Nehorai, A. & Paldi, E. “Electromagnetic Vector-Sensor Array Processing” Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton: CRC Press LLC, 1999 c  1999byCRCPressLLC 65 Electromagnetic Vector-Sensor Array Processing 1 Arye Nehorai The University of Illinois at Chicago Eytan Paldi Haifa, Israel 65.1 Introduction 65.2 The Measurement Model Single-Source Single-Vector Sensor Model • Multi-Source Multi-Vector Sensor Model 65.3 Cramer-Rao Bound for a Vector Sensor Array Statistical Model • The Cramer-Rao Bound 65.4 MSAE, CVAE, and Single-Source Single-Vector Sensor Analysis The MSAE • DST Source Analysis • SST Source (DST Model) Analysis • SST Source (SST Model) Analysis • CVAE and SST Source Analysis in the Wave Frame • A Cross-Product-Based DOA Estimator 65.5 Multi-Source Multi-Vector Sensor Analysis Results for Multiple Sources, Single-Vector Sensor 65.6 Concluding Remarks Acknowledgment References Appendix A: Definitions of Some Block Matrix Operators Dedicated to the memory of our physics teacher, Isaac Paldi 65.1 Introduction This article (see also [1, 2]) considers new methods for multiple electromagnetic source localization usingsensorswhoseoutput isa vectorcorrespondingtothecompleteelectric and magneticfields atthe sensor. These sensors, which will be called vector sensors, can consist for example of two orthogonal triads of scalar sensors that measure the electric and magnetic field components. Our approach is in contrast to other articles in this chapter that employ sensor arrays in which the output of each sensor is a scalar corresponding, for example, to a scalar function of the electric field. The main advantage of the vector sensors is that they make use of all available electromagnetic information and hence should outperform the scalar sensor arrays in accuracy of direction of arrival (DOA) estimation. Vector sensors should also allow the use of smaller array apertures while improving performance. 1 This work was supported by the U.S. Air Force Office of Scientific Research under Grant no. F49620-97-1-0481, the Office of Naval Research under Grant no. N00014-96-1-1078, the National Science Foundation under Grant no. MIP-9615590, and the HTI Fellowship. c  1999 by CRC Press LLC (Note that we use the term “vector sensor” for a device that measures a complete physical vector quantity.) Section 65.2 derives the measurement model. The electromagnetic sources considered can origi- nate from two types of transmissions: (1) Single signal transmission (SST), in which a single signal message istransmitted, and(2)dual signal transmission (DST), in whichtwoseparate signal messages are transmitted simultaneously (from the same source), see for example [3, 4]. The interest in DST is due to the fact that it makes full use of the two spatial degrees of freedom present in a transverse electromagnetic plane wave. This is particularly important in the wake of increasing demand for economical spectrum usage by existing and emerging modern communication technologies. Section 65.3 analyzes the minimum attainable variance of unbiased DOA estimators for a general vector sensor array model and multi-electromagnetic sources that are assumed to be stochastic and stationary. A compact expression for the corresponding Cram ´ er-Rao bound (CRB) on the DOA estimation error that extends previous results for the scalar sensor array case in [5] (see also [6]) is presented. A significant property of the vector sensors is that they enable DOA (azimuth and elevation) estimation of an electromagnetic source with a single vector sensor and a single snapshot. This result is explicitly shown by using the CRB expression for this problem in Section 65.4. A bound on the associated normalized mean-square angular error (MSAE, to be defined later) which is invariant to the reference coordinate system is used for an in-depth performance study. Compact expressions for this MSAE bound provide physical insight into the SST and DST source localization problems with a single vector sensor. The CRB matrix for an SST source in the sensor coordinate frame exhibits some nonintrinsic singularities (i.e., singularities that are not inherent in the physical model while being dependent on the choice of the reference coordinate system) and has complicated entry expressions. Therefore, we introduce a new vector angular error defined in terms of the incoming wave frame. A bound on the normalized asymptotic covariance of the vector angular error (CVAE) is derived. The relationship between the CVAE and MSAE and their bounds is presented. The CVAE matrix bound for the SST source case is shown to be diagonal, easy to interpret, and to have only intrinsic singularities. WeproposeasimplealgorithmforestimatingthesourceDOAwith asingle vectorsensor, motivated by the Poynting vector. The algorithm is applicable to various types of sources (e.g., wide-band and non-Gaussian); it does not require a minimization of a cost function and can be applied in real time. Statistical performance analysis evaluates the variance of the estimator under mild assumptions and compares it with the MSAE lower bound. Section 65.5 extends these results to the multi-source multi-vector sensor case, with special atten- tion to the two-source single-vector sensor case. Section 65.6 summarizes the main results and gives some ideas of possible extensions. The main difference between the topics of this article and other articles on source direction estima- tion is in our use of vector sensors with complete electric and magnetic data. Most papers have dealt with scalar sensors. Other papers that considered estimation of the polarization state and source direction are [7]–[12]. Reference [7] discussed the use of subspace methods to solve this problem using diversely polarized electric sensors. References [8]–[10] devised algorithms for arrays with two dimensional electric measurements. Reference [11] provided performance analysis for arrays with two types of electric sensor polarizations (diversely polarized). An earlier reference, [12], proposed an estimation method using a three-dimensional vector sensor and implemented it with magnetic sensors. All these references used only part of the electromagnetic information at the sensors, thereby reducing the observability of DOAs. In most of them, time delays between distributed sensors played an essential role in the estimation process. For a plane wave (typically associated with a single source in the far-field) the magnitude of the electric and magnetic fields can be found from each other. Hence, it may be felt that one (complete) field is deducible from the other. However, this is not true when the source direction is unknown. c  1999 by CRC Press LLC Additionally, the electric and magnetic fields are orthogonal to each other and to the source DOA vector, hence measuring both fields increasessignificantly theaccuracyof the sourceDOAestimation. This is true in particular for an incoming wave which is nearly linearly polarized, as will be explicitly shown by the CRB (see Table 65.1). The use of the complete electromagnetic vector data enables source parameter estimation with a single sensor (even with a single snapshot) where time delays are not used at all. In fact, this is shown to be possible for at least two sources. As a result, the derived CRB expressions for this problem are applicable to wide-band sources. The source DOA parameters considered include azimuth and elevation. This section also considers direction estimation to DST sources, as well as the CRB on wave ellipticity and orientation angles (to be defined later) for SST sources using vector sensors, which were first presented in [1, 2]. This is true also for the MSAE and CVAE quality measures and the associated bounds. Their application is not limited to electromagnetic vector sensor processing. We comment that electromagnetic vector sensors as measuring devices are commercially available and actively researched. EMC Baden Ltd. in Baden, Switzerland, is a company that manufactures them for signals in the 75 Hz to 30 MHz frequency range, and Flam and Russell, Inc. in Horsham, Pennsylvania, makes them for the 2 to 30 MHz frequency band. Lincoln Labs at MIT has performed some preliminary localization tests with vector sensors [13]. Some examples of recent research on sensor development are [14] and [15]. Following the recent impressive progress in the performance of DSP processors, there is a trend to fuse as much data as possible using smart sensors. Vector sensors, which belong to this category of sensors, are expected to find larger use and provide important contribution in improving the performance of DSP in the near future. 65.2 The Measurement Model This section presents the measurement model for the estimation problems that are considered in the latter parts of the article. 65.2.1 Single-Source Single-Vector Sensor Model Basic Assumptions Throughout the article it will be assumed that the wave is traveling in a nonconductive, homo- geneous, and isotropic medium. Additionally, the following will be assumed: A1: Plane wave at the sensor: This is equivalent to a far-field assumption (or maximum wave- length much smaller than the source to sensor distance), a point source assumption (i.e., the source size is much smaller than the source to sensor distance) and a point-like sensor (i.e., the sensor’s dimensions are small compared to the minimum wave-length). A2: Band-limitedspectrum: Thesignalhas aspectrumincluding onlyfrequencies ω satisfying ω min ≤|ω|≤ω max where 0 <ω min <ω max < ∞. This assumption is satisfied in practice. The lower and upper limits on ω are also needed, respectively, for the far-field and point-like sensor assumptions. Let E(t) and H(t) be the vector phasor representations (or complex envelopes, see e.g., [16, 17] and [1, Appendix A]) of the electric and magnetic fields at the sensor. Also, let u be the unit vector at the sensor pointing towards the source, i.e., u =   cos θ 1 cos θ 2 sin θ 1 cos θ 2 sin θ 2   (65.1) c  1999 by CRC Press LLC whereθ 1 andθ 2 denote,respectively,theazimuthandelevationanglesofu,seeFig.65.1.Thus, θ 1 ∈[0,2π)and|θ 2 |≤π/2. FIGURE65.1:Theorthonormalvectortriad(u,v 1 ,v 2 ). In[1,AppendixA]itisshownthatforplanewavesMaxwell’sequationscanbereducedtoan equivalentsetoftwoequationswithoutanylossofinformation.Undertheadditionalassumption ofaband-limitedsignal,thesetwoequationscanbewrittenintermsofphasors.Theresultsare summarizedinthefollowingtheorem. THEOREM65.1 UnderassumptionA1,Maxwell’sequationscanbereducedtoanequivalentsetof twoequations.Withtheadditionalband-limitedspectrumassumptionA2,theycanbewrittenas: u×E(t) =−ηH(t) (65.2a) u·E(t) = 0 (65.2b) whereηistheintrinsicimpedanceofthemediumand“×”and“·”arethecrossandinnerproductsof R 3 appliedtovectorsinC 3 .(Thatis,ifv,w∈C 3 thenv·w=  i v i w i .Thisisdifferentthanthe usualinnerproductofC 3 ). PROOF65.1 See[1,AppendixA].(Notethatu=−κwhereκistheunitvectorinthedirection ofthewavepropagation). Thus,undertheplaneandband-limitedwaveassumptions,thevectorphasorequations(65.2) providealltheinformationcontainedintheoriginalMaxwellequations.Thisresultwillbeusedin thefollowingtoconstructmeasurementmodelsinwhichtheMaxwellequationsareincorporated entirely. c  1999byCRCPressLLC TheMeasurementModel Supposethatavectorsensormeasuresallsixcomponentsoftheelectricandmagneticfields. (Itisassumedthatthesensordoesnotinfluencetheelectricandmagneticfields).Themeasurement modelisbasedonthephasorrepresentationofthemeasuredelectromagneticdata(withrespectto areferenceframe)atthesensor.Lety E (t)bethemeasuredelectricfieldphasorvectoratthesensor attimetande E (t)itsnoisecomponent.Thentheelectricpartofthemeasurementwillbe y E (t)=E(t)+e E (t) (65.3) Similarly,fromEq.(65.2a),afterappropriatescaling,themagneticpartofthemeasurementwillbe takenas y H (t)=u×E(t)+e H (t) (65.4) InadditiontoEq.(65.3)and(65.4),wehavetheconstraint(65.2b). Definethematrixcrossproductoperatorthatmapsavectorv∈R 3×1 to(u×v)∈R 3×1 by (u×)  =   0 −u z u y u z 0 −u x −u y u x 0   (65.5) whereu x ,u y ,u z arethex,y,zcomponentsofthevectoru.Withthisdefinition,Eqs.(65.3)and(65.4) canbecombinedto  y E (t) y H (t)  =  I 3 (u×)  E(t)+  e E (t) e H (t)  (65.6) whereI 3 denotesthe3×3identitymatrix.Fornotationalconveniencethedimensionsubscriptof theidentitymatrixwillbeomittedwheneveritsvalueisclearfromthecontext. Theconstraint(65.2b)impliesthattheelectricphasorE(t)canbewritten E(t)=Vξ(t) (65.7) whereVisa3×2matrixwhosecolumnsspantheorthogonalcomplementofuandξ(t)∈C 2×1 . Itiseasytocheckthatthematrix V=   −sinθ 1 −cosθ 1 sinθ 2 cosθ 1 −sinθ 1 sinθ 2 0 cosθ 2   (65.8) whosecolumnsareorthonormal,satisfiesthisrequirement.Wenotethatsinceu 2 =1thecolumns ofV,denotedbyv 1 andv 2 ,canbeconstructed,forexample,fromthepartialderivativesofuwith respecttoθ 1 andθ 2 andpost-normalizationwhenneeded.Thus, v 1 = 1 cosθ 2 ∂u ∂θ 1 (65.9a) v 2 = u×v 1 = ∂u ∂θ 2 (65.9b) and(u,v 1 ,v 2 )isarightorthonormaltriad,seeFig.65.1.(Observethatthetwocoordinatesystems showninthefigureactuallyhavethesameorigin).Thesignalξ(t)fullydeterminesthecomponents ofE(t)intheplanewhereitlies,namelytheplaneorthogonaltouspannedbyv 1 ,v 2 .Thisimplies thattherearetwodegreesoffreedompresentinthespatialdomain(orthewave’splane),ortwo independentsignalscanbetransmittedsimultaneously. c  1999byCRCPressLLC CombiningEq.(65.6)andEq.(65.7)wenowhave  y E (t) y H (t)  =  I (u×)  Vξ(t)+  e E (t) e H (t)  (65.10) ThissystemisequivalenttoEq.(65.6)withEq.(65.2b). Themeasuredsignalsinthesensorreferenceframecanbefurtherrelatedtotheoriginalsource signalatthetransmitterusingthefollowinglemma. LEMMA65.1 Everyvectorξ= [ ξ 1 ,ξ 2 ] T ∈C 2×1 hastherepresentation ξ=ξe iϕ Qw (65.11) where Q =  cosθ 3 sinθ 3 −sinθ 3 cosθ 3  (65.12a) w =  cosθ 4 isinθ 4  (65.12b) andwhereϕ∈(−π,π],θ 3 ∈(−π/2,π/2],θ 4 ∈[−π/4,π/4].Moreover,ξ,ϕ,θ 3 ,θ 4 in Eq.(65.11)areuniquelydeterminedifandonlyifξ 2 1 +ξ 2 2 =0. PROOF65.2 See[1,AppendixB]. Theequalityξ 2 1 +ξ 2 2 =0holdsifandonlyif|θ 4 |=π/4,correspondingtocircularpolarization (definedbelow).Hence,fromLemma65.1therepresentation(65.11),(65.12)isnotuniqueinthis caseasshouldbeexpected,sincetheorientationangleθ 3 isambiguous.Itshouldbenotedthatthe representation(65.11),(65.12)isknownandwasused(see,e.g.,[18])withoutaproof.However, Lemma65.1ofexistenceanduniquenessappearstobenew.Theexistenceanduniquenessproperties areimportanttoguaranteeidentifiabilityofparameters. Thephysicalinterpretationsofthequantitiesintherepresentation(65.11),(65.12)areasfollows. ξe iϕ :Complexenvelopeofthesourcesignal(includingamplitudeandphase). w:Normalizedoveralltransfervectorofthesource’santennaandmedium,i.e.,fromthe sourcecomplexenvelopesignaltotheprincipalaxesofthereceivedelectricwave. Q:Arotationmatrixthatperformstherotationfromtheprincipalaxesoftheincoming electricwavetothe(v 1 ,v 2 )coordinates. Letω c bethereferencefrequencyofthesignalphasorrepresentation,see[1,AppendixA].Inthe narrow-bandSSTcase,theincomingelectricwavesignalRe  e iω c t ξ(t)e iϕ(t) Qw  movesonaqua- sistationaryellipsewhosesemi-majorandsemi-minoraxes’lengthsareproportional,respectively,to cosθ 4 andsinθ 4 ,seeFig.65.2and[19].Theellipse’seccentricityisthusdeterminedbythemagnitude ofθ 4 .Thesignofθ 4 determinesthespinsignordirection.Moreprecisely,apositive(negative)θ 4 correspondstoapositive(negative)spinwithright-(left)handedrotationwithrespecttothewave propagationvectorκ=−u.AsshowninFig.65.2,θ 3 istherotationanglebetweenthe(v 1 ,v 2 ) coordinatesandtheelectricellipseaxes(v 1 ,v 2 ).Theanglesθ 3 andθ 4 willbereferredto,respectively, c  1999byCRCPressLLC FIGURE 65.2: The electric polarization ellipse. as the orientation and ellipticity angles of the received electric wave ellipse. In addition to the electric ellipse, there is also a similar but perpendicular magnetic ellipse. It should be noted that if the transfer matrix from the source to the sensor is time invariant, then so are θ 3 and θ 4 . The signal ξ (t) can carry information coded in various forms. In the following we discuss briefly both existing forms and some motivated by the above representation. Single Signal Transmission (SST) Model Suppose that a single modulated signal is transmitted. Then, using Eq. (65.11), this is a special case of Eq. (65.10) with ξ(t) = Qws(t) (65.13) where s(t) denotes the complex envelope of the (scalar) transmitted signal. Thus, the measurement model is  y E (t) y H (t)  =  I (u×)  VQws(t)+  e E (t) e H (t)  (65.14) Special cases of this transmission are linear polarization with θ 4 = 0 and circular polarization with |θ 4 |=π/4. Recall that since there are two spatial degrees of freedom in a transverse electromagnetic plane wave, one could, in principle, transmit two separate signals simultaneously. Thus, the SST method does not make full use of the two spatial degrees of freedom present in a transverse electromagnetic plane wave. Dual Signal Transmission (DST) Models Methods of transmission in which two separate signals are transmitted simultaneously from the same source will be called dual signal transmissions. Various DST forms exist, and all of them can be modeled by Eq. (65.10) with ξ(t) being a linear transformation of the two-dimensional source signal vector. One DST form uses two linearly polarized signals that are spatially and temporally orthogonal with an amplitude or phase modulation (see e.g., [3, 4]). This is a special case of Eq. (65.10), where c  1999 by CRC Press LLC thesignalξ(t)iswrittenintheform ξ(t)=Q  s 1 (t) is 2 (t)  (65.15) wheres 1 (t)ands 2 (t)representthecomplexenvelopesofthetransmittedsignals.Toguarantee uniquedecodingofthetwosignals(whenθ 3 isunknown)usingLemma65.1,theyhavetosatisfy s 1 (t)=0,s 2 (t)/s 1 (t)∈(−1,1).(Practicallythiscanbeachievedbyusingaproperelectronic antennaadapterthatyieldsadesirableoveralltransfermatrix.) AnotherDSTformusestwocircularlypolarizedsignalswithoppositespins.Inthiscase ξ(t) = Q[ws 1 (t)+ ws 2 (t)] (65.16a) w = (1/ √ 2)[1,i] T (65.16b) where wdenotesthecomplexconjugateofw.Thesignalss 1 (t),s 2 (t)representthecomplexenvelopes ofthetransmittedsignals.Thefirsttermonther.h.s.ofEqs.(65.16)correspondstoasignalwith positivespinandcircularpolarization(θ 4 =π/4),whilethesecondtermcorrespondstoasignalwith negativespinandcircularpolarization(θ 4 =−π/4).TheuniquenessofEqs.(65.16)isguaranteed withouttheconditionsneededfortheuniquenessofEq.(65.15). Theabove-mentionedDSTmodelscanbeappliedtocommunicationproblems.Assumingthat uisgiven,itispossibletomeasurethesignalξ(t)andrecovertheoriginalmessagesasfollows. ForEq.(65.15),anexistingmethodresolvesthetwomessagesusingmechanicalorientationofthe receiver’santenna(see,e.g.,[4]).Alternatively,thiscanbedoneelectronicallyusingtherepresentation ofLemma65.1,withouttheneedtoknowtheorientationangle.ForEqs.(65.16),notethat ξ(t)=we iθ 3 s 1 (t)+ we −iθ 3 s 2 (t),whichimpliestheuniquenessofEqs.(65.16)andindicatesthat theorientationanglehasbeenconvertedintoaphaseanglewhosesigndependsonthespinsign. Theoriginalsignalscanbedirectlyrecoveredfromξ(t)uptoanadditiveconstantphasewithout knowledgeoftheorientationangle.Insomecases,itisofinteresttoestimatetheorientation angle.LetWbeamatrixwhosecolumnsarew, w.ForEqs.(65.16)thiscanbedoneusingequal calibratingsignalsandthenpremultiplyingthemeasurementbyW −1 andmeasuringthephase differencebetweenthetwocomponentsoftheresult.Thiscanalsobeusedforrealtimeestimation oftheangularvelocitydθ 3 /dt. IngeneralitcanbestatedthattheadvantageoftheDSTmethodisthatitmakesfulluseofthe spatialdegreesoffreedomoftransmission.However,theaboveDSTmethodsneedtheknowledge ofuand,inaddition,maysufferfrompossiblecrosspolarizations(see,e.g.,[3]),multipatheffects, andotherunknowndistortionsfromthesourcetothesensor. Theuseoftheproposedvectorsensorcanmotivatethedesignofnewimprovedtransmissionforms. Herewesuggestanewdualsignaltransmissionmethodthatusesonlineelectroniccalibrationin ordertoresolvetheaboveproblems.Similartothepreviousmethodsitalsomakesfulluseofthe spatialdegreesoffreedominthesystem.However,itovercomestheneedtoknowuandtheoverall transfermatrixfromsourcetosensor. Supposethetransmittedsignalisz(t)∈C 2×1 (thissignalisasitappearsbeforereachingthe source’santenna).Themeasuredsignalis  y E (t) y H (t)  =C(t)z(t)+  e E (t) e H (t)  (65.17) whereC(t)∈C 6×2 istheunknownsourcetosensortransfermatrixthatmaybeslowlyvaryingdue to,forexample,thesourcedynamics.Tofacilitatetheidentificationofz(t),thetransmittercansend calibratingsignals,forinstance,transmitz 1 (t)=[1,0] T andz 2 (t)=[0,1] T separately.Sincethese c  1999byCRCPressLLC inputs are in phasor form, this means that actually constant carrier waves are transmitted. Obviously, one can then estimate the columns of C(t) by averaging the received signals, which can be used later for finding the original signal z(t ) by using, for example, least-squares estimation. Better estimation performance can be achieved by taking into account a priori information about the model. The use of vector sensors is attractive in communication systems as it doubles the channel capacity (compared with scalar sensors) by making full use of the electromagnetic wave properties. This spatial multiplexing has vast potential for performance improvement in cellular communications. Infutureresearchitwouldbeof interesttodevelopoptimalcodingmethods(modulationforms)for maximum channel capacity while maintaining acceptable distortions of the decoded signals despite unknown varying channel characteristics. It would also be of interest to design communication systems that utilize entire arrays of vector sensors. Observe that actually any combination of the variablesξ, ϕ, θ 3 and θ 4 can be modulated to carry information. A binary signal can be transmitted using the spin sign of the polarization ellipse (sign of θ 4 ). Lemma 65.1 guarantees the identifiability of these signals from ξ (t). 65.2.2 Multi-Source Multi-Vector Sensor Model Suppose that waves from n distant electromagnetic sources are impinging on an array of m vector sensors and that assumptions A1 and A2 hold for each source. To extend the model (65.10) to this scenario we need the following additional assumptions, which imply that A1, A2 hold uniformly on the array: A3: Plane wave across the array: In addition to A1, for each source the array size d A has to be much smaller than the source to array distance, so that the vector u is approximately independent of the individual sensor positions. A4: Narrow-band signal assumption: The maximum frequency of E(t), denoted by ω m , satisfies ω m d A /c  1,wherec is the velocity of wave propagation (i.e., the minimum modulating wave-length is much larger than the array size). This implies that E(t − τ) E(t) for all differential delays τ of the source signals between the sensors. Note that(under the assumption ω m <ω c )since ω m = max{|ω min −ω c |,|ω max −ω c |},itfollowsthat A4 is satisfied if (ω max − ω min )d A /2c  1 and ω c is chosen to be close enough to (ω max + ω min )/2. Let y EH (t) and e EH (t) be the 6m × 1 dimensional electromagnetic sensor phasor measurement and noise vectors, y EH (t)  =  (y (1) E (t)) T ,(y (1) H (t)) T , ··· ,(y (m) E (t)) T ,(y (m) H (t)) T  T (65.18a) e EH (t)  =  (e (1) E (t)) T ,(e (1) H (t)) T , ··· ,(e (m) E (t)) T ,(e (m) H (t)) T  T (65.18b) where y (j) E (t) and y (j) H (t) are, respectively, the measured phasor electric and magnetic vector fields at the jth sensor and similarly for the noise components e (j) E (t) and e (j) H (t). Then, under assumptions A3 and A4 and from Eq. (65.10), we find that the array measured phasor signal can be written as y EH (t) = n  k=1 e k ⊗  I 3 (u k ×)  V k ξ k (t) + e EH (t) (65.19) where⊗ is the Kronecker product, e k denotes the kth column of the matrix E ∈ C m×n whose (j, k) entry is E jk = e −iω c τ jk (65.20) c  1999 by CRC Press LLC [...]... [34] to their attention References [1] Nehorai, A and Paldi, E., Vector-sensor array processing for electromagnetic source localization, IEEE Trans on Signal Processing, SP-42, 376–398, Feb 1994 [2] Nehorai, A and Paldi, E., Vector sensor processing for electromagnetic source localization, Proc 25th Asilomar Conf Signals, Syst Comput., Pacific Grove, CA, Nov 1991, 566–572 [3] Schwartz, M., Bennett,... unconditional direction-ofarrival estimation, IEEE Trans Acoust., Speech, Signal Processing, ASSP-38, 1783–1795, Oct 1990 [6] Ottersten, B., Viberg, M and Kailath, T., Analysis of subspace fitting and ML techniques for parameter estimation from sensor array data, IEEE Trans Signal Processing, SP-40, 590–600, March 1992 [7] Schmidt, R.O., A signal subspace approach to multiple emitter location and spectral... 1994 [29] Hochwald, B and Nehorai, A., Identifiability in array processing models with vector-sensor applications, IEEE Trans Signal Process., SP-44, 83–95, Jan 1996 [30] Ho, K.-C., Tan, K.-C and Ser, W., An investigation on number of signals whose direction-ofarrival are uniquely determinable with an electromagnetic vector sensor, Signal Processing, 47, 41–54, Nov 1995 [31] Tan, K.-C., Ho, K.-C and... study of measurements obtainable with arrays of electromagnetic vector sensors, IEEE Trans Signal Process., SP-44, 1036–1039, Apr 1996 [32] Nehorai, A and Paldi, E., Acoustic vector-sensor array processing, IEEE Trans on Signal Processing, SP-42, 2481–2491, Sept 1994 A short version appeared in Proc 26th Asilomar Conf Signals, Syst Comput., Pacific Grove, CA, Oct 1992, 192–198 [33] Hawkes, M and Nehorai,... known or unknown CR 65.4.3 SST Source (DST Model) Analysis Consider the MSAE for a single signal transmission source when the estimation is done under the assumption that the source is of a dual signal transmission type In this case, the model (65.10) has to be used but with a signal in the form of (65.13) The signal covariance is then P = σs2 Qw(Qw)∗ (65.32) where σs2 = E s 2 (t) and Q and w are defined... models then become special cases of Eq (65.21) as follows For DST signals, the block columns Ak ∈ C6m×2 and the signals x(t) ∈ C2n×1 are Ak = x(t) = ek ⊗ rI3 (uk ×) Vk ξ T (t), · · · , ξ T (t) n 1 (65.45a) T (65.45b) The parameter vector of the kth source includes here its azimuth and elevation For the SST case, the columns Ak ∈ C6m×1 and the signals x(t) ∈ Cn×1 are rI3 (uk ×) Vk Qk wk Ak = ek ⊗ x(t) =... The main considerations in communication are transmission of signals over channels with limited bandwidth and their recovery at the sensor Vector sensors naturally fit these c 1999 by CRC Press LLC considerations as they have maximum observability to incoming signals and they double the channel capacity (compared with scalar sensors) with DST signals This has vast potential for performance improvement... summarized in the following: • Rotation around u: Singular only for a circularly polarized signal • Rotation around v 1 (electric ellipse’s major axis): Singular only for a linearly polarized signal and no magnetic measurement • Rotation around v 2 (electric ellipse’s minor axis): Singular only for a linearly polarized signal and no electric measurement • The rotation variances around v 1 and v 2 are symmetric... frame, have been derived and can be used for performance analysis Some generalizations of the bounds appear in [26] These bounds are not limited to electromagnetic vector sensor processing Performance comparisons of vector sensor processing with scalar sensor counterparts are of interest Identifiablity: The derived bounds and the identifiability analysis of [29] and [30] were used to show that the fusion... polarization sensitive array, IEEE Trans Antennas Propagat., AP-39, 1376–1383, Sept 1991 [11] Weiss, A.J and Friedlander, B., Performance analysis of diversely polarized antenna arrays, IEEE Trans Signal Processing, SP-39, 1589–1603, July 1991 [12] Means, J.D., Use of three-dimensional covariance matrix in analyzing the polarization properties of plane waves, J Geophys Res., 77, 5551–5559, Oct 1972 . A. & Paldi, E. “Electromagnetic Vector-Sensor Array Processing Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca. (1) Single signal transmission (SST), in which a single signal message istransmitted, and(2)dual signal transmission (DST), in whichtwoseparate signal messages