Tài liệu Digital Signal Processing Handbook P65 docx

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
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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.
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(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.
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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)
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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.
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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.
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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,
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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
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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
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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
where τ
jk
is the differential delay of the kth source signal between the jth sensor and the origin of
some fixed reference coordinate system (e.g., at one of the sensors). Thus, τ
jk
=−(u
k
· r
j
)/c,where
u
k
is the unit vector in the direction from the array to the kth source and r
j
is the position vector of
the jth sensor in the reference frame. The rest of the notation in Eq. (65.19) is similar to the single
source case, cf. Eqs. (65.1), (65.8), and (65.10). The vector ξ
k
(t) can have either the SST or the DST
form described above.
Observe that the signal manifold matrix in Eq. (65.19) can be written as the Khatri-Rao product
(see, e.g., [20, 21]) of E and a second matrix whose form depends on the source transmission type
(i.e., SST or DST), see also later.
65.3 Cram
´
er-Rao Bound for a Vector Sensor Array

65.3.1 Statistical Model
Consider the problem of finding the parameter vector θ in the following discrete-time vector sensor
array model associated with n vector sources and m vector sensors:
y(t) = A(θ)x(t) + e(t ) t = 1,2,...
(65.21)
where y(t) ∈ C
µ×1
are the vectors of observed sensor outputs (or snapshots), x(t) ∈ C
ν×1
are the
unknown source signals, and e(t ) ∈ C
µ×1
are the additive noise vectors. The transfer matrix A(θ)
∈ C
µ×ν
and the parameter vector θ ∈ R
q×1
are given by
A(θ) =

A
1
(θ
(1)
) ··· A
n
(θ
(n)
)


(65.22a)
θ =

(θ
(1)
)
T
, ··· ,(θ
(n)
)
T

T
(65.22b)
where A
k
(θ
(k)
)∈ C
µ×ν
k
and the parameter vector of the kth source θ
(k)
∈ R
q
k
×1
,thusν =

n

k=1
ν
k
and q =

n
k=1
q
k
. The following notation will also be used:
y(t) =

(y
(1)
(t))
T
, ··· ,(y
(m)
(t))
T

T
(65.23a)
x(t) =

(x
(1)
(t))
T
, ··· ,(x

(n)
(t))
T

T
(65.23b)
where y
(j)
(t) ∈ C
µ
j
×1
is the vector measurement of the jth sensor, implying µ =

m
j=1
µ
j
, and
x
(k)
(t) ∈ C
ν
k
×1
is the vector signal of the kth source. Clearly µ and ν correspond, respectively, to
the total number of sensor components and source signal components.
The model (65.21) generalizes the commonly used multi-scalar source multi-scalar sensor one
(see, e.g., [7, 22]). It will be shown later that the electromagnetic multi-vector source multi-vector
sensor data models are special cases of Eq. (65.21) with appropriate choices of matrices.

For notational simplicity, the explicit dependence on θ and t will be occasionally omitted.
We make the following commonly used assumptions on the model (65.21):
A5: The source signal sequence {x(1), x(2),...} is a sample from a temporally uncorrelated
stationary (complex) Gaussian process with zero mean and
E x(t)x
∗
(s) = Pδ
t,s
E x(t)x
T
(s) = 0 (for all t and s).
where E is the expectation operator, the superscript “
∗
” denotes the conjugate transpose,
and δ
t,s
is the Kronecker delta.
c

1999 by CRC Press LLC