specific range cell containingthe target can be written as

(14)

where is the target scattering coefficient at the operating fre-quency along the th path, and and are roundtrip delayand relative Doppler shift, respectively, along the th path.

Then, as before assuming that all the multipath delays areapproximately equal, i.e., for , andrepresenting , , we cansimplify (14) into

(15)

where• represents the scattering coeffi-

cients of the target at the carrier frequency over all mul-tipaths;

• contains theDoppler information of the target over all multipathswhere

(16)

Then, concatenating all the temporal data columnwise into anvector, we obtain the CE-OFDM measurement model as

follows:

(17)

where•

;• is a

matrix containing the Doppler information of thetarget through the parameter ;

• is an vector com-prising clutter returns, noise, and interference.

From the structures of (11) and (17), it is quite evident thatthe CE-OFDM measurement model resembles a similar formas that of the OFDM case when . However, note thatthe transmitting weights influence the OFDM measurementsthrough the matrix whereas CE-OFDM measurementsthrough the matrix .

C. Statistical Model

In our problem, the clutter could be the contribution of anyundesired reflections from the environment surrounding and/orbehind the target, or any random multipath reflections, from the

irregularities on the reflecting surface (e.g., windows and bal-conies of the buildings in an urban scenario), that cannot bemodeled as specular components. Therefore, for both OFDMand CE-OFDM measurement models, we assume that the clutterand noise are temporally white and circularly symmetric zero-mean complex Gaussian process with unknown covariances. In(10), the noise vector contains the clutter returns, noise, andco-channel interference at the output of subchannels, whichwe assume to be correlated with unknown positive definite co-variance matrix . Hence, the OFDM measurements are dis-tributed as

(18)

In (15), we assume that variance of is , and therefore theCE-OFDM measurements are distributed as

(19)

In these formulations, when the parameter is known, both(11) and (17) comply with the generalized multivariate analysisof variance (GMANOVA) structure [28], [36], which has beenstudied extensively in statistics and applied to a number of ap-plications in signal processing [29].

III. DETECTION TEST

In this section, we develop statistical detection tests forthe OFDM and CE-OFDM measurement models presented inSection II. Our goal is to decide whether a target is present ornot in the range cell under consideration.

We construct the decision problem to choose between twopossible hypotheses: the null hypothesis (target-free hypoth-esis) or the alternate hypothesis (target-present hypothesis).The tests can be expressed as

for OFDM modelunknown

unknown(20)

for CE-OFDM modelunknown

unknown.(21)

Because of the lack of knowledge about and (or ) we usethe generalized likelihood ratio (GLR) test [30, Ch. 6] in whichthe unknown parameters are replaced with their maximum like-lihood estimates (MLE). This approach also provides the infor-mation about the unknown parameters since the first step is tofind the MLEs.

Assuming that the parameter is known in (11), the GLR testfor (20) compares the ratio of the likelihood functions under thetwo hypotheses with a threshold as follows [30, Ch. 6.4.2]:

(22)

where and are the likelihood functions under and, and are the MLEs of under and , is the

MLE of under , and is the detection threshold. After

82 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 1, JANUARY 2011

some algebraic manipulations, it can be shown that the teststatistic of this problem is [29]

(23)

However, having a block-diagonal structure, the scattering ma-trix does not yield a closed-form MLE expression [35]. So,in this work, we use an approximate ML (AML) estimator for

, defined as [35, eq. (28)]

(24)

where

(25a)

(25b)

(25c)

(25d)

(25e)

(25f)

In (25e), is the th column of an matrix ,and in (25f), is a matrix representing the th block of

rows of . It is important to note here that the conditionmentioned in [35] is a typographical error. Because,

the actual assumption is that the columns of the individual sub-matrices (of [35]) need to be linearly independent and thatrequires , which transforms into in our work.

In our problem, however, the parameter is un-known, and therefore we compare the GLR test statistic

with a threshold.Similarly, when is known in (17), the GLR test statistic for

(21) can be expressed as (26), shown at the bottom of the page,where the exact ML expression of is given by

(27)

As before, since in our problem is unknown, the GLR testcompares witha threshold.

IV. ADAPTIVE WAVEFORM DESIGN

In this section, we develop an adaptive waveform design tech-nique to improve the target-detection performance. To derive amathematical formulation for optimal waveform selection, wefirst create a utility function according to certain criteria andthen determine the parameters for the next transmitting wave-form by optimizing this utility function. To construct such autility function we first consider the detection performance as-suming known target velocity parameter . Then, we explicitlystate the optimization problem and its solution.

A. Distributions of the Test Statistic for Known Target Velocity

In this subsection, we derive the distributions ofand when the target velocity

is known. The motivation behind these derivations is to look fora criterion of adaptive waveform design and not to characterizethe detection performance with known target velocity, becausein our problem the target velocity is unknown.

1) OFDM Detector: Under our OFDM measurementmodel is and the correspondingGLR test statistic is given by [37, Th. 3.10], [38, eq. (4)–(33)]

(28)

where ’s are mutually independent complex beta distributedrandom variables with and complexdegrees of freedom, written as

(29)and .

Under we have where. However, the distribution of

does not have a closed-form expression. In general we canexpress the GLR test statistic as the ratio of determinantsof two random matrices and

,

(30)

where and follow noncentral complex Wishart dis-tributions, denoted as and

, respectively; and ,

(26)

SEN AND NEHORAI: ADAPTIVE OFDM RADAR FOR TARGET DETECTION IN MULTIPATH SCENARIOS 83

are the noncentrality parameters [39, Th.7.8.1, Cor. 7.8.1.1]. Since

(31)

(32)

we find that follows a complex Wishart distribution of order, parameter , and complex degrees of freedom, de-

noted as

(33)

and follows a noncentral complex Wishart distribution withorder , parameter , complex degrees of freedom, and

as the noncentrality parameter, denoted as

(34)

We cannot simplify (30) any further such that it has a distribu-tion with closed-form expression.

However, in a special case, when (which istermed as the “linear case” after Anderson [40], [41] in somestatistical literature) the test statistic undercan be written as a product of independent complex betarandom variables where one of the beta variables is noncen-tral [42], [43]. The noncentrality parameter is given as the singlenonzero root of the equation , which is sameas . Here we remark that to achieve

in our problem we have to use single frequency signal insteadof multi-frequency OFDM signal.

Since, the distribution of the GLR test statistic for the OFDMmeasurement model does not have a closed-form expression fora finite value of , we explore the asymptotic performance char-acteristics of (23) assuming known target velocity. Following ananalogous discussion on real Gaussian variables from [44, Ch.8], [45] we find that as , under ,has a complex chi-square distribution with complex degreesof freedom, denoted as

(35)

Under , the limiting distribution of isa complex noncentral chi-square distribution with complexdegrees of freedom, denoted as

(36)

where is the noncentrality parameter andare the roots of . Obviously

another way to represent the same noncentrality parameter is. We may call the matrix as

the “signal-to-noise ratio matrix,” and hence the trace of it canbe considered as a sum of squared Mahalanobis distances [46].

2) CE-OFDM Detector: The distributions of the GLR teststatistic of the CE-OFDM model assuming known target ve-locity, , can be evaluated in a compara-tively easy way either through a direct computation of complexchi-square distributions using (19) in (26), or as a special case ofthe OFDM model with . Under the GLR test statisticfollows complex beta distribution with and complexdegrees of freedom, written as

(37)

where . Under the GLR teststatistic follows a complex noncentral beta distribution with thesame complex degrees of freedom and and noncentralityparameter , denoted as

(38)

where .

B. Waveform Design

From the discussion of the previous subsection, it is clearthat the GLR test results in a constant false alarm rate (CFAR)detector under both the OFDM (when ) and CE-OFDM(for finite ) models when the target velocity is known andthe detection performance depends on the system parametersthrough the noncentrality parameters and , respectively.Therefore, it is possible to improve the detection performanceby maximizing these noncentrality parameters.

However, in our problem the target velocity parameter isunknown. Moreover, in the OFDM measurement model thetarget scattering matrix is block diagonal. Still in our adaptivewaveform design problem we maximize the same expressionof the noncentrality parameter subject to a predefined energyconstraint. Thus, we formulate the optimization problem as

(39)

After some algebraic manipulations (see Appendix) we canrewrite this problem as

(40)

Hence, our optimization problem reduces to a simple eigen-value–eigenvector problem and the optimal solution, ,is the eigenvector corresponding to the largest eigenvalue of

.Note that in our problem , , and are not known.

Hence, we use their estimated values to obtain forthe next CPI. First, a nonoptimal is transmitted and thecorresponding measurements are stored over one particularCPI. Then, we estimate . Sub-stituting into (9) we compute and subsequently

84 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 1, JANUARY 2011

using (24) and (25). Then, the estimate of is evaluated as.

Finally, we choose as the eigenvector corresponding to

the largest eigenvalue of . Thecomputation of such an eigenvector is done using Matlab inreal time.

In a similar fashion, we can formulate the optimizationproblem for the CE-OFDM measurement model as follows:

(41)However, rewriting (16) as , where

it can be easily shown that the th entry of hasthe following form:

(42)

Hence, the noncentrality parameter in the CE-OFDM modeldoes not depend either on or . This implies that the perfor-mance characteristic of the CE-OFDM detector cannot be im-proved by any efficient choice of .

V. TEMPORALLY CORRELATED NOISE

In this section, we extend our models to include temporalcorrelation among the measurements. Recall that so far wehave assumed that the clutter and measurement noise are theindependent realizations of the same Gaussian random processfrom pulse to pulse. However, this assumption may not be validat high pulse repetition frequency [32]. In the following, wefirst present the statistical assumptions of temporally correlatednoise for both the OFDM and CE-OFDM measurement models,presented in (10) and (15), respectively, and then discuss thedetection tests.

A. Statistical Model

To develop a statistical model for the temporally correlatednoise from pulse to pulse measurements, we look into a branchof statistics known as the nearest neighbor analysis [33], [34].Instead of choosing any unstructured covariance matrix, we as-sume a Kronecker product structure of the form forOFDM measurement model (or for CE-OFDM model),where is an unknown positive definite temporal covari-ance matrix. Based on this assumption, the modified versions of(18) and (19) can be written, respectively, as

(43)

and

(44)

The matrix can have any structure or can even be un-structured. In the statistical literature two structured covariancematrices are very common. These are the compound sym-metric (CS) structure [47] and autoregressive structure of order1 (AR(1)) [48]. The CS covariance structure assumes thatall the temporal measurements are equicorrelated and do notdepend on the duration between the two time points, i.e.,

whenotherwise

(45)where is the coefficient of temporal correlation be-tween two time points. In AR(1) covariance structure, the tem-poral measurements are assumed to be more highly correlatedif they are close to each other in time duration, i.e.,

(46)

In our work, we consider the AR(1) model as it is suitable forpulsed radar applications in which the measurements are col-lected at equispaced time intervals. Note that in (45) and (46)we explicitly write to stress the fact that the covariancematrix (and also its inverse) is completely characterized by asingle parameter .

B. Detection Under Temporal Correlations

We construct the decision problem in terms of two possiblehypotheses and , to detect the presence of a target in therange cell under consideration, as

for OFDM modelunknown

unknown(47)

for CE-OFDM modelunknown

unknown.(48)

As before, since the parameters , , and (or ) are unknownwe apply the GLR test.

For the OFDM model the test compares the ratio of the likeli-hood functions maximized, with respect to the unknown param-eters, under the two hypotheses with a threshold as follows:

(49)where the MLEs of and are computed by replacingand with and , re-spectively.

Similarly, for the CE-OFDM model the test performs the fol-lowing comparison:

(50)

SEN AND NEHORAI: ADAPTIVE OFDM RADAR FOR TARGET DETECTION IN MULTIPATH SCENARIOS 85

Fig. 2. A schematic representation of the multipath scenario considered fornumerical examples.

where we use andinstead of and , respectively, to compute theMLEs of and .

VI. NUMERICAL RESULTS

In this section, we present the results of several numerical ex-amples to illustrate the performance characteristics of our pro-posed detectors, presented in Section III, as the associated GLRtest statistic with unknown target velocity does not have anyclosed-form analytical expression. For simplicity, we considera 2-D scenario, where both the radar and target are in the sameplane, as shown in Fig. 2. Our analyses can easily be extendedto 3-D scenarios. First, we provide a description of the simula-tion setup, and then discuss different numerical examples.

• Target and multipath parameters:— The target is moving with velocity

. This implies that.

— Throughout a given CPI, the target remains within a par-ticular range cell. We simulated the situation of a rangecell centered at 2 km North and 5 m East with respect tothe radar (positioned at the origin).

— There exists three different paths (i.e., ) betweenthat particular range cell and the radar. These are one di-rect path and two specular multipaths due to a couple ofreflecting surfaces oriented along North-South directionat 10 m East and 10 m West.

— The scattering coefficients of the target (i.e., the entriesof or ) are generated from a distribution.

• Radar parameters;— Carrier frequency 1 GHz.— Available bandwidth 100 MHz.— Pulse repetition interval 20 s.— Pulse width 50 ns.— Number of coherent pulses .— OFDM signal operates with active subcarriers

and the subcarrier spacing of 20 MHz.

Fig. 3. Effects of different SNR values on detection probability as a functionof probability of false alarm.

— CE-OFDM employs modulation index .— All the transmit weights are unity, i.e., .

We performed Monte Carlo simulations based on 20 000 inde-pendent trials to realize the following results. For the OFDMmeasurement model, the entries of were realized from the

distribution and then were scaled to satisfy the re-quired signal-to-noise ratio (SNR), defined as

Similarly, for the CE-OFDM model the value of was chosento achieve the required SNR value, defined as

A. Detector Performance

Fig. 3 depicts the variations of probability of detectionas a function of probability of false alarm at three

different SNR values for the OFDM and CE-OFDM signalingschemes. As expected, for both the measurement models thedetection performance improves as SNR is increased. However,at a fixed SNR, the detection performance of the widebandOFDM model is much better than that of the narrowbandCE-OFDM model. Because being a wideband signal the con-ventional OFDM can resolve the multipath and overcomefading. Additionally, it can exploit the target response at mul-tiple frequencies.

To show the advantage of using multi-frequency signalingsystem, we compared the detection performance at three dif-ferent values of , while keeping the SNR fixed at 5 dB.The results are presented in Fig. 4. Hence, it is evident that thefrequency diversity improves the target-detection performancein an OFDM system. However, this is not true in the case of

86 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 1, JANUARY 2011

Fig. 4. Effects of different number of subcarriers on detection probability as afunction of probability of false alarm.

Fig. 5. Effects of different directions of target velocity vector on detectionprobability as a function of probability of false alarm.

CE-OFDM model. As we showed in Section IV, the perfor-mance of the CE-OFDM detector does not depend on the valueof .

We studied the effects of different directions of the velocityvector on the detection performance at three different condi-tions: [0,10], [7.07, 7.07], [10, 0] m/s. The resultsare depicted in Fig. 5. For this simulation we kept the SNRfixed at 5 dB. As the angle between the target velocity vectorand radar LOS increases the performance of both the OFDMand CE-OFDM detectors deteriorate. However, the widebandOFDM model can resolve and exploit the multipath reflectionsbetter than the CE-OFDM model, and hence it shows improvedperformance even close to the LOS scenario.

B. Importance of Multipath Modeling

To understand the importance of proper exploitation ofmultipath reflections we devised the following simulations.We changed the velocity of the target such that it moves

Fig. 6. Effects of exploiting the multipath reflections on detection probabilityas a function of probability of false alarm in (a) OFDM and (b) CE-OFDMmodels.

perpendicular to the LOS direction. Then, we compared thedetection performances of the two systems: one of them con-siders all multipath reflections, and the other considersonly the LOS return. Fig. 6(a) and (b) shows the results at

5 dB for OFDM and CE-OFDM models, respec-tively. The OFDM signal, being a wideband, can better exploitthe multipath reflections to improve the detection performancecompared to the narrowband CE-OFDM signal. Moreover, thepresence of multipath reflections causes a small performancedegradation for CE-OFDM model as it cannot resolve themultipath returns.

A similar conclusion can also be drawn from Fig. 7. For thissimulation, we generated the measurements, for both the OFDMand CE-OFDM signaling schemes, in the presence of multipathusing the same setup as described before. However, in the de-tector we ignored the presence of multipaths in the tested rangecell. This may happen when the radar does not have a knowl-edge of the possible multipath scenario. From Fig. 7(a), we see

SEN AND NEHORAI: ADAPTIVE OFDM RADAR FOR TARGET DETECTION IN MULTIPATH SCENARIOS 87

Fig. 7. Effects of ignoring the multipath reflections at the detector on detec-tion probability as a function of probability of false alarm in (a) OFDM and (b)CE-OFDM models.

that the performance deteriorates when the detector in OFDMmodel ignores the multipath reflections. On the contrary, sincethe CE-OFDM model cannot resolves the multipath returns, itperforms a little better [see Fig. 7(b)] when the detector ignoresthe multipath.

C. Adaptive OFDM Waveform Design

To study the improvement in target-detection performancedue to the proposed adaptive waveform design technique, wedevised a simple problem. We assumed a system in which wetransmit in the first pulses. Then, based on thecorresponding measurements we solved (40) to compute the op-timized values of ’s for the next pulses. We compared thissystem with another system in which both the two sets ofpulses transmit . We fixed the SNR at 5 dB forthis simulation. From Fig. 8(a), we observe that the detectionperformance of the adaptive system is considerably improvedfor the OFDM model. However, this is not the case with the

Fig. 8. Gain due to adaptive waveform design of detection probability as afunction of probability of false alarm in (a) OFDM and (b) CE-OFDM models.

CE-OFDM model, as evident from Fig. 8(b). This again con-forms our derivation in Section IV that the noncentrality param-eter in the CE-OFDM model does not depend on the values of .

D. Under Temporal Correlation

Fig. 9 depicts the detection performance at three differentvalues of temporal correlation coefficient, , for the OFDM andCE-OFDM signaling schemes. We used the AR(1) covariancestructure, described in (46), to introduce temporal correlationamong the measurements. For this simulation we kept the SNRfixed at 5 dB. It is evident from this analysis that the target-de-tection performance deteriorates as the level of temporal corre-lation increases.

E. Detector Performance for Known Target Velocity

Finally, in Fig. 10, we show comparative performance resultsof the OFDM and CE-OFDM detectors for known and unknowntarget velocity . We plot these results at two different SNRvalues. The detection performance for known target velocity

88 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 1, JANUARY 2011

Fig. 9. Effects of different temporal correlations on detection probability as afunction of probability of false alarm.

Fig. 10. Loss due to the target-velocity estimation of detection probability as afunction of probability of false alarm in (a) OFDM and (b) CE-OFDM models.

may be looked upon as a hypothetical scenario, because in theunderlying detection problem we try to decide about the pres-ence or absence of a target whose velocity is known beforehand.

However, this analysis shows us quantitatively to what extentthe detection performance (for unknown ) degrades due to theestimation of . For example, for the OFDM detector, the valueof drops from 0.56 to 0.3 at and 0 dBdue to the velocity estimation process.

VII. CONCLUSION

In this paper, we addressed the problem of detecting a movingtarget by exploiting multipath reflections. First, we developed themeasurement model accounting for only a finite number of spec-ular multipath reflections. We considered two different schemesto modulate the transmitting carrier: amplitude modulationby employing an orthogonal frequency division multiplexing(OFDM) signal and phase modulation with a constant-envelopeOFDM (CE-OFDM) signal. The use of OFDM signal increasesthe frequency diversity of our system as different scatteringcenters of a target resonate variably at different frequencies. Weformulated the detection problem as a statistical hypothesis testand employed generalized likelihood ratio test to decide aboutthe presence of a moving target in a particular range cell. Then,we proposed an algorithm to optimally design the parametersof the transmitting waveform for the next coherent processinginterval. We showed that for OFDM radar the solution of this op-timization problem results in an eigenvector corresponding to thelargest eigenvalue of a matrix that depends on the target, clutter,and noise parameters. However, for the CE-OFDM model wecannot improve the detection performance in this way becausethe noncentrality parameter does not depend on the choice of thesignal structure (i.e., number of subcarriers and coefficients).In addition, we extended our models to consider the aspects oftemporal correlations in the measurement noise.

Our numerical results demonstrate the performance charac-teristics of the proposed detectors. Though the use of CE-OFDMallows one to overcome the peak to average power ration (PAPR)problem, but it results in worse detection performance com-pared to the conventional OFDM. Because, being a widebandsignal, OFDM outperforms the narrowband CE-OFDM systemby exploiting the multipath reflections. Moreover, since theCE-OFDM signaling does not have different subcarriers, it is notpossible to improve the detection performance by increasing thenumber of subcarriers, as happens in the case of OFDM detector.We also numerically demonstrated the performance improve-ment due to adaptive OFDM waveform design. This couldenable the radar to operate in a closed-loop fashion resultingan improved performance in the presence of nonstationaryclutter or noise. However, the CE-OFDM signal lacks such anadaptive design. Therefore, in future we plan to evaluate thedetection performance of different modulation schemes, otherthan CE-OFDM, that reduce the PAPR problem.

Additionally, in our future work, we will extend our model toincorporate more realistic physical effects, such as diffractions,refractions, and attenuations, which exist, for example, due tosharp edges and corners of the buildings or rooftops in an urbanenvironment. We will expand the detection procedure over mul-tiple rangecells toconsiderother significantmultipath reflectionsand realistically model the clutter returns that depend on trans-mitted signal [32]. As polarization allows identification of cor-related source signals (e.g., multipath) with small angle separa-tion [49], we will also study the detector performance employing

SEN AND NEHORAI: ADAPTIVE OFDM RADAR FOR TARGET DETECTION IN MULTIPATH SCENARIOS 89

polarized transceivers. We will integrate our detection procedurewith a target tracking algorithm and explore other criteria, e.g.,ambiguity function, mutual information, etc., to optimally designthe transmit waveform to improve the system performance. Wewill validate the performance of our proposed detector with realdata.

APPENDIX

In this Appendix, we show how we simplify the objectivefunction in (39) to that in (40).

First, using the relationship between a trace and vec operator,, we get

(A1)

Here, for simplicity of notations, we represent andwith and , respectively.

Then, we apply one of the properties of the vec operator,, to get

(A2)

Note that in our problem is a diagonal matrix.Therefore, can be written as

...(A3)

where is an matrix that has a 1 only at th positionand zero elsewhere. Similarly, we have

...(A4)

Additionally, from [50, Th. 1], we have

......

(A5)

Finally, substituting the results of (A3)–(A5) into (A2)

(A6)

which is same as the objective function of (40).

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Satyabrata Sen (S’07) received the B. E. degree inelectronics and telecommunication engineering fromJadavpur University, India, in 2002 and the M.Tech.degree in electrical engineering (with specializationin communication and signal processing) from IndianInstitute of Technology Bombay, India, in 2005. Cur-rently, he is working towards the Ph.D. degree in theDepartment of Electrical and Systems Engineering atWashington University in St. Louis.

His research interests are in the area of statisticalsignal processing, detection and estimation theory,

and their applications in radar, communications, and sensor arrays.Mr. Sen received the second place award in the student paper competition at

the Fifth International Waveform Diversity & Design (WDD) Conference 2010.

Arye Nehorai (S’80–M’83–SM’90–F’94) receivedthe B.Sc. and M.Sc. degrees from the Technion,Haifa, Israel, and the Ph.D. degree from StanfordUniversity, Stanford, CA.

He was previously a faculty member at YaleUniversity and the University of Illinois at Chicago.He is currently the Eugene and Martha LohmanProfessor and Chair of the Department of Electricaland Systems Engineering at Washington Universityin St. Louis (WUSTL). He serves as the Directorof the Center for Sensor Signal and Information

Processing at WUSTL.Dr. Nehorai has served as Editor-in-Chief of the IEEE TRANSACTIONS ON

SIGNAL PROCESSINGfrom 2000 to 2002. From 2003 to 2005, he was Vice-Pres-ident (Publications) of the IEEE Signal Processing Society (SPS), Chair of thePublications Board, and member of the Executive Committee of this Society.He was the Founding Editor of the special columns on Leadership Reflectionsin the IEEE Signal Processing Magazine from 2003 to 2006. He received the2006 IEEE SPS Technical Achievement Award and the 2010 IEEE SPS Merito-rious Service Award. He was elected Distinguished Lecturer of the IEEE SPS forthe term 2004 to 2005. He was corecipient of the IEEE SPS 1989 Senior Awardfor Best Paper coauthor of the 2003 Young Author Best Paper Award and core-cipient of the 2004 Magazine Paper Award. In 2001, he was named UniversityScholar of the University of Illinois. He is the Principal Investigator of the Mul-tidisciplinary University Research Initiative (MURI) project entitled AdaptiveWaveform Diversity for Full Spectral Dominance. He has been a fellow of theRoyal Statistical Society since 1996.