172 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, …...Many adaptive radar signal processing methods that assume the ... [13], [14], multiple input multiple output (MIMO) target

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

Fractional QCQP With Applications in ML SteeringDirection Estimation for Radar Detection

Antonio De Maio, Senior Member, IEEE, Yongwei Huang, Member, IEEE, Daniel P. Palomar, Senior Member, IEEE,Shuzhong Zhang, and Alfonso Farina, Fellow, IEEE

Abstract—This paper deals with the problem of estimating thesteering direction of a signal, embedded in Gaussian disturbance,under a general quadratic inequality constraint, representing theuncertainty region of the steering. We resort to the maximum like-lihood (ML) criterion and focus on two scenarios. The former as-sumes that the complex amplitude of the useful signal componentfluctuates from snapshot to snapshot. The latter supposes that theuseful signal keeps a constant amplitude within all the snapshots.We prove that the ML criterion leads in both cases to a fractionalquadratically constrained quadratic problem (QCQP). In order tosolve it, we first relax the problem into a constrained fractionalsemidefinite programming (SDP) problem which is shown equiva-lent, via the Charnes-Cooper transformation, to an SDP problem.Then, exploiting a suitable rank-one decomposition, we show thatthe SDP relaxation is tight and give a procedure to construct (inpolynomial time) an optimal solution of the original problem froman optimal solution of the fractional SDP. We also assess the qualityof the derived estimator through a comparison between its perfor-mance and the constrained Cramer Rao lower Bound (CRB). Fi-nally, we give two applications of the proposed theoretical frame-work in the context of radar detection.

Index Terms—Constrained maximum likelihood steering direc-tion estimation, fractional QCQP, radar applications.

I. INTRODUCTION

T HE problem of estimating the steering direction vector isof relevant interest in some applications concerning radar

detection and beamforming. In fact, there are several physicalphenomena why the received steering vector is quite often notaligned with the nominal expected direction: such as pointing er-rors caused by the radar antenna which forms a beam not pointedin the exact desired direction; imperfect array calibration and

Manuscript received March 13, 2010; accepted September 26, 2010. Dateof publication October 14, 2010; date of current version December 17, 2010.The associate editor coordinating the review of this manuscript and approvingit for publication was Prof. Dominic K. C. Ho. The work of D. P. Palomar andY. Huang was supported by the Hong Kong RGC 618709 research grant. Thework of S. Zhang was supported by National Natural Science Foundation ofChina (Grant 10771133).

A. De Maio is with the Dipartimento di Ingegneria Biomedica, Elettronica edelle Telecomunicazioni, Università degli Studi di Napoli “Federico II”, Napoli,Italy (e-mail: [email protected]).

Y. Huang and D. P. Palomar are with the Department of Electronic and Com-puter Engineering, Hong Kong University of Science and Technology, ClearWater Bay, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]).

S. Zhang is with the Industrial and Systems Engineering Program, Universityof Minnesota, Minneapolis, MN 55455 USA, and also with the Department ofSystems Engineering and Engineering Management, The Chinese University ofHong Kong, Shatin, Hong Kong (e-mail: [email protected]).

A. Farina is with the SELEX Sistemi Integrati, Rome, Italy (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/TSP.2010.2087327

distorted antenna shape because of array imperfections; spatialmultipath causing severe distortions in the presumed steeringvector due to the presence of signals from several paths (co-herent and incoherent local scattering); source wavefront dis-tortion implied by the signal propagation through nonhomoge-neous media; in-phase and quadrature components errors due tounbalanced lowpass filters; A/D sampling errors; dc bias; non-linearities; and intermodulation products. In other words, thereis some imperfect knowledge on the actual steering directionwhich is usually characterized in terms of an uncertainty region.Many adaptive radar signal processing methods that assume theexact knowledge of the signal array response vector often suffera performance degradation, when the actual steering vector isnot perfectly aligned with the nominal one (even small varia-tions in the array manifold can reduce their performance).

In order to account for the quoted uncertainty, several esti-mation techniques have been proposed in open literature. Forinstance, with reference to beamforming applications, steeringestimation is explicitly or implicitly addressed in [1]–[5]. As tothe case of radar detection, steering vector estimation can befound in [6]–[12].

All the mentioned approaches focus on a specific uncertaintyregion and exploit some different objective functions to opti-mize. Actually, to the authors best knowledge, the problem ofmaximum likelihood (ML) joint estimation of signal amplitudeand steering direction under a general steering uncertaintyregion, represented by a quadratic inequality constraint, has notyet been solved. To this end, in this paper, we consider steeringdirection ML estimation under a general quadratic inequalityconstraint (either convex or nonconvex) and the unit normconstraint which characterizes a direction vector. We focus ontwo distinct situations. One assumes that complex amplitude ofthe received useful signal changes from snapshot to snapshot,whereas the other accounts for a constant signal amplitude inall the received snapshots. Both the situations are of interest forradar signal processing applications concerning detection ofrange-spread targets [13], [14], multiple input multiple output(MIMO) target detection [15], [16], and radar detection usinga general antenna array configuration with a mix of high-and low-gain beams [17]. We show that, upon concentrationof the likelihood function over the amplitudes of the signalof interest, the steering direction estimation problem can beformulated as a fractional quadratically constrained quadraticproblem (QCQP). Hence, in order to solve it, we first relax theproblem into a constrained fractional semidefinite program-ming (SDP) which is proved equivalent to an SDP throughthe Charnes-Cooper transformation [18]. Then, we show thatthe relaxation is tight. Precisely, exploiting a suitable rank-onedecomposition theorem [19], [20], we devise a technique aimed

1053-587X/$26.00 © 2010 IEEE

DE MAIO et al.: FRACTIONAL QCQP WITH APPLICATIONS IN ML STEERING DIRECTION ESTIMATION FOR RADAR DETECTION 173

at recovering a solution of the original fractional QCQP from asolution of the fractional SDP relaxation. The overall procedureinvolves a polynomial computational complexity and, from theoptimization theory point of view, this paper provides an effi-cient procedure to solve a general fractional QCQP with threehomogeneous constraints, incorporating the specific rank-onedecomposition technique [19].

We also provide considerations concerning the identifiabilityof the parameters in the considered estimation problem and eval-uate the constrained Cramér–Rao lower bound (CRB) [21]. Fi-nally, we apply the estimation framework to two radar detectionproblems. The former refers to radar detection of distributed tar-gets; the latter to point-target detection with a radar using a gen-eral antenna array configuration with a mix of high- and low-gainbeams. The analysis of the resulting decision rules in comparisonwiththeclassicone,assumingalignednominalandactualsteeringdirections, confirms the effectiveness of the approach.

The paper is organized as follows. In Section II, we formu-late the steering direction estimation problem. In Section III, wediscuss the identifiability of the model parameters and proposethe new estimation procedure. Additionally, we derive the con-strained CRB and assess, through an example, the quality of theestimates by the proposed algorithm. In Section IV, we apply thenew estimation technique to two radar problems and analyze theperformance of the receivers exploiting the robust estimator ofthe steering direction in place of the nominal one. Finally, con-clusionsandpossiblefutureresearchtracksaregiveninSectionV.

Notation

We adopt the notation of using boldface for vectors (lowercase), and matrices (upper case). The transpose and the con-jugate transpose operators are denoted by the symbols and

, respectively. , , , , and arerespectively the trace, the determinant, the rank, the minimumeigenvalue and the maximum eigenvalue of the square matrix ar-gument. and denote, respectively, the identity matrix and thematrix with zero entries (their size is determined from the con-text). , , and are respectively the sets of -dimen-sional vectors of real numbers, -dimensional vectors of com-plex numbers, and Hermitian matrices. The curled in-equality symbol (and its strict form ) is used to denote gener-alized matrix inequality: for any , means that isa positive semidefinite matrix ( for positive definiteness).The Euclidean norm of the vector is denoted by . The letter

represents the imaginary unit (i.e., ), while the letteroften serves as index in this paper. For any complex number ,

we use and to denote, respectively, the real and theimaginary part of , is the modulus of , and is the conju-gate of . Finally, denotes the Kronecker product and for anyoptimization problem , represents its optimal value.

II. MAXIMUM LIKELIHOOD STEERING ESTIMATION: PROBLEM

FORMULATION

We assume that data are collected from sensors (forinstance the receiving elements of an antenna array) and denoteby , , the complex vectors of the received

samples which can be expressed as

(1)

where ’s are unknown complex parameters accounting forthe channel propagation and signal strength, is the unit-normsteering vector of the signal of interest (i.e., , with

), and ’s are the disturbance compo-nents, modeled as statistically independent zero-mean complexcircular Gaussian vectors with positive definite covariance ma-

trix . As to the , we consider the followingtwo different scenarios:

1) Scenario 1. It models as a vector of andis of relevant interest for radar detection of range-spreadtargets [13], [14], where the useful target might be spreadin more than one range cell. In this case, accounts forthe backscattering coefficient of the scattering center in theth range cell (if a high resolution radar is considered) or

the radar cross section from the target in the th range binwith reference to a formation of targets to be detected witha low-medium resolution radar.

2) Scenario 2. It assumes , , withknown complex numbers, while is an unknown pa-rameter. This signal model is useful in MIMO radar withcolocated antennas [16, p. 11, eq. 1.29] and in modern radarsystems [17] that have general antenna array configura-tions, containing a mix of high- and low-gain beams.

Additionally, we force to comply with a general (not neces-sarily convex) quadratic constraint, i.e.

with(2)

with , , and .The estimation of the steering direction is performed re-

sorting to the maximum likelihood (ML) criterion. Thistechnique is usually applied for radar detection applica-tions in the presence of steering vector mismatches and/orwhen rejection capabilities (namely the possibility to rejectsignals which are outside an acceptance region centeredaround the expected steering direction) are required to theradar receiver1. Otherwise stated, we are faced with thefollowing optimization problem: [see (3) at the bottom ofthe next page], where and

are the joint probabilitydensity functions (pdf’s) of the vectors underScenario 1 and Scenario 2, respectively.

Exploiting the Gaussian assumption, the pdf for Scenario 1can be written as

(4)

1The joint application of the ML criterion and the optimization theory to com-munications problems can be found in [22]–[24].


whereas that for Scenario 2, it is given by

(5)

This implies that the ML estimation problem can be written asshown in (6) at the bottom of the page.

Before presenting the solution method to the above problems,it is necessary to highlight their practical importance and itsrelation to other existing methods. In radar detection, typicalquadratic constraints that the vector has to comply with are

• Conic Constraint: ,where is the nominal steering vector (assumed withunit norm) and . This is tantamount to limitingthe minimum squared cosine angle between the actual andthe nominal steering directions, namely has to lie in aconic region with axis and whose aperture is ruled by

. This kind of constraint is encountered in [6]–[8], [10],[12]. A pictorial representation of the constraint set isgiven in Fig. 1(a).

• Elliptical Constraint:

, where is a positive semidefi-nite matrix such that .2 This is equivalentto assuming that lies into an ellipsoid whose center is theunit norm vector and whose shape is ruled by the matrix

. This constraint is commonly encountered in robustbeamforming applications [2], [3], [5] and, for ,it reduces to the similarity constraint [3],[4], [25], where rules the size of the simi-larity region. A pictorial representation of the resultingconstraint set is given in Fig. 1(b). Finally, in [26],some interesting procedures to determine the uncertaintyellipsoid are proposed and discussed.

• Exterior Conic Constraint:

, namely the useful signal lies outside a conic

2Otherwise � � � � � reduces to � . Indeed, suppose that � ��

and �� . Then ��

�� , which implies�� ; in other words, � �� and �� imply �� .

Fig. 1. Pictorial description of the set � for � � � and real observations.

region whose axis is the vector . This constraint is ofrelevance in radar detection problems where rejectioncapabilities must be conferred to the receiver [9], [27],[28]. Specifically, it arises when it becomes necessary todiscriminate between a useful signal lying in a conic re-gion ( hypothesis) and an interfering signal lying in thecomplement of the quoted region, namely the exterior ofthe cone (null hypothesis ). The graphic representationof the constraint for this specific situation is given inFig. 1(c).

• Exterior Elliptical Constraint:

, ,3 namely the usefulsignal has to lie outside of an ellipsoid whose center is thevector [see Fig. 1(d) for the corresponding representa-tion of ].

Evidently, the aforementioned four classes of quadratic con-straints may or may not be convex, and the feasible regionsare nonconvex, since they consist of one of the four classes of

3Otherwise, the set � � � � � is empty. Indeed, suppose that� �� , �� and �� , and then one gets the con-tradiction: ��

�� .

Scenario 1

Scenario 2 (3)

Scenario 1

Scenario 2.(6)


quadratic constraints and the norm constraint . Further-more, the objective function of problem is neither convexnor quadratic. In fact, consider the simplest case: , andthe objective function (for Scenario 1) becomes

(set for Scenario 2). Evaluating the Hessian matrix

(7)

it is neither a constant matrix nor a positive semidefinite ma-trix for some .4 In other words, the objectivefunction is neither quadratic nor convex; thus problemis a nonconvex program (neither a QCQP), and seems diffi-cult to solve. However, we will show that the problem can besolved efficiently. Precisely, in the next section, we will showthat problem , after concentration over the complex ampli-tudes ’s for Scenario 1 (over for Scenario 2) is equivalent toa fractional SDP, whose solutions can be obtained by solving itsequivalent SDP via the so-called Charnes-Cooper transforma-tion [18]. Then, we retrieve an optimal solution of from anoptimal solution to the equivalent fractional SDP through spe-cific rank-one decompositions [19].

III. IDENTIFIABILITY, SOLUTION TO THE ML STEERING

DIRECTION ESTIMATION PROBLEM, AND CONSTRAINED CRB

In this section, we first address the identifiability of the modelparameters, then we introduce the new algorithm which in poly-nomial time provides a solution to the estimation problem, and,finally, we assess the quality of the derived estimate througha comparison with the constrained CRB for the consideredproblem.

A. Identifiability of the Model Parameters

Identifiability refers to the study of the solution to the esti-mation problem in the noiseless case. It is basically a consis-tency aspect which allows to understand whether the solutionis unique in the ideal case of no noise. With reference to ourproblem, the identifiability equations can be written as

Scenario 1 (8)

4For example, at the points �� , �� , ��, the Hessian matrix (7) isnot positive semidefinite.

( for Scenario 2). Assuming that and ’s (for Scenario 2) are complex numbers, the identifiability of theparameters , , and holds if (8) has a uniquesolution with . Evidently this is not the case since both

’s ( for Scenario 2) and ’s ( for Scenario 2)with

ifif

[where is defined in (2)] and , are solu-tions of (8). Nevertheless, we can easily obtain identifiabilityimposing a further constraint on , i.e.

if

if (9)

With this additional constraint the phase ambiguity is eliminatedand the solution of (8) is unique thanks to the unit norm condi-tion.

It is worth pointing out that if the last constraint (9) is addedto the estimation problem of Section II, the algorithm, we willpropose in the next subsection, can be still applied to obtainthe ML estimates of the parameters provided that its output

( for Scenario 2) is phaserotated in order to comply with (9). In the following, the setaugmented with (9) is denoted by .

B. Solution to the ML Steering Direction Estimation Problem

In this subsection, we show how the solution to the MLsteering estimation problem can be obtained in polynomialtime. To this end, we observe that problem can be equiva-lently rewritten as shown in (10) at the bottom of the page. Theinner minimization is direct, i.e., for Scenario 1

(11)

and the minimal value is attained at

(12)

Scenario 1

Scenario 2.(10)


For Scenario 2

(13)

with

and the minimal value is attained at

(14)

Therefore, amounts to

(15)

where

Scenario 1

Scenario 2.

(16)

Now, let us focus on solving the fractional QCQP

(17)

where is defined by (2). The homogenized version of isshown in

(18)

We highlight that the two problems and have the same op-timal value (i.e., ), and is optimal to if

solves . In fact, it is evident that ; onthe other hand, the objective function of evaluated atis equal to the optimal value of , provided that is anoptimal solution for .

The SDP relaxation of is

(19)

where ’s, are defined as follows:

(20)

and

(21)

It is known that the single-ratio fractional SDP can be itera-tively solved by either the Dinkelbach algorithm (for instancesee [29] and [30]) or the bisection search (see [31]). In thispaper, we will solve in one single shot by resorting to avariation of the so-called Charnes-Cooper variable transforma-tion which was originally introduced in [18]. By the Charnes-Cooper variable transformation, one can replace a linear frac-tional program with at most two straightforward linear programsthat differ from each other by only a change of sign in the objec-tive function and in one constraint, and thus achieves a globaloptimal solution of the linear fractional program by solving atmost two linear programs. By using the idea of Charnes andCooper’s transformation, and considering that the denominatorof the fractional SDP is always positive, we can convert thefractional SDP into an equivalent SDP. Precisely, let us definethe transformed variable , where complies with

. Hence, multiplying by the numerator andthe denominator of the objective function in , we obtain theSDP problem

.

(22)

In the following, we show that the fractional SDP is equiv-alent to the SDP (22), in the sense that they have the equal op-timal value and optimal solutions differ from one to another by aconstant. In order to claim the equivalence between and ,we first prove three lemmas showing that the two problems are


solvable5 (Lemmas 3.1 and 3.2), and that they share the sameoptimal value (Lemma 3.3) even if the optimal solutions differup to a scalar whose expression is specified in Lemma 3.3.

Lemma 3.1: The fractional SDP problem is solvable.Proof: See Appendix A.

Lemma 3.2: The SDP problem is solvable.Proof: See Appendix B.

Lemma 3.3: Problems and have the equaloptimal value. Furthermore, if solves , then

solves ; if solves, then solves .

Proof: See Appendix C.Once an optimal solution of is obtained, we can check

the rank of . If is of rank one, then the solution is a globaloptimal solution of the fractional QCQP since is a re-laxation of by dropping the rank-one constraint. If isof rank higher than one, we will construct a rank-one optimalsolution of by a recent matrix decomposition the-orem [19]6. Then, the solution is an optimal solution of .In other words, the SDP relaxation (19) is tight. Specifically,in order to construct a rank-one optimal solution to , we usethe rank-one matrix decomposition theorem [19, Theorem 2.3],which is cited as the following lemma.

Lemma 3.4: Let be a nonzerocomplex Hermitian positive semidefinite matrix andbe Hermitian matrix, , and suppose that

, forany nonzero complex Hermitian positive semidefinite matrix

of size . Then,• if , one can find, in polynomial time, a

rank-one matrix such that (synthetically denoted as) is in , and

• if , for any not in the range space of , onecan find a rank-one matrix such that (syntheticallydenoted as ) is in the linearsubspace spanned by , and

Let us check the applicability of the lemma to both andthe matrix parameters of . Indeed, the condition ismild and practical. Now, in order to verify

for any nonzero

it suffices to prove that there is such that

5By “solvable,” we mean that the problem is feasible and bounded, and theoptimal value is attained; see [32, page 13].

6Note that once getting an optimal rank-one solution of (19) (as stated inAlgorithm 1 herein), we can give an optimal rank-one solution of (22) too,according to Lemma 3.3.

where

But this is evident for the matrix parameters7 of .Algorithm 1 summarizes the procedure leading to an optimal

solution of .

Algorithm 1: Algorithm for Fractional QCQP

Input: , , .

Output: An optimal solution of .

1: solve SDP finding an optimal solution andthe optimal value ;

2: let ;3: if then

4: perform an eigen-decomposition ,

where ; output and terminate.

5: else if then6: find

7: else8: find

9: end

10: let ; output .

The computational complexity, connected with the imple-mentation of Algorithm 18 includes the complexity of solvingSDP , which is of order (see [32, p. 250]),where is a prefixed accuracy, and the complexity of thespecific rank-one decomposition procedure which is .

Given an optimal solution of , it follows by (12) that forScenario 1,

and for Scenario 2,

7In fact, taking � � � � � and � � � � �, then � ��

� �� .8We highlight that Algorithm 1 includes solving the SDP relaxation problem

� , performing a similar Charnes-Cooper transformation, and some matrixrank-one decompositions which are specified in Lemma 3.4. In [19, Algorithm1] demonstrates how to decompose a positive semidefinite matrix according to[19, Th. 2.1]. In fact, [19, Th. 2.1] and [20, Th. 2.1] are the core of the rank-onedecompositions described in [19, Th. 2.3] (cited by Lemma 3.4 herein).


is an optimal solution of problem . Algorithm 2 describesthe procedure to find the ML estimates (i.e., the optimal solutionto ).

Algorithm 2: Finding an Optimal Solution of

Input: , , .

Output: An optimal solution forScenario 1 ( for Scenario 2).

1: define as (16);2: Algorithm 1 .

3: let ,, and for Scenario 1 (

, andfor Scenario 2).

C. Constrained CRB and Accuracy of the Proposed Estimator

In this subsection, we evaluate the constrained CRB for theestimation of and analyze the quality of the estimator proposedin the previous subsection through a comparison between itsaccuracy and the CRB. Assuming that is a regular point ofthe inequality constraints involved in , the CRB under theconstraint is identical to that under the constraintsand (with if and otherwise)[33]. In other words, only the equality constraints have to beconsidered. In the following, we derive the CRB exploiting thetechnique proposed in [21].

Let us define ,for Scenario 1

( for Scenario 2), and .After some standard algebra, the Fisher Information Matrix(FIM) [34] can be written as

where

with for Scenario 1 andfor Scenario 2;

with for Scenario 1 andfor Scenario 2.

Additionally, define , and con-sider the gradient vector of the constraints [21]

Scenario 1

Scenario 2

with . The above matrix has a full rowrank if and are not proportional. Focusing on situationswhere such assumption is satisfied, there exists a matrix suchthat (see the equation at the bottom of the page). Hence, by [21,Th. 1], the error covariance of an unbiased estimate ofcomplies with

for all such that . As a byproduct, theerror covariance of an unbiased estimate of fulfills

(23)

with the matrix formed by the firstrows and columns of .

Example: In this example, we study the mean squareerror (MSE) of the proposed algorithm, i.e.

(24)

(where denotes the number of independent data realiza-tions used to perform the MSE estimate and is the th MLestimate of ) in comparison with the trace of the CRB matrix[right-hand side (RHS) of (23)]. To this end, we assume ,

, covariance disturbance exponentially shaped withone-lag coefficient , elliptical constraint with ,

. The convex optimization MATLABtoolbox SElf-DUal-MInimization (SeDuMi) [35] is exploitedfor solving the SDP problem involved in Algorithm 1.9

In Fig. 2, MSE is plotted versus the signal-to-noise ratio(SNR), i.e.

(25)

for and several values of the parameter . We ob-serve that, as the SNR increases, the method’s MSE approachesthe trace of the constrained CRB matrix. Additionally, the lower

9The same solver is also exploited in the remaining simulations of the paper.

Scenario 1

Scenario 2


Fig. 2. MSE versus SNR for � � �, � � ��, perfectly matched steeringdirection, and several values of �. Trace of the constrained CRB matrix [RHSof (23)] solid line. � � �� o-marked curve. � � ��-marked curve. � � �

star-marked curve.

the uncertainty (in this case the volume of the ellipsoid aroundthe true steering direction), the better the performance.

IV. APPLICATIONS

In this section, we present two applications of the estimationtechnique developed in the previous section. Precisely, we willfocus on the problems of radar detection of range distributed tar-gets with an elliptical steering constraint and robust point-targetdetection with a radar using a general antenna array configura-tion with a mix of high- and low-gain beams.

A. Detection of Range Distributed Targets With an EllipticalSteering Constraint

Assume that data are collected by an array of sensorsand deal with the problem of detecting the presence of a targetacross range cells . Let , , bethe -dimensional vector of the received signal from the -thrange cell (primary data). Assume that a secondary data set, ,

, is available and each of such snapshots doesnot contain any useful target echo.

The detection problem to be solved can be formulated interms of the following binary hypotheses test:

(26)

where• ’s are unknown parameters accounting for the useful

target reflectivity and channel propagation effects;• ’s are -dimensional disturbance vectors modeled as

independent, zero-mean, complex Gaussian vectors withthe same unknown covariance matrix, i.e.

• is the actual steering vector, which due to sev-eral effects might not be aligned with the nominalsteering . In particular, in this application, we sup-pose that belongs to the set where

, and isa positive definite (or possibly semidefinite) matrix.

We investigate three possible statistical tests for the hypothesestest (26), i.e., the one-step generalized likelihood ratio test(GLRT), the two-step GLRT, and the modified two-step GLRT,which in the absence of steering mismatches have been derivedin [14].

One-Step GLRT: The one-step GLRT is the followingdecision rule shown in (27) at the bottom of the page, where

andare the data pdf’s under and , respectively, and isthe detection threshold set according to a design false alarmProbability . Performing the maximizations over ,

, after some standard algebra, we obtain the fol-lowing decision rule:

(28)

where , , and the samesymbol has been used to denote the modified threshold. Evi-dently, the optimization problem involved in the computation ofthe decision statistic can be solved resorting to the frameworkdeveloped in the previous section based on the Charnes-Coopertransformation.

Two-Step GLRT: This two-step procedure first derives theGLRT based on primary data, assuming that the covariance ma-trix is known (step 1). Then, a fully adaptive detector is obtainedby substituting the unknown matrix with the sample covariancematrix based on secondary data only (step 2). Step 1 can be ac-complished evaluating

(29)

(27)


which, after some algebra leads to

(30)

Hence, a completely adaptive detector (step 2) can be obtainedplugging in place of , i.e.,

(31)

The last maximization can be performed with the procedure ofSection II.

Modified Two-Step GLRT: This modified two-step proce-dure first derives the GLRT based on primary data, assumingthat the covariance structure is known (step 1).Then, a fully adaptive detector is obtained by substituting thesample covariance matrix in place of (step 2).

Step 1 requires the computation of (32) at the bottom of thepage, which, after some algebra, can be recast as

(33)

Plugging in place of , we get the modified two-step GLRT,i.e.

(34)

Again, Algorithm 1 of Section II can be used to obtain the op-timal value of the maximization problem in (34).

The new detectors (28), (31), and (34), will be respectivelyreferred to as Elliptically Constrained One-Step GLRT (EC-1S-GLRT), EC Two-Step GLRT (EC-2S-GLRT), and EC Modified2S-GLRT (EC-M2S-GLRT). Their counterparts, derived in [14]and assuming the perfect knowledge of , are instead respec-tively referred to as 1S-GLRT, 2S-GLRT, and M2S-GLRT.

In the following, we analyze the performance of the six con-sidered receivers assuming , , , targetuniformly spread within the range cells, covariance distur-bance exponentially shaped with one-lag coefficient ,false alarm probability , , and . Asto the actual steering direction , we simulate it as follows:

where denotes a (random) unit norm vector orthogonal tothe nominal direction and is a parameter which rules thedegree of mismatch.

In Fig. 3(a), we plot the detection Probability (evaluatedthrough Monte Carlo techniques) versus SNR (25) for

, namely actual steering direction perfectly matched with thenominal one. The curves show that the receivers which knowexactly the steering direction outperform the elliptically con-strained detectors which assume some uncertainty on the actualsteering direction. Indeed, for , the detection loss is al-ways kept within 4 dB for all the three considered design strate-gies.

In Fig. 3(b), we consider the case where some mismatch ispresent. Precisely, we set . Evidently, theclassic receivers show a performance degradation which mightalso be severe for the cases of the 1S-GLRT and the M2S-GLRT.As to the elliptically constrained receivers, they exhibit a morerobust behavior with respect to directional mismatches thanksto the uncertainty on the steering direction that they suppose atthe design stage.

Finally, in Fig. 3(c), we analyze the effect of the parameteron the detection performance for . The plots show

that the higher the higher the loss of each elliptically con-strained receiver with respect to its counterpart which exactlyknows the steering direction. This behavior can be explained ob-serving that the higher the wider the uncertainty region wherethe actual steering vector is supposed to belong to. Additionally,it seems that the EC-M2S-GLRT is the most sensitive among thenew receivers, whereas the EC-1S-GLRT and EC-2S-GLRT ex-hibit almost the same degree of sensitivity.

B. Robust Point-Target Detection With a Radar Using aGeneral Antenna Array Configuration With a Mix of High-and Low-Gain Beams

Let us consider a radar operating with a general antenna arrayconfiguration [17] which includes a mix of high- and low-gainbeams, obtained by suitable linear combinations of basic arrayelements. The radar is equipped with independent receivingchannels, each connected with one of the beams. According tothe model developed in [17], the echoes collected by thechannels at the th sampling instant are stacked to form the

-dimensional vector , , referredto as -th snapshot (for notational simplicity we assume thatcontains the echo from the cell under test). As to the spatial be-havior of the expected target, it is described by the -dimen-sional steering vector whose components are the complexamplitudes of the echo received at the -th channel from a nor-malized target with direction of arrival . If the radar contains

high-gain beams then ( is a -dimensionalcolumn vector), since the usual signal received at the low-gainbeams can be considered with a negligible power level [17].

The temporal behavior of the expected target echois described by the -dimensional vector

corresponding to the samples of thecoded transmitted waveform (assumed, without loss of gen-erality, with unit norm). Finally, the global disturbance is acombination of thermal noise and narrowband directional jam-

(32)


Fig. 3. (a) � versus SNR for � � �� , � � �, � � ��, � � �,� � ��, and perfectly matched steering direction. 1S-GLRT (dashedstar-marked curve), 2S-GLRT (dashed o-marked curve), M2S-GLRT (dashed+-marked curve), EC-1S-GLRT (solid star-marked curve), EC-2S-GLRT (solido-marked curve), EC-M2S-GLRT (solid +-marked curve). (b) � versus SNRfor � � �� , � � �, � � ��, � � �, � � ��, mismatched steeringdirection with � � ��

�� . 1S-GLRT (dashed star-marked curve),

2S-GLRT (dashed o-marked curve), M2S-GLRT (dashed +-marked curve),EC-1S-GLRT (solid star-marked curve), EC-2S-GLRT (solid o-marked curve),EC-M2S-GLRT (solid +-marked curve).

mers. Hence, the snapshots are modeled as independentcomplex Gaussian vectors with unknown covariance matrixand mean under (0 under ) with an unknownparameter accounting for the target backscattering and channelpropagation effects.

Let us denote by

Fig. 3. (Continued.) (c) � versus SNR for � � �� , � � �, � �

��, � � �, perfectly matched steering direction, and several values of � �� . Subplot a) 1S-GLRT (dashed curve) and EC-1S-GLRT (solidcurves). Subplot b) 2S-GLRT (dashed curve) and EC-2S-GLRT (solid curves).Subplot c) M2S-GLRT (dashed curve) and EC-M2S-GLRT (solid curves).

where is a matrix containing the high-gainbeams of each snapshot. The generalized likelihood ratio (GLR)for known can be written as [17]

(35)

where

Assuming that some uncertainty is available on , for instancedue to array pointing errors or miscalibration effects, one can re-sort to a further GLR over in order to cope with the additionalunknown parameter. If an elliptical model, as that of the pre-vious subsection, is considered for the uncertainty region, thenthe GLR becomes

(36)which can be equivalently written as shown (37), shown at thebottom of the page. The maximization in (37) can be accom-plished through Algorithm 1 of Section II. From the above equa-tion, we can construct the GLRT obtained comparing the GLR

(37)


with a threshold, set in order to ensure the design . The re-sulting detector will be referred to as the EC-GLRT and is givenby

(38)

Its counterpart, assuming the perfect knowledge of , will beinstead referred to as the GLRT. In the following, we assess theperformance of the EC-GLRT and the GLRT assuming that theactual steering direction is modeled as10

where is a sequence of independent zero-mean Gaussianrandom variables with variance and is another se-quence (statistically independent of ) of independent zero-mean Gaussian random variables with variance . As to thedisturbance scenario, we suppose the presence of noise plus twojammers impinging from 10 and 20 degrees, respectively, witha jammer to noise ratio of 20 dB.

In Fig. 4(a), we plot (evaluated through Monte Carlo tech-niques) of the two detectors versus SNR, i.e.

where is the nominal direction (i.e.,), , for ,

, , , , , , and twovalues of . The transmitted waveform is a Barker codeof length 13. For comparison purposes, the case of a perfectmatched steering direction is plotted as well.

The analysis of the curves shows that if a useful signal,perfectly matched to the nominal steering direction, impingeson the array, then the EC-GLRT exhibits almost the sameas the GLRT and, for , the respective performancecurves overlap. When a mismatch is present, the EC-GLRTachieves a performance level better than the GLRT; specificallythe stronger the mismatch the higher the detection gain.

The effect of the parameter is analyzed in Fig. 4(b), whereis plotted versus SNR for several values of , , and

the remaining simulation parameters as in Fig. 4(a). The plotshighlight that the performance gain of the EC-GLRT over theGLRT depends on the entity of the amplitude mismatch; pre-cisely, the higher the amplitude mismatch the higher the gainof the receiver which compensates for the amplitude mismatch.Summarizing, both the figures are a confirmation that, the ellip-tically constrained receiver is more robust than the counterpartwith respect to directional mismatches.

V. CONCLUSION

In this paper, we have considered the problem of constrainedsteering direction estimation of a signal embedded in Gaussiandisturbance. The constraint set has been represented in termsof a general quadratic inequality constraint which includes

10We are accounting for both amplitude and phase mismatch.

Fig. 4. (a) � versus SNR for � � �� , � � ��, � � �, � � ��,� � ��, mismatched steering direction with � � �� and two values of �(� � � star-marked curves and � � � o-marked curves). GLRT (solidcurves), EC-GLRT (dashed curves). (b) � versus SNR for � � �� , � �

��, � � �, � � ��, � � ��, mismatched steering direction with � � �

and three values of � (� � ��-marked curves), � � �� star-markedcurves, and � � �� o-marked curves.

many situations of practical relevance such as conic con-straints, elliptical constraints, exterior conic constraints, andexterior elliptical constraints. Additionally, a norm constraintaccounting for the unitary norm of the steering direction hasbeen considered. The estimation problem is attacked resortingto the ML criterion under two different assumptions for thereceived signal amplitude:

• the complex amplitude of the useful signal componentchanges from snapshot to snapshot;

• the useful signal keeps a constant amplitude within all thesnapshots.

We have proved that the ML criterion leads in both cases toa fractional QCQP. In order to solve it, we have first relaxedthe problem into a constrained fractional SDP problem whichwe prove equivalent to an SDP problem through the Charnes-Cooper transformation. Then, exploiting some rank-one decom-positions, we have shown that the SDP relaxation is actually


tight. Additionally, we have devised a procedure which con-structs (in polynomial time) an optimal solution of the originalproblem from an optimal solution of the fractional SDP. Wehave also analyzed the quality of the derived estimator througha comparison of its performance with the constrained CRB. Fi-nally, we have given two applications of the proposed theoret-ical framework in the context of radar detection of range dis-tributed targets and robust point-target detection with a radarusing a general antenna array configuration with a mix of high-and low-gain beams. In both the situations, the new procedureapplies successfully.

Possible future research tracks might concern the applicationof the framework to additional statistical signal processing prob-lems involving over the horizon (OTH) radar or synthetic aper-ture radar (SAR) processing (including atmospheric effects) aswell as the extension of the procedure to account for the pres-ence of a specific structure in the steering direction.

APPENDIX

A. Proof of Lemma 3.1

Proof: Observe that the feasible region of

(39)is a closed subset of the compact set

thus the set is compact as well. Also note that the denomi-nator and numerator of the objective function of the fractionalSDP are continuous, and that the denominator is positiveover the feasible region (because for any be-longing to the set which includes thefeasible region). Let . Then, forany , we have that

and

This implies that the objective function of problem is fi-nite-valued over the feasible region. It follows from Weierstrass’Theorem (for instance, see [36, Ch. 2]) that problem has al-ways an optimal solution. Hence, the problem is solvable.

B. Proof of Lemma 3.2

Proof: The dual of SDP is shown in

.(40)

Since has an optimal solution, say , hence it is easily seenthat is feasible for . There-fore, by weak duality, Problem is bounded below. Assume

that and are given such that. Then see,

which is equivalent to

Now, we can set to be sufficiently large and set so thatis close enough to

zero, andand . This means the SDP problemis strictly feasible. It follows from the strong duality theorem

(for example, see [32, Th. 1.7.1]) that Problem is solvable.

C. Proof of Lemma 3.3

Proof: Suppose that is an optimal solution of andis the optimal value of , and that is the optimal

value of . It is verified easily that withis feasible for , and the objective function

value at the feasible point is .Thus, we have .

On the other hand, let be an optimal solution of .We claim that . Indeed, if , then

, which implies .This is impossible since . It is checked that

is feasible for , and the objective function value ofat the feasible point is

. Therefore,we have , which yields .

ACKNOWLEDGMENT

The authors thank the Associate Editor and the Reviewersfor their constructive comments toward the improvement of theoriginal submitted paper.

REFERENCES

[1] S. A. Vorobyov, A. B. Gershman, and Z. Luo, “Robust adaptive beam-forming using worst-case performance optimization: A solution to thesignal mismatch problem,” IEEE Trans. Signal Process., vol. 51, no. 2,pp. 313–324, Feb. 2003.

[2] P. Stoica, Z. Wang, and J. Li, “Robust Capon beamforming,” IEEESignal Process. Lett., vol. 10, no. 6, pp. 172–175, Jun. 2003.

[3] J. Li, P. Stoica, and Z. Wang, “On robust Capon beamforming anddiagonal loading,” IEEE Trans. Signal Process., vol. 51, no. 7, pp.1702–1715, Jul. 2003.

[4] J. Li, P. Stoica, and Z. Wang, “Doubly constrained robust Capon beam-former,” IEEE Trans. Signal Process., vol. 52, no. 9, pp. 2407–2423,Sep. 2004.

[5] A. Beck and Y. C. Eldar, “Doubly constrained robust Capon beam-former with ellipsoidal uncertainty sets,” IEEE Trans. Signal Process.,vol. 55, no. 2, pp. 753–758, Jan. 2007.

[6] S. Ramprashad, T. W. Parks, and R. Shenoy, “Signal modeling anddetection using cone classes,” IEEE Trans. Signal Process., vol. 44,no. 2, pp. 329–338, Feb. 1996.

[7] M. N. Desai and R. S. Mangoubi, “Adaptive robust constrainedmatched filter and subspace detection,” in Proc. 36th Asilomar Conf.Signals, Syst., Comput., Pacific Grove, CA, Nov. 2002, pp. 768–772.

[8] M. N. Desai and R. S. Mangoubi, “Robust Gaussian and non-Gaussianmatched subspace detection,” IEEE Trans. Signal Process., vol. 51, no.12, pp. 3115–3127, Dec. 2003.


[9] F. Bandiera, A. De Maio, and G. Ricci, “Adaptive CFAR radar detec-tion with conic rejection,” IEEE Trans. Signal Process., vol. 55, no. 6,pp. 2533–2541, Jun. 2007.

[10] O. Besson, “Detection of a signal in linear subspace with boundedmismatch,” IEEE Trans. Aerosp. Electron. Syst., vol. 42, no. 3, pp.1131–1139, Jul. 2006.

[11] O. Besson, “Adaptive detection with bounded steering vectorsmismatch angle,” IEEE Trans. Signal Process., vol. 55, no. 4, pp.1560–1564, Apr. 2007.

[12] A. De Maio, S. De Nicola, Y. Huang, S. Zhang, and A. Farina, “Adap-tive detection and estimation in the presence of useful signal and inter-ference mismatches,” IEEE Trans. Signal Process., vol. 57, no. 2, pp.436–450, Feb. 2009.

[13] K. Gerlach and M. J. Steiner, “Adaptive detection of range distributedtargets,” IEEE Trans. Signal Process., vol. 47, no. 7, pp. 1844–1851,Jul. 1999.

[14] E. Conte, A. De Maio, and G. Ricci, “GLRT-based adaptive detectionalgorithms for range spread targets,” IEEE Trans. Signal Process., vol.49, no. 7, pp. 1336–1348, Jul. 2007.

[15] L. Xu, J. Li, and P. Stoica, “Target detection and parameter estimationfor MIMO radar systems,” IEEE Trans. Aerosp. Electron. Syst., vol. 44,no. 3, pp. 927–939, Jul. 2008.

[16] J. Li and P. Stoica, MIMO Radar Signal Processing. New York:Wiley, 2008.

[17] A. Farina, P. Lombardo, and L. Ortenzi, “A unified approach to adap-tive radar processing with general antenna array configuration,” SignalProcess., vol. 84, no. 9, pp. 1593–1623, Sep. 2004.

[18] A. Charnes and W. W. Cooper, “Programming with linear fractionalfunctionals,” Naval Res. Logist. Quarter., vol. 9, pp. 181–186, 1962.

[19] W. Ai, Y. Huang, and S. Zhang, “Further results on rank-one matrixdecomposition and its applications,” Math. Program., 2009, to be pub-lished.

[20] Y. Huang and S. Zhang, “Complex matrix decomposition and quadraticprogramming,” Math. Operat. Res., vol. 32, no. 3, pp. 758–768, Aug.2007.

[21] P. Stoica and B. C. Ng, “On the Cramér-Rao bound under parametricconstraints,” IEEE Signal Process. Lett., vol. 5, no. 7, pp. 177–179,1998.

[22] W.-K. Ma, T. N. Davidson, K. M. Wong, Z.-Q. Luo, and P. C. Ching,“Quasi-maximum-likelihood multiuser detection using semi-definiterelaxation with applications to synchronous CDMA,” IEEE Trans.Signal Process., vol. 50, no. 4, pp. 912–922, Apr. 2002.

[23] Y. Yang, C. Zhao, P. Zhou, and W. Xu, “MIMO detection of 16-QAMsignaling based on semidefinite relaxation,” IEEE Signal Process. Lett.,vol. 14, no. 11, pp. 797–800, Nov. 2007.

[24] N. D. Sidiropoulos and Z.-Q. Luo, “A semidefinite relaxation approachto MIMO detection for high-order QAM constellations,” IEEE SignalProcess. Lett., vol. 13, no. 9, pp. 525–528, Sep. 2006.

[25] J. Li, J. R. Guerci, and L. Xu, “Signal waveform’s optimal-under-Re-striction design for active sensing,” IEEE Signal Process. Lett., vol. 13,no. 9, pp. 565–568, Sep. 2006.

[26] R. G. Lorentz and S. Boyd, “Robust minimum variance beam-forming,” IEEE Trans. Signal Process., vol. 53, no. 5, pp. 1684–1696,May 2005.

[27] N. B. Pulsone and C. M. Rader, “Adaptive beamformer orthogonal re-jection test,” IEEE Trans. Signal Process., vol. 49, no. 3, pp. 521–529,Mar. 2001.

[28] F. Bandiera, D. Orlando, and G. Ricci, “Advanced radar detectionschemes under mismatched signal models,” in Synthesis Lectures onSignal Processing N. 8. New York: Morgan and Claypool, 2009.

[29] J. P. Crouzeix and J. A. Ferland, “Algorithms for generalized fractionalprogramming,” Math. Program., vol. 52, pp. 191–207, 1991.

[30] A. I. Barros, J. B. G. Frenk, S. Schaible, and S. Zhang, “A new algo-rithm for generalized fractional programs,” Math. Program., vol. 72,pp. 147–175, 1996.

[31] S. Boyd and L. Vandenberghe, Convex Optimization. New York:Cambridge Univ. Press, 2008.

[32] A. Nemirovski, Lectures on Modern Convex Optimization, ClassNotes. Atlanta, GA: Georgia Inst. Technol., 2005.

[33] J. D. Gorman and A. O. Hero, “Lower bounds for parametric estimationwith constraints,” IEEE Trans. Inf. Theory, vol. 26, pp. 1285–2301,Nov. 1990.

[34] S. M. Kay, Fundamentals of Statistical Signal Processing: EstimationTheory. Upper Saddle River, NJ: Prentice-Hall, 1993, vol. 1.

[35] J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimizationover symmetric cones,” Optimization Methods Softw., vol. 11–12, pp.625–653, Aug. 1999.

[36] D. P. Bertsekas, A. Nedic, and A. E. Ozdaglar, Convex Analysis andOptimization. New York: Athena Scientific, 2003.

Antonio De Maio (S’01–A’02–M’03–SM’07) wasborn in Sorrento, Italy, on June 20, 1974. He receivedthe Dr.Eng. degree (with honors) and the Ph.D. de-gree in information engineering, both from the Uni-versity of Naples Federico II, Naples, Italy, in 1998and 2002, respectively.

From October to December 2004, he was a VisitingResearcher with the U.S. Air Force Research Labora-tory, Rome, NY. From November to December 2007,he was a Visiting Researcher with the Chinese Uni-versity of Hong Kong, Hong Kong. Currently, he is an

Associate Professor with the University of Naples Federico II. His research in-terest lies in the field of statistical signal processing, with emphasis on radar de-tection, optimization theory applied to radar signal processing, and multiple-ac-cess communications.

Dr. De Maio is the recipient of the 2010 IEEE Fred Nathanson MemorialAward as the young (less than 40 years of age) AESS Radar Engineer 2010whose performance is particularly noteworthy as evidenced by contributions tothe radar art over a period of several years, with the following citation for “ro-bust CFAR detection, knowledge-based radar signal processing, and waveformdesign and diversity.”

Yongwei Huang (M’09) received the B.Sc. degreein computational mathematics in 1998, and theM.Sc. degree in operations research in 2000, bothfrom Chongqing University. In 2005, he receivedthe Ph.D. degree in operations research from theChinese University of Hong Kong, Hong Kong.

He was a Postdoctoral Fellow with the Depart-ment of Systems Engineering and EngineeringManagement, Chinese University of Hong Kong,and a Visiting Researcher with the Department ofBiomedical, Electronic, and Telecommunication En-

gineering, University of Napoli “Federico II”, Italy. Currently, he is a ResearchAssociate with the Department of Electronic and Computer Engineering, HongKong University of Science and Technology. His research interests are relatedto optimization theory and algorithms, including conic optimization, robustoptimization, combinatorial optimization, and stochastic optimization, andtheir applications in signal processing for radar and wireless communications.

Daniel P. Palomar (S’99–M’03–SM’08) receivedthe electrical engineering and Ph.D. degrees (bothwith honors) from the Technical University of Cat-alonia (UPC), Barcelona, Spain, in 1998 and 2003,respectively.

He is an Associate Professor with the Depart-ment of Electronic and Computer Engineering,Hong Kong University of Science and Technology(HKUST), Hong Kong, which he joined in 2006. Hehad previously held several research appointments,namely, at King’s College London (KCL), London,

U.K.; UPC; Stanford University, Stanford, CA; Telecommunications Techno-logical Center of Catalonia (CTTC), Barcelona; Royal Institute of Technology(KTH), Stockholm, Sweden; University of Rome “La Sapienza,” Rome, Italy;and Princeton University, Princeton, NJ. His current research interests includeapplications of convex optimization theory, game theory, and variationalinequality theory to signal processing and communications.

Dr. Palomar is an Associate Editor of the IEEE TRANSACTIONS ON

INFORMATION THEORY, and has been an Associate Editor of the IEEETRANSACTIONS ON SIGNAL PROCESSING, a Guest Editor of the IEEE SIGNAL

PROCESSING MAGAZINE 2010 Special Issue on Convex Optimization forSignal Processing, a Guest Editor of the IEEE JOURNAL ON SELECTED AREAS

IN COMMUNICATIONS 2008 Special Issue on Game Theory in Communica-tion Systems, the Lead Guest Editor of the IEEE JOURNAL ON SELECTED

AREAS IN COMMUNICATIONS 2007 Special Issue on Optimization of MIMOTransceivers for Realistic Communication Networks. He serves on the IEEESignal Processing Society Technical Committee on Signal Processing forCommunications (SPCOM). He was the General Co-Chair of the 2009 IEEEWorkshop on Computational Advances in Multi-Sensor Adaptive Processing(CAMSAP). He is a recipient of a 2004/2006 Fulbright Research Fellowship;the 2004 Young Author Best Paper Award by the IEEE Signal ProcessingSociety; the 2002/2003 Best Ph.D. prize in Information Technologies andCommunications by UPC; the 2002/2003 Rosina Ribalta first prize for theBest Doctoral Thesis in Information Technologies and Communications bythe Epson Foundation; and the 2004 prize for the Best Doctoral Thesis inAdvanced Mobile Communications by the Vodafone Foundation and COIT.


Shuzhong Zhang received the B.Sc. degree inapplied mathematics from Fudan University in 1984,and the Ph.D. degree in operations research andeconometrics from the Tinbergen Institute, ErasmusUniversity, in 1991.

He is a Professor with the Industrial and SystemsEngineering Program, University of Minnesota, andis currently on leave from the Department of SystemsEngineering and Engineering Management, The Chi-nese University of Hong Kong. He had held facultypositions with the Department of Econometrics, Uni-

versity of Groningen (1991–1993), and the Econometric Institute, Erasmus Uni-versity (1993–1999). Since 1999, he has been with Department of Systems Engi-neering and Engineering Management, The Chinese University of Hong Kong.

Dr. Zhang received the Erasmus University Research Prize in 1999, theCUHK Vice-Chancellor Exemplary Teaching Award in 2001, the SIAMOutstanding Paper Prize in 2003, and the IEEE Signal Processing Society BestPaper Award in 2010. He was an elected Council Member at Large of theMathematical Programming Society for 2006–2009, and is Vice President ofthe Operations Research Society of China. He serves on the Editorial Board offive academic journals, including Operations Research and the SIAM Journalon Optimization.

Alfonso Farina (M’95–SM’98–F’00) received thedoctor degree in electronic engineering from theUniversity of Rome (I), Italy, in 1973.

In 1974, he joined Selenia, now SELEX SistemiIntegrati, where he has been a Manager since May1988. He was Scientific Director in the Chief Tech-nical Office. Currently, he is Director of the Analysisof Integrated Systems Unit. In his professional life,he has provided technical contributions to detection,signal, data, image processing and fusion for the mainradar systems conceived, designed, and developed in

the Company. He has provided leadership in many projects—also conducted inthe international arena—in surveillance for ground and naval applications, inairborne early warning and in imaging radar. From 1979 to 1985, he has alsobeen a Professor of radar techniques with the University of Naples; in 1985 hewas appointed Associate Professor. He is the author of more than 500 peer-re-viewed technical publications and the author of books and monographs: RadarData Processing (Vol. 1 and 2) (translated in Russian and Chinese), (U.K.: Re-searches Studies Press, and New York: Wiley, 1985–1986); Optimized RadarProcessors, (on behalf of IEE, London, U.K.: Peter Peregrinus, 1987); and An-tenna Based Signal Processing Techniques for Radar Systems, 1992. He wrotethe chapter “ECCM Techniques” in the Radar Handbook (2nd ed., 1990, and3rd ed., 2008), edited by Dr. M. I. Skolnik.

Dr. Farina has been session chairman at many international radar conferences.In addition to lecturing at universities and research centers in Italy and abroad, he

also frequently gives tutorials at the International Radar Conferences on signal,data, and image processing for radar; in particular on multisensor fusion, adap-tive signal processing, space-time adaptive processing (STAP), and detection. In1987, he received the Radar Systems Panel Award of IEEE Aerospace and Elec-tronic Systems Society (AESS) for development of radar data processing tech-niques. He is the Italian representative of the International Radar Systems Panelof the IEEE AESS. He is the Italian industrial representative (Panel Member atLarge) at the Sensor and Electronic Technology (SET) of Research TechnologyOrganisation (RTO) of NATO. He has been on the BoD of the InternationalSociety for Information Fusion (ISIF). He has been the Executive Chair of theInternational Conference on Information Fusion (Fusion) 2006, Florence, Italy,July 10–13, 2006. He has been nominated Fellow of IEEE with the following ci-tation: “For development and application of adaptive signal processing methodsfor radar systems.” Recently, he has been nominated international fellow of theRoyal Academy of Engineering, U.K.; this fellowship was presented to himby HRH Prince Philip, the Duke of Edinburgh. He is a referee of numerouspublications submitted to several journals of IEEE, IEE, Elsevier, etc. He hasalso cooperated with the Editorial Board of the IEE Electronics & Communi-cation Engineering Journal. More recently, he has served as a member of theEditorial Board of Signal Processing (Elsevier) and has been Co-Guest Editorof its Special Issue on New Trends and Findings in Antenna Array Processingfor Radar, September 2004. He is the corecipient of the following Best PaperAwards: entitled to B. Carlton of the IEEE TRANSACTIONS ON AEROSPACE AND

ELECTRONIC SYSTEMS for 2001 and 2003 and also of the International Confer-ence on Fusion 2005. He has been the leader of the team that received the 2002AMS CEO award for Innovation Technology. He has been the corecipient ofthe AMS Radar Division award for Innovation Technology in 2003. Moreover,he has been the corecipient of the 2004 AMS CEO Award for Innovation Tech-nology. He has been the leader of the team that won the 2004 First Prize Awardfor Innovation Technology of Finmeccanica, Italy. This award context has seenthe submission of more than 320 projects. This award has been set for the firsttime in 2004. In September 7, 2006, he received the Annual European GroupTechnical Achievement Award 2006 by the European Association for Signal,Speech and Image Processing (EURASIP), with the citation: “For developmentand application of adaptive signal processing technique in practical radar sys-tems.” In 2006 and 2009, he was a corecipient of the annual Innovation Tech-nology award of SELEX Sistemi Integrati. He has been appointed member inthe Editorial Boards of IET Radar, Sonar and Navigation and of Signal, Image,and Video Processing Journal (SIVP). He has been the General Chairman ofthe IEEE Radar Conference 2008, Rome, May 26–30, 2008. He is a Fellow ofthe Institution of Engineering and Technology (IET), UK. He has been recentlynominated Fellow of EURASIP, the citation reads: “For contributions to radarsystem design, signal, data and image processing, data fusion and particularlyfor the development of innovative algorithms for deployment into practical radarsystems.” He is also the recipient of the 2010 IEEE Dennis J. Picard Gold Medalfor Radar Technologies and Applications with the following citation: “For con-tinuous, innovative, theoretical and practical contributions to radar systems andadaptive signal processing techniques.”

Documents

172 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, …...Many adaptive radar signal processing methods that assume the ... [13], [14], multiple input multiple output (MIMO) target