CapacityofMultiple-AntennaFadingChannels:SpatialFadingCorrelation,DoubleScattering,
andKeyhole
HyundongShin,StudentMember,IEEE,andJaeHongLee,Member,IEEE
Abstract—Thecapacityofmultiple-inputmultiple-output(MIMO)wirelesschannelsislimitedbyboththespatialfadingcorrelationandrankdeficiencyofthechannel.Whilespatialfadingcorrelationreducesthediversitygains,rankdeficiencyduetodoublescatteringorkeyholeeffectsdecreasesthespatialmultiplexinggainsofmultiple-antennachannels.Inthispaper,takingintoaccountrealisticpropagationenvironmentsinthepresenceofspatialfadingcorrelation,doublescattering,andkeyholeeffects,weanalyzetheergodic(ormean)MIMOcapacityforanarbitraryfinitenumberoftransmitandreceiveantennas.Weassumethatthechannelisunknownatthetransmitterandperfectlyknownatthereceiversothatequalpowerisallocatedtoeachofthetransmitantennas.Usingsomestatisticalprop-ertiesofcomplexrandommatricessuchasGaussianmatrices,Wishartmatrices,andquadraticformsintheGaussianmatrix,wepresentaclosed-formexpressionfortheergodiccapacityofindependentRayleigh-fadingMIMOchannelsandatightupperboundforspatiallycorrelated/doublescatteringMIMOchannels.Wealsoderiveaclosed-formcapacityformulaforkeyholeMIMOchannels.Thisanalyticformulaexplicitlyshowsthattheuseofmultipleantennasinkeyholechannelsonlyoffersthediversityadvantage,butprovidesnospatialmultiplexinggains.Numericalresultsdemonstratetheaccuracyofouranalyticalexpressionsandthetightnessofupperbounds.
IndexTerms—Channelcapacity,distributionsofrandomma-trices,doublescattering,keyhole,multiple-inputmultiple-output(MIMO)systems,multipleantennas,spatialfadingcorrelation.
I.INTRODUCTION
ULTIPLE-inputmultiple-output(MIMO)communica-tionsystemsusingmultiple-antennaarraysatboththetransmitterandthereceiverhavedrawnconsiderableattentioninresponsetotheincreasingrequirementsonhighspectralefficiencyandreliabilityinwirelesscommunications[1]–[10].Recentseminalworkin[1]and[2]hasshownthattheuseofmultipleantennasatbothendssignificantlyincreasesthein-formation-theoreticcapacityfarbeyondthatofsingle-antennasystemsinrichscatteringpropagationenvironments.Asthenumberofantennasatboththetransmitterandthereceivergetslarger,thecapacityincreaseslinearlywiththeminimumofthenumberoftransmitandreceiveantennasforfixedpowerandbandwidth,assumingindependentandidenticallydistributed(i.i.d.)Rayleighfadingbetweenantennapairs[1]–[3].
ManuscriptreceivedOctober29,2002;revisedMay4,2003.ThisworkwassupportedinpartbytheNationalResearchLaboratory(NRL)ProgramandtheBrainKorea21Project,Korea.
TheauthorsarewiththeSchoolofElectricalEngineering,SeoulNationalUniversity,Seoul151-742,Korea(e-mail:shd71@snu.ac.kr).CommunicatedbyT.L.Marzetta,GuestEditor.DigitalObjectIdentifier10.1109/TIT.2003.817439
M
AnimportantopenprobleminMIMOcommunicationtheoryistoobtainclosed-formanalyticformulasforthecapacityormutualinformationofwirelessMIMOchannels.However,itisamathematicallychallengingtaskinthatcalculationsoftheMIMOcapacityrequiretakingexpectationswithrespecttoarandomchannelmatrixratherthanascalarrandomvariable(RV)forthesingle-antennacase.Inrandommatrixtheory[28]–[31],itiswellknownthattheeigenvaluesofalargeclassofrandommatrixensembleshavefewerrandomfluctuationsasthematrixdimensiongetslarger—thatis,therandomdistributionofeigenvaluesconvergestoadeterministiclimitingdistributionforalargematrixsize.Anotherusefulresultoftherandommatrixtheoryisacentrallimittheoremforrandomdeterminants[29],[32],whichstatesthatthedistributionoftherandomMIMOcapacityisasymptoticallyGaussianasthenumberofantennastendstoinfinitywithacertainlimitingratiobetweenthenumbersoftransmitandreceiveantennas.Theseresultsoftherandommatrixtheorywereappliedforthecaseofuncorrelatedchannelsin[1]and[10]–[13],andforspatiallycorrelatedchannelsin[17]and[18].Althoughthisasymptoticanalysisisonlyanapproximationtothecaseoffinitematrixsize,itcircumventsthedifficultprobleminanalyticalcalculationoftheMIMOcapacityandprovidesimportantinsightsintoimpactsoftheuseofmultipleantennasonthecapacitybehavior.
Forthefinitenumberoftransmitandreceiveantennas,Telatar[1]derivedtheanalyticalexpressionfortheergodic(ormean)capacityofi.i.d.Rayleighflat-fadingMIMOchannelsbyusingtheeigenvaluedistributionoftheWishartmatrixinintegralforminvolvingtheLaguerrepolynomials.In[12],SmithandShafifurtherderivedthevarianceofcapacitybyextendingtheanal-ysisin[1]andobtainedthecomplementarycumulativedistribu-tionfunctionofthecapacityusingtheGaussianapproximationtorandomMIMOcapacity.Similarresultsarealsofoundin[13]inwhichthedensityfunctionofarandommutualinformationfori.i.d.MIMOchannelswasderivedintheformoftheinverseLaplacetransformandthesameGaussianapproximationresultasin[12]waspresented.
Inrealisticpropagationenvironments,rankdeficiencyofachannelmatrixduetopinholeorkeyholeeffects[19]–[21]mayseverelydegradethecapacityofMIMOchannelsaswellasspa-tialfadingcorrelation[14]–[18].Whilespatialcorrelationre-ducesdiversityadvantages,rankdeficiencyofthechannelde-creasesaspatialmultiplexingability,i.e.,theslopeofacapacitycurveoverasignal-to-noiseratio(SNR).Recently,Gesbertetal.[19]introducedadoublescatteringMIMOchannelmodel
0018-9448/03$17.00©2003IEEE
SHINANDLEE:CAPACITYOFMULTIPLE-ANTENNAFADINGCHANNELSthatincludesboththefadingcorrelationandrankdeficiency,andpointedouttheexistenceofpinholechannelsthatexhibituncor-relatedspatialfadingbetweenantennasbutstillhaveapoorrankproperty.Also,in[20]and[21],theoccurrenceofarank-defi-cientchannel,calledakeyholechannel,hasbeenproposedanddemonstratedthroughphysicalexamples.Inthepresenceofthekeyhole,thechannelhasonlyasingledegreeoffreedomal-thoughthespatialfadingisuncorrelated,andeachentryofthechannelmatrixisaproductoftwocomplexGaussianRVs,incontrasttothecomplexGaussiannormallyassumedinwirelesschannels.Infact,keyholechannelsmaybeviewedasaspe-cialcaseofdoublescatteringMIMOchannels.Thesedegen-eratechannelssignificantlydeviatefromtheidealisticcapacitybehaviorofi.i.d.channelsandareofinterestbecauseofrecentvalidationthroughphysicalmeasurements.
Inthispaper,takingintoaccountrealisticpropagationenvi-ronmentssuchasspatialfadingcorrelation,doublescattering,andkeyholephenomena,weprovideanalyticalexpressionsfortheergodiccapacityofMIMOchannelswithfinitetransmitandreceiveantennas.Inparticular,wederiveaclosed-formex-pressionforthecapacityofi.i.d.Rayleigh-fadingMIMOchan-nels.IncontrasttoTelatar’sintegralexpression,thiscapacityformulaisintermsofafinitesumofthewell-knownspecialfunctions(exponentialintegralfunctions,orincompletegammafunctions)andcanbecalculatedwithoutexplicitnumericalin-tegration.ForspatiallycorrelatedanddoublescatteringMIMOchannels,wedeveloptightupperboundsonthecapacitybyusingJensen’sinequalityandelementarypropertiesofdeter-minants,suchastheprincipalminordeterminantexpansionforthecharacteristicpolynomialofamatrixandtheBinet–Cauchyformulaforthedeterminantofaproductmatrix.1Finally,forkeyholeMIMOchannelsweprovideaclosed-formsolutionforthecapacityandshowthatincreasingthenumberofantennasservesonlytoeliminatetheeffectoffading,butprovidesnofurtherbenefits(e.g.,spatialmultiplexinggains).
Therestofthepaperisorganizedasfollows.SectionIIgives,forreference,thedefinitionsofthecomplexGaussianmatrix,Wishartmatrix,andthepositive-definitequadraticforminthecomplexGaussianmatrix,andpresentssomenewresultsconcerningexpectationsofcertain(logarithmic)determinantalformsofthemwithafinitematrixsize.UsingsomeoftheresultsinSectionII,wederiveaclosed-formexpressionoftheergodicMIMOcapacityforthei.i.d.caseandupperboundsforspatiallycorrelatedanddoublescatteringcasesinSectionIII.We,finally,deriveaclosed-formcapacityformulaforkeyholeMIMOchannelsinSectionIII-D.SectionIVconcludesthepaper.
Weshallusethefollowingnotationsinthispaper.Thesu-perscripts
,andwedenotethat
ispositivedefinite.
1ThisapproachwasfirstintroducedbyGrantin[10]toobtaintheupperbound
ontheergodicMIMOcapacityforthei.i.d.Rayleigh-fadingcase.
2637
II.REVIEWANDSOMERESULTSONRANDOMMATRICESMuchattentionhasbeengivenovertheyearstothedistri-butiontheoryofrandommatricesbecausetheyappearinmany
applicationsinstatisticsandcommunicationtheory.Thedistri-butionofthecovariancematrixofsamplesfromamultivariateGaussiandistribution,whichisknownastheWishartdistri-bution,wasperhapsthebeginningofatheoryofdistributionsofrandommatrices[36],[28].Inthissection,byfocusingonthecomplexcases,webrieflyreviewthedefinitionsanddistri-butionsofGaussianmatrices,Wishartmatrices,andpositive-definitequadraticformsintheGaussianmatrix—whichgener-alizetheunivariateGaussianRV,centralchi-squareRV,andcen-tralpositive-definitequadraticformintheGaussianRV,respec-tively—andderivesomenewresultsonthem,whichareusedtocalculatetheergodiccapacityofMIMOchannelsinthenextsection.
Inderivingthestatisticsofacertainrandommatrix,theJaco-biansofmatrixtransformationsareneededandfunctionsofamatrixargumentarealsowidelyusedincalculationsinvolvingmatrix-variatedistributions.Theintroductionstothemareprovidedin[28],[33],[34],[37],[38],and[40].Inparticular,[33]hasdealtwithawideclassofmatrix-variatedistributions.Althoughthistextbookconcentratesonmatricesofrealrandomvariates,onecaneasilydevelopthecorrespondingcomplexcases.
A.ComplexGaussianMatricesLetusdenotethecomplex
issaidtohaveamatrix-variateGaussiandistribution
withmeanmatrixwhere
(1)
Inthefollowinglemma,wegiveapreliminaryresultonthecomplexGaussianmatrix.LemmaII.1:If
,thenwehavefor
and
otherwise
(2)
where
,
2638IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.49,NO.10,OCTOBER2003
and
isaminordeterminantof,i.e.,ade-terminantofthematrixlyinginthe
otherwise
where
,andthepermutationsand
havethesameorder—otherwise,wealwaysgettermsmul-tipliedbyotherindependentzero-meanGaussianentries—andthecorrespondingexpectedvaluesareequaltoonebecauseall
entriesof
areindependentwithunitvariance.ThereareB.ComplexWishartMatrices
Wishartdistributions,firstobtainedbyFisher[35]inthebi-variatecaseandgeneralizedbyWishart[36]usingageometrical
argument,areofgreatinterestinmultivariatestatisticalanalysis,arisingnaturallyinappliedresearchandasabasisfortheoret-icalmodels(see[28],[30],[33]andreferencestherein).DefinitionII.2([33,Definition3.2.1]):ArandomHermi-tianpositive-definitematrix
,
,,and
and
[45].
Proof:SeeAppendixA.
(5)
where
isthehypergeometricfunctionoftwoHer-mitianmatricesdefinedby(51).Notethatif
SHINANDLEE:CAPACITYOFMULTIPLE-ANTENNAFADINGCHANNELS2639
TheoremII.2(MomentsofGeneralizedVariance):If
thmomentofthegeneralizedvari-ance
wherethe
,
-rowedprincipalminordeterminantofas
istheGausshypergeo-metricfunctionofaHermitianmatrixdefinedby(50).
,thenInparticular,if
(10)
whereif
or
,thenthe
,then
TheoremII.3:If
positivereal-valuedconstant,thenwehave
isanarbitrary
Weremarkthatif
:
(9)
2640IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.49,NO.10,OCTOBER2003
boundontheergodiccapacityofMIMOchannelsinthenextsection.
III.CAPACITYOFMIMOWIRELESSCHANNELSWeconsiderapoint-to-pointcommunicationlinkwith
and
regardlessofthenumberof
antennas,i.e.,
(14)
where
isacomplex,
of
arethecomplexchannelgainsbetween
transmitantenna
sta-tisticallyindependentequalpowercomponentseachwithacir-cularlysymmetriccomplexGaussiandistribution,thechannelcapacityundertransmitpowerconstraint
(bits/s/Hz)
withrespecttothe
randomchannelmatrix
,i.e.,[1]
.Thefollowingresultgivesaclosed-form
formulafortheergodiccapacityofi.i.d.Rayleigh-fadingMIMOchannels.
2The
capacityforfadingchannelscanbedefinedinanumberofways,de-pendingontheamountofchannelknowledge,delayconstraints,signalingcon-straints,andstatisticalnatureofthechannel.Thevariouscapacitymeasuresforfadingchannelscanbefoundin[23].
TheoremIII.1:If
,i.e.,forani.i.d.
Rayleigh-fadingMIMOchannelwith
andequalpowerallocationisgivenby
,then
.
From(16)andTheoremII.1,wehavethetheorem.
forthei.i.d.
Rayleigh-fadingcasewasfirstderivedbyTelatar[1]insingleintegralforminvolvingtheLaguerrepolynomials.Incontrast,TheoremIII.1providesaclosed-formexpression
for
intermsoftheexponentialintegralfunctions(orincompletegammafunctions).Moreover,itgeneralizesthepreviouslyknownresultofclosed-formcapacityformulasforRayleigh-fadingchannelswithreceptiondiversity[22]toMIMOcases.
Example1:Consider
.From(18)with
and
antennasatboththetransmitterandthereceiverisgivenby
transmitantennasisgivenby
receiveantennasisgivenby
SHINANDLEE:CAPACITYOFMULTIPLE-ANTENNAFADINGCHANNELSFig.1.Ergodiccapacityofi.i.d.Rayleigh-fadingMIMOchannelswithttransmitandrreceiveantennas.
Furthermore,using[10,LemmaA.1],wehaveathighSNR
—thatis,thecapacityincreases
;
b)
,
;c)
,
;d)
,
andwhentheSNRandtheratiobetween
.Forexample,thecapacity
ofthechannelwith
.
B.SpatiallyCorrelatedMIMOChannels
WeconsidercorrelatedRayleigh-fadingMIMOchannelswiththecorrelationstructureofaproductform[17],i.e.,
(24)
where
,andaretransmitandreceivecorrelationmatrices,respectively.From(1)andmakingthetransformationfrom
2641
itiseasytoseethat.Thefollowingresultgivesanupperboundontheergodiccapacityofsuchachannel.
TheoremIII.2:If
,i.e.,foracor-relatedRayleigh-fadingMIMOchannelwith
Notethattheupperbound(25)isthelogarithmofapolyno-mialofdegree
th-ordercoefficientofthepoly-nomialdependsonlyonsumsofall
forevencorrelatedchannels.Inparticular,if,using(7),
wehaveathighSNR
th-order
termin(25).From(23)and(26),wecanseethattheca-pacityreductionduetothespatialfadingcorrelationis
bits/s/HzathighSNR.Example4(ConstantCorrelationModel):Acorrela-
tionmatrixiscalledthe
...
...
.....
.
...
Thiscorrelationmodelmayapproximatecloselyspacedan-tennasandmaybeusedfortheworstcaseanalysisorforsome
roughapproximationsusingtheaveragevalueofcorrelationcoefficientsforalloff-diagonalentriesofthecorrelationmatrix.
Sinceeigenvaluesof
andmultiplicities,itsdeterminantcanbewrittenas
(27)
2642IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.49,NO.10,OCTOBER2003
Fig.2.ErgodiccapacityofspatiallycorrelatedRayleigh-fadingMIMOchannelswithttransmitandrreceiveantennas.Thetransmitandreceivecorrelationsareconstantcorrelationswithcorrelationcoefficients
=0:7,i.e.,8=2
(28)
(25)(28)(29)(a)20Example5(JakesModel[15],[27]):Ingeneral,thefadingcorrelationdependsonboththeantennaspacingandthean-gularspectrumoftheincomingradiowave.Ifweemploya
Fig.3.Ergodiccapacityasafunctionofcorrelationcoefficientforspatially
correlatedRayleigh-fadingMIMOchannelswithttransmitandrreceive
antennas.8
=2and8
dB.
((i0j)12d=)and=20
lineararraywithequallyspacedantennasandtheclassicalJakescorrelationmodelwiththeuniformangularspectrum[27],the
isthewavelength,and
SHINANDLEE:CAPACITYOFMULTIPLE-ANTENNAFADINGCHANNELS2643
of20dBwhen
(30)
where
,
,
fl
fl=fl
fl=fl
=
,and
,thenusing(28)andThe-oremIII.3,wehaveanupperboundontheergodiccapacityasshownin(32)atthebottomofthepage.
Fig.5showsthesimulationresultsandupperboundsfortheergodiccapacityofdoublescatteringMIMOchannelswith
,,,and.Theupperboundisplottedusing(32).ThisexampleservestodemonstratetheeffectofrankdeficiencyofthechannelontheMIMOcapacitybehavior.Wecanseethattheasymptoticslopes
,ofcapacitycurvesfor
,because
inallthreecases.Inotherwords,sincethe
rankof
and
,
SimilartocorrelatedMIMOchannels,theupperbound(31)
inisthelogarithmofapolynomialofdegree
,(33)reducesto
(34)
2644IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.49,NO.10,OCTOBER2003
andif
(35)
(33)becomes(22).
D.KeyholeMIMOChannels[20],[21]
IncertainMIMOpropagationenvironments,adegeneratechannelwithonlyasingledegreeoffreedom(i.e.,one-rankchannelmatrix)mayariseduetothekeyholeeffect[20],[21].Insuchachannel,theonlywayfortheradiowavetopropagatefromthetransmittertothereceiveristopassthroughthe
keyhole,andeachentryof
isaproductoftwoindependentcomplexGaussianRVsratherthanthecomplexGaussian.
Then,thechannelmatrix
forkeyholeMIMOchannelsisgivenby[20],[21]
(36)
whereand
.Sinceisofonerank,weneednot
usetherandommatrixresultsinanalysisforkeyholechannels.Thefollowingtheoremprovidesaclosed-formsolutionfortheergodiccapacityofkeyholeMIMOchannels.TheoremIII.4:If
and
(37)
where
istheEuler’sdigammafunction
[45,eq.(8.360.1)]and
Weremarkthatas,thelasttermin(37)vanishesandthecapacitybecomesasymptotically
(38)
whichshowsthattheuseofmultipleantennasinkeyholechan-nelscannotprovidethespatialmultiplexinggainandonlyof-3The
Meijer’sG-functionisprovidedasthebuilt-infunctionincommon
mathematicalsoftwarepackagessuchasMAPLEandMATHEMATICA.
Fig.6.ErgodiccapacityofkeyholeMIMOchannelswithttransmitandrreceiveantennas.
fersthediversitygaindeterminedby
whichagreeswith(34).
Fig.6showstheergodiccapacityofkeyholeMIMOchannelswhen
and
and
th-ordercoefficientdependsonlyonsumsofall
SHINANDLEE:CAPACITYOFMULTIPLE-ANTENNAFADINGCHANNELSAPPENDIXA
PROOFOFTHEOREMII.1
TheproofofTheoremII.1requiresthefollowingresultontheeigenvaluedensityofWishartmatrices.LemmaA.1(Bronk[39]):If
,thentheden-
sityfunctionofanunorderedeigenvalue
definedas
[45,eq.(8.970.1)]
in(39)canberewrittenas
2645
(47)
Substituting(47)into(43),wecompletetheproofofthetheorem.
isthePochhammersymbol,
suchthat
(50)
isthezonal
polynomialofaHermitianmatrix[38,eq.(85)].Fromthedensityof
2646IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.49,NO.10,OCTOBER2003
where
,using
fora
Hermitianmatrix
APPENDIXC
PROOFOFTHEOREMIII.4
Notethat
,,and.Since
independentexponentialRVs,respectively,
theyarecentralchi-squaredistributedwith
is,therefore,givenby[24]
isthe
thmomentof
(64)
wherethelastequalityfollowsfrom[45,eq.(9.31.2)].Substi-tuting(61)and(64)into(60)givestheresult(37).
SHINANDLEE:CAPACITYOFMULTIPLE-ANTENNAFADINGCHANNELS[8]B.L.Hughes,“Differentialspace-timemodulation,”IEEETrans.In-form.Theory,vol.46,pp.2567–2578,Nov.2000.
[9]B.M.Hochwald,T.L.Marzetta,andB.Hassibi,“Space-timeau-tocoding,”IEEETrans.Inform.Theory,vol.47,pp.2761–2781,Nov.2001.
[10]A.Grant,“Rayleighfadingmultiple-antennachannels,”EURASIPJ.
Appl.SignalProcessing(SpecialIssueonSpace-TimeCoding(PartI)),vol.2002,no.3,pp.316–329,Mar.2002.
[11]S.VerdúandS.Shamai(Shitz),“SpectralefficiencyofCDMAwith
randomspreading,”IEEETrans.Inform.Theory,vol.45,pp.622–640,Mar.1999.
[12]P.J.SmithandM.Shafi,“OnaGaussianapproximationtothecapacity
ofwirelessMIMOsystems,”inProc.IEEEInt.Conf.Communications(ICC’02),NewYork,Apr.2002,pp.406–410.
[13]Z.WangandG.B.Giannakis,“Outagemutualinformationofspace-time
MIMOchannels,”inProc.40thAllertonConf.Communication,Control,andComputing,Monticello,IL,Oct.2002,pp.885–894.
[14]D.-S.Shiu,G.J.Foschini,M.J.Gans,andJ.M.Kahn,“Fadingcorre-lationanditseffectonthecapacityofmultielementantennasystems,”IEEETrans.Commun.,vol.48,pp.502–513,Mar.2000.
[15]D.Chizhik,F.Rashid-Farrokhi,J.Ling,andA.Lozano,“Effectofan-tennaseparationonthecapacityofBLASTincorrelatedchannels,”IEEECommun.Lett.,vol.4,pp.337–339,Nov.2000.
[16]S.L.LoykaandA.Kouki,“NewcompoundupperboundonMIMO
channelcapacity,”IEEECommun.Lett.,vol.6,pp.96–98,Mar.2002.[17]C.-N.Chuah,D.N.C.Tse,J.M.Kahn,andR.A.Valenzuela,“Capacity
scalinginMIMOwirelesssystemsundercorrelatedfading,”IEEETrans.Inform.Theory,vol.48,pp.637–650,Mar.2002.
[18]A.L.Moustakas,S.H.Simon,andA.M.Sengupta,“MIMOcapacity
throughcorrelatedchannelsinthepresenceofcorrelatedinterferersandnoise:A(notso)largeNanalysis,”preprint,May2002.
[19]D.Gesbert,H.Bölcskei,D.A.Gore,andA.J.Paulraj,“OutdoorMIMO
wirelesschannels:Modelsandperformanceprediction,”IEEETrans.Commun.,vol.50,pp.1926–1934,Dec.2002.
[20]D.Chizhik,G.J.Foschini,andR.A.Valenzuela,“Capacitiesofmulti-el-ementtransmitandreceiveantennas:Correlationsandkeyholes,”Elec-tron.Lett.,vol.36,no.13,pp.1099–1100,June2000.
[21]D.Chizhik,G.J.Foschini,M.J.Gans,andR.A.Valenzuela,“Key-holes,correlations,andcapacitiesofmultielementtransmitandreceiveantennas,”IEEETrans.WirelessCommun.,vol.1,pp.361–368,Apr.2002.
[22]M.-S.AlouiniandA.J.Goldsmith,“CapacityofRayleighfadingchan-nelsunderdifferentadaptivetransmissionanddiversity-combiningtech-niques,”IEEETrans.Veh.Technol.,vol.48,pp.1165–1181,July1999.[23]E.Biglieri,J.Proakis,andS.Shamai(Shitz),“Fadingchannels:In-formation-theoreticandcommunicationsaspects,”IEEETrans.Inform.Theory,vol.44,pp.2619–2692,Oct.1998.
[24]H.ShinandJ.H.Lee,“Effectofkeyholesonthesymbolerrorrateof
space-timeblockcodes,”IEEECommun.Lett.,vol.7,pp.27–29,Jan.2003.
[25]E.T.Browne,IntroductiontotheTheoryofDeterminantsandMa-trices.ChapelHill,NC:Univ.NorthCarolinaPress,1958.
2647
[26]P.LancasterandM.Tismenetsky,TheTheoryofMatrices,2nded.San
Diego,CA:Academic,1985.
[27]W.C.Jakes,Jr.,MicrowaveMobileCommunications.NewYork:
Wiley,1974.
[28]R.J.Muirhead,AspectsofMultivariateStatisticalTheory.NewYork:
Wiley,1982.
[29]V.L.Girko,TheoryofRandomDeterminants.Dordrecht,TheNether-lands:Kluwer,1990.
[30]A.Edelman,“Eigenvaluesandconditionnumbersofrandommatrices,”
Ph.D.dissertation,MIT,Cambridge,MA,May1989.
[31]J.W.Silverstein,“Strongconvergenceoftheempiricaldistributionof
eigenvaluesoflargedimensionalrandommatrices,”J.MultivariateAnal.,vol.55,pp.331–339,1995.
[32]V.L.Girko,“Arefinementofthecentrallimittheoremforrandomdeter-minants,”TheoryProbab.ItsApplic.,vol.42,no.1,pp.121–129,1998.[33]A.K.GuptaandD.K.Nagar,MatrixVariateDistributions.Boca
Raton,FL:Chapman&Hall/CRC,2000.
[34]A.M.Mathai,JacobiansofMatrixTransformationsandFunctionsof
MatrixArguments.Singapore:WorldScientific,1997.
[35]R.A.Fisher,“Frequencydistributionofthevaluesofthecorrelation
coefficientinsamplesfromindefinitelylargepopulation,”Biometrika,vol.10,pp.507–521,1915.
[36]J.Wishart,“Thegeneralizedproductmomentdistributioninsamples
fromanormalmultivariatepopulation,”Biometrika,vol.A20,pp.32–52,1928.
[37]N.R.Goodman,“Statisticalanalysisbasedonacertainmultivariate
complexGaussiandistribution(anintroduction),”Ann.Math.Statist.,vol.34,pp.152–177,1963.
[38]A.T.James,“Distributionsofmatrixvariatesandlatentrootsderived
fromnormalsamples,”Ann.Math.Statist.,vol.35,pp.475–501,1964.[39]B.V.Bronk,“Exponentialensemblesforrandommatrices,”J.Math.
Phys.,vol.6,pp.228–237,1965.
[40]C.G.Khatri,“Oncertaindistributionproblemsbasedonpositivedefi-nitequadraticfunctionsinnormalvectors,”Ann.Math.Statist.,vol.37,pp.468–479,1966.
[41]B.K.Shah,“Distributiontheoryofapositivedefinitequadraticform
withmatrixargument,”Ann.Math.Statist.,vol.41,pp.692–697,1970.[42]C.G.Khatri,“Seriesrepresentationsofdistributionsofquadraticform
inthenormalvectorsandgeneralizedvariance,”J.MultivariateAnal.,vol.1,pp.199–214,1971.
[43]H.GaoandP.J.Smith,“Adeterminantrepresentationforthedistribution
ofquadraticformsincomplexnormalvectors,”J.MultivariateAnal.,vol.73,pp.155–165,2000.
[44]I.Olkin,“The70thanniversaryofthedistributionofrandommatrices:
Asurvey,”LinearAlgebraAppl.,vol.354,pp.231–243,Oct.2002.[45]I.S.GradshteynandI.M.Ryzhik,TableofIntegrals,Series,andProd-ucts,6thed.SanDiego,CA:Academic,2000.
[46]A.P.Prudnikov,Y.A.Brychkov,andO.I.Marichev,IntegralsandSe-ries.NewYork:GordonandBreach,1990,vol.3.
因篇幅问题不能全部显示，请点此查看更多更全内容