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1、Chapter2Chapter2DtDetermiiitinisticanddRdRandomSignalAnalysisSignalAnalysisTheContentofChapter2TheContentofChapter22.1Bandpassandlowpasssignalrepresentation2.2Signalspacerepresentationofwaveforms2.3Someusefulrandomvariables2.4Boundsontailprobabilities25Limittheoremsforsumsofrandomvariables2.5Limi
2、ttheoremsforsumsofrandomvariables2.6Complexrandomvariables2.7Randomprocesses2.8Seriesexpppansionofrandomprocesses2.9Bandpassandlowpassrandomprocesses2.2Signalspacerepresentationofwaveforms2.2Signalspacerepresentationofwaveforms2.2-1Vectorspaceconceptsnv=∑veiii=1(linearcombinationofunitvectorsin(l
3、inearcombinationofunitvectors,inn-dimensionspace).n*Hv,v=vv1212⋅=∑vv12ii=vv21i=1If0,vv⋅=orthogonal12n1/2‖v‖=(v·v)=normv2,,gvectorlength∑ii=1‖v+v‖≤‖v‖+‖v‖triangleinequality1212
4、v·v
5、≤‖v‖‖v‖Cauchy-Schwartzinequality1212=,whenVV=a12‖v+v‖2≤‖v1‖2+‖v‖2+2Re[v·v]12212matrixalgebramatrixalgebra:Lineartrans
6、formation:V′=AV,ifV′=λV,wehaveAV=λV,iseigenvalueandλViseigenvectorofthetransformation.Constructingasetoforthonorminal(orthogonaln1andeachwithunitnorm)vectorsandeachwithunitnorm)vectorsfromasetofmvectorsv,(1≤≤im)withndimensions,iGenerallywehavenn<,and1nalllways≤smallleroneoffnanddm,i.e,1Ifmnthennm
7、<,≤1Ifmnthenn,≥≤n1Then，howtorealizetheorthogonalexpansionsforaspacevector?spacevector?Gram-SchmidtProcedure(constructingasetofnorthonorminalvectors1fromasetof-dimensionalvectornisv),≤mivisarbitrarilyselectedoneofthevectorsiv1u=1v1uvvu′=−⋅()u22211u′2u=2u′2uvvuuvuu′=−⋅()()−⋅33311322u′3u=,..........
8、.......3u′32.2-2SignalspaceconceptsAparalleltreatmentasavectorforsignalsdefinedoninterval[a,b].Ifsignalsarecomplex-valued,b*<>xtxt(),()=xtxtdt()()((innerpproduct))((2.2-15))12∫a121/2b2xt()=
9、()
10、xtdt=ξ((norm,vectorleng)gth)(∫a)xxtxt()+≤+()xt()xt()(triangleinequality)12121/21/2bbb*22xtxtdt()()≤
11、()
12、x
13、tdt
14、()
15、xtdt(2.2-19)∫∫∫1212aaa(CauchySchwarzinequality)2.2-3Theoremoforthogonalexpansionsofsignalss(t)s(t)sdeteisdeterministc,eastic,real-vavauedsga,wtluedsignal,withfiniteenergy.areorthonormalvectorsftn(),1,...,=K,i.e.
