GAUSSIAN RANDOM VECTORS AND PROCESSES

1 , , ,wellknown, ,theseassumptionsareoftenapproximatelysa tisfied,sotheresults,ifusedwithinsightan dcare, ,butstartswithastudyofGaussian(normal1)r andomvariablesandvectors, (rv)WisdefinedtobeanormalizedGaussianrvi fithasthedensityfW(w)=1p2 exp w22 ;forallw2R.( )1 Gaussianrv sareoftencallednormalrv ,firstbecausethecorrespondingprocessesar eusuallycalledGaussian,secondbecauseGaus sianrv s(whichhavearbitrarymeansandvariances)ar eoftennormalizedtozeromeanandunitvarianc e,andthird,becausecallingthemnormalgives thefalseimpressionthatotherrv (w)integratesto1( ,itisaprobabilitydensity), , ,ifweconsiderthervZ= W,thenthedistributionfunctionsofZandWare relatedbyFZ( w)=FW(w).

2 Thismeansthattheprobabilitydensitiesarer elatedby fZ( w)=fW(w).ThusthePDFofZisgivenbyfZ(z)=1 fW z =1p2 exp z22 2 .( )ThusthePDFforZisscaledhorizontallybythe factor ,andthenscaledverticallyby1/ ( ).Thisscalingleavestheintegralofthedensi tyunchangedwithvalue1andscalesthevarianc eby approach0,thisdensityapproachesanimpulse , ,ZbecomestheatomicrvforwhichPr{Z=0}= ,weuse( )asthedensityforZforall 0,withtheaboveunderstandingaboutthe = ( ),forany 0, {|Z|> }=.318,Pr{|Z|>3 }=.0027,andPr{|Z|>5 }= 10 (w)fZ(w) :GraphofthePDFofanormalizedGaussianrvW(t hetallercurve)andofazero-meanGaussianrvZ withstandarddeviation2(theflattercurve).

3 IfweshiftZbyanarbitrary 2 RtoU=Z+ ,thenthedensityshiftssoastobecenteredatE [U]= ,andthedensitysatisfiesfU(u)=fZ(u ).ThusfU(u)=1p2 exp (u )22 2 .( )ArandomvariableUwiththisdensity,forarbi trary and 0,isdefinedtobeaGaussianrandomvariablean disdenotedU N( , 2).Theaddedgeneralityofameanoftenobscure sformulas;weusuallyassumezero-meanrv sandrandomvectors(rv s) canberegardedasaconstant plusthefluctuation,U , ,gZ(r),ofaGaussianrvZ N(0, 2), :gZ(r)=E[exp(rZ)]=1p2 Z1 1exp(rz)exp z22 2 dz=1p2 Z1 1exp z2+2 2rz r2 42 2+r2 22 dz( )=exp r2 22 1p2 Z1 1exp (z r )22 2 dz ( )=exp r2 22.

4 ( )Wecompletedthesquareintheexponentin( ).Wethenrecognizedthattheterminbracesin( ) (r)existsforallrealr,althoughitincreases rapidlywith|r|. ,themomentsforZ N(0, 2),canbecalculatedfromtheMGFtobeEhZ2ki=( 2k)! 2kk!2k=(2k 1)(2k 3)(2k 5)..(3)(1) 2k.( )Thus,E Z4 =3 4,E Z6 =15 6, + N( , 2),letZ=U ,ThenZ N(0, 2)andgU(r)isgivenbygU(r)=E[exp(r( +Z))]=er E erZ =exp(r +r2 2/2).( )Thecharacteristicfunction,gZ(i )=E ei Z forZ N(0, 2)andi imaginarycanbeshowntobe( , [27]).gZ(i )=exp 2 22 ,( )Theargumentin( )to( )doesnotshowthissincetheterminbracesin( ) ,thecharacteristicfunctionisusefulfirstb ecauseitexistsforallrv sandsecondbecauseaninversionformula(esse ntiallytheFouriertransform) `matrix[A]isanarrayofn`elementsarrangedi nnrowsand`columns; , [AT]ofann `matrix[A]isan` nmatrix[B]withBkj= , `andasquarematrix[A]issymmetricif[A]=[A] [A]and[B]areeachn `matrices,[A]+[B]isann `matrix[C]withCjk=Ajk+Bjkforallj, [A]isn `and[B]is` r,thematrix[A][B]isann rmatrix[C]withelementsCjk= (orcolumnvector) ,wedenoteavectoraas(a1.)

5 ,an) (column)vectorofdimensionn,thenaaTisann ,[K](ifitexists)ofanarbitraryzero-meann- rvZ=(Z1,..,Zn)Tisthematrixwhosecomponent sareKjk=E[ZjZk].Foranon-zero-meann-rvU,l etU=m+Zwherem=E[U]andZ=U [K]ofUisdefinedtobethesameasthecovarianc ematrixofthefluctuationZ, ,Kjk=E[ZjZk]=E[(Uj mj)(Uk mk)].Itcanbeseenthatifann ncovariancematrix[K]exists,itmustbesymme tric, ,itmustsatisfyKjk=Kkjfor1 j,k (MGF)ofann-rvZisdefinedasgZ(r)=E[exp(rTZ )]wherer=(r1,..,rn) ( ).Aswewillsoonsee,however,theMGFexistsev erywhereforGaussiann-rv ,gZ(i )=Ehei TZi,ofann-rvZ,where =( 1,.., n)Tisarealn-vector, ,thecharacteristicfunctionalwaysexistsfo rallreal , (IID), ,1 j n,isnormalizedGaussian,Wj N(0,1).

6 Bytakingtheproductofndensitiesasgivenin( ),thejointdensityofW=(W1,W2,..,Wn)TisfW( w)=1(2 )n/2exp w21 w22 w2n2 =1(2 )n/2exp wTw2 .( )2 SeeShiryaev,[27], , ,fW(w)issphericallysymmetricaroundtheori gin,andpointsofequalprobabilitydensityli eonconcentricspheresaroundtheorigin( ).&%'$ :gW(r)=E[exprTW)]=E[exp(r1W1+ +rnWn]=E24 Yjexp(rjWj)35=YjE[exp(rjWj)]=Yjexp r2j2!=exp rTr2 .( )Theinterchangeoftheexpectationwiththepr oductaboveisjustifiedbecause,first,therv sWj(andthustherv sexp(rjWj))areindependent,and,second,the expectationofaproductofindependentrv ( ).ThecharacteristicfunctionofWissimilarl ycalculatedusing( ),gW(i )=exp T 2 ,( )Nextconsiderrv sthatarelinearcombinationsofW1.

7 ,Wn, ,rv softheformZ=aTW=a1W1+ + , ,Z N(0, 2)where 2=Pnj=1a2j, ,Z N(0,Pja2j). {Z1,Z2,..,Zn}isasetofjointly-Gaussianzer o-meanrv s,andZ=(Z1,..,Zn)TisaGaussianzero-meann- rv,if,forsomefinitesetofIIDN(0,1)rv s, ,..,Wm,eachZjcanbeexpressedasZj=mX`=1aj` W` ,Z=[A]W( )where{aj`,1 j n,1 ` m,} ,U=(U1,..,Un)TisaGaussiann-rvifU=Z+ ,whereZisazero-meanGaussiann-rvand (0,1)rv ,..,ZntobejointlyGaussianifallofthemarel inearcombinationsofacommonsetofIIDnormal izedGaussianrv sfarbeyondbeingindividuallyGaussian, , sisthatinmanyphysicalsituationstherearem ultiplerv seachofwhichisalinearcombinationofacommo nlargesetofsmallessen-tiallyindependentr v ,and,moretothepointhere, ,whenabroadbandnoisewaveformispassedthro ughanarrowbandlinearfilter,theoutputatan ygiventimeisusuallywellapproximatedasthe sumofalargesetofessentiallyindependentrv erentlinearcombinationsofthesamesetofund erlyingsmall,essentiallyindependent,rv (Z1.)

8 ,Zn) (Y1,..,Yk)Tbeak-rvsatisfyingY=[B] :SinceZisazero-meanGaussiann-rv,itcanber epresentedasZ=[A]WwherethecomponentsofWa reIIDandN(0,1).ThusY=[B][A] [B][A]isamatrix, , (Z1,..,Zn) (a1,..,an)T, s,Z1,Z2thatareeachzero-meanGaussianbutfo rwhichZ1+ ,then,Z1andZ2arenotjointlyGaussianandthe 2-rvZ=(Z1,Z2) N(0,1),andletXbeindependentofZ1andtakeeq uiprobablevalues N(0,1)andE[Z1Z2]= ,fZ1Z2(z1,z2),however,isimpulsiveonthedi agonalswherez2= +Z2cannotbeGaussian, ,asweseelater,isthatuncorrelatedjointlyG aussianrv (MGF) [K].Essentially,asdevelopedlater,Zischar acterizedbyaprobabilitydensitythatdepend sonlyon[K].

9 [K].ThentheMGF,gZ(r)=E[exp(rTZ)]andthech aracteristicfunctiongZ(i )=E[exp(i TZ)]aregivenbygZ(r)=exp rT[K]r2 ;gZ(i )=exp T[K] 2 .( )Proof:Foranygivenrealn-vectorr=(r1,..,r n)T,letX= ,Xiszero-meanGaussianandfrom( ),gX(s)=E[exp(sX)]=exp( 2Xs2/2).( )Thusforthegivenr,gZ(r)=E[exp(rTZ)]=E[ex p(X)]=exp( 2X/2),( )wherethelaststepuses( )withs= ,sinceX=rTZ,wehave 2X=E |rTZ|2 =E[rTZZTr]=rTE[ZZT]r=rT[K]r.( )Substituting( )into( ),yields( ).Theproofisthesameforthecharacteristicf unctionexcept( )isusedinplaceof( ).Sincethecharacteristicfunctionofann-rv uniquelyspecifiestheCDF, ,wewillshowlaterthatforanypossiblecovari ancefunctionforanyn-rv, ( ),letUbeaGaussiann-rvwithanarbitrarymean , ,U=m+ [K]ofUisthesameasthatforZ,yieldinggU(r)= exp rTm+rT[K]r2 ;gU(i )=exp i Tm T[K] 2.

10 ( )WedenoteaGaussiann-rvUofmeanmandcovaria nce[K]asU N(m,[K]). s(specialcase)Azero-meanGaussiann-rv,byd efinition,hastheformZ=[A]WwhereWisN(0,[I n]).Inthissectionwelookatthespecialcasew here[A]isn [K]=E[ZZT]=E[[A]WWT[A]T]=[A]E[WWT][A]T=[ A][A]T( )sinceE[WWT]istheidentitymatrix,[In].Tof indfZ(z)inthiscase,wefirstconsiderthetra nsformationofreal-valuedvectors,z=[A] ( ,thevectorwhosejthcomponentis1andwhoseot hercomponentsare0).Then[A]ej=aj,whereaji sthejthcolumnof[A].Thus,z=[A]wtransforms eachunitvectorejintothecolumnajof[A].For n=2, [A] , onasideiscarriedintoanparallelogramwithc orners0,a1 ,a2 ,and(a1+a2).