17 Electromagnetic Vector Sensors with Beamforming Applications*

Transcription

1 17 Electromagnetic Vector Senor with Beamforming Application* Arye Nehorai Univerity of Illinoi at Chicago Kwok-Chiang o Addet Technovation Pte. Ltd. B. T. G. Tan National Univerity of Singapore 17.1 Introduction Advantage of Uing Electromagnetic Vector Senor itorical Development Content of thi Chapter 17. Beamforming Problem Formulation and Preliminary Dicuion Abbreviation and Notation Beamforming Problem Formulation for Single-Meage Signal Beamforming Problem Formulation for Dual-Meage Signal Aumption Performance Meaure A Ueful Reult 17.3 Signal to Interference-plu-Noie Ratio for Single-Meage Signal Comparion with Scalar-Senor Array Comparion with a Single Tripole 17.4 Signal to Interference-plu-Noie Ratio for Dual-Meage Signal 17.5 Numerical Reult 17.6 Beampattern of an Electromagnetic Vector Senor 17.7 Concluion 17.1 Introduction Direction-of-arrival (DOA) etimation and beamforming for electromagnetic (EM) wave are two common objective of array proceing. Early work on DOA etimation and beam forming ha been baed on calar enor, each of which provide meaurement of only one component of the electric or magnetic field induced. Subequent reearch ha invetigated the ue of enor that meaure two component of the electric or magnetic field, and tripole enor that meaure three complete component of the electric field. In recent year, reearcher have propoed the ue of EM vector enor that meaure the three complete component of the electric field and three component of the magnetic field at one point, for DOA etimation. 1, *The work of A. Nehorai wa upported by the Air Force Office of Scientific Reearch under Grant F , F , and F , the National Science Foundation under Grant MIP ; and the Office of Naval Reearch under Grant N by CRC Pre LLC

2 Advantage of Uing Electromagnetic Vector Senor EM vector enor are enitive to both the DOA and polarization information in the incoming wave. The polarization provide a crucial criterion for ditinguihing and iolating ignal that may otherwie overlap in conventional calar-enor array. When a ingle EM vector enor i ued for DOA etimation or beamforming, it ha the following advantage and capabilitie: DOA etimation/beamforming in three dimenional (3D) while occupying very little pace Reolution of very cloely paced (even coincident) ource baed on polarization difference Ability to proce wideband ignal in the ame way a narrowband ignal andling of ource with either ingle- or dual-meage ignal Iotropic repone No need for location calibration and time ynchronization among different component Some of thee advantage reult from the fact that no time delay are ued. In contrat, conventional calar enor method require a two-dimenional (D) array for DOA etimation/beamforming in a 3D pace, need accurate location calibration and time ynchronization, and require much higher computational cot to proce wideband intead of narrowband ignal. An array of patially ditributed EM vector enor can additionally exploit time delay among the enor. In general, array of EM vector enor can achieve better performance than calar enor array, while occupying le pace. They can alo be paced farther apart o increae aperture and hence performance without introducing ambiguitie itorical Development The ubject of electromagnetic vector enor wa firt introduced to the ignal proceing community by Nehorai and Paldi. 1, EM vector enor are commercially available and actively reearched. EMC Braden Ltd. in Baden, Switzerland, manufacture them for a 75z to 30Mz frequency range, and Flam and Ruell, Inc. in orham, Pennylvania, for to 30Mz. Lincoln Laboratorie at MIT ha performed preliminary localization tet with the EM vector enor of Flam and Ruell, Inc. 3 Other reearch on enor development i reported by Kanda 4 and Kanda and ill. 5 The optimum accuracy of ource parameter etimation for vector enor array i analyzed by Nehorai and Paldi 1, in term of Cramér Rao bound (abolute limit on the accuracy of the cla of unbiaed parameter etimator). Quality meaure are defined for etimating DOA and orientation in 3D pace, including mean-quare angular error and covariance of vector-angular error. Lower bound on thee meaure give concrete reult on expected performance. A fat algorithm for DOA etimation uing an EM vector enor ha been propoed. 1, Inpired by the Poynting theorem, it form the cro product of the electric field vector with the complex conjugate of the magnetic vector and average over time. The aymptotic performance under general condition i hown to be cloe to optimum. Some cro-product algorithm for ource tracking uing an EM vector enor have alo been propoed. 6 A minimum-noie-variance beamformer for interference cancellation employing a ingle EM vector enor ha been propoed. 7 It aume that the DOA and polarization of the ource are known. Thi enable uppreion of uncorrelated interference, even if it come from the ame direction a the ource, baed on polarization a well a location difference. Analyi of the ignal to interference-plu-noie ration (SINR) howed the beamformer to be very effective, particularly when the ignal and interference are differently polarized. Some high-reolution DOA etimation algorithm have been developed for EM vector-enor array for variou cenario. Thee include thoe algorithm cutomized for cenario with only completely polarized ignal, 8,10 thoe for cenario with only incompletely polarized ignal, 9 and thoe for more general cenario where completely and incompletely polarized ignal may coexit. 11 On a eparate note, 00 by CRC Pre LLC

3 ome article reported preliminary reult on the application of EM vector enor to communication problem. 1,13 The identifiability and uniquene iue aociated with EM vector enor have been rigorouly tudied in Reference 14 to 18. In particular, it ha been etablihed that one EM vector enor can etimate uniquely the DOA and polarization ellipe of up to three ource. 15,16 In comparion, one would need at leat four appropriately paced calar enor to determine uniquely the DOA of one ource. An active and a paive model for etimating polarimetric parameter aociated with EM wave have alo been propoed, 19 in which EM vector enor are ued a a receiving device Content of Thi Chapter Thi chapter preent a minimum-noie-variance type beamformer for a ingle EM vector enor 7 (which firt appear in Reference 7) and alo analye the performance of the beamformer. Such a beamformer require the knowledge of the DOA and polarization parameter of the ignal, and aume that the ignal, inference, and noie are mutually uncorrelated. The beamformer minimize the output variance while maintaining the gain in the direction of the ignal. Thi ha the effect of preerving the ignal while minimizing contribution to the output due to interference and noie arriving from direction other than the direction of the ignal. The invetigation of the performance of the beamformer i retricted to cenario where there exit one ignal and one interference that are uncorrelated. Two type of ignal are conidered: one carrie a ingle meage, and the other carrie two independent meage imultaneouly 1, The former i called a ingle-meage (SM) ignal, and the latter a dual-meage (DM) ignal. On the other hand, the interference under conideration take a general form that can be completely polarized (CP) or incompletely polarized (IP). Note that SM ignal are CP, while DM ignal are IP. Firt, explicit expreion for the SINR are obtained for both SM ignal and DM ignal. Then ome phyical implication aociated with the SINR expreion are dicued. In particular, it i deduced that for the two type of ignal of interet, the SINR rie with an increae in the eparation between the DOA and/or the polarization of the ignal and the interference for all DOA and polarization (calarenor array and a ingle tripole do not have uch propertie). Moreover, a trategy for effectively uppreing an interference with an EM vector enor i identified. The SINR expreion for the SM ignal alo provide a bai for generating a DM ignal in which the two meage ignal have minimum interference effect on one another. The outline of thi chapter i a follow. Section 17. preent problem formulation for beamforming and ome preliminary dicuion. The analye concerning SM and DM ignal are preented in Section 17.3 and 17.4, repectively. Section 17.5 preent numerical example. Section 17.6 analye the characteritic of the mainlobe and idelobe of the beampattern of an EM vector enor, and compare them with other type of enor array. Section 17.7 preent concluion. 17. Beamforming Problem Formulation and Preliminary Dicuion Abbreviation and Notation Abbreviation EM SINR DOA DM SM electromagnetic ignal to interference-plu-noie ratio direction of arrival dual meage ingle meage 00 by CRC Pre LLC

4 DOP CP IP UP PD ULA UCA degree of polarization completely polarized incompletely polarized unpolarized polarization difference uniform linear array uniform circular array Notation ( ) T, ( ), * tranpoe, ermitian and complex conjugate I n n n identity matrix y (t), y d (t) 6 1 complex envelope (phaor) meaurement (both electric and magnetic field) received at an EM vector enor at time t aociated with SM ignal or DM ignal y E (t) 3 1 complex envelope (phaor) electric field meaurement y (t) 3 1 complex envelope (phaor) magnetic field meaurement e E (t), e (t) 3 1 complex envelope (phaor) electric and magnetic noie (t) complex envelope of an SM ignal d,1 (t), d, (t) complex envelope of the firt and econd meage ignal of a DM ignal R, R d covariance matrice of y (t), y d (t) R i interference covariance matrix w weight vector of the minimum-noie-variance beamformer φ, ψ azimuth and elevation aociation with a DOA α, β orientation and ellipticity angle aociated with a polarization of a CP ignal θ vector denoting [φ, ψ, α, β] T a(θ) teering vector of an EM vector enor for a CP ignal with parameter θ B(φ, ψ) teering matrix of an EM vector enor for a UP ignal with DOA (φ, ψ) Q(α) rotation matrix with angle α h(β) 1 unit-norm vector repreenting ellipticity of a polarization σ i,c, σ i,u power of the CP and UP component of an interference σ power of an SM ignal σ d,1, σ d, power of the firt and econd meage ignal of a DM ignal σ power of the electric/magnetic noie i difference between the polarization of an SM ignal and interference d,1 i difference between the polarization of the firt meage ignal (of a DM ignal) and interference d, i difference between the polarization of the econd meage ignal (of a DM ignal) and interference γ angular eparation between the DOA of the ignal and interference λ wavelength Note that the ubcript, d, d,1, d,, and i are ued to aociate ome ymbol with SM ignal, DM ignal, the firt and econd meage of a DM ignal, and interference, repectively. For example, the ymbol β, β d,1, β d,, and β i denote the ellipticity angle aociated with SM ignal, the firt and econd meage of a DM ignal, and interference, repectively Beamforming Problem Formulation for Single-Meage Signal Thi and the next ubection decribe the data model a propoed by Nehorai and Paldi 1, for a ingle EM vector enor receiving an SM ignal or a DM ignal. Note that an SM ignal and a DM ignal are two type of ignal encountered in communication application, and they are reult of two different method of ignal tranmiion. For an SM ignal, one meage ignal i tranmitted from a ource uing a fixed polarization, uch a a linear polarization or a circular polarization. For DM ignal, two independent meage are tranmitted imultaneouly from the ame ource uing two different polarization. [We hall call uch tranmiion method dual-ignal tranmiion (DST)]. Variou DST form exit. One DST form ue two linearly polarized ignal that are patially and temporally orthogonal with an amplitude or phae modulation (ee, e.g., Reference 9 and 30). Another DST form ue two circularly 00 by CRC Pre LLC

5 polarized ignal with oppoite pin. In general, it can be tated that the advantage of the DST method i that it double the bandwidth of a communication ytem. owever, the preceding DST method may uffer from poible cro-polarization (ee, e.g., Reference 9), multipath effect, and other unknown ditortion from the ource to the enor. Note that an SM ignal i common in mot communication application, and tranmiion of a DM ignal i ued in ome communication application uch a digital televiion (TV) broadcating decribed in Reference 31, microwave radio ytem manufactured by elio, 3 cellular radio network, 33 and optical communication ytem. 34 With the notation tated in Subection 17..1, the complex (phaor) enor meaurement obtained by an EM vector enor at time t, induced by an SM ignal in the preence of an interference and additive noie i given by y E () t y ()= t a t B i i i t e t y t = ( θ ) ( )+ ( φ, ψ ) ξ ( )+ () () (17.1) where ( )= ( ) ( ) ( ) a θ B φ, ψ Q α h β θ= [ φ, ψ, α, β] v φψ v φψ B( (, ) (, ) φψ, )= v φψ, v φψ, ( ) ( ) inφ coφinψ ( v( φψ, ) v ( φψ, ))= coφ inφinψ 0 coψ T coα Q( α)= inα β h( β)= co jinβ inα coα (17.) e(t) = [e ET (t), e T (t)] T, (t) 1, ξ i (t) 1, y E (t), y (t), e E (t), e (t), 3 1. The firt, econd, and third term on the right-hand ide of Eq. (17.1) correpond to meaurement induced by the ignal, interference, and noie, repectively. Phyically, y E (t) and y (t) are the threecomponent meaurement of the electric and magnetic field at the enor at time t, repectively, and e E (t) and e (t) are the noie component in thee meaurement. The parameter φ ( π, π] and ψ [ π/, π/] are the azimuth and elevation of the ignal, repectively; and α ( π/, π/] and β [ π/4, π/4] are the polarization parameter referred to a the orientation angle and ellipticity, repectively. The vector a(θ) i the teering vector of an EM vector enor aociated with an SM ignal with parameter θ, and v(φ, ψ) and ṽ(φ, ψ) are unit vector that pan the ame plane a the electric and magnetic field vector of the incoming ignal with DOA (φ, ψ). The variable (t) i the complex envelope of the ignal and ξ i (t) the complex envelope of the interference. The covariance of ξ i (t) determine the tate of polarization of the interference. Indeed, the interference covariance matrix, R i = E(ξi(t)ξ i (t)), can be expreed a by CRC Pre LLC

6 σiu, Ri I σic, Q αi h βi h βi Q α = + ( ) ( ) ( ) ( i) (17.3) The firt term on the right-hand ide of Eq. (17.3) i the UP component with power σ i,cu, and the econd i the CP component with power σ i,c. The degree of polarization (DOP) of the interference i defined a the ratio between the power of the CP component and the total power of the interference (i.e., σ i,c /(σ i,u + σ i,c ). The interference i aid to be CP if σ i,c 0 but σ i,u = 0, IP if σ i,c 0 and σ i,cu 0, and UP if σ i,u 0 but σ i,c = 0. The output of a beamformer in thi cae i ŝ()= t w y() t (17.4) where w 6 1 i a weight vector. Suppoe the DOA and polarization parameter of the ignal are known; then, for the minimum-noie-variance beamformer, the weight vector i obtained through the following contrained minimization: ubject to w w = arg min w R w a = 1 w 6 1 (17.5) where R = E(y (t)y (t)) i the data covariance matrix, and a denote a(θ ). The beamformer attempt to uppre all incoming interference except for the deired ignal with teering vector a Beamforming Problem Formulation for Dual-Meage Signal The complex (phaor) enor meaurement obtained by an EM vector enor at time t, induced by a DM ignal in the preence of an interference and additive noie i given by where y t a θ t a θ t B φ ψ ξ t e t ()= ( ) ()+ ( ) ()+ ( ) ( )+ () d d, 1 d, 1 d, d, i, i i θ φ, ψ, α, β and θ φ, ψ, α, β = ( ) = ( ) d, 1 d d d, 1 d, 1 d, d d d, d, (17.6) The firt and econd term on the right-hand ide of Eq. (17.6) correpond to meaurement induced by the firt and econd meage ignal, repectively, aociated with the DM ignal, wherea the third and fourth term correpond to the interference and noie, repectively. The variable d,k (t) and a(θ d,k ), where k = 1,, are the complex envelope and teering vector of the kth meage ignal. Note that the two teering vector a(θ d,1 ) and a(θ d, ) have the ame DOA (φ d, ψ d ) but different polarization (i.e., (α d,1, β d,1 ) (α d,, β d, )). In Section 17.4, an appropriate choice of (α d,1, β d,1 ) and (α d,, β d, ) that minimize the interference effect on one meage ignal reulting from the other will be propoed. The output of a beamformer for the firt and econd meage ignal are, repectively and ˆ d, 1 wd, 1 d t y t ()= () ˆ d, wd, d t y t ()= () where w d,1, w d, 6 1 are the correponding weight vector. Note that to optimize the recovery of the meage ignal, a pecific weight vector i ued for each meage ignal eparately. Suppoe the DOA 00 by CRC Pre LLC

7 and polarization parameter of the ignal are known. Then, for the minimum-noie-variance beamformer, the weight vector for the kth meage ignal, where k = 1, i obtained through the following contrained minimization: ubject to w w = arg min w R w a dk, dk, = 1 w 6 1 d (17.7) where R d = E(y d (t)y d (t)) i the data covariance matrix, and a d,k denote a(θ d,k ) Aumption The analye to be carried out are baed on the following: ASSUMPTION 1. The DOA and polarization parameter of the ignal are known. ASSUMPTION. The complex envelope of (t), d,1 (t) and d, (t), and of each component of e E (t) and e (t) are all zero-mean Gauian random variable. ASSUMPTION 3. The ignal i uncorrelated with the interference. ASSUMPTION 4. The variou component of the noie are uncorrelated among themelve, and alo uncorrelated with both the ignal and interference. ASSUMPTION 5. The power of the electric noie and magnetic noie are all equal to σ (i.e., the noie covariance matrix i equal to σ I 6 ). Under Aumption to 5, the data covariance matrix i R = σ a a + B R B + σ I i i i for the cae of SM ignal, where σ = E( (t) *(t)) i the power of the ignal, and 6 for the cae of DM ignal, where σ d,k Performance Meaure R = σ a a + σ a a + B R B + σ I d d, 1 d, 1 d, 1 d, d, d, = E( d,k (t) d,k *(t)) i the power of kth meage ignal, k = 1,. The ratio between the output power of the ignal and output power of the interference and noie (SINR) i ued to evaluate the beamformer performance. The SINR meaure ha been ued a performance indicator for beamformer in many tudie. For an SM ignal, the SINR i given by i i i 6 σw aa w SINR = w R σ a a w ( ) (17.8) For a DM ignal, the SINR for the kth meage ignal, ŝ d,k (t), i σ w a a w dk, dk, dk, dk, dk, SINR dk, = wdk, Rd σd, kad, kad, k wdk, ( ) (17.9) where k = 1,. 00 by CRC Pre LLC

8 In thi chapter, explicit expreion for SINR, SINR d,1, and SINR d, are obtained, and their characteritic in term of the variou parameter of the ignal, interference, and noie are invetigated. To interpret the SINR expreion, a parameter that provide a meaure for the difference between the polarization of two ignal (uing the Poincaré phere polarization repreentation) 3 i introduced. Firt, let (φ 1, ψ 1, α 1, β 1 ) and (φ, ψ, α, β ) be the DOA/polarization of two ignal, and conider a new coordinate ytem where the DOA of the two ignal both lie in the xy plane (uch a coordinate ytem can alway be obtained with an appropriate coordinate rotation). In uch a new coordinate ytem, the ellipticity angle β k of the ignal remain unchanged. owever, the orientation angle α k will change, which hall be denoted by α k. According to the Poincaré phere repreentation, a polarization (α k, β k ) i repreented by a point (referred to a Poincaré point for convenience) on a phere whoe center i at the origin and radiu i 1. The poition vector of that point i P k [ ] = co α co β, in α co β, in β T (17.10) Such a repreentation ha two deirable propertie. Firt, for two polarization with the ame orientation angle (with repect to the new coordinate ytem), the larger the difference i in their ellipticity angle, the larger the ditance between two Poincaré point aociated with the two polarization. Second, for two polarization with the ame ellipticity angle (with repect to the new coordinate ytem), the larger the difference in their orientation angle*, the larger the ditance between two Poincaré point aociated with the two polarization. Thu, it i meaningful to take the difference between the polarization of the two ignal to be 1, the horter arc length joining p 1 and p, where p 1 and p are the repreentation for the polarization (α 1, β 1 ) and (α, β ) on the Poincaré phere, repectively. REMARKS. (1) To obtain the difference between the polarization of two ignal, there i a need to know the polarization, a well a the DOA of thee ignal. () It can be hown that the difference between the polarization of two ignal i independent of the coordinate ytem. (3) When dealing with the difference between the polarization of two ignal, only the polarization of the CP component of the ignal are of concern. (4) The range of 1 i [0, π]. (5) The arc length 1 i related to the orientation and ellipticity angle through Lemma 1. LEMMA 1 (Compton 1 ). Conider polarization (α 1, β 1 ) and (α, β ) aociated with two ignal. Let (α 1, β 1 ) and (α, β ) be the polarization in a coordinate ytem uch that the DOA of thee ignal both lie in the xy plane. Then co 1 = h ( ) Q β ( α ) Q ( α1 ) h ( β1) (17.11) A Ueful Reult Under Aumption 1 to 5, it can be hown that the weight vector atifying repectively Eq. (17.5) and (17.7) are w 1 1 R a R a d d, k = and w dk= k = 1, for 1, 1 a R a a R a dk, d d, k (17.1) *The increae in difference between the orientation angle i valid within a certain (ueful) range of the orientation angle. 00 by CRC Pre LLC

9 By ubtituting Eq. (17.1) directly into the expreion for SINR given by Eq. (17.8) and (17.9), the DOA and polarization of the ignal and interference, a well a the noie power, are hidden in two matrice whoe invere need to be evaluated. For eae of interpreting the dependence of SINR on the ignal, interference, and noie, the following reult, which i ueful for implifying the analyi of SINR expreion, i needed. LEMMA (Cox 4 ). Let R = α k aa + G 6 6, and wˆ = arg min w Rw w 6 1 ubject to w a = 1 where a 6 1 i a defined in Eq. (17.), and σ k i a real contant. If G i noningular, then σk wˆ aa wˆ 1 = σ k a G a wˆ Gwˆ To further implify the analyi of the SINR expreion to be preented later, it i aumed hereafter that (φ, ψ ) = (φ d,1, ψ d,1 ) = (φ d,, ψ d, ) = (0, 0) and ψ i = 0 (i.e., the DOA of the ignal i parallel to the x-axi and that of the interference i in the xy plane)*. With uch a etup, the eparation between the DOA of the ignal and interference i imply φ i. In addition, the difference between the polarization of the ignal and interference, i, atifie co ( i /) = h (β i )Q (α i )Q(α )h(β ) Signal to Interference-plu-Noie Ratio for Single-Meage Signal For convenience, the angular eparation between the DOA of the ignal and interference i referred to a DOA eparation, and i denoted by γ. Moreover, the difference between the polarization of the ignal and interference i referred to a polarization difference (PD). Theorem 1 below expree the SINR explicitly in term of the DOA eparation, PD, and power of the ignal, interference, and noie. TEOREM 1. The expreion of SINR, a given in Eq. (17.8), can be expreed a SINR = + γ 1 co σ σ σ + σ, ( ) ( iu) σ σ iu, i σic, co + σ + σ + σ ic, iu, (17.13) PROOF. See Reference 7. REMARK. For UP interference, the PD i i undefined and can take any value within [0, π]. owever, σ i,c = 0 in thi cae and the lat term of Eq. (17.13) i zero regardle of the value of i. Clearly, SINR increae with an increae in the ignal power, σ, but decreae with an increae in the noie power, σ, a well a the power of the CP (i.e., σ i,c ) or UP (i.e., σ i,u ) component of the interference. owever, the dependencie of SINR on PD and DOA eparation are nontrivial, and are etablihed in the following corollarie. *Thi can be achieved with an appropriate coordinate rotation. It can be hown that with uch a coordinate rotation, SINR, SINR d,1, and SINR d, a defined in Eq. (17.8) and (17.9) remain invariant. Moreover, the eparation between the DOA and difference between the polarization of the ignal and interference remain unchanged (the latter follow from the definition of the difference between two polarization preented in Section 17..5). 00 by CRC Pre LLC

10 COROLLARY 1. If σ i,c 0 and γ π, then SINR i an increaing function of i. COROLLARY. If σ i,u 0 or i π, then SINR i an increaing function of γ. COROLLARY 3. If σ i,c = 0, then SINR i independent of i. COROLLARY 4. SINR attain the maximum value, SINRmax = σ /σ, when either γ = π, or both i = π and σ i,u = 0 are true. Moreover, SINR max imply take the value of SINR in the abence of interference. COROLLARY 5. For given (fixed) σ, σ i,u + σ i,c, σ, and γ π, the minimum of SINR i attained when i = 0 and i,u = 0. PROOF. See Reference 7. REMARKS. 1. Corollary 1 mean that SINR generally increae with an increae in the PD i, except for two pecial cae: (a) σ i,c = 0, or (b) γ = π. Note that cae (a) correpond to cenario where the interference i UP, and cae (b) to cenario where the DOA of the ignal i exactly oppoite to that of the interference. For cae (a), the interference ha no CP component and thu the PD hould not affect SINR (ee Corollary 3). On the other hand, by Corollary 4, SINR for cae (b) alway attain the maximum max value SINR regardle of the other ignal parameter.. A pecial cae of Corollary 1 i that even if the DOA of the ignal and interference are identical, one can till increae the value of SINR by increaing the PD i. Thi i a feature that calar-enor array lack. Indeed, for a calar-enor array, if the DOA of the interference i identical to that of the ignal, interference uppreion i impoible regardle of the PD, the number of enor, and array aperture. 3. Corollary mean that SINR generally increae with an increae in the DOA eparation γ, except for the cae where both σ i,u = 0 and i = π hold. For the cae where σ i,u = 0 and i = π, SINR max can alway be attained regardle of the other parameter (ee Corollary 4). Note that σ i,u = 0 mean that the interference i CP, and i = π mean that the PD i the larget poible. For a coordinate ytem where the DOA of the ignal and interference both lie in the xy plane, uch a PD arie when the polarization aociated with the ignal and interference atify (α, β ) = (α i ± π/, β i ). Phyically, the two polarization ellipe aociated with the polarization (α, β ) and (α i, β i ) have the ame hape but have their major axe orthogonal to each other, and at the ame time, the direction of pin of the electric field aociated with the two polarization are oppoite. 4. By Corollary 3, if the interference i UP, then it i not poible to increae the SINR by varying the polarization of the ignal (α, β ). 5. Corollary 4 mean that SINR attain the larget poible value, SINR max, when either the DOA of the ignal and interference are oppoite, or when the interference i CP with larget poible polarization difference, π. In either cae, SINR max obtained i equivalent to the SINR when there i no interference regardle of the interference power (i.e., the interference become completely ineffective). 6. Corollary 5 mean that, for any given DOA eparation, SINR attain it lowet value when the interference i CP with polarization difference equal to 0. The fact that SINR increae with an increae in the DOA eparation or PD for all DOA and polarization (ee Corollarie 1 and ) i an important feature aociated with an EM vector enor. Indeed, thi feature i deirable becaue it i natural to expect a higher SINR with a larger DOA eparation or PD. In contrat, for calar enor array and a ingle tripole, SINR doe not necearily increae with an increae in the eparation in DOA or polarization (the cae of a tripole i elaborated in Section 17.3.). The preceding corollarie are potentially ueful in ome application. For example, one can exploit the fact that SINR increae with an increae in the PD (Corollary 1) to effectively uppre an interference if the DOA and polarization of the interference are known. Indeed, for a fixed DOA eparation γ, one can maximize SINR by tranmitting the ignal with polarization uch that the PD i the larget poible (i.e., i = π). Thi would lead to SINR = σ (1 + co γ) σ i,u /(σ + σ i,u )]/σ. Clearly, if the interference i CP (i.e., σ i,u = 0), then SINR attain SINR max = σ /σ, which i the value when there i no interference, regardle of the DOA eparation and the interference power. 00 by CRC Pre LLC

11 Comparion with Scalar-Senor Array Beamformer uing calar-enor array have been addreed in the literature. 0 Thi ubection hall dicu ome advantage of uing an EM vector enor a compared with calar-enor array for beamforming in 3D pace. Firt, for a calar-enor array, at leat three enor are needed to perform beamforming, which mean that it occupie a larger pace than an EM vector enor. Second, when the DOA of the interference i identical to that of the ignal, interference uppreion i impoible regardle of the number of calar enor and the array aperture. In contrat, a ingle vector enor can uppre an interference if the difference between the polarization of the ignal and interference i nonzero (ee Remark of the corollarie to Theorem 1). Third, conider a ignal and an interference with ufficiently large DOA eparation. Then, to uppre the interference with arbitrary DOA, only one EM vector enor i needed. owever, for the cae of a calar enor array, at leat four appropriately paced calar enor are needed. Indeed, to uppre an interference, the teering vector aociated with the interference mut be linearly independent of that aociated with the ignal. In thi connection, it ha been hown that to enure every two teering vector with ditinct DOA to be linearly independent, one EM vector enor i ufficient, 18 but at leat four calar enor with inter-enor pacing all le than a half wavelength are needed for the cae of calar-enor array. 5 Thi i a reult of the fact that an EM vector enor earche in both the polarization and DOA domain, wherea a calar-enor array ue only time delay information. Fourth, the SINR for a vector enor i iotropic, wherea for a calar-enor array, it very much depend on the array geometry, and doe not necearily increae with an increae in the DOA eparation Comparion with a Single Tripole A beamformer uing a ingle tripole ha been addreed by Compton. 1 Compton invetigated the performance of a ingle tripole in uppreing a CP interference on receiving an SM ignal. From the reult obtained by Compton, one can deduce that, unlike the cae of an EM vector enor, the SINR for a ingle tripole doe not necearily increae with an increae in the DOA eparation or the PD. Two example hall be ued to illutrate thi property. Firt, conider a ignal and an interference with DOA lying in the xy plane and vertically and linearly polarized. Then, the electric field induced by the ignal and interference are identical (except for a cale contant); thu, it i poible to dicriminate the ignal and interference regardle of their DOA eparation. Thu, the SINR remain unchanged (which i the mallet poible) regardle of the DOA eparation. Next, conider a ignal and an interference with oppoite DOA and both lying in the xy plane, and uppoe that both of them are circularly polarized. Then, the SINR when the ignal and interference have the ame pin (the PD i 0) i larger than when they have oppoite pin (the PD i π). (Thi i becaue the electric field induced at a tripole due to the ignal and interference, with oppoite DOA, are identical (except for a cale contant) if their direction of pin are oppoite, but are ditinct if their direction of pin are identical. Conequently, the SINR doe not necearily increae with an increae in the PD. Compton ha alo etablihed that the SINR for a ingle tripole i the lowet (with SINR being equal to σ /(σ + σ i,c )) if one of the following three condition hold: 1. The interference ha the ame DOA and polarization a the ignal.. The DOA of the ignal i oppoite to that of the interference, and the polarization of the ignal and interference atify α = α i and β = β i. 3. The ignal and interference are both linearly polarized, and their electric field are parallel to each other. Now let u examine the preceding three condition for an EM vector enor (for cenario where there exit a CP interference and an SM ignal). SINR for an EM vector enor i lowet only if condition 1 i atified, and the lowet SINR equal to σ /(σ + σ i,c ), which i higher than the lowet SINR obtained with a ingle tripole. A for condition, Corollary 4 tate that a long a the DOA of the ignal i oppoite to that of the interference, SINR for an EM vector enor alway attain the maximum value, SINR max, regardle of the polarization of the ignal and interference. On the other hand, high SINR can be 00 by CRC Pre LLC

12 obtained for an EM vector enor even when condition 3 i met. Indeed when condition 3 i met, the DOA eparation may range from 0 to π. By Corollary, SINR can be increaed by increaing the DOA eparation, and by Corollary 4, SINR attain the maximum value, SINR max, when the DOA of the ignal i oppoite to that of the interference. Thu, a ingle EM vector enor genrally outperform a ingle tripole in uppreing a CP interference when receiving an SM ignal Signal to Interference-plu-Noie Ratio for Dual-Meage Signal A DM ignal conit of two SM ignal (or CP ignal) with the ame DOA but different polarization. The effective polarization of uch a DM ignal varie with time, and thu, the tate of polarization of a DM ignal can either be IP or UP. To tranmit a DM ignal (coniting of two uncorrelated meage ignal), it i deirable that the interference effect of one meage ignal on the other be minimal. Becaue the DOA parameter aociated with the two meage ignal are identical, it i poible to exploit the difference only in the polarization parameter to reduce the interference effect. In thi connection, Corollary 4 of Theorem 1 provide a good way for chooing the polarization. Indeed, conider the cenario where there i no external interference and view one meage ignal a the deired CP ignal, and the other meage ignal a a CP interference. Then, by Corollary 4 of Theorem 1, both SINR d,1 and SINR d, attain their maximum value if the difference between the polarization of the two meage ignal i equal to π (i.e., when extracting one meage ignal, there i theoretically no interference effect from the other). Therefore, it i aumed hereafter that the polarization of the two meage ignal are choen in uch a way that the PD i π, meaning that the polarization atify (α d,1, β d,1 ) = (α d, ± π/, β d, ) (refer to Remark 3 of the corollarie to Theorem 1 for a relevant phyical meaning). For convenience, the difference between the polarization of the firt meage ignal and the interference (i.e., i d,1 ) i referred to a the firt PD, and the difference between the polarization of the econd meage ignal and the interference (i.e., i d, ) i referred to a the econd PD. Theorem that follow expree SINR d,1 and SINR d, explicitly in term of the DOA eparation, the firt and the econd PD, and the power of the two meage ignal, interference and noie. TEOREM : If (α d,1, β d,1 ) = (α d, ± π/, β d, ), then where SIN R SIN R = σ co d, 1 1 i + γ ν co co in 4 µ ν 4 ( 1 γ) σ σ 4σδ 1 = σ co d, ( 1 i + γ) ν µ + νco ( coγ) in σ σ 4σδ d, 1 d, 1 d, d, ( ) σσiu, σσic, µ =, ν = σ + σ σ + σ σ σ σ δ δ 1 ( iu, ) σ + σ 1 γ = + co d, 4 σ σ σ d, σ + σ 1 γ 1 = + co d, 4 σ σ σ d, 1 ( iu, )( ic, + + iu, ) ( ) + ( ) + µ νin d, 1 i d, i µ νin d, 1 i d, i (17.14) (17.15) (17.16) 00 by CRC Pre LLC

13 PROOF. See Reference 7. REMARK. Theorem i derived baed on the aumption that σ d,1 and σ d,, the power of the firt and econd meage ignal, repectively, are nonzero. If σ d,1 or σ d, i equal to zero, Theorem reduce to the cae of SM ignal that have been addreed in Section 3, and the derivation of the SINR expreion i omewhat different from thoe of SINR d,1 and SINR d,. COROLLARY 1. If σ i,c 0 and γ π, then SINR d,k i an increaing function of d,k i, for k = 1,. COROLLARY. If σ d,k i,u 0 or i π, then SINR d,k i an increaing function of γ for k = 1,. COROLLARY 3. If σ i,c = 0, then SINR d,k i independent of d,k i, for k = 1,. COROLLARY 4. SINR d,k attain the maximum value, SINR max d,k = σ d,k /σ, when either γ = π, or both d,k i = π and σ i,u = 0 are true, for k = 1,. Moreover, SINR max d,k imply take the value of SINR d,k in the abence of intereference. COROLLARY 5. For given (fixed) σ d,1, σ d,, σ i,u + σ i,c, σ, and γ π, the minimum of SINR d,k i attained when d,k i = 0 and σ i,u = 0, for k = 1,. PROOF. See Reference 7. REMARKS. 1. The dependence of SINR d,k on d,k i, γ, σ d,k, σ, σ i,c, and σ i,u a preented in Corollarie 1 to 5 of Theorem i baically identical to that of SINR on i, γ, σ, σ, σ i,c, and σ i,u a preented in Corollarie 1 to 5 of Theorem 1. Therefore, the dicuion concerning Corollarie 1 to 5 of Theorem 1 in Section 17.3 i applicable to Corollarie 1 to 5 of Theorem.. Becaue p d,1 and p d, a defined in Eq. (17.10) (which correpond to the repreentation of (α d,1, β d,1 ) and (α d,, β d, ), repectively, on the Poincaré phere) are antipodal point on the Poincaré phere, it can be hown that the um of the firt PD, i d,1, and the econd PD, i d,, i equal to a contant π. Thu, an increae in the firt PD lead to a decreae in the econd PD, and vice vera. Thi ha two implication. Firt, by Corollary 1 of Theorem, increaing the value of the firt (or econd) PD lead to an increae in SINR d,1 (or SINR d, ), but a decreae in SINR d, (or SINR d,1 ). Conequently, the value of both SINR d,1 and SINR d, cannot be increaed imultaneouly with a change in the polarization of the interference. Second, by Corollary 4 of Theorem, if the DOA eparation i not equal to π, then SINR d,1 attain it maximum value when the interference i CP and the firt PD i equal to π. owever, the econd PD become zero, and thu by Corollary 5, SINR d, attain it minimum value. Thu, for each DOA eparation that i not equal to π, SINR d,1 attain it maximum value if and only if SINR d, attain it minimum value. Becaue SINR d,1 and SINR d, are generally not identical, it i not eay to addre the SINR for the DM ignal a a whole. ere, let u conider the wort cae and ued SINR min = min{sinr d,1 SINR d, } (which give the maller value between SINR d,1 and SINR d, ) a a meaure of the effective SINR. Next, becaue it i reaonable to aume that both meage ignal are equally important, one may take the power of the two meage ignal to be identical. With thee conideration, one can eaily verify from Corollarie 1 to 3 of Theorem that 1. SINR min increae with an increae in DOA eparation.. SINR min increae when the firt (or the econd) PD increae from 0 to π/, but decreae when the firt (or the econd) PD increae form π/ to π. 3. SINR min i independent of the firt and econd PD if the interference i UP. Comparable reult are not available for calar enor, imply becaue calar enor cannot receive two (independent) meage ignal imultaneouly Numerical Reult Thi ection preent ome numerical example to ae the reliability of the theoretical prediction of the performance of an EM vector enor a preented in Section 17.3 and Firt, exact data covariance matrice R and R d were ued in the experiment for checking the SINR expreion derived in Section 17.3 and Next, to make the experiment realitic, R and R d generated from finite number of naphot were ued. 00 by CRC Pre LLC

14 SINR (db) SINR (db) DOP = 1 Infinite naphot available DOA Separation (deg) (a) DOP = 0.5 Infinite naphot available DOA Separation (deg) (b) PD= 0 PD= π/ PD= π PD= 0 PD= π/ PD= π SINR (db) DOP = 0 Infinite naphot available DOA Separation (deg) (c) PD= 0, π/, π FIGURE 17.1 (a) Graph of SINR v. DOA eparation for one SM ignal and one interference uncorrelated with SNR = INR = 10 db. The three curve correpond to PD = 0, PD = π/ and PD = π. The DOP of the interference i 1 and true covariance i ued. (b) A in Fig. 17.1a, but the DOP of the interference i 0.5. (c) A in Fig. 17.1a, but the DOP of the interference i IEEE. With permiion. The cae of one SM ignal and one interference impinging on an EM vector enor wa invetigated. The ignal wa circularly polarized with poitive pin (i.e., β = π/4). The ignal, interference, and noie were uncorrelated, and the ignal-to-noie ratio (SNR) and interference-to-noie ratio (INR) were both 10 db. The three graph of Figure 17.1 how the reult for the cae where infinitely many naphot (i.e., exact R and R d ) were ued, the SINR wa computed baed on the raw expreion (without any implification) given by Eq. (17.8). Figure 17.1a how the value of SINR a a function of DOA eparation when the DOP of the interference wa 1 (i.e., σ i,u = 0 and hence the interference wa CP). The value of the PD conidered were 0, π/, and π, which correpond to interference with polarization that were circular with poitive pin (i.e., β i = π/4), linear (i.e., β i = 0), and circular with negative pin (i.e., β i = π/4), repectively. The cenario in Fig. 17.1b and c were identical to thoe of Fig. 17.1a except 00 by CRC Pre LLC

15 that the DOP of the interference were 0.5 (correponding to IP interference with σ i,c = σ i,u ) and 0 (i.e., = 0 and hence the interference wa UP), repectively. The SINR in Fig. 17.1a to c confirm Corollarie 1 σ i,c to 4 of Theorem 1, that i, that SINR increae with an increae in the DOA eparation or the PD, SINR i independent of the PD when the interference i UP (ee Fig. 17.1c), and SINR attain the maximum value, σ /σ, when the DOA eparation i π or when the interference i CP (DOP = 1) and the PD i π. Alo, imulation for cenario identical to thoe of Fig. 17.1a to c, but with 00 naphot, were conducted (ee Reference 7 for more detail). Two obervation were obtained. Firt, the SINR obtained uing 00 naphot differ from that obtained uing infinitely many naphot by le than db. Moreover, the dependencie of SINR on the variou parameter are imilar. Finally, imulation for cenario imilar to thoe of Fig. 17.1a to c, but for a DM ignal, were alo conducted, and the reult were imilar (ee Reference 6 for more detail) Beampattern of an Electromagenetic Vector Senor Thi ection firt analyze the beampattern of an EM vector enor, and then make a comparion with two other type of enor array. Firt, conider an EM vector enor that ha been teered toward (or focued in) the direction/polarization θ F, and aume that there i no noie and interference (an aumption adopted in ome relevant tudie,7 ). Then the normalized repone (or beampattern) of the EM vector enor reulting from an incident ignal with direction/polarization θ k i given by ( )= ( ) ( ) g θ θ θ θ F, k a F a k 4 (17.17) (The function g(θ F, θ k ) reache the maximum when θ k = θ F, and the maximum value attained i 1. Becaue the magnitude quared of a(θ) i 4, a denominator i introduced on the right-hand ide of Eq. (17.17) o that the magnitude of g(θ F, θ k ) i normalized to 1 when θ k = θ F ). Note that, unlike calarenor array with beampattern that are only function of DOA, the beampattern of an EM vector enor i dependent on both the DOA and polarization. To facilitate the analyi of the beampattern, let u conider the coordinate ytem where φ F = ψ F = φ k = 0. Let the eparation between the DOA (φ F, ψ F ) and (φ k, ψ k ) be γ k F. Then Eq. (17.17) can be expreed a ( )= g θf, θk F ( 1+ coγ k ) F k co 4 (17.18) where k F i the difference between the polarization toward which the EM vector enor i teered and the polarization of the incident ignal. Although Eq. (17.18) i derived uing the coordinate ytem where φ F = ψ F = φ k = 0, it hold for any (φ F, ψ F ) and (φ k, ψ k ). Thi i becaue Eq. (17.18) i a function of only two parameter γ k F and k F, which are both independent of the actual coordinate ytem. From the expreion of g(θ F, θ k ) given by Eq. (17.18), everal propertie of the beampattern of an EM vector enor can be deduced. Firt, the repone of an EM vector enor in the direction/polarization θ k decreae with an increae in γ k F or k F. Second, when γ k F = π (i.e., at the direction oppoite to the beam teer direction), or when k F = π (i.e., if the difference in polarization i the larget poible), an EM vector enor doe not have any repone. Finally, becaue g(θ F, θ k ) attain it maximum if and only if θ k = θ F, the beampattern of an EM vector enor doe not contain grating lobe* (i.e., idelobe that i a high a the mainlobe). In contrat, the beampattern for calar enor with uniform linear or uniform circular array geometry contain grating lobe (thi property i demontrated in the latter part of thi ection), and uch calar enor are not able to uppre interference arriving in the direction of the grating lobe. *Such a property i alo een in acoutic vector enor, which meaure the acoutic preure and all three component of the acoutic particle velocity induced by acoutic ignal by CRC Pre LLC

16 FIGURE 17. Polar plot of a cro ection of the beampattern of an EM vector enor IEEE. With permiion. FIGURE 17.3 The 3-dB beamwidth of an EM vector enor againt the polarization difference, F k IEEE. With permiion. Figure 17. how the polar plot of any cro ection of the beampattern that contain the beam teer direction, for k F = 0, π/4, π/, 3π/4. Note that regardle of the beam teer direction/polarization, the hape of the beampattern i identical to that hown in Fig Now the 3-dB (or half-power) beamwidth of the mainlobe i analyzed. Note that the beampattern of an EM vector enor i dependent on polarization in addition to DOA, and thu i very complex. ere, the analyi of the 3-dB beamwidth hall be retricted to the cae of a fixed value of k F (a a reult, the beampattern depend only on the eparation in DOA). It can be deduced from Eq. (17.18) that, for a fixed k F, the 3-dB beamwidth i given by F 1 k F co co 1 0, if k [ π ] F 0 if [, k π π] Figure 17.3 plot the 3-dB beamwidth a a function of k F. Thi beamwidth decreae gradually from 13π/18 to 0 if k F [0, π/]. Beyond thi interval (i.e., k F [π/, π]), it i identically zero. Thi indicate the excellent ability of the EM vector enor to ditinguih ignal and interference that have ufficiently large difference in polarization. 00 by CRC Pre LLC

17 F FIGURE 17.4 Beampattern of an EM vector enor with beam teer direction (φ F, ψ F ) = (π/, 0). The value of k i 7 π/1. The 3 db plane i alo hown IEEE. With permiion. The performance of an EM vector enor i further illutrated in Fig Indeed, Fig how the beampattern when an EM vector enor i teered toward the direction (φ F, ψ F ) = (π/, 0) and an arbitrary polarization, with k F being fixed at 7π/1. For eae of viualizing the beampattern variation with repect to a reference fixed at 3 db, the horizontal plane cutting the z-axi at 3 db i alo hown. Regardle of the beam teer direction, the hape of the beampattern i identical to that hown in Fig. 17.4, except for a hift in poition. (In Fig. 17.4, a well a the other figure to be preented ubequently, the value of a beampattern repone i truncated to 40 db if it i maller than 40 db). Note that becaue the beampattern of an EM vector enor i dependent on both DOA and polarization, it i not obviou how to define a idelobe for thi enor. owever, for a fixed k F, the beampattern i a decreaing function of γ k F (with maximum value at (φ k, ψ k ) = (φ F, ψ F )). Conequently, there i effectively no idelobe for a fixed k F. Now let u compare the beampattern of an EM vector enor with two type of enor/array. Conider array of ix iotropic calar enor that meaure only one component of the electric or magnetic field induced. Two common enor configuration are conidered: a ix-enor uniform linear array (ULA) lying along the y-axi with inter-enor pacing equal to λ/, where λ i the wavelength of the ignal of concern, and a ix-enor uniform circular array (UCA) with enor coordinate (co π τ/3, in π τ/3, 0)λ, for τ = 0,, 5. Note that, unlike an EM vector enor, the hape of the beampattern of the ULA and UCA are dependent on the beam teer direction. Thu, to analyze the beampattern, imulation for many different beam teer direction are conducted. An undeirable property of the beampattern of the ULA and UCA i that they have grating lobe or idelobe. Moreover, many grating lobe/idelobe occur at direction that are very far from the beam teer direction. For example, Fig and 17.6 plot the beampattern of, repectively, the ULA and UCA, when the array are teered to (π/, 0). In each figure, FIGURE 17.5 Beam pattern of a ix-calar enor ULA lying along the y-axi with interenor pacing equal to half wavelength. The beam-teer direction (φ F, ψ F ) i (π/, 0). The 3-dB horizontal plane i alo hown IEEE. With permiion. 00 by CRC Pre LLC

18 FIGURE 17.6 Same a Fig except that the array i a ix-calar enor UCA with enor coordinate (co π τ/3, in π τ/3,0)λ, for τ = 0,, IEEE. With permiion. a 3 db plane i alo plotted. For ULA, it can be een that the beampattern contain a grating lobe. A for UCA, there i a idelobe with trength greater than 3 db Concluion The chapter dicued a minimum-noie-variance type of beamformer employing an EM vector enor for one ignal and one interference that are uncorrelated. Both SM and DM ignal were conidered, and the tate of polarization of the interference under conideration ranged from completely polarized to unpolarized. To analyze the beamformer performance, an explicit expreion for the SINR of an SM ignal wa firt obtained. It wa then deduced that the SINR of an SM ignal increae with an increae in the eparation between the DOA and/or the polarization, for all DOA and polarization (calarenor array and a ingle tripole 1 do not have uch propertie). It wa alo deduced that a ingle EM vector enor can uppre an (uncorrelated) interference that ha the ame DOA a the ignal and ditinct polarization (thi i impoible for calar enor regardle of the number of enor and the array aperture). In addition, a trategy for effectively uppreing the interference wa identified. Moreover, the SINR expreion for the SM ignal alo provided a bai for generating DM ignal of which the two meage ignal have the minimum interference effect on one another. An explicit expreion for the SINR of uch DM ignal wa alo derived. Subequently, it wa deduced that the previouly mentioned characteritic for the SINR of an SM ignal were alo valid for the SINR of a DM ignal. Fairly extenive computer imulation were conducted, and the reult obtained were in good agreement with the analyi preented in thi chapter. Finally, the chapter alo analyzed the characteritic of the mainlobe and idelobe of the beampattern of an EM vector enor, and demontrated the advantage of an EM vector enor over ome calar-enor array. In particular, it ha been hown that the beampattern of an EM vector enor doe not contain grating lobe. In contrat, the beampattern of a ix-enor uniform linear array and a ix-enor uniform circular array have grating lobe. Moreover, many grating lobe occur at direction that are very far from the beam teer direction. The propoed beamformer can be extended eaily to handle multiple ource with divere polarization uing multiple vector enor a receiver. Some poible follow-up tudie are (1) invetigation of the beamforming performance of an EM vector enor for multiple ignal and multiple interference; () performance with multiple EM vector enor; (3) performance for the ignal and interference that are correlated; (4) performance when the power of the electric noie and magnetic noie at EM vector enor are not identical; (5) effect of channel depolarization on the ignal; and (6) comparion of an EM vector enor with other type of EM enor. Acknowledgment We are grateful to Bert ochwald of Lucent Technologie, United State, and Kah-Chye Tan of Addet Technovation Pte. Ltd. Singapore, for their ueful uggetion. 00 by CRC Pre LLC