Optimal Polarized Beampattern Synthesis Using a Vector Antenna Array

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Optimal Polarized Beampattern Synthesis Using a Vector Antenna Array Jin-Jun Xiao, Member, IEEE, and Arye Nehorai, Fellow, IEEE

Abstract—Using polarized waveforms increases the capacity of communication systems and improves the performance of active sensing systems such as radar. We consider the optimal synthesis of a directional beam with full polarization control using an array of electromagnetic vector antennas (EMVA). In such an array, each antenna consists of p  2 orthogonal electric or magnetic dipole elements. The control of polarization and spatial power patterns is achieved through carefully designing the amplitudes and phases of the weights of these dipole antennas. We formulate the problem in a convex form, which is thus efficiently solvable by existing solvers such as the interior point method. Our results indicate that vector antenna arrays not only enable full polarization control of the beampattern, but also improve the power gain of the main beam (over the sidelobes), where the gain is shown numerically to be linearly proportional to vector antenna dimensionality p. This implies that EMVA not only offers the freedom to control the beampattern polarization, but also virtually increases the array size by exploiting the full electromagnetic (EM) field components. We also study the effect of polarization on the spatial power pattern. Our analysis shows that for arrays consisting of pairs of electrical and magnetic dipoles, the spatial power pattern is 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 increases the capacity of communication systems and improves the performance of active sensing systems such as radar. In radar, polarimetric information in the backscattered waveforms contains the target features, such as geometrical structure, shape, reflectivity, and orientation, and can thus be exploited to significantly improve the sensing performance [1]–[9]. Capturing the backscattered polarimetric information requires the radar to measure both horizontal and vertical components of the received waveforms, which enables the so-called reception polarization diversity [3]. However, to efficiently acquire complete polarimetric information on the target, it is desirable for

Manuscript received December 04, 2007; revised July 30, 2008. First published October 31, 2008; current version published January 30, 2009. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Shahram Shahbazpanahi. The work was supported in part by the Department of Defense under Air Force Office of Scientific Research MURI Grant FA9550-05-0443, AFOSR Grant FA 9550-05-1-0018, and by DARPA funding by NRL Grant N00173-06-1G006. This work was presented in part at the 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: [email protected]. edu; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.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 that transmits waveforms with both horizontal and vertical orientations has been developed and adopted in a wide range of applications [12]. In fact, polarization diversity not only provides more complete information about the target, but also is a degree of freedom of the transmit waveforms that can be exploited in response to the dynamic targets and environments. Adaptive waveform polarization has been recently studied in the literature and has been shown to be an efficient approach to improve radar sensing performance in various scenarios [5], [13]–[15]. The focus of the aforementioned work is on the analysis of performance improvement bought by adaptively controlling the transmit waveform polarization, but the issues regarding how to generate such polarized waveforms have not been addressed. In wireless communications, waveform polarization has been shown to significantly improve the capacity of communication systems [16]. Polarization diversity in communications has also been well studied in the literature [17]–[22]. Classical wireless communications relies on one channel per frequency, although it is well understood that the two polarization states of planar waves allow two distinct information channels; techniques such as polarization diversity already take advantage of this. In [16] the authors further show that in a scattering environment, an extra factor of three in channel capacity can be obtained, relative to the limit imposed by using dual-polarized radio signals. The extra capacity arises because there are six distinguishable electric and magnetic states of polarization at a given point. In addition, polarized arrays can match the polarization of a desired signal and null an interferer with the same direction of arrival. The potential of polarized arrays for interference rejection in wireless communication systems has been investigated in recent years [23], [24]. In this paper, we study beampattern synthesis using an array of vector antennas. Each vector antenna in the array consists of orthogonal electric or magnetic dipole elements. Our aim is to synthesize beampatterns with not only a desired spatial power pattern, but also a desired polarization. In traditional beampattern synthesis of a scalar array (i.e., an array of scalar antennas), radio signals from a set of small nondirectional antenna elements are combined with different weights to achieve the beam spatial directionality. The beam emitted by such arrays is of a fixed polarization and cannot be controlled. To obtain a beam with full polarization control, we use an array of vector antennas. We design complex weights for individual array elements to achieve a beam with both the desired spatial power density and the desired polarizations. Our goal is to explore the potential of electromagnetic vector antennas (EMVA) from the

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XIAO AND NEHORAI: OPTIMAL POLARIZED BEAMPATTERN SYNTHESIS

transmitter point of view. By exploiting the full EM field components, we study how vector antennas can achieve polarization control and improve the spatial power pattern of the array. There has been a long history of research on the problem of beampattern synthesis [25]–[28]. The main difference between the present paper and the previous work is that our work jointly designs the waveform polarization and the spatial power pattern, whereas work reported in the existing literature focuses on the latter only. We consider the joint design of the vector antenna array and compare its performance with the scalar arrays. We design the amplitudes and phases of the electric fields emitted from these dipole antennas to achieve the desired polarization control and spatial power pattern. By formulating the problem in an efficiently solvable convex optimization form, we can easily adopt various design criteria on the spatial power pattern and polarization by adding corresponding constraints. Our results indicate that our vector antenna not only allows full polarization control of the beampattern but also improves the power gain of the main beam (over the sidelobes), where the gain is linearly proportional to vector antenna dimensionality . This implies that EMVA has the advantage of not only providing the freedom to control the beampattern polarization but also virtually increasing the array size by exploiting full EM fields components. We also show that for arrays consisting of pairs of electrical and magnetic dipoles, the spatial power pattern is independent of the mainbeam polarization constraint. Finally, we note that the performance of the minimum-noisevariance beamformer of a single EM vector antenna has been studied in [29]. The work therein considers the beamforming of the measurements of the complete electric and magnetic fields induced by EM signals from a single EM vector antenna. Our paper on the other hand, studies the optimal beamforming of an array of vector antennas. 1) Organization of the Paper: The rest of the paper is organized as follows. In Section II, we give a brief overview of EM waveform polarization by introducing the polarization ellipse and a polarization transformation. Section III presents the problem formulation. In this section we first introduce the EMVA array and then give a convex formulation of the polarized waveform synthesis using an EMVA array. In Section IV, we study the cases for which each antenna in the array consists of a pair of electric and magnetic dipoles. We show that in such cases, the achievable beampattern spatial power gain is independent of the mainbeam polarization constraint. We also compare the vector array’s performance with its scalar counterpart. Section V is devoted to numerical simulations. Through numerical examples, we show that when , the spatial power gain (mainbeam over sidelobes) of the polarized beampattern is proportional to the antenna dimension. Section VI summarizes and concludes the paper. Notations: Throughout this paper we adopt the following notations. A lower case letter (e.g., ) denotes a scalar, a boldface/ lowercase letter (e.g., ) denotes a vector, and a boldface/uppercase letter (e.g., ) denotes a matrix. For a complex matrix , and denote the transpose and Hermitian of , 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 otherwise noted.

577

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

II. WAVEFORM POLARIZATION A plane wave (which is typically associated with a single source in the far field) radiated by an antenna comprises an electric field and a magnetic field that are orthogonal to the direction of propagation. The electric and magnetic fields are orthogonal to each other, and the magnitude of the electric and magnetic fields can be found from each other. Hence, only the electric field of the EM wave is usually used to describe its polarization. A. Polarization Ellipse Consider a narrowband plane wave traveling in a uniform medium in the 3-D space. We assume that its electric field can be represented by (1) where and are two complex numbers and is the scalar complex envelope of the waveform. The polarization is the locus of the electric field vector as a function of time; see Fig. 1 (in the between the and figure, there is a phase difference -components). It describes the direction of wave oscillation in the plane perpendicular to the direction of propagation [30]. For ease of visualization, one efficient method of describing the waveform polarization is the so-called polarization ellipse1 [2], [31]. Given an electric field of a plane EM wave in (1), it turns out that the wave polarization status is fully determined by the . The following theorem gives the relationship vector . between the polarization status of and Theorem 1 ([2]): Every nonzero has a unique representation:

(2) where Moreover, if

. in (2) are uniquely determined if and only .

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

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It is easy to verify that and in addition

is an orthogonal transformation,

Thus, for any pairs of polarizations the transformation

and

,

(5) is orthogonal. Further, it maps is verified here

Fig. 2. Polarization ellipse.

In (2), is called the orientation angle (the angle between the major semi-axis of the ellipse and the -axis), and is the ellipticity angle (the angle measuring the ratio of the two , the resultant semi-axes); see Fig. 2. For example, when gives a horizontal polarpolarization is linear; moreover, leads to a vertical polarization. However, ization and , the resultant polarization is circular for any orifor entation angle . In the next section, we will use the relationship between and to design the amplitudes and phases of each antenna sub-element to synthesize the polarized beampattern.

B. Polarization Transformation to denote For convenience, we introduce the notation . More specifically, for any pair of the polarization status , we introduce the angles notation:

to

We will use such transformations in Sections IV and V to simplify the design of a polarized beampattern and show the property of polarized beampattern synthesis.

III. PROBLEM FORMULATION In this section we formulate the problem of the optimal beampattern synthesis with full polarization control. We first introduce the configurations of the EMVA and the EMVA array used in this paper, and then formulate the beampattern synthesis problem 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

(3) denotes a rotation matrix of angle . It turns out that where can be mathematically transformed to any polarization by an orthogonal transformation. To show that, we introduce a linear transformation in which

, which

.

(6)

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

(4)

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Fig. 5. A 6D EM vector antenna.

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

Fig. 3. The coordinate system.

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

It is easy to see that forms a right-hand coordinate system; see Fig. 3. For a plane wave traveling along , the electric 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 consist of up to six dipoles: three electric dipoles and three magnetic dipoles along the , , and axis; see Fig. 5. For each of the six dipoles, given a driven antenna current in the dipole, the radiated electric 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 is determined by the antenna parameters and the distance to the antenna):

This is a key property that will be used in Section IV to discuss the relationship between the beampattern spatial power gain and the beam polarization constraint. For a vector antenna consisting of dipole elements, we use to denote the antenna response. In this paper, we will consider antenna types with . The scalar antenna array will also be considered for comparison purcase with poses. For example, if an antenna consists of one electric dipole . If the element at the -axis only, its response antenna consists of both electric and magnetic dipole elements along the axis, the response

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

(10) Equivalently,

(7) (8) (9) Above , , and denote the response of the three electric dipoles located at the , , and axis, respectively. Similar definitions follow for , , and .

, we and

and

are the two columns of

.

B. Vector Antenna Array The antenna array we consider in this paper consists of EMVAs located at positions in with coordinates . Each antenna in the array has dipole elements. The antennas 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)

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(11)

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Let us use along

and

and to denote the decomposition of . We immediately have

Along , the wave polarization is determined by the ratio beand , and the radiated energy can be extween . pressed as C. Polarized Beampattern Synthesis

Fig. 6. Vector antenna array.

We further introduce

to be the concatenation of all

:

It is easy to see that the dimensionality of is . of the antennas, the array Given location response, as a function of spatial direction , can be expressed as

where and is the wave number. (which Thus, in terms of the vector antenna response ), we further obtain that the vector antenna has dimension array response is

The problem of beampattern synthesis is to design the antenna weights to achieve a desired wave pattern. In a scalar array, the goal is merely to control the spatial power pattern. However, for the array of vector antennas, we can further achieve the control of beam polarization. Specifically, our goal is 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 in a vector array enforces an additional polarization constraint on the main beam. There are various criteria for suppressing the sidelobe power while maintaining the mainlobe power and polarization. We focus on the sidelobe power minimization in this paper, and this leads to the following optimization problem:

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

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

and

where and are defined as in (10). The normalized electrical field emitted from the antenna array ) can (ignoring the common carrier and the baseband signal be expressed as

(12) It is easy to see that the above problem is convex, and is in fact a second-order cone programming (SOCP) problem [32]. SOCP is a special class of convex optimization problems and therefore enjoys all the advantages of convexity. There are well-developed numerical methods to solve a general convex optimization problem, the best known of which is the interior point method. For the numerical examples in Section V, we adopt an optimization toolbox, SeDuMi [33], to solve the SOCP formulated above. SeDuMi, which stands for Self-Dual-Minimization, is a software package that solves optimization problems over symmetric cones using the primal-dual interior-point methods.

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

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

In the next two sections, we discuss the polarized beampattern synthesis and analyze its performance for two cases and where is the number of dipole elements in each , we will compare its pervector antenna. Specifically, for , we formance with its scalar array counterpart, and for will analyze the mainbeam power gain versus .

(14) where

(15) Using (13), we obtain

IV. DISCUSSION AND PERFORMANCE ANALYSIS WHEN In this section, we study the beampattern synthesis using 2-D vector antenna arrays, where each vector antenna has one electric dipole and one magnetic dipole co-aligned along either the , , or axis. Our aims are twofold: (i) to find the relationship between beampattern power gain and its polarization constraint; and (ii) to compare the performance of such 2-D vector arrays with their scalar counterparts. Depending on which pair of dipoles is chosen, there are three . In possibilities for the vector antenna response matrix has the following structure: each case,

In terms of and problem in (12) becomes

, the original optimization

(16) where • for dipoles along the

axis,

; • for dipoles along the axis, ; • for dipoles along the axis, . For all these cases, it holds that

(13)

We will show that for the above problem, the following results hold. , which measures the beampati) The optimal solution tern power gain over the sidelobes, does not depend on ; and the main beam polarization ii) The beampattern spatial power gain is the same when compared to its scalar array counterpart in which each sensor only consists of only one electrical (or magnetic) dipole. However, the latter scalar array does not have the ability to control the beampattern polarization. Proof of Independence of the Polarization

Following the notations introduced in (11), the weights at the two dipole elements for the th vector antenna are and . in this section to denote For convenience, we use the weight on the electric dipole, and to denote the weight on the magnetic dipole; see Fig. 7. We further introduce the following notations:

For any arbitrary pair to (16) is denoted by lemma holds. Lemma 2: Assume that (16). For any and

Notice that

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

and

are row vectors.

, we assume the optimal solution . We will show that the following is the optimal solution to , it holds that

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

For a given direction

, we further define the transformation

where

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

It is easy to see that is an orthogonal transformation. is an optimal solution to Suppose (16) when the mainbeam points at direction and has polariza. The corresponding optimal objective tion constraint value is denoted by . Recalling the first and third constraints in (16), we obtain that

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

is due to (18), , and rive from (19) that

where

is because of the definition of is from (17). Further, we can de-

for any . We conclude from (20) and (21) that is a solution to (16) when the mainbeam polarization constraint is . Moreover, the corresponding objective value is . This implies that

.

(18) is a feaWe first claim that the above chosen sible solution to problem (16) with mainbeam polarization constraint . For that purpose, we verify that

We can similarly prove that

. Thus, . Essentially Lemma 2 reveals the fact that the mainbeam power gain is independent of the polarization constraint when each vector sensor in the array consists of a pair of coaligned electric and magnetic dipoles. A. Comparison With Scalar Antenna Arrays

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

In this section, we compare the performance of the 2-D antenna array (as displayed in Fig. 7) with its scalar counterpart. In the corresponding scalar array, each antenna consists of either an electric or a magnetic dipole; see Fig. 8. Such scalar arrays do not allow mainbeam polarization control due to the lack of degree of freedom. In fact, for any spatial direction , the polarizations of the wave radiated by all antennas in the scalar array are identical. Thus, the resultant wave radiated by the array remains fixed no matter how the weights for the antenna elements are designed. We, hence, discard the polarization constraint for the problem of beampattern synthesis using such scalar arrays and consider only the spatial power distribution. The following two problems formulate the beampattern synthesis using the array of electric dipoles and the array of magnetic dipoles, respectively. • 1-D array with electric dipoles: The antenna response and the optimal antenna weights can be obtained by solving

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(21) • 1-D array with magnetic dipoles: The antenna response and the optimal antenna weights can be obtained by solving

(22) It is easy to see that the above two problems have the same optimal solution because

holds for (21) and (22), respectively, and, in addition, the two . arrays share the same array response , antenna response , For given array response to denote the value and mainbeam direction , we use of the objective function at the solutions for (21) and (22). We further have the following lemma, which states the relationship of the power gain achieved by these two scalar arrays and the vector array in (16). , it holds that Lemma 3: For any

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 in only. Similar results Lemma 2 and 3 hold for the case of are still under investigation. for V. NUMERICAL RESULTS In this section we give numerical examples to show the performance of the proposed beampattern synthesis method. We and . Since all formuwill consider two cases: lated problems in (12), (16), (21), and (22) are convex, we adopt the optimization toolbox, SeDuMi [33] to solve the formulated problems. A. 2-D Arrays: Vector Versus Scalar The analysis in Section IV implies that for the 2-D vector arrays with coaligned electric and magnetic dipoles, the achievable spatial power pattern is identical to that achieved by the scalar arrays consisting of either electric or magnetic dipoles. However, the vector array has the advantage of enabling the control of the polarization. In addition, as revealed by (25) in

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

the Appendix, designing the weights for the vector array can be achieved by designing the two subarrays separately and then combining these weights properly without loss of optimality. Fig. 9 shows the computer simulation of a linear array of 15 elements that are located on the axis and separated by a half wavelength, i.e.,

In the vector array, each antenna consists of one electric dipole axis. and one magnetic dipole, both pointing along the The scalar array compromises 18 electric dipoles. The main with a beamwidth 7.5 . The beam is targeted at mainbeam of the vector array has a polarization constraint . As can be seen, both scalar and vector arrays achieve the same spatial power pattern, whereas the beam from the vector array has the desired polarization property.

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TABLE I VECTOR ANTENNA DIPOLE ELEMENTS FOR 2 USED IN THE SIMULATIONS



p



6

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

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

(23) where is the wavelength. Still in the vector array, each antenna consists of one electric dipole and one magnetic dipole, both pointing along the axis. The two scalar arrays compromise electric dipoles only or magnetic dipoles only. The main with beam points at direction beamwidth for both the elevation and azimuth angles. As can be 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 Sensor Dimensionality In this section, we analyze by simulation the performance of beampattern synthesis using arrays of high-dimensional antenna arrays. We still use array size 18 with antennas located at the , , and axis as given in (23). We will consider arrays of vectors antennas with . For each , we choose dipole elements

Fig. 11. Illustration of mainbeam power gain: p = 3.

according to Table I. We note that at each row, the symbol “ ” denotes that a certain dipole element is included in the antenna. , one electric dipole and one magnetic For example, when dipole at the axis are selected. Correspondingly, the antenna [cf., (7)–(9)]. response 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 plots the two-dimensional power pattern in the plane of the eleva. In the bottom figure, each curve tion and azimuth angles shows the slice of the power pattern versus for fixed . As can be seen, once the sensor dimensionality increases, the mainbeam power gain versus the sidelobes increases. As plotted in Fig. 14, when increases from 1 to 2, polarization control is , we enabled while the power gain remains the same (for

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Fig. 12. Illustration of mainbeam power gain: p

= 4.

choose the array of electric dipoles pointing along the axis). ), all three curves for For the vector sensor arrays (when 6, 12, 18 show that such gain is almost different array size linearly proportional to . We, thus, conclude that the EMVA array 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 components radiated at the same physical location by a vector antenna, we cannot only obtain the full control of the waveform polarization, but also improve the array resolution. The resolution increase of the EMVA array is due to the almost independent responses from antenna elements from the same location. In applications where antenna array physical size is limited by space, the result above shows that the array resolution can go beyond the limit imposed by a scalar array if on top of the space dimension, the polarization dimension is exploited.

VI. CONCLUSION In this paper, we have considered the beampattern synthesis with polarization constraints using an array of vector antennas

585

Fig. 13. Illustration of mainbeam power gain: p

= 6.

Fig. 14. Power gain versus antenna dimensionality p (note that polarization control is enabled if and only if p ).

2

consisting of electric and magnetic dipole elements. We formulated the problem in a convex form, for which the design variables are the amplitudes and phases of the weights in the antenna elements, and the polarization condition is cast in a linear constraint. We have shown that arrays consisting of 2-D vector antennas, where each antenna compromises a pair of coaligned

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electric and magnetic dipoles, have the same capability of suppressing the sidelobe power density as the corresponding scalar arrays of electric or magnetic dipoles. However, the vector array has the additional capability of controlling the beam polarization. For arrays of vector antennas with the number of dipole , we have shown by simulation elements in each antenna that the power gain achieved by the array is linearly proportional to . This reveals that vector antenna arrays not only enable polarization control, but also virtually increase the array size by exploiting multiple EM fields at each physical point. Note that although we have considered only the transmit beampattern synthesis, similar results can be generalized to the receive beamforming of vector antenna arrays. Also the beampattern synthesis problem in this paper is formulate based upon the assumption of a narrowband plane wave traveling in a uniform medium. Its extension to the wideband waveforms in inhomogeneous medium is part of our future work.

in which

It is easy to see that is independent of , satisfies (27), and further 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 conto be , where is the straint at . It then polarization determined by the vector follows from Theorem 1 that

APPENDIX PROOF OF LEMMA 3

(29)

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

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:

it holds that

In addition

(24) Let us construct a

(independent of ) such that

Due to the orthogonality between and , we obtain that . It, there, fore, holds that at any feasible solution, . This implies that if for which we can choose is an optimal solution to (16), then the

(25)

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

where (31) (26) with . Such a is a solution to (16), which gives an for those further satisfying the following: objective value (27) Obviously, there are infinitely many such can choose a diagonal

. For example, we

Combining (28) and (31) completes the proof. ACKNOWLEDGMENT The first author is indebted to Dr. M. Hurtado for his helpful comments on earlier versions of this manuscript. REFERENCES [1] D. Giuli, L. Facheris, M. Fossi, and A. Rossettini, “Simultaneous scattering matrix measurement through signal coding,” in Proc. IEEE Int. Radar Conf., Arlington, VA, May 1990, pp. 258–262. [2] A. Nehorai and E. Paldi, “Vector-sensor array processing for electromagnetic source localization,” IEEE Trans. Signal Process., vol. 42, pp. 376–398, Feb. 1994.

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XIAO AND NEHORAI: OPTIMAL POLARIZED BEAMPATTERN SYNTHESIS

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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. degree in mathematics and the Ph.D. degree in electrical engineering from the University of Minnesota, Twin Cities, in 2003 and 2006, respectively. He is currently a Postdoctoral Fellow and Adjunct Professor with the Department of Electrical and Systems Engineering, Washington University, St. Louis, MO. In the summer of 2002, he worked on barcode signal processing at Symbol Technologies (currently Motorola Enterprise Mobility Division), Holtsville, NY. His current research interests are in statistical signal processing, information theory, convex optimization, and their applications to sensors/radar, sensor networking, and bioimaging.

Arye Nehorai (S’80–M’83–SM’90–F’94) received the B.Sc. and M.Sc. degrees in electrical engineering from the Technion, Israel, and the Ph.D. degree in electrical engineering from Stanford University, CA. From 1985 to 1995, he was a faculty member with the Department of Electrical Engineering, Yale University, New Haven, CT. In 1995, he joined the Department of Electrical Engineering and Computer Science, The University of Illinois at Chicago (UIC), as a Full Professor . From 2000 to 2001, he was Chair 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, he became Chairman of the Department of Electrical and Systems Engineering, Washington University, St. Louis (WUSTL). He is the inaugural holder of the Eugene and Martha Lohman Professorship and the Director of the Center for Sensor 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 Publications Board, member of the Board of Governors, and member of the Executive Committee of this Society. From 2003 to 2006, he was the Founding Editor of the special columns on Leadership Reflections in the IEEE Signal Processing Magazine. He was corecipient of the IEEE SPS 1989 Senior Award for Best Paper with P. Stoica, coauthor of the 2003 Young Author Best Paper Award, and corecipient of the 2004 Magazine Paper Award with A. Dogandzic. He was elected Distinguished Lecturer of the IEEE SPS for the term 2004 to 2005 and received the 2006 IEEE SPS Technical Achievement Award. He is the Principal Investigator of the new multidisciplinary university research initiative (MURI) project entitled Adaptive Waveform Diversity for Full Spectral Dominance. He has been a Fellow of the Royal Statistical Society since 1996.

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