576 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 2, FEBRUARY 2009

Optimal Polarized Beampattern SynthesisUsing a Vector Antenna Array

Jin-Jun Xiao, Member, IEEE, and Arye Nehorai, Fellow, IEEE

Abstract—Using polarized waveforms increases the capacity ofcommunication systems and improves the performance of activesensing systems such as radar. We consider the optimal synthesisof a directional beam with full polarization control using an arrayof electromagnetic vector antennas (EMVA). In such an array,each antenna consists of � � � orthogonal electric or magneticdipole elements. The control of polarization and spatial powerpatterns is achieved through carefully designing the amplitudesand phases of the weights of these dipole antennas. We formulatethe problem in a convex form, which is thus efficiently solvableby existing solvers such as the interior point method. Our resultsindicate that vector antenna arrays not only enable full polariza-tion control of the beampattern, but also improve the power gainof the main beam (over the sidelobes), where the gain is shownnumerically to be linearly proportional to vector antenna dimen-sionality �. This implies that EMVA not only offers the freedom tocontrol the beampattern polarization, but also virtually increasesthe array size by exploiting the full electromagnetic (EM) fieldcomponents. We also study the effect of polarization on the spatialpower pattern. Our analysis shows that for arrays consisting ofpairs of electrical and magnetic dipoles, the spatial power patternis independent of the mainbeam polarization constraint.

Index Terms—Beampattern synthesis, convex optimization,vector antenna, waveform polarization.

I. INTRODUCTION

I T is well known that exploiting waveform polarization in-creases the capacity of communication systems and im-

proves the performance of active sensing systems such as radar.In radar, polarimetric information in the backscattered wave-forms contains the target features, such as geometrical structure,shape, reflectivity, and orientation, and can thus be exploitedto significantly improve the sensing performance [1]–[9]. Cap-turing the backscattered polarimetric information requires theradar to measure both horizontal and vertical components ofthe received waveforms, which enables the so-called receptionpolarization diversity [3]. However, to efficiently acquire com-plete polarimetric information on the target, it is desirable for

Manuscript received December 04, 2007; revised July 30, 2008. First pub-lished October 31, 2008; current version published January 30, 2009. The asso-ciate editor coordinating the review of this manuscript and approving it for pub-lication was Dr. Shahram Shahbazpanahi. The work was supported in part bythe Department of Defense under Air Force Office of Scientific Research MURIGrant FA9550-05-0443, AFOSR Grant FA 9550-05-1-0018, and by DARPAfunding by NRL Grant N00173-06-1G006. This work was presented in part atthe IEEE Statistical Signal Processing Workshop, Madison, WI, August 2007.

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

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

Digital Object Identifier 10.1109/TSP.2008.2007107

radar to be able to control the transmit waveform polarization,which in addition enables the transmit polarization diversity [6],[10], [11]. Among current radar systems, polarimetric radar thattransmits waveforms with both horizontal and vertical orienta-tions has been developed and adopted in a wide range of ap-plications [12]. In fact, polarization diversity not only providesmore complete information about the target, but also is a de-gree of freedom of the transmit waveforms that can be exploitedin response to the dynamic targets and environments. Adaptivewaveform polarization has been recently studied in the litera-ture and has been shown to be an efficient approach to improveradar sensing performance in various scenarios [5], [13]–[15].The focus of the aforementioned work is on the analysis ofperformance improvement bought by adaptively controlling thetransmit waveform polarization, but the issues regarding how togenerate such polarized waveforms have not been addressed.

In wireless communications, waveform polarization has beenshown to significantly improve the capacity of communicationsystems [16]. Polarization diversity in communications has alsobeen well studied in the literature [17]–[22]. Classical wirelesscommunications relies on one channel per frequency, althoughit is well understood that the two polarization states of planarwaves allow two distinct information channels; techniques suchas polarization diversity already take advantage of this. In [16]the authors further show that in a scattering environment, anextra factor of three in channel capacity can be obtained, rel-ative to the limit imposed by using dual-polarized radio signals.The extra capacity arises because there are six distinguishableelectric and magnetic states of polarization at a given point. Inaddition, polarized arrays can match the polarization of a de-sired signal and null an interferer with the same direction of ar-rival. The potential of polarized arrays for interference rejectionin wireless communication systems has been investigated in re-cent years [23], [24].

In this paper, we study beampattern synthesis using an arrayof vector antennas. Each vector antenna in the array consistsof orthogonal electric or magnetic dipole elements. Ouraim is to synthesize beampatterns with not only a desired spa-tial power pattern, but also a desired polarization. In traditionalbeampattern synthesis of a scalar array (i.e., an array of scalarantennas), radio signals from a set of small nondirectional an-tenna elements are combined with different weights to achievethe beam spatial directionality. The beam emitted by such arraysis of a fixed polarization and cannot be controlled. To obtain abeam with full polarization control, we use an array of vectorantennas. We design complex weights for individual array el-ements to achieve a beam with both the desired spatial powerdensity and the desired polarizations. Our goal is to explore thepotential of electromagnetic vector antennas (EMVA) from the

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XIAO AND NEHORAI: OPTIMAL POLARIZED BEAMPATTERN SYNTHESIS 577

transmitter point of view. By exploiting the full EM field com-ponents, we study how vector antennas can achieve polarizationcontrol and improve the spatial power pattern of the array.

There has been a long history of research on the problem ofbeampattern synthesis [25]–[28]. The main difference betweenthe present paper and the previous work is that our work jointlydesigns the waveform polarization and the spatial power pat-tern, whereas work reported in the existing literature focuseson the latter only. We consider the joint design of the vectorantenna array and compare its performance with the scalar ar-rays. We design the amplitudes and phases of the electric fieldsemitted from these dipole antennas to achieve the desired po-larization control and spatial power pattern. By formulating theproblem in an efficiently solvable convex optimization form,we can easily adopt various design criteria on the spatial powerpattern and polarization by adding corresponding constraints.Our results indicate that our vector antenna not only allows fullpolarization control of the beampattern but also improves thepower gain of the main beam (over the sidelobes), where thegain is linearly proportional to vector antenna dimensionality

. This implies that EMVA has the advantage of not only pro-viding the freedom to control the beampattern polarization butalso virtually increasing the array size by exploiting full EMfields components. We also show that for arrays consisting ofpairs of electrical and magnetic dipoles, the spatial power pat-tern is independent of the mainbeam polarization constraint.

Finally, we note that the performance of the minimum-noise-variance beamformer of a single EM vector antenna has beenstudied in [29]. The work therein considers the beamforming ofthe measurements of the complete electric and magnetic fieldsinduced by EM signals from a single EM vector antenna. Ourpaper on the other hand, studies the optimal beamforming of anarray of vector antennas.

1) Organization of the Paper: The rest of the paper isorganized as follows. In Section II, we give a brief overviewof EM waveform polarization by introducing the polarizationellipse and a polarization transformation. Section III presentsthe problem formulation. In this section we first introducethe EMVA array and then give a convex formulation of thepolarized waveform synthesis using an EMVA array. In Sec-tion IV, we study the cases for which each antenna in the arrayconsists of a pair of electric and magnetic dipoles. We showthat in such cases, the achievable beampattern spatial powergain is independent of the mainbeam polarization constraint.We also compare the vector array’s performance with its scalarcounterpart. Section V is devoted to numerical simulations.Through numerical examples, we show that when , thespatial power gain (mainbeam over sidelobes) of the polarizedbeampattern is proportional to the antenna dimension. Sec-tion VI summarizes and concludes the paper.

Notations: Throughout this paper we adopt the following no-tations. A lower case letter (e.g., ) denotes a scalar, a boldface/lowercase letter (e.g., ) denotes a vector, and a boldface/up-percase letter (e.g., ) denotes a matrix. For a complex matrix

, and denote the transpose and Hermitianof , respectively. We use “ ” to denote the complex symbol,i.e., . The letter denotes an identity matrix of size

. In addition, all vectors are in column form unless other-wise noted.

Fig. 1. Illustration of the electric fields of a plane wave.

II. WAVEFORM POLARIZATION

A plane wave (which is typically associated with a singlesource in the far field) radiated by an antenna comprises an elec-tric field and a magnetic field that are orthogonal to the directionof propagation. The electric and magnetic fields are orthogonalto each other, and the magnitude of the electric and magneticfields can be found from each other. Hence, only the electricfield of the EM wave is usually used to describe its polarization.

A. Polarization Ellipse

Consider a narrowband plane wave traveling in a uniformmedium in the 3-D space. We assume that its electric field canbe represented by

(1)

where and are two complex numbers and is the scalarcomplex envelope of the waveform. The polarization is the locusof the electric field vector as a function of time; see Fig. 1 (in thefigure, there is a phase difference between the and

-components). It describes the direction of wave oscillationin the plane perpendicular to the direction of propagation [30].For ease of visualization, one efficient method of describing thewaveform polarization is the so-called polarization ellipse1 [2],[31].

Given an electric field of a plane EM wave in (1), it turnsout that the wave polarization status is fully determined by thevector . The following theorem gives the relationshipbetween the polarization status of and .

Theorem 1 ([2]): Every nonzero hasa unique representation:

(2)

where .Moreover, in (2) are uniquely determined if and onlyif .

1Besides the polarization ellipse, there are other parametrizations for thewaveform polarization, such as the Jones vector [31].

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578 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 2, FEBRUARY 2009

Fig. 2. Polarization ellipse.

In (2), is called the orientation angle (the angle betweenthe major semi-axis of the ellipse and the -axis), and isthe ellipticity angle (the angle measuring the ratio of the twosemi-axes); see Fig. 2. For example, when , the resultantpolarization is linear; moreover, gives a horizontal polar-ization and leads to a vertical polarization. However,for , the resultant polarization is circular for any ori-entation angle .

In the next section, we will use the relationship betweenand to design the amplitudes and phases of each

antenna sub-element to synthesize the polarized beampattern.

B. Polarization Transformation

For convenience, we introduce the notation to denotethe polarization status . More specifically, for any pair ofangles , we introduce thenotation:

(3)

where denotes a rotation matrix of angle . It turns out thatany polarization can be mathematically transformed to

by an orthogonal transformation. To show that, we in-troduce a linear transformationin which

(4)

It is easy to verify that is an orthogonal transformation,and in addition

Thus, for any pairs of polarizations and ,the transformation

(5)

is orthogonal. Further, it maps to , whichis verified here

We will use such transformations in Sections IV and V to sim-plify the design of a polarized beampattern and show the prop-erty of polarized beampattern synthesis.

III. PROBLEM FORMULATION

In this section we formulate the problem of the optimalbeampattern synthesis with full polarization control. We firstintroduce the configurations of the EMVA and the EMVA arrayused in this paper, and then formulate the beampattern synthesisproblem using tools from convex optimization.

A. Vector Antenna Response

Let be a unit vector representing a spatial direction in .We can write in the form

(6)

where is the azimuth angle andis the elevation angle. For each , we further choose

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XIAO AND NEHORAI: OPTIMAL POLARIZED BEAMPATTERN SYNTHESIS 579

Fig. 3. The coordinate system.

Fig. 4. Dipole antennas: magnetic dipole (left); electric dipole (middle); dipolepair (right).

It is easy to see that forms a right-hand coordinatesystem; see Fig. 3. For a plane wave traveling along , the elec-tric field is orthogonal to and lies in the plane spanned by

.We consider vector antennas consisting of orthogonal electric

and magnetic dipoles; see Fig. 4. Such vector antennas can con-sist of up to six dipoles: three electric dipoles and three magneticdipoles along the , , and axis; see Fig. 5. For each of the sixdipoles, given a driven antenna current in the dipole, the radiatedelectric field in the far field is orthogonal to radiation direction. In the coordinate system , each of the six dipoles has

the following responses [2] (ignoring a common constant that isdetermined by the antenna parameters and the distance to theantenna):

(7)

(8)

(9)

Above , , and denote the response of the threeelectric dipoles located at the , , and axis, respectively. Sim-ilar definitions follow for , , and .

Fig. 5. A 6D EM vector antenna.

It is readily seen that there is a reciprocal property betweenthe electric dipole and magnetic dipole along the same axis.Specifically, we have

This is a key property that will be used in Section IV to discussthe relationship between the beampattern spatial power gain andthe beam polarization constraint.

For a vector antenna consisting of dipole elements, we useto denote the antenna response. In this paper, we will con-

sider antenna types with . The scalar antenna arraycase with will also be considered for comparison pur-poses. For example, if an antenna consists of one electric dipoleelement at the -axis only, its response . If theantenna consists of both electric and magnetic dipole elementsalong the axis, the response

Moreover, for a given antenna response , weuse and to denote its response to the anddimensions, respectively, or as a formula,

(10)

Equivalently, and are the two columns of .

B. Vector Antenna Array

The antenna array we consider in this paper consists ofEMVAs located at positions in with coordinates

. Each antenna in the array has dipole elements. Theantennas are driven by the same carrier signal with wavelength

and convex envelope . The antenna currents (or weights)on the dipoles in the th EMVA are denoted by (see Fig. 6)

(11)

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580 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 2, FEBRUARY 2009

Fig. 6. Vector antenna array.

We further introduce to be the concatenation of all :

It is easy to see that the dimensionality of is .Given location of the antennas, the array

response, as a function of spatial direction , can be expressedas

where and is the wave number.Thus, in terms of the vector antenna response (which

has dimension ), we further obtain that the vector antennaarray response is

...

which has dimensionality .We introduce the antenna array response along the and

directions

where and are defined as in (10).The normalized electrical field emitted from the antenna array

(ignoring the common carrier and the baseband signal ) canbe expressed as

Let us use and to denote the decomposition ofalong and . We immediately have

Along , the wave polarization is determined by the ratio be-tween and , and the radiated energy can be ex-pressed as .

C. Polarized Beampattern Synthesis

The problem of beampattern synthesis is to design the an-tenna weights to achieve a desired wave pattern. In a scalararray, the goal is merely to control the spatial power pattern.However, for the array of vector antennas, we can furtherachieve the control of beam polarization. Specifically, our goalis to synthesize a beampattern with the following properties.

i) The mainbeam, assumed to be pointing at a direction ,has desired power and polarization ;

ii) The power of sidelobes in a region of interest (denoted by) are suppressed.

Compared with the scalar array, the beampattern synthesis ina vector array enforces an additional polarization constraint onthe main beam.

There are various criteria for suppressing the sidelobe powerwhile maintaining the mainlobe power and polarization. Wefocus on the sidelobe power minimization in this paper, and thisleads to the following optimization problem:

To obtain the optimal weights for the antenna elements,we need to solve the above optimization problem. To removethe “max” operation in the objective function and transform itinto a standard function, we introduce an auxiliary variable .We further assume without loss of generality. Thenwe obtain the following problem:

(12)

It is easy to see that the above problem is convex, and is in facta second-order cone programming (SOCP) problem [32]. SOCPis a special class of convex optimization problems and thereforeenjoys all the advantages of convexity. There are well-devel-oped numerical methods to solve a general convex optimizationproblem, the best known of which is the interior point method.For the numerical examples in Section V, we adopt an opti-mization toolbox, SeDuMi [33], to solve the SOCP formulatedabove. SeDuMi, which stands for Self-Dual-Minimization, is asoftware package that solves optimization problems over sym-metric cones using the primal-dual interior-point methods.

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XIAO AND NEHORAI: OPTIMAL POLARIZED BEAMPATTERN SYNTHESIS 581

Fig. 7. Array of 2-D paired dipoles.

In the next two sections, we discuss the polarized beampat-tern synthesis and analyze its performance for two casesand where is the number of dipole elements in eachvector antenna. Specifically, for , we will compare its per-formance with its scalar array counterpart, and for , wewill analyze the mainbeam power gain versus .

IV. DISCUSSION AND PERFORMANCE ANALYSIS WHEN

In this section, we study the beampattern synthesis using 2-Dvector antenna arrays, where each vector antenna has one elec-tric dipole and one magnetic dipole co-aligned along either the

, , or axis. Our aims are twofold: (i) to find the relationshipbetween beampattern power gain and its polarization constraint;and (ii) to compare the performance of such 2-D vector arrayswith their scalar counterparts.

Depending on which pair of dipoles is chosen, there are threepossibilities for the vector antenna response matrix . Ineach case, has the following structure:

where• for dipoles along the axis,

;• for dipoles along the axis,

;• for dipoles along the axis,

.For all these cases, it holds that

(13)

Following the notations introduced in (11), the weights at thetwo dipole elements for the th vector antenna are and .

For convenience, we use in this section to denote

the weight on the electric dipole, and to denotethe weight on the magnetic dipole; see Fig. 7. We further intro-duce the following notations:

Notice that and are row vectors.

For this 2-D case, we further calculate the electrical fieldemitted from the antenna array

(14)

where

(15)

Using (13), we obtain

In terms of and , the original optimizationproblem in (12) becomes

(16)

We will show that for the above problem, the following resultshold.

i) The optimal solution , which measures the beampat-tern power gain over the sidelobes, does not depend onthe main beam polarization ; and

ii) The beampattern spatial power gain is the same whencompared to its scalar array counterpart in which eachsensor only consists of only one electrical (or magnetic)dipole. However, the latter scalar array does not have theability to control the beampattern polarization.

Proof of Independence of the Polarization

For any arbitrary pair , we assume the optimal solutionto (16) is denoted by . We will show that the followinglemma holds.

Lemma 2: Assume that is the optimal solution to(16). For any and , it holds that

Proof: Suppose and are two arbitrarywaveform polarizations; see (3). As verified in Section II-B,

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582 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 2, FEBRUARY 2009

if we introduce an orthogonal transformation[cf., (5)], it holds that

For a given direction , we further define the transformation

where

It is easy to see that is an orthogonal trans-formation.

Suppose is an optimal solution to(16) when the mainbeam points at direction and has polariza-tion constraint . The corresponding optimal objectivevalue is denoted by . Recalling the first and thirdconstraints in (16), we obtain that

(17)

Consider (16) with another polarization constraint .We choose the following antenna weights:

(18)

We first claim that the above chosen is a fea-sible solution to problem (16) with mainbeam polarization con-straint . For that purpose, we verify that

(19)

where the first equality is due to (15).Therefore, at , the resultant electrical field

(20)

Fig. 8. Scalar arrays. (a) Array of electric dipoles. (b) Array of magneticdipoles.

where is due to (18), is because of the definition of, and is from (17). Further, we can de-

rive from (19) that

for any .We conclude from (20) and (21) that is

a solution to (16) when the mainbeam polarization constraintis . Moreover, the corresponding objective value is

. This implies that

We can similarly prove that . Thus,.

Essentially Lemma 2 reveals the fact that the mainbeampower gain is independent of the polarization constraint wheneach vector sensor in the array consists of a pair of coalignedelectric and magnetic dipoles.

A. Comparison With Scalar Antenna Arrays

In this section, we compare the performance of the 2-D an-tenna array (as displayed in Fig. 7) with its scalar counterpart. Inthe corresponding scalar array, each antenna consists of eitheran electric or a magnetic dipole; see Fig. 8. Such scalar arraysdo not allow mainbeam polarization control due to the lack ofdegree of freedom. In fact, for any spatial direction , the polar-izations of the wave radiated by all antennas in the scalar arrayare identical. Thus, the resultant wave radiated by the array re-mains fixed no matter how the weights for the antenna elementsare designed. We, hence, discard the polarization constraint forthe problem of beampattern synthesis using such scalar arraysand consider only the spatial power distribution.

The following two problems formulate the beampattern syn-thesis using the array of electric dipoles and the array of mag-netic dipoles, respectively.

• 1-D array with electric dipoles: The antenna responseand the optimal antenna weights

can be obtained by solving

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XIAO AND NEHORAI: OPTIMAL POLARIZED BEAMPATTERN SYNTHESIS 583

(21)

• 1-D array with magnetic dipoles: The antenna responseand the optimal antenna weights

can be obtained by solving

(22)

It is easy to see that the above two problems have the sameoptimal solution because

holds for (21) and (22), respectively, and, in addition, the twoarrays share the same array response .

For given array response , antenna response ,and mainbeam direction , we use to denote the valueof the objective function at the solutions for (21) and (22). Wefurther have the following lemma, which states the relationshipof the power gain achieved by these two scalar arrays and thevector array in (16).

Lemma 3: For any , it holds that

Proof: See the Appendix.The results given in Lemmas 2 and 3 will be illustrated in the

next section by numerical examples. We note that the results inLemma 2 and 3 hold for the case of only. Similar resultsfor are still under investigation.

V. NUMERICAL RESULTS

In this section we give numerical examples to show the per-formance of the proposed beampattern synthesis method. Wewill consider two cases: and . Since all formu-lated problems in (12), (16), (21), and (22) are convex, we adoptthe optimization toolbox, SeDuMi [33] to solve the formulatedproblems.

A. 2-D Arrays: Vector Versus Scalar

The analysis in Section IV implies that for the 2-D vectorarrays with coaligned electric and magnetic dipoles, the achiev-able spatial power pattern is identical to that achieved by thescalar arrays consisting of either electric or magnetic dipoles.However, the vector array has the advantage of enabling thecontrol of the polarization. In addition, as revealed by (25) in

Fig. 9. Comparison of power gain of the beampattern synthesis of a lineararray: scalar array (top); vector array (bottom).

the Appendix, designing the weights for the vector array canbe achieved by designing the two subarrays separately and thencombining these weights properly without loss of optimality.

Fig. 9 shows the computer simulation of a linear array of 15elements that are located on the axis and separated by a halfwavelength, i.e.,

In the vector array, each antenna consists of one electric dipoleand one magnetic dipole, both pointing along the axis.The scalar array compromises 18 electric dipoles. The mainbeam is targeted at with a beamwidth 7.5 . Themainbeam of the vector array has a polarization constraint

. As can be seen, both scalar and vector ar-rays achieve the same spatial power pattern, whereas the beamfrom the vector array has the desired polarization property.

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584 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 2, FEBRUARY 2009

Fig. 10. Comparison of power gain of the beampattern synthesis: (a) Vectorarray. (b) Array of electric dipoles. (c) Array of magnetic dipoles.

Another example is given in Fig. 10. There are 18 antennasin each array. The antennas are located at the three axes withcoordinates:

(23)

where is the wavelength. Still in the vector array, each an-tenna consists of one electric dipole and one magnetic dipole,both pointing along the axis. The two scalar arrays compro-mise electric dipoles only or magnetic dipoles only. The mainbeam points at direction withbeamwidth for both the elevation and azimuth angles. As canbe seen, the three arrays achieve the same spatial power pattern,but the vector array also achieves a mainbeam control where,in this example, the polarization of the mainbeam is set to be

.

B. High-Dimensional Arrays: Power Gain Versus SensorDimensionality

In this section, we analyze by simulation the performance ofbeampattern synthesis using arrays of high-dimensional antennaarrays. We still use array size 18 with antennas located at the ,

, and axis as given in (23). We will consider arrays of vectorsantennas with . For each , we choose dipole elements

TABLE IVECTOR ANTENNA DIPOLE ELEMENTS FOR � � � � �

USED IN THE SIMULATIONS

Fig. 11. Illustration of mainbeam power gain: � � �.

according to Table I. We note that at each row, the symbol “ ”denotes that a certain dipole element is included in the antenna.For example, when , one electric dipole and one magneticdipole at the axis are selected. Correspondingly, the antennaresponse [cf., (7)–(9)].

In addition, in these examples, we choose. The polarization of

the mainbeam is set to be .The achieved spatial power patterns for , 4, 6 are given

in Figs. 11–13, respectively. For each case, the top figure plotsthe two-dimensional power pattern in the plane of the eleva-tion and azimuth angles . In the bottom figure, each curveshows the slice of the power pattern versus for fixed . As canbe seen, once the sensor dimensionality increases, the main-beam power gain versus the sidelobes increases. As plotted inFig. 14, when increases from 1 to 2, polarization control isenabled while the power gain remains the same (for , we

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XIAO AND NEHORAI: OPTIMAL POLARIZED BEAMPATTERN SYNTHESIS 585

Fig. 12. Illustration of mainbeam power gain: � � �.

choose the array of electric dipoles pointing along the axis).For the vector sensor arrays (when ), all three curves fordifferent array size 6, 12, 18 show that such gain is almostlinearly proportional to . We, thus, conclude that the EMVAarray has the advantage of

i) enabling control of the beampattern polarization;ii) virtually increasing the array size, since multiple EM

fields at each antenna are exploited.In other words, by exploiting the electrical and magnetic com-

ponents radiated at the same physical location by a vector an-tenna, we cannot only obtain the full control of the waveformpolarization, but also improve the array resolution. The resolu-tion increase of the EMVA array is due to the almost indepen-dent responses from antenna elements from the same location.In applications where antenna array physical size is limited byspace, the result above shows that the array resolution can gobeyond the limit imposed by a scalar array if on top of the spacedimension, the polarization dimension is exploited.

VI. CONCLUSION

In this paper, we have considered the beampattern synthesiswith polarization constraints using an array of vector antennas

Fig. 13. Illustration of mainbeam power gain: � � �.

Fig. 14. Power gain versus antenna dimensionality � (note that polarizationcontrol is enabled if and only if � � �).

consisting of electric and magnetic dipole elements. We formu-lated the problem in a convex form, for which the design vari-ables are the amplitudes and phases of the weights in the an-tenna elements, and the polarization condition is cast in a linearconstraint. We have shown that arrays consisting of 2-D vectorantennas, where each antenna compromises a pair of coaligned

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586 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 2, FEBRUARY 2009

electric and magnetic dipoles, have the same capability of sup-pressing the sidelobe power density as the corresponding scalararrays of electric or magnetic dipoles. However, the vector arrayhas the additional capability of controlling the beam polariza-tion. For arrays of vector antennas with the number of dipoleelements in each antenna , we have shown by simulationthat the power gain achieved by the array is linearly proportionalto . This reveals that vector antenna arrays not only enable po-larization control, but also virtually increase the array size byexploiting multiple EM fields at each physical point.

Note that although we have considered only the transmitbeampattern synthesis, similar results can be generalized tothe receive beamforming of vector antenna arrays. Also thebeampattern synthesis problem in this paper is formulate basedupon the assumption of a narrowband plane wave traveling in auniform medium. Its extension to the wideband waveforms ininhomogeneous medium is part of our future work.

APPENDIX

PROOF OF LEMMA 3

As we demonstrated in Section IV-B, (21) and (22) share thesame set of optimal solutions. Take one of them, which we de-note by and , respectively. We use to de-note the achieved optimal objective value. This implies that forany and

it holds that

In addition

(24)

Let us construct a (independent of ) such that

(25)

where

(26)

with . Such a is a solution to (16), which gives anobjective value for those further satisfying the following:

(27)

Obviously, there are infinitely many such . For example, wecan choose a diagonal

in which

It is easy to see that is independent of , satisfies (27), andfurther satisfies (26) because of (24). Therefore, the given

in (25) is a solution to (16), which gives an objective value .This implies that

(28)

On the other hand, for (16), we choose the polarization con-straint at to be , where is thepolarization determined by the vector . It thenfollows from Theorem 1 that

(29)

for some . Since the polarization constraint at is, it holds that for any feasible solution to (16)

(30)

The equalities in (29)–(30) lead to the following:

Due to the orthogonality between and, we obtain that . It, there-

fore, holds that at any feasible solution, ,for which we can choose . This implies that if

is an optimal solution to (16), then the

electric component is a solution to the 1-D electricdipole array problem in (21). Thus

(31)

Combining (28) and (31) completes the proof.

ACKNOWLEDGMENT

The first author is indebted to Dr. M. Hurtado for his helpfulcomments on earlier versions of this manuscript.

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Jin-Jun Xiao (S’04–M’07) received the B.Sc.degree in applied mathematics from Jilin University,Changchun, China, in 1997, and the M.Sc. degreein mathematics and the Ph.D. degree in electricalengineering from the University of Minnesota, TwinCities, in 2003 and 2006, respectively.

He is currently a Postdoctoral Fellow and AdjunctProfessor with the Department of Electrical and Sys-tems Engineering, Washington University, St. Louis,MO. In the summer of 2002, he worked on barcodesignal processing at Symbol Technologies (currently

Motorola Enterprise Mobility Division), Holtsville, NY. His current research in-terests are in statistical signal processing, information theory, convex optimiza-tion, and their applications to sensors/radar, sensor networking, and bioimaging.

Arye Nehorai (S’80–M’83–SM’90–F’94) receivedthe B.Sc. and M.Sc. degrees in electrical engineeringfrom the Technion, Israel, and the Ph.D. degree inelectrical engineering from Stanford University, CA.

From 1985 to 1995, he was a faculty memberwith the Department of Electrical Engineering, YaleUniversity, New Haven, CT. In 1995, he joined theDepartment of Electrical Engineering and ComputerScience, The University of Illinois at Chicago (UIC),as a Full Professor . From 2000 to 2001, he wasChair of the department’s Electrical and Computer

Engineering (ECE) Division, which then became a new department. In 2001,he was named University Scholar of the University of Illinois. In 2006, hebecame Chairman of the Department of Electrical and Systems Engineering,Washington University, St. Louis (WUSTL). He is the inaugural holder of theEugene and Martha Lohman Professorship and the Director of the Center forSensor Signal and Information Processing (CSSIP) at WUSTL since 2006.

Dr. Nehorai was Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL

PROCESSING during 2000–2002. During 2003–2005 he was Vice President(Publications) of the IEEE Signal Processing Society, Chair of the PublicationsBoard, member of the Board of Governors, and member of the ExecutiveCommittee of this Society. From 2003 to 2006, he was the Founding Editor ofthe special columns on Leadership Reflections in the IEEE Signal ProcessingMagazine. He was corecipient of the IEEE SPS 1989 Senior Award for BestPaper with P. Stoica, coauthor of the 2003 Young Author Best Paper Award,and corecipient of the 2004 Magazine Paper Award with A. Dogandzic. He waselected Distinguished Lecturer of the IEEE SPS for the term 2004 to 2005 andreceived the 2006 IEEE SPS Technical Achievement Award. He is the PrincipalInvestigator of the new multidisciplinary university research initiative (MURI)project entitled Adaptive Waveform Diversity for Full Spectral Dominance. Hehas been a Fellow of the Royal Statistical Society since 1996.

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