1490 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. …vorobyov/TSP14b.pdf · the DOA estimation performance offered by using the proposed joint and SDD transmit beamspace design methods

1490 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 62, NO. 6, MARCH 15, 2014

Efficient Transmit Beamspace Design for Search-FreeBased DOA Estimation in MIMO RadarArash Khabbazibasmenj, Student Member, IEEE, Aboulnasr Hassanien, Member, IEEE,

Sergiy A. Vorobyov, Senior Member, IEEE, and Matthew W. Morency

Abstract—In this paper, we address the problem of transmitbeamspace design for multiple-input multiple-output (MIMO)radar with colocated antennas in application to direction-of-ar-rival (DOA) estimation. A new method for designing the transmitbeamspace matrix that enables the use of search-free DOA esti-mation techniques at the receiver is introduced. The essence ofthe proposed method is to design the transmit beamspace matrixbased on minimizing the difference between a desired transmitbeampattern and the actual one while enforcing the constraint ofuniform power distribution across the transmit array elements.The desired transmit beampattern can be of arbitrary shape andis allowed to consist of one or more spatial sectors. The numberof transmit waveforms is even but otherwise arbitrary. To allowfor simple search-free DOA estimation algorithms at the receivearray, the rotational invariance property is established at thetransmit array by imposing a specific structure on the beamspacematrix. Semidefinite programming relaxation is used to approxi-mate the proposed formulation by a convex problem that can besolved efficiently. We also propose a spatial-division based design(SDD) by dividing the spatial domain into several subsectorsand assigning a subset of the transmit beams to each subsector.The transmit beams associated with each subsector are designedseparately. Simulation results demonstrate the improvement inthe DOA estimation performance offered by using the proposedjoint and SDD transmit beamspace design methods as comparedto the traditional MIMO radar technique.

Index Terms—Direction-of-arrival estimation, parameterestimation, phased-MIMO radar, semidefinite programmingrelaxation, transmit beamspace design.

Manuscript received May 09, 2013; revised October 02, 2013; accepted De-cember 23, 2013. Date of publication January 10, 2014; date of current versionFebruary 25, 2014. The associate editor coordinating the review of this man-uscript and approving it for publication was Prof. Antonio Napolitano. Thiswork is supported in part by the Natural Science and Engineering ResearchCouncil (NSERC) of Canada. Some preliminary results relevant to this paperhave been presented by the authors at ICASSP, Prague, Czech Republic, 2011and Asilomar, Pacific Grove, CA, USA, 2012.A. Khabbazibasmenj and M. W. Morency are with the Department of Elec-

trical and Computer Engineering, University of Alberta, Edmonton, AB T6G2V4, Canada (e-mail: [email protected]; [email protected]).A. Hassanien is with the Department of Electrical and Computer Engineering,

University of Alberta, Edmonton, AB T6G 2V4, Canada, and also with the Elec-trical Engineering Department, Aswan University, Aswan, Egypt (e-mail: [email protected]).S. A. Vorobyov is with the Department of Electrical and Computer Engi-

neering, University of Alberta, Edmonton, AB T6G 2V4, Canada, on leave andcurrently with the Department of Signal Processing and Acoustics, Aalto Uni-versity, Espoo, Finland (e-mail: [email protected]).This paper has supplementary downloadable multimedia material available at

http://ieeexplore.ieee.org provided by the authors. This includes Matlab codesthat can generate simulation results and figures shown in this paper. This mate-rial is 319KB in size.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.2014.2299513

I. INTRODUCTION

I N array processing applications, the direction-of-arrival(DOA) parameter estimation problem is the most fun-

damental one [1]. Many DOA estimation techniques havebeen developed for the classical array processing single-inputmultiple-output (SIMO) setup [1], [2]. The development of anovel array processing configuration that is best known as mul-tiple-input multiple-output (MIMO) radar [3], [4] has openednew opportunities in parameter estimation. Many works haverecently been reported in the literature showing the benefitsof applying the MIMO radar concept using widely separatedantennas [5]–[8] as well as using colocated transmit and receiveantennas [9]–[16]. We focus on the latter case in this paper.In MIMO radar with colocated antennas, a virtual array with

a larger number of virtual antenna elements can be formed andused for improved DOA estimation performance as comparedto the performance of SIMO radar [17], [18] for relativelyhigh signal-to-noise ratios (SNRs), i.e., when the benefits ofincreased virtual aperture start to show up. The SNR gain forthe traditional MIMO radar (with the number of orthogonalwaveforms being the same as the number of transmit antennaelements), however, decreases as compared to the phased-arrayradar where the transmit array radiates a single waveformcoherently from all antenna elements [12], [13]. A trade-offbetween the phased-array and the traditional MIMO radarcan be achieved [12], [14], [19] which gives the best of bothconfigurations, i.e., the increased number of virtual antennaelements due to the use of waveform diversity together withSNR gain due to subaperture based coherent transmission.Several transmit beamforming techniques have been devel-

oped in the literature to achieve transmit coherent gain inMIMOradar under the assumption that the general angular locationsof the targets are known a priori to be located within a certainspatial sector. The increased number of degrees of freedom forMIMOradar, due to theuseofmultiplewaveforms, is used for thepurpose of synthesizing a desired transmit beampattern based onoptimizing the correlation matrix of the transmitted waveforms[4], [20], [21]. To apply the designs obtained using the aforemen-tionedmethods, theactualwaveformsstillhave tobefoundwhichcan be a difficult and computationally demanding problem [22].One of themajormotivations for designing the transmit beam-

pattern is realizing thepossibility of achievingSNRgain togetherwith increased aperture for improved DOA estimation in a widerange of SNRs [15], [23]. In particular, it has been shown in [15]that the performance of a MIMO radar system with a number oforthogonal waveforms less than the number of transmit antennas

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KHABBAZIBASMENJ et al.: EFFICIENT TRANSMIT BEAMSPACE DESIGN 1491

and with transmit beamspace design capability is better thanthe performance of a MIMO radar system with full waveformdiversity and no transmit beamforming gain. Remarkably, usingMIMO radar with proper transmit beamspace design, it is pos-sible to guarantee the satisfaction of such desired property forDOA estimation as the rotational invariance property (RIP) atthe receive array [15]. This is somewhat similar in effect to theproperty of orthogonal space-time block codes, in that the shapeof the transmitted constellation does not change at the receiverindependentof the channel.The latter allows for a simpledecoder[24]. Similarly, here the RIP allows for simple DOA estimationtechniques at the receiver although theRIP is actually enforced atthe transmitter, and the propagationmedia cannot break it thanksto the proper design of transmit beamspace. Since the RIP holdsat the receive array independent of the propagation media andreceive antenna array configuration, the receive antenna arraycan be of arbitrary configuration. The possibility to satisfy theRIP as a general capability of the transmit beamspace-based de-sign methods has been discussed in [15]. However, the methodsdeveloped in [15] suffer from the shortcomings that there is nocontrol on the shape of the transmit beampattern, the transmitpower distribution across the antenna array elements is notuniform, and the achieved phase rotations come with variationsin the magnitude of different transmit beams that degrades theperformance of DOA estimation at the receiver, i.e., the RIP isnot satisfied precisely in general.In this paper, we consider the problem of transmit beamspace

design for DOA estimation in MIMO radar with colocated an-tennas. Motivated by the aforementioned shortcomings of themethods in [15], we propose a new method for designing thetransmit beamspace that enables the use of search-free DOA es-timation techniques at the receive antenna array.1 The essenceof the proposed method is to design the transmit beamspacematrix based on minimizing the difference between a desiredtransmit beampattern and the actual one while enforcing theuniform power distribution constraint across the transmit arrayantenna elements as well as RIP. The desired transmit beam-pattern can be of arbitrary shape and is allowed to consist ofone or more spatial sectors. The case of even but otherwise ar-bitrary number of transmit waveforms is considered. To allowfor simple search-free DOA estimation algorithms at the re-ceiver, the RIP is established at the transmit antenna array byimposing a specific structure on the transmit beamspace matrix.The proposed structure is based on designing the transmit beamsin pairs where the transmit weight vector associated with a cer-tain transmit beam is the conjugate flipped version of the weightvector associated with another beam, i.e., one transmit weightvector is designed for each pair of transmit beams. All pairs aredesigned jointly while satisfying the requirement that the twotransmit beams associated with each pair enjoy rotational in-variance with respect to each other. Semidefinite programming(SDP) relaxation is used to approximate the proposed formula-tion by a convex problem that can be solved efficiently using, forexample, interior point methods. In comparison to our previousmethod [23] that achieves phase rotation between two transmit

1An early and very preliminary exposition of this work has been presented inparts in [25] and [26].

beams, the proposed method enjoys the following advantages.(i) It ensures that the magnitude response of the two transmitbeams associated with one pair of transmit beams is exactly thesame at all spatial directions, a property that improves the DOAestimation performance. (ii) It ensures uniform power distribu-tion across transmit antenna array elements. (iii) It enables es-timating the DOAs via estimating the accumulated phase rota-tions over all transmit beams instead of only two beams. (iv) Itonly involves optimization over half the entries of the transmitbeamspace matrix which decreases the computational load. Wealso propose an alternative formulation based on splitting theoverall transmit beamspace design problem into several smallerproblems. The alternative formulation is referred to as the spa-tial-division based design (SDD) which involves dividing thespatial domain into several subsectors and assigning a subsetof the transmit beamspace pairs to each subsector. The SDDmethod enables post processing of data associated with differentsubsectors independently with estimation performance compa-rable to the performance of the joint transmit beamspace design.Simulation results demonstrate the improvement in the DOAestimation performance that is achieved by using the proposedjoint transmit beamspace design and SDDmethods as comparedto the traditional MIMO radar technique.The rest of the paper is organized as follows. Section II

introduces the system model for mono-static MIMO radarsystem with transmit beamspace. The problem formulation isdeveloped in Section III while the transmit beamspace designproblem for even but otherwise arbitrary number of transmitwaveforms is developed in Section IV. Section V gives simula-tion examples for the proposed DOA estimation techniques andconclusions are drawn in Section VI. This paper is reproducibleresearch, and software needed to generate the simulation resultscan be obtained from the IEEE Xplore together with the paper.

II. SYSTEM MODEL AND MAIN IDEA

Consider a mono-static MIMO radar system equipped with atransmitter being uniform linear array (ULA) of colocatedantennas with inter-element spacing measured in wavelength,and a receive array of antennas configured in a random shape.The transmit and receive arrays are assumed to be close enoughto each other such that the spatial angle of a target in the far-field remains the same with respect to both arrays. Let

be the vector that contains the com-plex envelopes of the waveforms whichare assumed to be orthogonal, i.e.,

(1)

where is the pulse duration, and stand for the trans-pose and the conjugate, respectively, and is the Kroneckerdelta. The actual transmitted signals are taken as linear com-binations of the orthogonal waveforms. Therefore, thevector of the baseband representation of the transmitted signalscan be written as [15]

(2)


where is the signal transmitted from antenna andis the transmit beamspace matrix. It is worth notingthat each of the orthogonal waveforms istransmitted over one transmit beam where the th column of thebeamspace matrix corresponds to the transmit beamformingweight vector used to form the th beam.Let be the

transmit array steering vector. The transmit power dis-tribution pattern can be expressed as [20]

(3)

where stands for the conjugate transpose, ,and

(4)

is the cross-correlation matrix of the transmitted signals .One way to achieve a certain desired transmit beampattern is

to optimize over the cross-correlation matrix such as in [20],[21]. In this case, a complementary problem has to be solvedafter obtaining in order to find appropriate signal vectorthat satisfies (4). Solving such a complementary problem is ingeneral difficult and computationally demanding. However, inthis paper, we extend our approach of optimizing the transmitbeampattern via designing the transmit beamspace matrix.According to this approach, the cross-correlation matrix is ex-pressed as that holds due to the orthogonality ofthe waveforms (see (1) and (2)). Then the transmit beamspacematrix can be designed to achieve the desired beampatternwhile satisfying many other requirements mandated by prac-tical considerations such as equal transmit power distributionacross the transmit array antenna elements, achieving a desiredradar ambiguity function, etc. Moreover, this approach enablesenforcing the RIP which facilitates subsequent processing stepsat the receive antenna array, e.g., it enables applying accuratecomputationally efficient DOA estimation using search-freedirection finding techniques such as ESPRIT.The signal measured at the output of the receive array due to

echoes from narrowband far-field targets can be modeled as

(5)

where is the time index within the radar pulse, is the slowtime index, i.e., the pulse number, is the reflection coeffi-cient of the target located at the unknown spatial angle ,is the receive array steering vector, and is thevector of zero-mean white Gaussian noise with variance ,and is the identity matrix of size . In (5), the targetreflection coefficients are assumed to obeythe Swerling II model, i.e., they remain constant during the du-ration of one radar pulse but change from pulse to pulse. More-over, they are assumed to be drawn from a normal distributionwith zero mean and variance .

By matched filtering to each of the orthogonal basiswaveforms , the virtual data vectorscan be obtained as2

(6)

where is the th column of the transmit beamspace matrix, is the noise term whose

covariance is .Let be the noise free component of the virtual data

vector (6) associated with the th target, i.e.,. Then, one can easily observe that the

th and the th components associated with the th target arerelated to each other through the following relationship

(7)

where is the phase of the inner product . Theexpression (7) means that the signal component corre-sponding to a given target is the same as the signal component

corresponding to the same target up to a phase rotationand a gain factor.The RIP can be enforced by imposing the constraint

while designing the transmitbeamspace matrix . The main advantage of enforcing theRIP is that it allows us to estimate DOAs via estimating thephase rotation associated with the th and th pair of thevirtual data vectors using search-free techniques, e.g., ESPRIT.Moreover, if the number of transmit waveforms is more thantwo, the DOA estimation can be carried out via estimating thephase difference between two newly defined vectors as it willbe shown later in the paper.

III. PROBLEM FORMULATION

The main goal is to design a transmit beamspace matrixwhich achieves a spatial beampattern that is as close as possibleto a certain desired one. Substituting in (3), thespatial beampattern can be rewritten as

(8)

Therefore, we design the transmit beamspace matrix basedon minimizing the difference between the desired beampatternand the actual beampattern given by (8). Using the minimax

2Practically, this matched filtering step is performed for each Doppler-rangebin, i.e., the received data is matched filtered to a time-delayed Doppler-shifted version of the waveforms .


criterion, the transmit beamspace matrix design problem can beformulated as

(9)

(10)

where is the desired beampattern andis the total transmit power. The constraints enforced in

(10) are used to ensure that individual antennas transmit equalpowers given by . We incorporate the uniform power dis-tribution across the array antenna elements given by (10) whichis necessary from a practical point of view. In practice, each an-tenna in the transmit array typically uses the same power ampli-fier, and thus has the same dynamic power range. If the powerused by different antenna elements is allowed to vary widely,this can severely degrade the performance of the system due tothe nonlinear characteristics of the power amplifier.Another goal that we wish to achieve is to enforce the RIP

to enable for search-free DOA estimation. Enforcing the RIPbetween the th and th transmit beams for the evennumber of orthogonal waveforms is equivalent to ensuring thatthe following relationship holds

(11)

Ensuring (11), the optimization problem (9)-(10) can be refor-mulated as

(12)

(13)

(14)

It is worth noting that the constraints (13) as well as the con-straints (14) correspond to non-convex sets and, therefore, theoptimization problem (12)–(14) is a non-convex problem whichis difficult to solve in a computationally efficient manner. More-over, the fact that (14) should be enforced for every direction

makes it difficult to satisfy (14) unless a spe-cific structure on the transmit beamspace matrix is imposed.In the following section we propose a specific structure to

to overcome the difficulties caused by (14) and show how touse SDP relaxation to overcome the difficulties caused by thenon-convexity of (12)–(14).

IV. TRANSMIT BEAMSPACE DESIGN

A. Two Transmit Waveforms

We first consider a special, but practically important case oftwo orthogonal waveforms. Thus, the dimension of is. Then under the aforementioned assumption of ULA at the

MIMO radar transmitter, the RIP can be satisfied by choosingthe transmit beamspace matrix to take the form

(15)

where is the flipped version of vector , i.e.,, . Indeed, in this case,

and the RIP is clearly satisfied.In order to show the latter, the inner products and

can be, respectively, expressed as

(16)

(17)

Factoring out the term from the right handside of (17) and conjugating it, (17) can be equivalentlyrewritten as

(18)

From (18), it can be seen that the terms andare identical in magnitude. It is noteworthy to mention that, fora fixed beamforming vector , there may exist other beam-forming vectors that have the same beampattern as . However,we have only considered for reasons of simplicity which be-comes more clear later in Section IV-B.Substituting (15) in (12)–(14) and introducing the auxiliary

variable , the optimization problem (12)–(14) can be reformu-lated as follows for the case of two transmit waveforms andwhen the number of transmit antennas is even3

(19)

(20)

(21)

(22)

where is a set of directionsthat are properly chosen (uniform or nonuniform) to approx-imate the spatial domain . It is worth noting thatthe optimization problem (19)–(22) has significantly more de-grees of freedom than the beamforming problem in the case ofphased-array where the magnitudes of , arefixed. The constraints (14) are not shown in the optimizationproblem (19)–(22) as they are inherently enforced due to theuse of the specific structure of given in (15).The problem (19)–(22) belongs to the class of non-convex

quadratically-constrained quadratic programming (QCQP)

3The case when the number of transmit antennas is odd can be carried out ina straightforward manner.


problems which are in general NP-hard. However, a well devel-oped SDP relaxation technique can be used to approximatelysolve it [27]–[31]. Indeed, let us use the facts that

(23)

(24)

where stands for the trace and is an matrix suchthat with therest of the elements equal to zero. Then the problem (19)–(22)can be written as an SDP problem with rank constraint. Specif-ically, introducing the new variable and using theproperties (23) and (24), the problem (19)–(22) can be equiva-lently written as

(25)

(26)

(27)

(28)

(29)

where is a Hermitianmatrix and denotes the rank of amatrix. Note that the last two constraints in (25)–(29) imply thatthe matrix is positive semidefinite. The problem (25)–(29) isnon-convex with respect to because the last constraint is non-convex. However, by means of the SDP relaxation technique,we can drop this constraint and replace it by the constraint thatis positive semidefinite, i.e., . The resulting problem is

the relaxed version of (25)–(29) and it is a convex SDP problemwhich can be efficiently solved using, for example, interior pointmethods. When the relaxed problem is solved, extraction of thesolution of the original problem is typically done via so-calledrandomization techniques [27].Let denote the optimal solution of the relaxed problem.

If the rank of is one, the optimal solution of the originalproblem (19)–(22) can be obtained by simply finding the prin-cipal eigenvector of . However, if the rank of the matrix

is higher than one, the randomization approach can beused. Various randomization techniques have been developedin the literature and are generally based on generating a set ofcandidate vectors and then choosing the candidate which givesthe minimum of the objective function of the original problem.Our randomization procedure can be described as follows. Let

denote the eigen-decomposition of . Thecandidate vector can be chosen as whereis random vector whose elements are random variables uni-

formly distributed on the unit circle in the complex plane. Can-didate vectors are not always feasible and should be mapped to anearby feasible point. This mapping is problem dependent [31].In our case, if the condition

does not hold, we can map this vector to a nearbyfeasible point by scaling and to

satisfy this constraint. Among the candidate vectors we thenchoose the one which gives the minimum objective function,i.e., the one withminimum .

B. Even Number of Transmit Waveforms

Let us consider now the transmit beamspace matrixwhere and is an even

number. In the previous subsection, we saw that by consideringthe specific structure for the transmit beamspacematrixwith only two waveforms, the RIP is guaranteed at the receiveantenna array. In this part, we obtain the RIP for the more gen-eral case of more than two waveforms. It provides more degreesof freedom for obtaining a better performance. For this goal, wefirst show that if for some the following relation holds

(30)

then the two new sets of vectors defined as the summation ofthe first data vectors , and the lastdata vectors will satisfy the RIP. Morespecifically, by defining the following vectors

(31)

(32)

the corresponding signal component of target in the vectorhas the same magnitude as in the vector if the (30)

holds. In this case, the only difference between the signal com-ponents of the target in the vectors and is thephase which can be used for DOA estimation. Based on thisfact, for ensuring the RIP between the vectors and ,(30) needs to be satisfied for every angle .Considering the equation (18), it is clear that the RIP

between and holds provided that. Based on the latter fact, a simple structure on

the beamspace matrix which guarantees the satisfaction ofthe (30) for any arbitrary is as follows:• is an even number,• equals to ,• , .

More specifically, if the transmit beamspace matrix has the fol-lowing structure

(33)


then the signal component of associated with the th targetis the same as the corresponding signal component of upto phase rotation of

(34)

where denotes the phase operator and the resulted phasedifference can be used as a look-up table finding DOA. It isworth noting that our proposed structure for the beamspace ma-trix (33), decreases the available degrees of freedom for thebeampattern matching by a factor of two as compared to a gen-eral beamspace matrix of the same size. However, it is an ac-ceptable price for having the RIP which allows for low com-plexity DOA estimation for a receive array of any arbitrary con-figuration. By considering the aforementioned structure for thetransmit beamspace matrix , it is guaranteed that the RIP issatisfied and other additional design requirements can be satis-fied through the proper design of .Substituting (33) in (12)–(14), the optimization problem of

transmit beamspace matrix design can be reformulated as

(35)

(36)

where denotes the Euclidean norm.

Introducing the new variablesand following similar steps as in the case of

two transmit waveforms, the problem (35)–(36) can be equiv-alently rewritten as

(37)

(38)

(39)

where are Hermitian matrices. Theproblem (37)–(39) can be solved in a similar way as theproblem (25)–(29). Specifically, the problem (37)–(39) can beaddressed using SDP relaxation, that is, dropping the rank-oneconstraints and solving the resulting convex problem. Specifi-cally, the problem (37)–(39) without the rank-one constraintsand with constraints is convex and,therefore can be solved efficiently using interior point methods.Once the matrices are obtained,the corresponding weight vectors can

be obtained using randomization techniques. We use the ran-domization method introduced in Section IV-A over every

separately and then map the obtainedrank-one solutions to the closest feasible points. Among thecandidate solutions, the best one is then selected.

C. Optimal Rotation of the Transmit Beamspace Matrix

The solution of the optimization problem (35)–(36) isnot unique and, as it will be explained shortly in detail, anyspatial rotation of the optimal transmit beamspace matrix isalso optimal. Among the set of the optimal solutions of theproblem (35)–(36), the one with better energy preservation isfavorable. As a result, after the approximate optimal solutionof the problem (35)–(36) is obtained, we still need to find theoptimal rotation which results in the best possible transmitbeamspace matrix in terms of the energy preservation. Morespecifically, since the DOA of the target at is estimated basedon the phase difference between the signal components ofthis target in the newly defined vectors, i.e.,and , to obtain the best performance,

should be designed in a way that the magnitudes of thesummations and taketheir largest values.Since the phase of the product terms in

(or equivalently in )may be different for different waveforms, the terms in thesummation (or equivalently in the summation

) may add incoherently and, therefore,it may result in a small magnitude which in turn degrades theDOA estimation performance. In order to avoid this problem,we use the property that any arbitrary rotation of the transmitbeamspace matrix does not change the transmit beampattern.Specifically, if is atransmit beamspace matrix with the introduced structure, thenthe new beamspace matrix defined as

(40)

has the same beampattern and the same power distributionacross the antenna elements. Here

and is a unitary matrix. Basedon this property, after proper design of the beamspace matrixwith a desired beampattern and the RIP, we can rotate the beamsso that the magnitude of the summation isincreased as much as possible.Since the actual locations of the targets are not known a

priori, we design a unitary rotation matrix so that the integrationof the squared magnitude of the summationover the desired sector is maximized. As an illustrating exampleand because of space limitations, we consider the case whenis 4. In this case, we have

(41)

Integration of the squared magnitude of the summationover the desired sectors can be expressed


as shown by the following set of equations, where denotesthe desired sectors, stands for the real part of a complex

number, and .

(42a)

(42b)

Also note the last term inside the integral (42b) follows fromthe equation (41). We aim at maximizing the expression (42b)with respect to the unitary rotation matrix . Since the first twoterms inside the integral in (42b) are independent of the unitarymatrix, it suffices only to minimize the integration of the lastterm.Using the property that , where

denotes the Frobenius norm, and the cyclical property of thetrace, i.e., , the integral of the last termin (42b) can be equivalently expressed as

(43)

The only term in the integral (43) which depends on is. Therefore, the minimization of the inte-

gration of the last term in (42b) over a sector can be stated asthe following optimization problem

(44)

(45)

where and . Becauseof the unitary constraint, the optimization problem (44)-(45) be-longs to the class of optimization problem over the Stiefel man-ifold [32], [34]. Note that since the objective function in the op-timization problem (44)-(45) depends not just on the subspacespanned by , but rather on the basis as well, the correspondingmanifold is a Steifel manifold, in contrast to the more commonGrassmannian manifold. In order to address this problem, wecan use the existing steepest descent-based algorithm developedin [32].

D. Spatial-Division Based Design (SDD)

It is worth noting that instead of designing all transmitbeams jointly, an easy alternative for designing is todesign different pairs of beamforming vectors ,

separately. Specifically, in order to avoid

the incoherent summation of the terms in ,equivalently, in , the matrix can bedesigned in such a way that the corresponding transmit beam-patterns of the beamforming vectors do notoverlap and they cover different parts of the desired sector withequal energy. This alternative design is referred to as the SDDmethod. The design of one pair has been alreadyexplained in Section IV-A.

V. SIMULATION RESULTS

Throughout our simulations, we assume a uniform lineartransmit array with antennas spaced half a wavelengthapart, and a non-uniform linear receive array ofelements. The locations of the receive antennas are randomlydrawn from the set [0, 9] measured in half a wavelength. Noisesignals are assumed to be Gaussian, zero-mean, and white bothtemporally and spatially. In each example, targets are assumedto lie within a given spatial sector. From example to example thesector widths in which transmit energy is focused is changed,and, as a result, so does the optimal number of waveforms tobe used in the optimization of the transmit beamspace matrix.The optimal number of waveforms is calculated based onthe number of dominant eigenvalues of the positive definitematrix (see [15] for explanations andcorresponding Cramer-Rao bound derivations and analysis).We assume that the number of dominant eigenvalues is even;otherwise, we round it up to the nearest even number. Thereason that an odd number of dominant eigenvalues is roundedup, as opposed to down, is that overusing waveforms is lessdetrimental to the performance of DOA estimation than under-using, as it is shown in [15]. Three examples are chosen to testthe performance of our algorithm.In Example 1, a single centrally located sector of width 20

is chosen to verify the importance of the proposed structureon the beamspace matrix (33). In Example 2, two separatesectors each with a width of 20 degrees are chosen. Finally,in Example 3, a single, centrally located sector of width 30degrees is chosen. The optimal number of waveforms usedfor each example is two, four, and four, respectively. Themethods tested are the traditional MIMO radar with uniformtransmit power density and [17], the proposed jointlyoptimum transmit beamspace design method, the SDD methodwhen applicable, and the method of [15]. In [15], a convexoptimization based method is used to approximately map thetransmit steering vector to the steering vector of a uniformlinear array.Throughout the simulations, we refer to the proposed jointly

optimum transmit beamspace design method as the best achiev-able transmit beamspace design (since the solution obtainedthrough SDP relaxation and randomization is suboptimal in gen-eral) to distinguish it from the SDD method in which differentpairs of the transmit beamspace matrix columns are designedseparately. For the traditional MIMO radar, the following set oforthogonal baseband waveforms is used

(46)


Fig. 1. Example 1: Performance of the new proposed method with and withoutuniform power distribution across transmit waveforms.

while for the proposed transmit beamspace-based method, thefirst waveforms of (46) are employed. Throughout all sim-ulations, the total transmit power remains constant atand the number of radar pulses used is 50. The root mean squareerror (RMSE) and probability of target resolution are calculatedbased on 500 independent Monte-Carlo runs.

A. Example 1: Effect of the Proposed Structure on theBeamspace Matrix (33)

In this example, we aim at studying how the lack of theproposed structure on the beamspace matrix (33) affects theperformance of the new proposed method. For this goal, weconsider two targets that are located in the directions and5 and the desired sector is chosen as . Twoorthogonal waveforms are considered and the best achievabletransmit beamspace matrix denoted as is obtained bysolving the optimization problem (19)–(22) without rank-oneconstraint. To simulate the case in which the beamspace matrixdoes not enjoy the proposed beamspace structure (33), but itpreserves the same transmit beampattern of , we use therotated transmit beamspace matrix , where is a unitarymatrix defined as

Note that and lead to the same transmit beampat-tern and as a result the same transmit power within the desiredsector, however, compared to the former, the latter one does notenjoy the proposed structure on the beamspace matrix (33). TheRMSE curves of the proposed DOA estimation method for both

and versus are shown in Fig. 1. It can be seenfrom this figure that the lack of the proposed structure can de-grade the performance of DOA estimation severely.

Fig. 2. Example 2: Transmit beampatterns of the traditional MIMO and theproposed transmit beamspace design-based methods.

B. Example 2: Two Separated Sectors of Width 20 DegreesEach

In the second example, two targets are assumed to lie withintwo spatial sectors: one from and the otherfrom . The targets are located at and

. Fig. 2 shows the transmit beampatterns of the tradi-tional MIMOwith uniform transmit power distribution, both thebest achievable and SDD designs for , and the optimal beam-pattern obtained through the relaxed version of the optimizationproblem (25)–(29) before randomization. It can be seen fromthe figure that the best achievable transmit beamspace methodprovides the most even concentration of power in the desiredsectors, and it almost exactly coincides with the optimal beam-pattern before randomization. The beampattern before random-ization is shown for comparison against the beampattern of thebest achievable beamspace. The SDD technique provides con-centration of power in the desired sectors above and beyond tra-ditional MIMO; however, the energy is not evenly distributedwith one sector having a peak beampattern strength of 15 dB,with the other having a peak of no more than 12 dB. The per-formance of all three methods is compared in terms of the cor-responding RMSEs versus SNR as shown in Fig. 3. As we cansee in the figure, the best achievable beamspace and the SDDmethods have lower RMSEs as compared to the RMSE of thetraditional MIMO radar. It is also observed from the figure thatthe performance of the SDD method is very close to the perfor-mance of the best achievable beamspace design.To assess the proposed method’s ability to resolve closely

located targets, we move both targets to the locationsand . The performance of all three methods testedis given in terms of the probability of target resolution. Notethat the targets are considered to be resolved if the following issatisfied [2]


Fig. 3. Example 2: Performance comparison between the traditional MIMOand the proposed transmit beamspace design-based methods.

Fig. 4. Example 2: Performance comparison between the traditional MIMOand the proposed transmit beamspace design-based methods.

where and denotes estimation of . Theprobability of source resolution versus SNR for all methodstested are shown in Fig. 4. It can be seen from the figure thatthe SNR threshold at which the probability of target resolutiontransitions from very low values (i.e., resolution fail) to valuesclose to one (i.e., resolution success) is the lowest for thebest achievable transmit beamspace, second lowest for theSDD method, and finally, highest for the traditional MIMOradar method. In other words, the figure shows that the jointlyoptimal transmit beamspace design-based method has a higherprobability of target resolution at lower values of SNR than the

Fig. 5. Example 3: Transmit beampatterns of the traditional MIMO, method of[15], and the proposed methods.

SDD method, while the traditional MIMO radar method hasthe worst resolution performance.

C. Example 3: Single and Centrally Located Sector of Width30 Degrees

In the last example, a single wide sector is chosen as. The optimal number of waveforms for such

a sector is found to be four. In this example, we compare theperformance of the proposed method to that of the traditionalMIMO radar, and the method of [15]. Four transmit beams areused for designing the best achievable transmit beamspace,while only two are used for the method of [15], as suggestedtherein (see Example 3 in [15]). The SDD method is notconsidered in this example as the corresponding spatially di-vided sectors in this case are not spatially divided, but ratheradjacent. As a result, their overlapping side-lobes will resultin energy loss and performance degradation as compared toExample 2.Fig. 5 shows the transmit beampatterns for the methods

tested. Similar to Example 2, the beampattern before ran-domization is also shown in Fig. 5 for comparison with thebest achievable beamspace. Fig. 5 again shows an almostexact correspondence between the pre- and post-randomiza-tion beampatterns. In order to test the RMSE performance ofthe methods tested, two targets are assumed to be located at

and . Fig. 6 shows the RMSEs versus SNRfor the methods tested. As we can see in the figure, the RMSEfor the case of the best achievable transmit beamspace is lowerthan that of traditional MIMO, and the method of [15]. Thelatter can be attributed to the fact that the proposed method hasmore flexibility compared with that of [15]. Specifically, themethod of [15] suffers when the optimal number of waveformsis larger than 2, whereas the proposed method can be appliedwith an arbitrary but even number of waveforms. In order totest resolution capabilities of each method, the same criterion


Fig. 6. Example 3: Performance comparison between the traditional MIMO,method of [15], and the proposed methods.

Fig. 7. Example 3: Performance comparison between the traditional MIMO,method of [15], and the proposed methods.

as in Example 2 is used. From Fig. 7, it can be observed thatthe proposed methods outperforms the method of [15], andtraditional MIMO radar as expected.

VI. CONCLUSION

The problem of transmit beamspace design for MIMO radarwith colocated antennas with application to DOA estimationhas been considered. A new method for designing the transmitbeamspace matrix that enables the use of search-free DOAestimation techniques at the receiver has been introduced.The essence of the proposed method is to design the transmitbeamspace matrix based on minimizing the difference between

a desired transmit beampattern and the actual one. The caseof even but otherwise arbitrary number of transmit waveformshas been considered. The transmit beams are designed inpairs where all pairs are designed jointly while satisfying therequirements that the two transmit beams associated with eachpair enjoy rotational invariance with respect to each other.Unlike previous methods that achieve phase rotation betweentwo transmit beams while allowing the magnitude to be dif-ferent, a specific beamspace matrix structure achieves phaserotation while ensuring that the magnitude response of thetwo transmit beams is exactly the same at all spatial directionshas been proposed. The SDP relaxation technique has beenused to transform the proposed formulation into a convex op-timization problem that can be solved efficiently using interiorpoint methods. An alternative SDD method that divides thespatial domain into several subsectors and assigns a subset ofthe transmit beamspace pairs to each subsector has been alsodeveloped. The SDD method enables post processing of dataassociated with different subsectors independently with DOAestimation performance comparable to the performance of thejoint transmit beamspace design-based method. Simulationresults have been used to demonstrate the improvement in theDOA estimation performance offered by using the proposedjoint and SDD transmit beamspace design methods as com-pared to the traditional MIMO radar.

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Arash Khabbazibasmenj (S’08) received the B.Sc.and M.Sc. degrees in electrical engineering (commu-nications) from Amirkabir University of Technology,Tehran, Iran, and the University of Tehran, Tehran,Iran, in 2006 and 2009, respectively, and the Ph.D.degree in electrical engineering (signal processing)from the University of Alberta, Edmonton, Alberta,Canada, in 2013.He is currently working as a Postdoctoral Fellow

in the Department of Electrical and Computer Engi-neering of the University of Alberta. During spring

and summer of 2011, he was also a visiting student at Ilmenau University of

Technology, Germany. His research interests include signal processing and op-timization methods in radar, communications, and related fields.Dr. Khabbazibasmenj is a recipient of the Alberta Innovates Graduate Award

in ICT.

Aboulnasr Hassanien (M’08) received the B.Sc. de-gree in electronics and communications engineeringand the M.Sc. degree in communications engineeringfrom Assiut University, Assiut, Egypt, in 1996 and2001, respectively, and the Ph.D. degree in electricaland computer engineering from McMaster Univer-sity, Hamilton, ON, Canada, in 2006.From 1997 to 2001, he was a Teaching Assistant

with the Department of Electrical Engineering, SouthValley University, Egypt. FromMay to August 2003,he was a Visiting Researcher at the Department of

Communication Systems, University of Duisburg-Essen, Duisburg, Germany.From April to August 2006, he was a Research Associate with the Instituteof Telecommunications, Darmstadt University of Technology, Germany. FromSeptember 2006 to October 2007, he was an Assistant Professor with the De-partment of Electrical Engineering, South Valley University (later Aswan Uni-versity), Aswan, Egypt. Since November 2007, he has been with the Departmentof Electrical and Computer Engineering, University of Alberta, Edmonton, AB,Canada, where he is currently a Research Associate. His research interests arein statistical and array signal processing, MIMO radar/sonar, parameter esti-mation, robust adaptive beamforming, and applications of signal processing inexploration seismology.

Sergiy A. Vorobyov (M’02–SM’05) received theM.Sc. and Ph.D. degrees in systems and control fromKharkiv National University of Radio Electronics,Ukraine, in 1994 and 1997, respectively.He is a Professor with the Department of Signal

Processing and Acoustics, Aalto University, Finland,and is currently on leave from the Department ofElectrical and Computer Engineering, Universityof Alberta, Edmonton, Canada. He has been withthe University of Alberta as an Assistant Professorfrom 2006 to 2010, Associate Professor from 2010

to 2012, and Full Professor since 2012. Since his graduation, he also heldvarious research and faculty positions at Kharkiv National University ofRadio Electronics, Ukraine; the Institute of Physical and Chemical Research(RIKEN), Japan; McMaster University, Canada; Duisburg-Essen Universityand Darmstadt University of Technology, Germany; and the Joint ResearchInstitute between Heriot-Watt University and Edinburgh University, U.K.He has also held short-term visiting positions at Technion, Haifa, Israel andIlmenau University of Technology, Ilmenau, Germany. His research interestsinclude statistical and array signal processing, applications of linear algebra,optimization, and game theory methods in signal processing and communica-tions, estimation, detection and sampling theories, and cognitive systems.Dr. Vorobyov is a recipient of the 2004 IEEE Signal Processing Society Best

Paper Award, the 2007 Alberta Ingenuity New Faculty Award, the 2011 CarlZeiss Award (Germany), the 2012 NSERC Discovery Accelerator Award, andother awards. He served as an Associate Editor for the IEEE TRANSACTIONS ONSIGNAL PROCESSING from 2006 to 2010 and for the IEEE SIGNAL PROCESSINGLETTERS from 2007 to 2009. He was a member of the Sensor Array and Multi-Channel Signal Processing Committee of the IEEE Signal Processing Societyfrom 2007 to 2012. He is a member of the Signal Processing for Communi-cations and Networking Committee since 2010. He has served as the TrackChair for Asilomar 2011, Pacific Grove, CA, the Technical Co-Chair for IEEECAMSAP 2011, Puerto Rico, and the Tutorial Chair for ISWCS 2013, Ilmenau,Germany.

Matthew W. Morency received the Bachelors ofScience degree in electrical engineering from theUniversity of Alberta in 2013.He is currently a graduate student at the Univer-

sity of Alberta in the Department of Electrical andComputer Engineering. His research interests are insignals processing and MIMO radar.

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1490 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. …vorobyov/TSP14b.pdf · the DOA estimation performance offered by using the proposed joint and SDD transmit beamspace design methods