Adaptive OFDM Radar for Target Detection in ... - Semantic Scholar

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

Adaptive OFDM Radar for Target Detection in Multipath Scenarios Satyabrata Sen, Student Member, IEEE, and Arye Nehorai, Fellow, IEEE

Abstract—We develop methods for detecting a moving target in the presence of multipath reflections, which exist, for example, in urban environments. We take advantage of the multipath propagation that increases the spatial diversity of the radar system and provides different Doppler shifts over different paths. We employ a broadband orthogonal frequency division multiplexing (OFDM) signal to increase the frequency diversity of the system as different scattering centers of a target resonate variably at different frequencies. To overcome the peak-to-average power ratio (PAPR) problem of the conventional OFDM, we also use constant-envelope OFDM (CE-OFDM) signaling scheme. First, we consider a simple scenario in which the radar receives only a finite number of specularly reflected multipath signals. We develop parametric measurement models, for both the OFDM and CE-OFDM signaling methods, under the generalized multivariate analysis of variance (GMANOVA) framework and employ the generalized likelihood ratio (GLR) tests to decide about the presence of a target in a particular range cell. Then, we propose an algorithm to optimally design the parameters of the OFDM transmitting waveform for the next coherent processing interval. In addition, we extend our models to study the aspects of temporal correlations in the measurement noise. We provide a few numerical examples to illustrate the performance characteristics of the proposed detectors and demonstrate the achieved performance improvement due to adaptive OFDM waveform design. Index Terms—Adaptive waveform design, asymptotic performance analysis, multipath, OFDM radar, target detection, urban scenarios.

I. INTRODUCTION

T

HE problem of detection and tracking targets in the presence of multipath, particularly in urban environments, are becoming increasingly relevant and challenging to radar technologies. In [1], we have shown that the target detection capability can be significantly improved by exploiting multiple Doppler shifts corresponding to the projections of the target velocity on each of the multipath components. Furthermore, the multipath propagations increase the spatial diversity of the radar system by providing extra “looks” at the target and thus enabling target detection and tracking even beyond the line-of-sight (LOS) Manuscript received January 04, 2010; accepted October 04, 2010. Date of publication October 11, 2010; date of current version December 17, 2010. This work was supported by the Department of Defense under the Air Force Office of Scientific Research MURI Grant FA9550-05-1-0443 and ONR Grant N000140810849. The associate editor coordinating the review of this manuscript 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: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2010.2086448

[2], [3]. Other areas of application in which multipath effects are of primary interest are in low-angle tracking (sea-skimmers) [4]–[7], height finding [8], [9], and radar-aided navigation and landing systems [10]. Similar problems have been addressed in sonar literature due to bottom bounce in shallow waters [11], [12]. Note that in [13] we have demonstrated that the directionfinding capability of a radar system can be improved also by exploiting multipath reflections close to the sensors. To resolve and exploit the multipath components it is generally common to use short pulse, multi-carrier wideband radar signals. We consider the orthogonal frequency division multiplexing (OFDM) signaling scheme [14], [15], which is one of the ways to accomplish simultaneous use of several subcarriers. The use of OFDM signal mitigates the possible fading, resolves the multipath reflections, and provides additional frequency diversity as different scattering centers of a target resonate at different frequencies. Although OFDM has been elaborately studied and commercialized in the digital communication field [16], it has not so widely been studied by the radar community apart from a few recent efforts [17]–[19]. One of major reasons of such unpopularity is that OFDM has a time-varying envelope and that originates a potentially high peak-to-average power ratio (PAPR) [20], [21]. A high value of PAPR demands for system components (e.g., transmitter’s power amplifier) with a large linear region of operation. However, practical power amplifiers operate over limited linear region, beyond which they saturate causing nonlinear distortion to the signal [22]. Over the years, a number of approaches have been proposed to deal with the PAPR problem. A comprehensive survey of PAPR reduction techniques can be found in [20] and [23, Ch. 6]. One of such methods is to apply the phase modulation transform that achieves the lowest possible PAPR (0 dB). In this work, besides considering conventional OFDM, we also include the constant-envelope OFDM (CE-OFDM) signaling scheme [21], [24]–[27], which is based on using a real-valued baseband OFDM signal to phase modulate the carrier. First, we discuss a detection problem in which the radar has the complete knowledge of the first-order (or single bounce) specularly reflected multipath signals. We also assume that the clutter and measurement noise are temporally white. In Section II, we develop the measurement models, for both the OFDM and CE-OFDM signaling schemes, under the generalized multivariate analysis of variance (GMANOVA) framework [28], [29]. Based on these models, in Section III, we formulate the detection problem as a hypothesis test to decide about the presence of a target in a particular range cell. Due to the lack of knowledge of all the parameters in our models,

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

we employ the generalized likelihood ratio (GLR) test [30, Ch. 6]. We present numerical results to evaluate the performance of these proposed detectors, as we do not have any analytical expressions to evaluate their performances. Then, in Section IV, we propose a criterion to adaptively compute the parameters of the next transmitting waveform. To construct such a criterion we first look into the performance characteristics of the GLR test statistics for both OFDM and CE-OFDM models assuming that the target velocity is known. However, this analysis does not characterize the detection performance of our detectors, in which the target velocity is unknown. The analysis with known target velocity shows that the GLR test results in constant false alarm rate (CFAR) detectors for both OFDM (with large number of temporal samples) and CE-OFDM (with finite number of temporal sample) models, and the detection performances depend on the system parameters through the corresponding noncentrality parameters of the distributions under alternate hypothesis. This implies that it is possible to improve the detection performance by maximizing these noncentrality parameters. We apply this idea to our problem and formulate the optimization problem to select the parameters of the next transmitting waveform that maximizes the same expression of the noncentrality parameter subject to a fixed transmission-energy constraint. For the OFDM model, we show that the solution of this optimization problem results in an eigenvector corresponding to the largest eigenvalue of a matrix that depends on the target, clutter, and noise parameters. However, for the CE-OFDM model we cannot improve the detection performance in this way because the noncentrality parameter does not depend on the transmitting waveform. Later in the paper, in Section V, we relax the assumption of temporal whiteness to study the effects of temporally correlated measurement noise process on our models. Temporal correlations exist in certain radar applications, in particular at high pulse repetition frequencies (PRF) [31], [32]. To model the temporal correlation matrix, we look into a branch of statistics known as the nearest neighbor analysis [33], [34], and present the consequent detection tests. To illustrate the potential of our proposed detectors, we present numerical examples in Section VI. We find that the wideband OFDM model performs better than the narrowband CE-OFDM model in exploiting the multipath reflections. In addition, we achieve significant performance improvement due to adaptive OFDM waveform design. However, the CE-OFDM signal lacks such an adaptive design as the detection performance does not depend on the transmitting coefficients. Finally, we give concluding remarks and some thoughts on a few unaddressed issues in Section VII. Notations: We list here some notational convention that will be used throughout this paper. We use math italic for scalers, lowercase bold for vectors, and uppercase bold for matrices. For a matrix , , , , , , , , denote the transpose, conjugate-transpose, deterand th entry, generalized inverse (such that minant, ), trace, vec-operation, and block-diagonal vec-operation (defined in [35, eq. (7)]) of , respectively. represents an idenforms a square matrix with tity matrix of dimension . , nonzero entries only on the main diagonal. Additionally,

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Fig. 1. Schematic representation of the multipath scenario.

, and are the inner-product, Kronecker product, and element-wise Hadamard product operators, respectively. II. PROBLEM DESCRIPTION AND MODELING We consider a far-field point target moving with a constant relative velocity , with respect to the radar, in a multipath-rich environment, as shown in Fig. 1. At the operating frequency, we assume that the reflecting surfaces produce only specular reflections of the radar signal. We assume that the radar has the complete knowledge of the environment that is under surveillance. Hence, for every range cell the radar knows the number between the radar and target and the of possible multipath direction-of-arrival (DOA) unit-vectors ( , ) along each such path. Under this scenario, we first introduce the parametric measurement models for both OFDM and CE-OFDM signaling techniques. Then, we discuss our statistical assumptions on the clutter and noise. A. OFDM Measurement Model acWe consider an OFDM signaling system [15] with Hz, and pulse duration tive subcarriers, a bandwidth of of seconds. Let represents the complex weights transmitted over the subcarriers, satisfying . Then, the complex envelope of the transmitted signal can be represented as (1) where denotes the subcarrier spacing. Let be the carrier frequency of operation, then the transmitted signal is given by (2) where represents the th subcarrier frequency. Interchanging the real and summation operators, we can also rewrite (2) as (3)

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where

where (4)

(9)

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 written as

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

(5)

(10) where • • is an complex diagonal matrix that contains the transmitted weights ; is an complex • rectangular block-diagonal matrix where each nonzero , , block represents the scattering coefficients of the target at the th subchannel over all multipath; • is an complex vector where , , contains the Doppler information of the target at the th subchannel over all multipath; • is a column vector containing the unknown target-velocity components; is an vector • of clutter returns, measurement noise, and co-channel interference. Then, concatenating all the temporal data columnwise into an matrix we obtain the OFDM measurement model as follows:

is a complex quantity representing the scattering coefwhere ficient of the target along the th subchannel and th path; where is the relative Doppler shift along the th path and is the speed of propagation; is the roundtrip delay between the radar and target along the th path; represents the clutter and measurement noise along the th subchannel 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 between any two multipath signals are very small in comparison to the for . actual roundtrip delays, i.e., These assumptions can be justified in systems where the path lengths of multipath arrivals differ little (e.g., narrow urban canyon where the range is much greater than the width). Furas the roundtrip delay corresponding to ther, let us denote the range cell under consideration. Then, the information of the roundtrip delays can be automatically incorporated into the , , model by choosing where is the pulse repetition interval (PRI) and is the number of temporal measurements within a given coherent processing interval (CPI). Hence, corresponding to a specific range cell containing the target, the complex envelope of the received signal at the output of the th subchannel is

(8)

(11) where • ; is an • matrix containing the Doppler information of the target through 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 baseband OFDM signal to phase modulate the carrier. The complex envelope of a CE-OFDM transmitted signal is represented as [21] (12) where signal

is the modulation index in radians and message bears an OFDM signal structure (13)

are real-valued weights at difwhere ferent subcarriers. Assuming a narrowband signal model (which can be achieved with small modulation index [21]), the complex envelope of the


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

(14) is the target scattering coefficient at the operating frewhere are roundtrip delay quency along the th path, and and and relative Doppler shift, respectively, along the th path. Then, as before assuming that all the multipath delays are for , and approximately equal, i.e., , , we can representing simplify (14) into

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irregularities on the reflecting surface (e.g., windows and balconies of the buildings in an urban scenario), that cannot be modeled as specular components. Therefore, for both OFDM and CE-OFDM measurement models, we assume that the clutter and noise are temporally white and circularly symmetric zeromean complex Gaussian process with unknown covariances. In contains the clutter returns, noise, and (10), the noise vector co-channel interference at the output of subchannels, which we assume to be correlated with unknown positive definite covariance matrix . Hence, the OFDM measurements are distributed as (18) is , and therefore the In (15), we assume that variance of CE-OFDM measurements are distributed as (19)

(15) where represents the scattering coeffi• cients of the target at the carrier frequency over all multipaths; contains the • multipaths Doppler information of the target over all where

(16) Then, concatenating all the temporal data columnwise into an vector, we obtain the CE-OFDM measurement model as follows:

In these formulations, when the parameter is known, both (11) and (17) comply with the generalized multivariate analysis of variance (GMANOVA) structure [28], [36], which has been studied extensively in statistics and applied to a number of applications in signal processing [29]. III. DETECTION TEST In this section, we develop statistical detection tests for the OFDM and CE-OFDM measurement models presented in Section II. Our goal is to decide whether a target is present or not in the range cell under consideration. We construct the decision problem to choose between two (target-free hypothpossible hypotheses: the null hypothesis esis) or the alternate hypothesis (target-present hypothesis). The tests can be expressed as for OFDM model

(17) where • ; is a • matrix containing the Doppler information of the target through the parameter ; is an vector com• prising clutter returns, noise, and interference. From the structures of (11) and (17), it is quite evident that the CE-OFDM measurement model resembles a similar form . However, note that as that of the OFDM case when the transmitting weights influence the OFDM measurements whereas CE-OFDM measurements through the matrix through the matrix .

unknown unknown

(20)

unknown unknown.

(21)

for CE-OFDM model

Because of the lack of knowledge about and (or ) we use the generalized likelihood ratio (GLR) test [30, Ch. 6] in which the unknown parameters are replaced with their maximum likelihood estimates (MLE). This approach also provides the information about the unknown parameters since the first step is to find the MLEs. Assuming that the parameter is known in (11), the GLR test for (20) compares the ratio of the likelihood functions under the two hypotheses with a threshold as follows [30, Ch. 6.4.2]: (22)

C. Statistical Model In our problem, the clutter could be the contribution of any undesired reflections from the environment surrounding and/or behind the target, or any random multipath reflections, from the

where and are the likelihood functions under and , and are the MLEs of under and , is the under , and is the detection threshold. After MLE of

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

As before, since in our problem compares a threshold.

is unknown, the GLR test with

IV. ADAPTIVE WAVEFORM DESIGN

(23) However, having a block-diagonal structure, the scattering madoes not yield a closed-form MLE expression [35]. So, trix in this work, we use an approximate ML (AML) estimator for , defined as [35, eq. (28)]

(24) where (25a) (25b)

In this section, we develop an adaptive waveform design technique to improve the target-detection performance. To derive a mathematical formulation for optimal waveform selection, we first create a utility function according to certain criteria and then determine the parameters for the next transmitting waveform by optimizing this utility function. To construct such a utility function we first consider the detection performance assuming known target velocity parameter . Then, we explicitly state the optimization problem and its solution. A. Distributions of the Test Statistic for Known Target Velocity In this subsection, we derive the distributions of and when the target velocity is known. The motivation behind these derivations is to look for a criterion of adaptive waveform design and not to characterize the detection performance with known target velocity, because in our problem the target velocity is unknown. our OFDM measurement 1) OFDM Detector: Under and the corresponding model is GLR test statistic is given by [37, Th. 3.10], [38, eq. (4)–(33)]

(25c) (25d)

(28)

(25e) (25f)

In (25e), is the th column of an matrix , is a matrix representing the th block of and in (25f), rows of . It is important to note here that the condition mentioned in [35] is a typographical error. Because, the actual assumption is that the columns of the individual sub(of [35]) need to be linearly independent and that matrices , which transforms into in our work. requires is unIn our problem, however, the parameter 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)

where ’s are mutually independent complex beta distributed and complex random variables with degrees of freedom, written as

(29) and Under

.

we have where . However, the distribution of does not have a closed-form expression. In general we can express the GLR test statistic as the ratio of determinants and of two random matrices , (30) and follow noncentral complex Wishart diswhere tributions, denoted as and , respectively; and ,

(26)


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

(31)

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the “signal-to-noise ratio matrix,” and hence the trace of it can be considered as a sum of squared Mahalanobis distances [46]. 2) CE-OFDM Detector: The distributions of the GLR test statistic of the CE-OFDM model assuming known target ve, can be evaluated in a comparalocity, tively easy way either through a direct computation of complex chi-square distributions using (19) in (26), or as a special case of . Under the GLR test statistic the OFDM model with and complex follows complex beta distribution with degrees of freedom, written as (37)

(32) we find that follows a complex Wishart distribution of order , parameter , and complex degrees of freedom, denoted as

where . Under the GLR test statistic follows a complex noncentral beta distribution with the and and noncentrality same complex degrees of freedom , denoted as parameter (38)

(33) and follows a noncentral complex Wishart distribution with order , parameter , complex degrees of freedom, and as the noncentrality parameter, denoted as (34) We cannot simplify (30) any further such that it has a distribution with closed-form expression. However, in a special case, when (which is termed as the “linear case” after Anderson [40], [41] in some under statistical literature) the test statistic can be written as a product of independent complex beta random variables where one of the beta variables is noncentral [42], [43]. The noncentrality parameter is given as the single , which is same nonzero root of the equation as . Here we remark that to achieve in our problem we have to use single frequency signal instead of multi-frequency OFDM signal. Since, the distribution of the GLR test statistic for the OFDM measurement model does not have a closed-form expression for a finite value of , we explore the asymptotic performance characteristics of (23) assuming known target velocity. Following an analogous discussion on real Gaussian variables from [44, Ch. , under , 8], [45] we find that as has a complex chi-square distribution with complex degrees of freedom, denoted as (35) , the limiting distribution of Under a complex noncentral chi-square distribution with degrees of freedom, denoted as

is complex (36)

is the noncentrality parameter and are the roots of . Obviously another way to represent the same noncentrality parameter is . We may call the matrix as where

where

.

B. Waveform Design From the discussion of the previous subsection, it is clear that the GLR test results in a constant false alarm rate (CFAR) ) and CE-OFDM detector under both the OFDM (when (for finite ) models when the target velocity is known and the detection performance depends on the system parameters and , respectively. through the noncentrality parameters Therefore, it is possible to improve the detection performance by maximizing these noncentrality parameters. However, in our problem the target velocity parameter is unknown. Moreover, in the OFDM measurement model the target scattering matrix is block diagonal. Still in our adaptive waveform design problem we maximize the same expression of the noncentrality parameter subject to a predefined energy constraint. Thus, we formulate the optimization problem as

(39) After some algebraic manipulations (see Appendix) we can rewrite 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 . are not known. Note that in our problem , , and Hence, we use their estimated values to obtain for is transmitted and the the next CPI. First, a nonoptimal corresponding measurements are stored over one particular . SubCPI. Then, we estimate stituting into (9) we compute and subsequently

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using (24) and (25). Then, the estimate of Finally, we choose

is evaluated as . as the eigenvector corresponding to

. The the largest eigenvalue of computation of such an eigenvector is done using Matlab in real time. In a similar fashion, we can formulate the optimization problem for the CE-OFDM measurement model as follows:

(41) , where

However, rewriting (16) as

it can be easily shown that the the following form:

th entry of

and (44) can have any structure or can even be unThe matrix structured. In the statistical literature two structured covariance matrices are very common. These are the compound symmetric (CS) structure [47] and autoregressive structure of order 1 (AR(1)) [48]. The CS covariance structure assumes that all the temporal measurements are equicorrelated and do not depend on the duration between the two time points, i.e., when otherwise (45) is the coefficient of temporal correlation bewhere tween two time points. In AR(1) covariance structure, the temporal measurements are assumed to be more highly correlated if they are close to each other in time duration, i.e.,

has (46)

(42) Hence, the noncentrality parameter in the CE-OFDM model does not depend either on or . This implies that the performance characteristic of the CE-OFDM detector cannot be improved by any efficient choice of .

In our work, we consider the AR(1) model as it is suitable for pulsed radar applications in which the measurements are collected at equispaced time intervals. Note that in (45) and (46) we explicitly write to stress the fact that the covariance matrix (and also its inverse) is completely characterized by a single parameter . B. Detection Under Temporal Correlations We construct the decision problem in terms of two possible and , to detect the presence of a target in the hypotheses range cell under consideration, as for OFDM model

V. TEMPORALLY CORRELATED NOISE In this section, we extend our models to include temporal correlation among the measurements. Recall that so far we have assumed that the clutter and measurement noise are the independent realizations of the same Gaussian random process from pulse to pulse. However, this assumption may not be valid at high pulse repetition frequency [32]. In the following, we first present the statistical assumptions of temporally correlated noise for both the OFDM and CE-OFDM measurement models, presented in (10) and (15), respectively, and then discuss the detection tests.

unknown unknown

(47)

unknown unknown.

(48)

for CE-OFDM model

As before, since the parameters , , and (or ) are unknown we apply the GLR test. For the OFDM model the test compares the ratio of the likelihood functions maximized, with respect to the unknown parameters, under the two hypotheses with a threshold as follows:

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

(49) where the MLEs of and are computed by replacing and with and , respectively. Similarly, for the CE-OFDM model the test performs the following comparison:

(50) (43)


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

where we use instead of MLEs of and

and

and , respectively, to compute the

. VI. NUMERICAL RESULTS

In this section, we present the results of several numerical examples to illustrate the performance characteristics of our proposed detectors, presented in Section III, as the associated GLR test statistic with unknown target velocity does not have any closed-form analytical expression. For simplicity, we consider a 2-D scenario, where both the radar and target are in the same plane, as shown in Fig. 2. Our analyses can easily be extended to 3-D scenarios. First, we provide a description of the simulation 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 particular range cell. We simulated the situation of a range cell centered at 2 km North and 5 m East with respect to the radar (positioned at the origin). ) between — There exists three different paths (i.e., that particular range cell and the radar. These are one direct path and two specular multipaths due to a couple of reflecting surfaces oriented along North-South direction at 10 m East and 10 m West. — The scattering coefficients of the target (i.e., the entries distribution. of or ) are generated from a • Radar parameters; 1 GHz. — Carrier frequency 100 MHz. — Available bandwidth — Pulse repetition interval 20 s. 50 ns. — Pulse width . — Number of coherent pulses active subcarriers — OFDM signal operates with and the subcarrier spacing of 20 MHz.

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Fig. 3. Effects of different SNR values on detection probability as a function of 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 independent trials to realize the following results. For the OFDM were realized from the measurement model, the entries of distribution and then were scaled to satisfy the required signal-to-noise ratio (SNR), defined as

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

was chosen

A. Detector Performance Fig. 3 depicts the variations of probability of detection as a function of probability of false alarm at three different SNR values for the OFDM and CE-OFDM signaling schemes. As expected, for both the measurement models the detection performance improves as SNR is increased. However, at a fixed SNR, the detection performance of the wideband OFDM model is much better than that of the narrowband CE-OFDM model. Because being a wideband signal the conventional OFDM can resolve the multipath and overcome fading. Additionally, it can exploit the target response at multiple frequencies. To show the advantage of using multi-frequency signaling system, we compared the detection performance at three different values of , while keeping the SNR fixed at 5 dB. The results are presented in Fig. 4. Hence, it is evident that the frequency diversity improves the target-detection performance in an OFDM system. However, this is not true in the case of

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

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

CE-OFDM model. As we showed in Section IV, the performance of the CE-OFDM detector does not depend on the value of . We studied the effects of different directions of the velocity vector on the detection performance at three different condi[0,10], [7.07, 7.07], [10, 0] m/s. The results tions: are depicted in Fig. 5. For this simulation we kept the SNR fixed at 5 dB. As the angle between the target velocity vector and radar LOS increases the performance of both the OFDM and CE-OFDM detectors deteriorate. However, the wideband OFDM model can resolve and exploit the multipath reflections better than the CE-OFDM model, and hence it shows improved performance even close to the LOS scenario. B. Importance of Multipath Modeling To understand the importance of proper exploitation of multipath 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 probability as a function of probability of false alarm in (a) OFDM and (b) CE-OFDM models.

perpendicular to the LOS direction. Then, we compared the detection performances of the two systems: one of them conmultipath reflections, and the other considers siders all only the LOS return. Fig. 6(a) and (b) shows the results at 5 dB for OFDM and CE-OFDM models, respectively. The OFDM signal, being a wideband, can better exploit the multipath reflections to improve the detection performance compared to the narrowband CE-OFDM signal. Moreover, the presence of multipath reflections causes a small performance degradation for CE-OFDM model as it cannot resolve the multipath returns. A similar conclusion can also be drawn from Fig. 7. For this simulation, we generated the measurements, for both the OFDM and CE-OFDM signaling schemes, in the presence of multipath using the same setup as described before. However, in the detector we ignored the presence of multipaths in the tested range cell. This may happen when the radar does not have a knowledge of the possible multipath scenario. From Fig. 7(a), we see


Fig. 7. Effects of ignoring the multipath reflections at the detector on detection 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 OFDM model ignores the multipath reflections. On the contrary, since the CE-OFDM model cannot resolves the multipath returns, it performs a little better [see Fig. 7(b)] when the detector ignores the multipath. C. Adaptive OFDM Waveform Design To study the improvement in target-detection performance due to the proposed adaptive waveform design technique, we devised a simple problem. We assumed a system in which we in the first pulses. Then, based on the transmit corresponding measurements we solved (40) to compute the optimized values of ’s for the next pulses. We compared this system with another system in which both the two sets of pulses transmit . We fixed the SNR at 5 dB for this simulation. From Fig. 8(a), we observe that the detection performance of the adaptive system is considerably improved for the OFDM model. However, this is not the case with the

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

CE-OFDM model, as evident from Fig. 8(b). This again conforms our derivation in Section IV that the noncentrality parameter in the CE-OFDM model does not depend on the values of . D. Under Temporal Correlation Fig. 9 depicts the detection performance at three different values of temporal correlation coefficient, , for the OFDM and CE-OFDM signaling schemes. We used the AR(1) covariance structure, described in (46), to introduce temporal correlation among the measurements. For this simulation we kept the SNR fixed at 5 dB. It is evident from this analysis that the target-detection performance deteriorates as the level of temporal correlation increases. E. Detector Performance for Known Target Velocity Finally, in Fig. 10, we show comparative performance results of the OFDM and CE-OFDM detectors for known and unknown target velocity . We plot these results at two different SNR values. The detection performance for known target velocity

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However, this analysis shows us quantitatively to what extent the detection performance (for unknown ) degrades due to the estimation of . For example, for the OFDM detector, the value drops from 0.56 to 0.3 at and 0 dB of due to the velocity estimation process. VII. CONCLUSION

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

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

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

In this paper, we addressed the problem of detecting a moving target by exploiting multipath reflections. First, we developed the measurement model accounting for only a finite number of specular multipath reflections. We considered two different schemes to modulate the transmitting carrier: amplitude modulation by employing an orthogonal frequency division multiplexing (OFDM) signal and phase modulation with a constant-envelope OFDM (CE-OFDM) signal. The use of OFDM signal increases the frequency diversity of our system as different scattering centers of a target resonate variably at different frequencies. We formulated the detection problem as a statistical hypothesis test and employed generalized likelihood ratio test to decide about the presence of a moving target in a particular range cell. Then, we proposed an algorithm to optimally design the parameters of the transmitting waveform for the next coherent processing interval. We showed that for OFDM radar the solution of this optimization problem results in an eigenvector corresponding to the largest eigenvalue of a matrix that depends on the target, clutter, and noise parameters. However, for the CE-OFDM model we cannot improve the detection performance in this way because the noncentrality parameter does not depend on the choice of the signal structure (i.e., number of subcarriers and coefficients). In addition, we extended our models to consider the aspects of temporal correlations in the measurement noise. Our numerical results demonstrate the performance characteristics of the proposed detectors. Though the use of CE-OFDM allows one to overcome the peak to average power ration (PAPR) problem, but it results in worse detection performance compared to the conventional OFDM. Because, being a wideband signal, OFDM outperforms the narrowband CE-OFDM system by exploiting the multipath reflections. Moreover, since the CE-OFDM signaling does not have different subcarriers, it is not possible to improve the detection performance by increasing the number of subcarriers, as happens in the case of OFDM detector. We also numerically demonstrated the performance improvement due to adaptive OFDM waveform design. This could enable the radar to operate in a closed-loop fashion resulting an improved performance in the presence of nonstationary clutter or noise. However, the CE-OFDM signal lacks such an adaptive design. Therefore, in future we plan to evaluate the detection performance of different modulation schemes, other than CE-OFDM, that reduce the PAPR problem. Additionally, in our future work, we will extend our model to incorporate more realistic physical effects, such as diffractions, refractions, and attenuations, which exist, for example, due to sharp edges and corners of the buildings or rooftops in an urban environment. We will expand the detection procedure over multiple range cells to consider other significant multipath reflections and realistically model the clutter returns that depend on transmitted signal [32]. As polarization allows identification of correlated source signals (e.g., multipath) with small angle separation [49], we will also study the detector performance employing


polarized transceivers. We will integrate our detection procedure with a target tracking algorithm and explore other criteria, e.g., ambiguity function, mutual information, etc., to optimally design the transmit waveform to improve the system performance. We will validate the performance of our proposed detector with real data. APPENDIX In this Appendix, we show how we simplify the objective function in (39) to that in (40). First, using the relationship between a trace and vec operator, , we get

(A1) and Here, for simplicity of notations, we represent with and , respectively. Then, we apply one of the properties of the vec operator, , to get

(A2) Note that in our problem can be written as Therefore,

is a diagonal matrix.

.. .

(A3)

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

th position

.. .

(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).

89

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Satyabrata Sen (S’07) received the B. E. degree in electronics and telecommunication engineering from Jadavpur University, India, in 2002 and the M.Tech. degree in electrical engineering (with specialization in communication and signal processing) from Indian Institute of Technology Bombay, India, in 2005. Currently, he is working towards the Ph.D. degree in the Department of Electrical and Systems Engineering at Washington University in St. Louis. His research interests are in the area of statistical signal 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) received the B.Sc. and M.Sc. degrees from the Technion, Haifa, Israel, and the Ph.D. degree from Stanford University, Stanford, CA. He was previously a faculty member at Yale University and the University of Illinois at Chicago. He is currently the Eugene and Martha Lohman Professor and Chair of the Department of Electrical and Systems Engineering at Washington University in St. Louis (WUSTL). He serves as the Director of 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-President (Publications) of the IEEE Signal Processing Society (SPS), Chair of the Publications Board, and member of the Executive Committee of this Society. He was the Founding Editor of the special columns on Leadership Reflections in the IEEE Signal Processing Magazine from 2003 to 2006. He received the 2006 IEEE SPS Technical Achievement Award and the 2010 IEEE SPS Meritorious Service Award. He was elected Distinguished Lecturer of the IEEE SPS for the term 2004 to 2005. He was corecipient of the IEEE SPS 1989 Senior Award for Best Paper coauthor of the 2003 Young Author Best Paper Award and corecipient of the 2004 Magazine Paper Award. In 2001, he was named University Scholar of the University of Illinois. He is the Principal Investigator of the Multidisciplinary University Research Initiative (MURI) project entitled Adaptive Waveform Diversity for Full Spectral Dominance. He has been a fellow of the Royal Statistical Society since 1996.