New Aspects of Electromagnetic Information Theory for Wireless and Antenna Systems

15 Pages·0.52 MB·English

by  Gruber F.K., Marengo E.A.

3470 IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.56,NO.11,NOVEMBER2008
New Aspects of Electromagnetic Information Theory
for Wireless and Antenna Systems
FredK. Gruber,StudentMember,IEEE, and EdwinA. Marengo,SeniorMember,IEEE
Abstract—Thispaperinvestigatesinformation-theoreticcharac- methodologies for NDF determination such as eigenvalue de-
terization,viaShannon’sinformationcapacityandnumberofde- compositionsofintegraloperatorsandprolatespheroidalwave-
greesoffreedom,ofwaveradiation(antenna)andwirelessprop-
function theory [9], [10]. Related work from the same period
agation systems. Specifically, the paper derives, from the funda-
isthatofLinfoot[11]andFellgettandLinfoot[12]who liter-
mental physical point of view of Maxwell’s equations describing
allytreatedimagingsystemsascommunicationchannels.After
electromagneticfields,theShannoninformationcapacityofspace-
timewirelesschannelsformedbyelectromagneticsourcesandre- theseseminalinvestigations,furtherprogressonNDFofwave
ceiversinaknownbackgroundmedium.Thetheoryisdeveloped radiation,propagation,andscatteringsystemshasbeenreported
first for the case of sources working at a fixed frequency (time- fromtimetotimebyotherauthors[4],[13]–[18].
harmoniccase)andisexpandedlatertothemoregeneralcaseof
In recent years, the need for both fundamental theory and
temporallybandlimitedsystems(time-domainfields).Intheban-
computational methodology for estimating both NDF and
dlimited case we consider separately the two cases of time-lim-
ited and essentially bandlimited systems and of purely bandlim- Shannon’s information capacity [19], [20] of a variety of
itedsystems.Thedevelopmentstakeintoaccountthephysicalra- wave radiation and propagation systems, including random
diatedpowerconstraintinadditiontoaconstraintinthesource(cid:0)(cid:0) media channels, has been growing steadily, mostly due to
normwhichactstoavoidantennasuperdirectivity.Basedonsuch
its fundamental importance in space-time wireless channels
radiated power and current (cid:0)(cid:0) norm constraints we derive the
including multiple-input multiple-output (MIMO) systems
Shannoninformationcapacityofcanonicalwirelessandantenna
systemsinfreespace,foragivenadditiveGaussiannoiselevel,as [21]–[23]. The present paper builds from recent work with
wellasanassociatednumberofdegreesoffreedomresultingfrom thismotivation,beingofparticularrelevancetheinvestigations
suchcapacitycalculations.Thederivedresultsalsoillustrate,from carriedoutby:Hanlenetal.[24],[25]whoinvestigatedgeneral
anewinformation-theoreticpointofview,thetransitionfromnear
communication channels within a general abstract operator
tofarfields.
theory in Hilbert spaces and, within the scalar framework, the
IndexTerms—Antenna,degreesoffreedom,electromagneticin- information capacity in the formally tractable case of sources
formation,informationcapacity,waveinformation,wireless. and fields in spherical regions under the standard source
normconstraint;Huietal.[26]whosimilarlyprovidedgeneral
expressions for the information capacity of Gaussian wireless
I. INTRODUCTION links from a spatial region to another; Hanlen and Fu [27]
who proposed numerical techniques for estimating the NDF
T HISresearchisconcernedwiththeformulationoffunda- of general wave propagation systems including media formed
mentalwirelesscommunicationandantennaengineering by collections of scatterers embedded to a given background
problemsatthecrossroadsofthewell-establishedfieldsofelec- medium (e.g., free space); Jensen and Wallace [28] (see also
tromagnetic theory and information theory. The interdiscipli- their related past work in [29] and [30]) who studied the
naryfieldconstitutedbysuchwave-andinformation-theoretic information capacity of Gaussian wireless links between two
problemsand theirsolutions can be descriptivelytermedelec- volumes under a constraint in the radiated power and who
tromagneticinformationtheoryor,withintheantennafocus,an- also proposed an approach to limit antenna superdirectivity
tenna information theory. Work in this area, combining wave while making the optimal power allocations for the different
physicswithinformationtheory,hasalonghistory,datingback wavepropagativemodesconnectingthevolumes;Morrisetal.
totheoriginsofinformationtheoryand,inparticular,tothepi- [31] who also studied the effect of the superdirectivity on the
oneeringworkonlightandinformationbyGabor[1]–[3]who informationcapacityofMIMOsystemsunderaradiatedpower
studiedthenumberofdegreesoffreedom(NDF)[4]ofdiffrac- constraint;GustafssonandNordebo[32],[33]whostudiedthe
tion-limitedopticalimagingsystemsvia“essentialdimension” spectral efficiency of an antenna inside a sphere, as measured
considerations (e.g., “resolution cell” concepts). The NDF of by the information capacity, by means of a model of MIMO
imaging systems was investigated around the same time also channels combining information theory, antenna theory, and
by di Francia [5]–[8] who explored the role of the evanescent broadbandmatchingtheory;Migliore[34]whoconsideredthe
planewavespectrainsuperresolution,aswellasrigorousformal capacityandNDFofagroupofantennasandscatterersinside
a sphere by means of the multipole expansion and equivalent
electrical networks and who also proposed a way to take into
ManuscriptreceivedNovember10,2007;revisedMay15,2008.Currentver-
sionpublishedNovember14,2008.ThisworkwassupportedbytheNational accountthetemporalresourcesbymeansofasamplingtheory
ScienceFoundationbyGrant0746310. in space and time [35]; Chakraborty and Franceschetti [36]
TheauthorsarewiththeDepartmentofElectricalandComputerEngineering
whoalsoconsideredthecapacityofspace-timecommunication
NortheasternUniversity,Boston,MA02115USA(e-mail:[email protected]).
DigitalObjectIdentifier10.1109/TAP.2008.2005468 channelsbymeansofageneralizedsamplingtheoremapplied
0018-926X/$25.00©2008IEEE
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GRUBERANDMARENGO:NEWASPECTSOFELECTROMAGNETICINFORMATIONTHEORY 3471
tospaceandtime;Poonetal.[37]whoconsideredNDFandin- trol[31]),foragivennoiselevelatthefieldreceiver.Thus,the
formation capacity of indoor wave propagation systems based derivedcapacityandNDFresultsareimmediatelyapplicableto
on combined analytical and empirical, measurement-based sourcesandreceiversinfreespacethataresupportedinconcen-
models, while also illustrating, among other aspects, the role tricsphericalregions.However,manyofthederivedformalities
ofcooperativeversusnoncooperativeusersintheenvironment; haveapplicabilitytomoregeneralgeometries,asfollowsfrom
and Xu and Janaswamy [38] who proposed a noise-dependent ourdiscussioninSectionsII-AandBofthepaperwhereessen-
definitiontoselecttheNDFintheapproachin[18]andnumeri- tiallythegeneralindex inSectionII-Bwillingeneralsubsti-
callystudiedtheeffectofmultiplescatteringinthepropagation tute the particular multipole-mode index , , empha-
environment. sizedlaterinthepaper(SectionII-Candtherestofthepaper).
The overall goal of the present paper is the investigation of Animportantaspectoftheelectromagneticinformationtheory
certainfundamentalaspectsnotfullycoveredbypreviouswork reportedhereisthatourcapacityandNDFcalculationstakeinto
inthisfruitfularea,mostlyoninformation-theoreticcharacter- accountpracticalrestrictionsinboththesource normandthe
ization,viaShannon’sinformationcapacityandNDF,ofwave radiatedpower.Thisisincontrasttoallpastworkweareaware
radiation (antenna) and wireless propagation systems. Specifi- of where the main emphasis has been to either constrain the
cally,thepaperderives,fromthefundamentalphysicalpointof source norm[24],[25],[37]or,morerecently,toconstrain
viewofMaxwell’sequationsdescribingelectromagneticfields, theradiatedpoweronly,whileindirectlylimitingthesource
theinformationcapacityofspace-limited,time-limited,andes- norm by means of different approaches like the restriction of
sentiallybandlimitedwirelesschannelsaswellasspace-limited thetransmissioncurrentstospecificsubspaces[31],theuseof
and strictly bandlimited wireless channels formed by electro- channelmodelsthatincludetheeffectofthelossesandthermal
magneticsourcesandreceiversinaknownbackgroundmedium, noise[28],orthetruncationofthenumberoftermsintheexpan-
undertheassumptionofadditiveGaussiannoiseperturbations. sionsofthefield[32],[33].Infact,ouradoptionofthismethod-
Whileparticularattentionisgiventofreespace,theresultsare ology complements recently reported approaches in [28] and
basedonGreen’sfunctionssothatthegeneralprocedureandre- [31]toestimatewirelesselectromagneticcapacitywhilepenal-
sultsderivedinthepaperapplytomoregeneralknownmedia. izingantennasuperdirectivity,andourtreatmentalsoprovides
Amongotheraspects,thederivedinformation-theoreticresults adifferentandindependentreformulationofsomeoftheresults
arealsousedtoaddressrelatedquestionssuchasNDFfortime- in[28],[31]where,likethosepapers,weshowthattheradiated
domainfields,the transitionfromnear tofarfields inthetime poweraloneisgenerallyinsufficientasaconstraintforelectro-
domain,bothofwhicharetopicsofmuchimportanceunderthe magneticcapacitycalculationsandthat,instead,itmustbecom-
broader umbrella of time-domain electromagnetics, as well as plementedinpracticebyotherphysicallymotivatedconstraints
the effect of finite transmit/receive times on the capacity and (such as the norm constraint). Our treatment complements
NDF(inthetime-limitedandessentiallybandlimitedcase). the analysis in those papers byconsidering the two most typi-
Methodologically, the electromagnetic information results callyadoptedconstraintssimultaneously,andalsogoesbeyond
are discussed first for time-harmonic excitation in Section III the scope of that past work by extending the results to broad-
of the paper, and generalized later to more general time-do- bandfields.Also,regardingthecalculationoftheNDF,instead
mainfieldsinSectionIV.Theconnectionbetweenthegeneral ofassuminganasymptoticbehaviorofthefunctionsusedinthe
electromagneticinformationtheoryinSectionIV,applicableto analysisinorderto finda finiteNDFor truncatingthe dimen-
quite general broadband fields, and the time-harmonic theory sionalityaprioribyothercriteria(ashasbeendoneinpastwork
in Section III (from which narrowband information-theoretic inthisarea[4],[17],[18],[32]–[34],[37],[40]),weshowthat
concepts such as spectral efficiency can be derived) is also thisisnotnecessarysincetheoptimalpowerassignmentmaxi-
discussed, along with illustration of the broader applicability mizingthemutualinformationbetweentransmitterandreceiver
of the general approach in Section IV. The formulation in alreadydeterminesthenumberofchannelsthatareusefulfora
Section IV builds from the one-dimensional time-domain givennoiselevel.
treatment of (temporally) bandlimited channels by Gallager
[39], which we generalize to the four-dimensional space-time
domain in the particular physical context of electromagnetic II. ELECTROMAGNETICINFORMATIONCHANNELSINGENERAL
wave radiation and propagation systems. Our work on the GEOMETRIES
space-limited, time-limited and essentially bandlimited case
This section reviews the representation of general and par-
appears to be new, while our work on the strictly bandlimited
ticularspatialelectromagneticchannelsintermsofthesingular
caseisarigorousalternativethatcomplementsrelatedresearch
system ( , , ; ) of the corresponding
in[35],[36]whichisbasedonsamplingtheoremsinspaceand
source-to-fieldmapping(see,e.g.,[17]and[41,pp.234–239]).
time.
This representation is then used in the rest of the paper to
Particularattentionisgiveninthispapertothefundamental
characterizeinformationcapacityandNDF.
problem of sources confined within a given spherical volume
ofradius ,subjectedtoupperboundslimitingthemaximalal-
lowed radiated power, and/or the maximal source norm or
A. GeneralSourcesandFieldsinHilbertSpace
functionalenergy(relatedtoimpressedcurrentlevelsinanan-
tenna, and thereby also to the associated ohmic losses which Consider an electromagnetic source (a current den-
mustbeboundedinpracticeaswellastosuperdirectivitycon- sity)confinedtoavolume .Thefrequencydomainrepresen-
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3472 IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.56,NO.11,NOVEMBER2008
tationofthissourceisobtainedbytakingthetemporalFourier where is the respective singular value. The corresponding
transformof ,whichgivesthespectralcurrentdensity transmitter-sidesingularfunctionisgivenby[41]
(4)
Expandingthekernelin(1)intermsofthesingularfunctionsso
where .Intheremainderofthissectionandinthenext that
one, the focus will be on spatial electromagnetic information
foragivenfrequency sothatfornotationalsimplicitywewill (5)
suppress the dependence, e.g., we will use in place of
, with the understanding that all quantities depend on
anddefining
the frequency .Thissource isassumed tobelong tothe
Hilbert space of square integrable functions of
(6)
support withinnerproductdefinedby
and
(7)
where overaquantitymeansitscomplexconjugate.
Theelectricfield ismeasuredinavolume .Thismea- yieldsthealternative(singularsystemordiagonalizing)repre-
suredfieldbelongstotheHilbertspace withinner sentationof(1)
productdefinedby
(8)
C. SVDforaSphericalScanningGeometryinFreeSpace
B. SingularSystemRepresentationoftheSource-to-Field
Nowconsidertheparticularcaseofasphericalscanningge-
Mapping
ometry in free space where the source is confined to the
Themeasuredfield isgivenby sphericalvolume andwherethe
electricfield ismeasured inasphericalscanningsurface
(1) ofradius concentrictothesource.
Thisgeometryisofinterestforantennas(e.g.,onecanenvision
a spherical near-field scanning setup to test and compare an-
wherewehaveintroducedtheradiationoperator ,
tennas informationally) and enables the analytical calculation
isanindicatorfunctionparameterizedby withvalue
ofthesingularsystemof bymeansofsphericalwavefunction
1if or0if ,and representsthedyadic
expansions.
Greenfunctionsatisfying
1) The Multipole Expansion: In free space, the temporal
Fouriertransformofthemeasuredfieldisgivenfromthemulti-
poleexpansion[44],[45]by
where is the identity dyadic, is Dirac’s delta function, and
(9)
isthewavenumberwhere isthefree-space
permittivityand isthefree-spacepermeability.Forfiniteand
disjointtransmissionandreceptionvolumes,theGreenfunction
where ,where
kernelin(1)issquareintegrable[4],[18],[41],[42]sothatthe
operator is compactand canberepresentedviathe singular
(10)
valuedecomposition(SVD)[41]–[43].
TheSVDof canbeobtainedfromtheeigendecomposition
and
of the associated self-adjoint operator [41, pp. 237,238], [42,
pp.258,259] where representsthe
(11)
adjointof definedby sothat
are the electric and magnetic multipole fields, respectively,
(2)
where is the spherical Hankel function of the first kind
(correspondingtooutgoingwaves)andorder ,and isthe
Let be an eigenfunction of the operator (and re- vector spherical harmonic of degree and order defined by
ceiver-sidesingularfunctionof ),then [46],[47]
(3) (12)
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GRUBERANDMARENGO:NEWASPECTSOFELECTROMAGNETICINFORMATIONTHEORY 3473
where (the angular momentum operator),
(22)
representsthedirectionof ,and representsthe
spherical harmonics [48, p. 787]. The expansion coefficients
and
in(9)aregivenby
(23)
(13)
Theresultsaboveholdforbothnearandfarfields.Inthespecial
wherethesphericalwavefunctions aredefinedby far-field case, one can use the large-argument approximation
[48,Eq.(11.160a)]toarrivefrom(19)
(14)
and(20)attheapproximation whichin
turnyieldsthefar-fieldsingularvalues
and
(24)
(15)
The far-field singular values in (24) are well known to decay
where is the free-space impedance , and is rapidlyfor [4],[43],[45],[51]–[53].Since,asdiscussed
thesphericalBesselfunctionoforder . in(9), takesvalues1and2,while takesintegervaluesfrom
Thesphericalwavefunctions and obeytheor- to ,foreachindex thereare singularvaluesso
thogonalityrelations[see[45,Eq.(17)]] thatanestimateoftheNDForinformationcontent[4]ofthefar
field is . On the other hand, this rule of
(16) thumb does not hold in the near zone, where we must use the
generalresult(23).
where here and henceforth denotes Euclidean norm (so Replacingtheindex in(8)yieldstherepresenta-
that ), where denotes the tion
Kroneckerdeltaandwhere,see(17)atthebottomofthepage.
Similarly,itisnothardtoshowbyborrowingfrom[49,Eqs.
(5), (13), (16)], including the orthogonality property of the (25)
vectorfunctions , ,and [50,pp.
1898–1901]aswellastherecurrencerelationsfor ,thatthe It is important to emphasize that the , , indexes represent
multipolefields, and ,arealsoorthogonalinthe spatialmodesthatarephysicallydifferentiable(separable)due
sense that totheorthogonalityofthefunctions and [via(21)
and(22)or, alternatively,(16)and (18)]sotheycorrespondto
(18) aneffectivenumberoftransmitterandreceivermodesthatare
potentially communicable between the antenna or scatterer of
where[see(19)atthebottomofthepage] and spatial dimension and a receiver outside the antenna. Also
note that (without further information) in view of (7) and (8)
(20)
onlythescalarquantities associatedtoagivensourcecan
beknownaboutthatsource,i.e.,onlycertainprojectionsofthe
2) SingularSystemRepresentation: Using(9)and(13),the
sourceontothespaceoffunctions ,inparticular,canbe
kernelof in(1)isfoundtobegivenby(5)with substituted
measuredfrom the receivedsignals. Hence,we will formulate
by and
nextthecommunicationproblemintermsofthequantity
ascontainingalltheinformationaboutthetransmitterstate(as
(21)
determinedbythesource )thatisinprinciplerecoverable
(17)
(19)
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3474 IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.56,NO.11,NOVEMBER2008
fromthemeasurementofthereceiversignal.Nofurtherinfor- associatedohmiclossesandthelevelofsuperdirectivityofthe
mation or components of this source can be estimated from antenna [31]. This quantity is commonly known as the (func-
the exterior field measurements alone, so the communication tional)energyofthesourcewhetheritisdirectlyrelatedtothe
problem reduces to the mapping dictated (in singular system physical energy or not [41, p. 98]. For a deterministic source
representation)by(25). itisgivenby
III. SPACEELECTROMAGNETICINFORMATIONCAPACITY (28)
Inthissectionthediscreterepresentationof(1)intermsofan
infinite number of spatial modes in (25) is interpreted as a set where indicatesamperesandmmeters.Boundingthisquan-
of parallel communication channels where each output tity restricts the sources so that they are square integrable. At
is related to the input probabilistically due to the pres- thesametime,thisboundrestrictstheamountofpowerradiated
enceof noise(to beadded inthe following).The inputmodes by the antenna since integral operators with square integrable
representrandomvariablescontainingtheinformationto kernels, as in the present radiation analysis, are bounded
betransmitted,therefore,thesource ischaracterizedbya [41]. An insightful way of visualizing this in the framework
setofrandomstatesorcurrentsthataretobetransmittedtothe of linear inversion theory is as follows. Having a bounded
output inthepresenceofnoise. source norm requires satisfying the Picard condition
[41, Eq. (10.96)]. However,
A. NoiseModeling
Inpractice,errorsareintroducedtothemeasurementsmainly
fromimperfectantennalocation,distortionofthefieldduetothe where denotes the largest of the singular values ,
test equipment, imprecise measurements of the fields, and nu- so that the norm of the field (at the spherical surface) is also
mericalapproximations[54].Sincemosterrorsareindependent bounded.Byreferringto(31)whereitisshownthattheradiated
andadditive[54],anaturalchoiceforthenoisemodelisthatofa power is equal to the norm of the far field (apart from a
whiteGaussianrandomprocess definedbytheusualprop- constant factor) we conclude that the radiated power is also
erty that for any function the projection is bounded.
a zero mean Gaussian random variable with variance [39, Mathematically, the source norm is the electromagnetic
p. 365]. In particular, if is expanded in terms of a set of counterpart of the bounded power constraint of temporal
orthonormalfunctionsthentheresultingsetofcoefficientsare channel communication theory (see, e.g., [39, Ch. 8]) but for
independent[39,p.367].Thus,addingthisnoisetotheformu- antennasitdoesnotrepresenttheaverageradiatedpowerwhich
lationinSectionII(where becomes )onereadilyfinds is calculated from Poynting’s theorem [55, Eq. (4.21)] and is
thatthechannelmodelin(8)takestherevisedform generallygivenintermsofthemultipolemomentsby[51,Eq.
(4.24)]
(26) (29)
where
(30)
(27)
(31)
and where in-
dicatesvoltsand indicatestheexpectedvalue.
In some situations it may be convenient to modify the where here and henceforth denotes the far-field sin-
noise to allow different variances for different channels so gularvaluesin(24)and indicateswatts.Note,however,that
that . The develop- this constraint alone is not enough to find realistic solutions
ments that follow still apply in this case after the replacement sinceboundingtheradiatedpowerwillnotnecessarilyleadtoa
.Also,forthecaseofcolorednoisewheretheset boundedsource normin(28)(duetothedecayingbehavior
of coefficients have an arbitrary correlation matrix, an ofthesingularvaluessince as ;see,e.g.,[43,
alternative procedure involving the eigendecomposition of the p.41]and[41,p.252]).
correlationmatrixofthenoisecanbeemployedasdescribedin
[20, Ch.9]. C. SpaceCapacityWithMultipleConstraints
Considertheparallelchannelsin(26)where represents
B. Capacity Constraints and Connection to Physical
whiteGaussiannoisewithvariance andwherethe norm
Quantities
ofthesourceisconstrainedto ,i.e.
One of the most important and common constraints for ca-
pacitycalculationsistheboundingofthesquareofthe norm (32)
ofthesourcewhichboundsthecurrentlevelsand,therefore,the
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GRUBERANDMARENGO:NEWASPECTSOFELECTROMAGNETICINFORMATIONTHEORY 3475
where and the radiated power is con- Without loss of generality from this point forward we will in-
strainedto ,i.e. cludethefactor intheLagrangemultiplierssothat
and .
From(37)theoptimalsource normassignmentis
(33)
where (44)
(34)
where . Replacing in (35) yields the
Thecapacityisobtainedfrommaximizingthemutualinfor-
spacecapacitywithconstraintsinthesource normandradi-
mation [20] between the input and the output by
ated power
findingtheoptimalprobabilitydistributionfortheset .
As is well known [19], [20], for uncorrelated Gaussian noise
themutualinformationismaximizedwhentheinputprobability
(45)
distribution isan uncorrelatedGaussian distributionwithvari-
ance .Thus,thecapacityisobtainedbysolvingtheopti-
mizationproblem[56,p.182]
wherethemultipliers and arechosentoagreewithcondi-
tions(38)–(43).Specialcasesandillustrationsofthesegeneral
results willbe given later, butnext we wish tomake a few re-
(35)
marksonfourkeythemesthatareofmuchinteresttous,asboth
applicationandmotivationaspectsofthiseffort,andthatapply
to both the present space-information analysis as well as the
subjectto(32)–(34).
more general broadband-spatio–temporal information analysis
Adualoptimizationproblemassociatedtothiscapacitycal-
of Section III-C-I. These aspects are: 1) information-theoretic
culation can be found by means of the Lagrange multipliers
estimation of NDF of electromagnetic radiation and propaga-
technique [57, p.176] where the Lagrangian function is given
tionchannels;2)antennasuperdirectivitycontrolvia norm
by
constraint;3)near-to-far-fieldtransitionfromthepointofview
of Shannon information; and 4) applicability to both transmit
andreceiveconfigurations(informationalreciprocity).
1) Number of Degrees of Freedom: In this paper, the NDF
is defined as the number of channels with norm in
(44)greaterthanzero.ThevaluesoftheLagrangemultipliers
and thatminimizetheLagrangianin(36)assigntheavailable
source norm totheinfinitechannelsinanoptimalway,i.e.,
dependingonthechannelgainsdefinedbythesingularvalues.
Becauseofthefiniteamountofresources,onlyafinitenumber
(36)
ofchannels,outoftheinfinitenumberoriginallyavailable,are
used.Thisnumber(ofeffectivewavemodesthatcanbecommu-
and the necessary and sufficient conditions for optimality are nicated)dependsontheconstraintlevelofthesource norm
givenbytheKuhn-Tuckertheorem[57,p.138] ,ontheconstraintintheradiatedpower ,onthenoisevari-
ance , and on the behavior of the singular values which in
(37) turn depends on the geometry of the system. Our analysis in
(44) and associated expressions shows that it is not necessary
to artificially truncate the number of channels a priori, as has
(38) been done, however,in the majority of past works in this area
[4],[17],[18],[34],[36],[37],[40].Instead,thefinitenumber
ofchannelsappearsnaturallyasabyproductofmaximizingthe
(39)
capacity.ThisdefinitionoftheNDFdiffersfromthecommonly
used one in linear inverse problems where the NDF is associ-
(40)
ated to the number of “significant” singular values. This last
(41)
definition is particularly appropriate for the case of rank defi-
cient operators where the corresponding singular values have
(42) a step-like behavior with a clear gap between large and small
singularvalues.Forexample,asfollowsfromourdiscussionin
(24),forlargeantennasandobservationregionsinthefarzone
(43) oftheantennatheNDF isrelativelywellde-
finedandindependentofthenoiselevel(see,e.g.,[37],[58]).In
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3476 IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.56,NO.11,NOVEMBER2008
Fig.2. CapacityandNDFasafunctionofthesource(cid:8) normbound(cid:0)fora
fixednoisevariance(cid:9) (cid:2)(cid:6)(cid:6)(cid:6)(cid:7)(cid:8)(cid:10) anddifferentlevelsofthepowerconstraint
(cid:3)forasourcewith(cid:4)(cid:5)(cid:2)(cid:6)(cid:6)(cid:7)(cid:8)andconcentricsphericalreceiverwith(cid:7)(cid:2)(cid:9)(cid:4).
Fig.1. CapacityandNDFasafunctionoftheratio(cid:0)(cid:0)(cid:2)(cid:0)(cid:2)(cid:3)forafixedpower
(cid:3) (cid:2)(cid:3)(cid:4)(cid:5)andfordifferentlevelsoftheSNRforasourcewith(cid:4)(cid:5)(cid:2)(cid:6)(cid:6)(cid:7)(cid:8)
andconcentricsphericalreceiverwith(cid:7)(cid:2)(cid:9)(cid:4).
othercases,however,thesingularvaluesdecaymoregradually
withoutacleargap(e.g.,forthecaseofsmallantennasorob-
servationregionsinthenearzone),makingitnecessarytouse
regularizationtechniqueswhichusuallyinvolvethenoiselevel,
signalenergy,andotherfactors[41].Forexample,XuandHan-
swamy[38]consideredadefinitionoftheNDFdependentofthe
noise levelsand the source norm inthe context ofan elec-
tromagnetic communication system. In the present paper, we
follow this more general philosophy ofnot truncating apriori
theNDF,andinsteadlettheverycalculationofcapacityreveal
theNDFasfunctionofthephysicallymotivatedconstraintsof
powerandsource norm,thenoiselevel,andthesystemge-
ometry(includingthenear-fieldcase).
2) Superdirectivity: Foragivenmaximumallowedradiated
power,constrainingthe normofthesourcehastheeffectof
controllingthelevelofsuperdirectivityorsupergainofthean- Fig.3. Optimal(cid:8) normandradiatedpowerassignmentaswellassingular
values corresponding to each channel for a source with (cid:4)(cid:5) (cid:2) (cid:6)(cid:6)(cid:7)(cid:8),(cid:0) (cid:2)
tenna.Infact,acommonmetricofthesuperdirectivityisgiven
(cid:6)(cid:6)(cid:9)(cid:11)(cid:12) (cid:2)(cid:4),aradiatedpowerconstraint(cid:3) (cid:2)(cid:3)(cid:4)(cid:5),anda(cid:13)(cid:14)(cid:15) (cid:2)(cid:3)(cid:6)(cid:16)(cid:17).
by the geometric Q factor (see, e.g., [31], and similar factors Theconcentricsphericalreceiverwaslocatedat(cid:7)(cid:2)(cid:9)(cid:4).
used in antenna synthesis [59]–[61]) which is proportional to
theratiobetweensource normandradiatedpower.
Forexample,considerasourcewith andaconcen- (forthis variancetheSNR isbetween10and
tricsphericalreceiverwith .Fig.1showsthebehavior 17dBdependingonthevaluesofthepower ).
of the capacity and NDF as a function of the ratio It is interesting to note that the norm assignment is no
for a fixed power and for different levels of the longer as obvious as in the case with only the norm con-
signal-to-noiseratio(SNR)definedas straintwherethechannelswiththelargergains(largersingular
values)arealwaystheoptimaltouse.Duetotheinterplaybe-
tweenpowerandsource normconstraintsthismaynolonger
(46)
be the case. For example, Fig. 3 shows the norm assign-
ment for the system in the previous example with ,
The plot shows that capacity and NDF increase as the ratio , and . In this case, channels with
(thesource norm)increases,asexpected,butalsoaplateau alowergainareassignedmostoftheavailable normwhile
is eventually reached beyond which further increase is negli- thechannelswithhighergainareassignedmostoftheradiated
gible.Thesamegeneralbehaviorisillustratedfromacomple- power.
mentary point of view in Fig. 2, where we consider different 3) Near- to Far-Field Transition: An important factor that
valuesoftheradiatedpowerconstraint andafixedvariance affects the capacity and NDF is the distance between receiver
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GRUBERANDMARENGO:NEWASPECTSOFELECTROMAGNETICINFORMATIONTHEORY 3477
Fig.4. SpacecapacityandNDFforasourceofradius(cid:0) (cid:0) (cid:2)(cid:2)(cid:2)(cid:3)(cid:4)(cid:5)(cid:3)(cid:0) (cid:0)
(cid:2)(cid:2)(cid:6)(cid:7)(cid:8)asafunctionofthereceiverradius(cid:4)forseveralvaluesoftheSNR.Also
Fig.5. SpacecapacityandNDFforasourceofradius0.4m(cid:5)(cid:3)(cid:0)(cid:0)(cid:9)(cid:2)(cid:10)(cid:11)(cid:8)as
shownintheplotarethetworelevantfieldregions(seediscussionintext).
afunctionofthereceiverradius(cid:4)forseveralvaluesoftheSNR.Alsoshownin
theplotarethethreerelevantfieldregions(seediscussionintext).
and source.Thisbehavior arisesfromthe factor of
thesingularvaluesgivenby(19)and(20)whichdependsonthe
where satisfies
receiverradius .TheplotinFig.4shows,forthesamesource
with ,thecapacityandtheNDFcorrespondingasa
functionofthereceiverdistance forseveralvaluesoftheratio
SNR. In the plot the outer boundary of the reactive near-field (48)
region is marked as a dotted line at a distance of from
the surface of the antenna (a usual estimate for small radia-
tors [62]) where is the wavelength of the radiation. Fig. 5 Sincethefar-fieldsingularvalues decreasemorerapidly
shows the same plot for a larger source with
thanthenear-fieldsingularvalues thecapacityin(47)will
wherethethreerelevantfieldregionshavebeenin-
not converge in general. This unbounded behavior of the ca-
dicated.Theouterboundaryofthereactivenear-fieldregionis
pacity due tounconstrained superdirectivity hasbeen reported
takenas andtheouterboundaryoftheradiating
beforein[28]and[31].
near-field region is taken as [62]. For both sources,
Aspecialsituationoccurswhenthereceiverisinthefar-field
theNDFandcapacitydecreaserapidly,convergingtothecorre-
regionwhere sothatthesingularvaluesin(47)and
spondingfar-fieldvaluesasthereceiverdistance approximates
(48)cancel,yieldingacapacity
the outer boundary of the reactive near-field region. In partic-
ular, in both cases the NDF converges to a minimum value as
thereceiverexitsthereactivenear-fieldregion.
4) ReceiveMode: Itisimportanttomentionthattheprevious
developmentsalsoapplyforareceiverantennasurroundedbya
sphericaltransmitterifthemediumissuchthatthereciprocity (49)
condition holds, which is the case for
freespace.Inparticular,itcanbeshownthatforbothsituations
thatisindependentofthesingularvalues andthatincreases
(transmit versus receive) the singular values are the same im-
linearly with the SNR. This comes at the expense of a source
plyingthatthecapacitygivenby(45)alsoremainsunchanged.
with norm
Thisremarkholdsforallthecapacitycalculationsofthispaper.
D. SpaceCapacityWithRadiatedPowerConstraint
(50)
Consider now having an unconstrained source norm,
whichcanbeshowntocorrespondtousing in(36).The
capacityexpressionin(45)thenreducesto
thatisunboundedduetotheexponentiallydecayingbehaviorof
thesingularvaluesforlarge [4],[43],[45],[63].Notethatthe
lastresultin(49)occursbecauseoftheparticulargeometryof
(47) thereceiverandthatthismaynotbethecaseformorearbitrary
configurationsasillustrated,forexample,in[28].
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3478 IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.56,NO.11,NOVEMBER2008
E. SpaceCapacityWithSource NormConstraint
Consider now the complementary situation where only the
source normisdirectlybounded,whichcorrespondsto
.Thecapacityexpressionin(45)reducesto
Fig.6. Channelmodelwithatransmissiontimelimitation.
(51)
same developments for arbitrary , , and , will naturally
render the respective information capacity for bandlimited
electromagnetic systems by means of a limiting procedure
where satisfies
involving , , .
A. Spatio-TemporalAnalysis
(52)
AsinSectionII,foreachfrequency thesource can
be represented in terms of a set of equivalent independent cir-
cuits[33],[34],[66]representingeachofthesphericalmodesin
and can be computed via the well-known procedure called
themultipoleexpansionoftheelectricfieldin(9).Thisspher-
“water-filling”[20]or“water-pouring”[64].
icalexpansionleadstoaninfinitesetofwaveformsoftheform
IV. SPACE-TIMEELECTROMAGNETICINFORMATIONCAPACITY
In this section, we consider the practical scenario of (tem-
porally) bandlimited signals motivated by the requirement of
limiting the bandwidth of transmitted signals (to avoid inter- where , is given by (23),
ferences, follow regulations, and so on). It will be henceforth for inthebandofinterest, isdefinedin
(6),and isdefinedin(7).
assumed that this bandwidth is within the inherent bandwidth
limitationsofpracticalantennasystems.However,weshallnot ApplyingtheinverseFouriertransformsothat,forexample,
dwell on particular antenna matching networks and the Bode- and considering the pres-
Fano matching theory that has already been treated in the an- enceofwhiteGaussiannoiseatthereceiverwefindthemodel
tennacontextin[32],[33],[65].Ourfocusisthefirstprinciples
source-to-field channel within a rather broad context pertinent
towidebandradiating,propagatingandevenscatteringsystems.
Inthissection,thepreviousdevelopmentsfortime-harmonic
(53)
sourcesoperatingatfrequency areextendedtotime-varying
sources. To address this bandlimited electromagnetic channel
whereeachspatialmodel , , representsacontinuous-time
case,wederiveinthefollowingthespace-timegeneralization,
random process (waveform). For simplicity, all random pro-
within the particular electromagnetic physical framework, of
cesses are assumed wide-sense stationary [67] so that the
the classical information theory for temporally bandlimited
autocorrelation of the input signal is
channels developed by Shannon [19] and elaborated in detail
and the autocorrelation of the noise
alsobyGallager[39].Methodologically,theparticularapproach
adopted next borrowsfrom the time domain treatment ofban- signal is
dlimited channels by Gallager [39]. The main ingredients of
. It is also assumed that the noise signals corre-
thetheoryaredevelopedfirstfortheparticularcaseofasource
sponding to different spatial modes are independent so that
that radiates during the finite time period an
. It can
electricfieldthatthereceivermeasuresatalltimes(seeFig.6).
The bandlimitation constraint is modeled by a bandpass filter be shown from (17), (19), (20) and (23) that is a real
of bandwidth at the transmitter, that will be implemented and even function with respect to the frequency, so that the
after the companion time limitation consisting on masking inverseFouriertransformisalsoreal.
the temporaltransmit signal within the window .
B. Space-TimeCapacityWithSource NormConstraint
Laterweconsiderthemoregeneralscenariooftransmitterand
receiversystemsthataretimelimitedsuchthatthetransmitter We will first consider the case of a single constraint in the
operates (is turned on) during a time window source normindetail.Thenwewillshowthecorresponding
of duration , while the receiver operates (captures electro- results for the case of a radiated power constraint and for the
magnetic signals) during a time window . Given that simultaneousconstraining ofthesource norm andradiated
in this theory the transmitter and receiver are both space and power.
time limited (the spatial part is the finite size constraint), this 1) Finite Transmission Window: Consider a source that ra-
is an information theory for space-limited, time-limited, and diatesonlyduringafinitetimeperiod while
essentially bandlimited electromagnetic channels. Finally, the thereceivedsignalismeasuredatalltimes.AsshowninFig.6,
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GRUBERANDMARENGO:NEWASPECTSOFELECTROMAGNETICINFORMATIONTHEORY 3479
thisinputtimelimitationcanbemodeledbydefiningtheindi-
cator function (60)
where , ,
otherwise
and and where
andtheauxiliarytime-variantfilter[39,Sec.8.4] .
The normconstraintofthesource isgivenby
(54)
(61)
The singular functions of the linear mapping associated to the
filter arefoundfromtheintegralequations
Lettingthecapacityforatimeinterval bedenotedby
(55)
andfollowingtheprocedureinSectionIII-Eitisfoundthat
and
(62)
(56)
where
where and are the singular functions,
arethesingularvalues,and
(63)
with selectedsothat
Inviewof(54)
(64)
In general, the singular values are determined by
solvingtheeigenvalueproblemshownin(55).However,anim-
(57) portantspecialcaseoccursif isflatinthefrequencybandof
interest.Inthiscasethefunctions in(56)areessentially
givenbytheprolatespheroidalwavefunctions[9],[39](ascan
Animportantpropertyofthesingularvalues andthe
beshownbyconsideringthelowpassequivalentofthesystem
singularfunctions isthattheyarethesolutionofthe
[68,App.B]sothatthebandofinterestbecomes )
maximizationproblem[39,Eq.(8.4.28)]
whicharethebandlimitedfunctionswiththelargest normin
thetransmissiontimewindow.Furthermore,thesingularvalues
(58)
are essentially a product of two gains: a spatial gain
given by (which depends on the spatial limitation in size
whilethesingularfunctions arethesolutionto
ofthetransmitterandreceiver)andatemporalgainassociated
totheprolatespheroidalwavefunctions(whichisafunctionof
(59) thetime-frequencyproduct [9,p.45]).
The NDF is obtained from the number of channels with
Theorthonormalfunctions arethesetoftime-limited nonzero functional energy in (63) and represents the
functions [which follows from (55) and (57)] whose output to numberofdiscretespatio–temporalmodesthatarerelevantfor
thepropagationchannelhasthelargest norm[followingfrom the current geometry, constraints, and noise level. As an illus-
(58)]and,hence,areessentiallybandlimited.Therefore,theyare trationconsiderasourcewith operatingatacentral
theidealexpansionfunctionsfortheinputsignals .On frequency andanarrowbandwidthsuchthatthe
theotherhand,theorthonormalfunctions arethesetof quantities areessentiallyconstant(plotnotshown)andthe
bandlimitedfunctions[followingfrom(56)]withthelargest previousapproximationisapplicable.Fig.7showsthecapacity
normconcentratedinthetimeinterval [following in (62) and the NDF as a function of the SNR parameterized
from(59)].Thus,thereceivedsignal willberepresented by the parameter . As expected, increasing
bymeansofthissetoffunctions. the time-bandwidth product increases the number of
Theexpansionintermsofthesingularfunctions and spatio–temporaldegreesoffreedomandthecapacity.
yieldsthediscreteindependentparallelGaussianchan- Theseresultscanalsoshedlightontheneartofarfieldtransi-
nels tionfortime-domainfields.Forexample,considertheprevious
sourcewith ,anda .Fig.8shows
the capacity and NDF as a function of the receiver distance
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