6. Frequency-Domain Analysis of Discrete-Time Sign

6.Frequency-DomainAnalysisofDiscrete-TimeSignalsandSystems6.1.PropertiesofSequenceAexp(jwn)(1.3.3)6.2.DefinitionofDiscrete-TimeFourierSeries(3.6)6.3.PropertiesofDiscrete-TimeFourierSeries(3.7)6.4.DefinitionofDiscrete-TimeFourierTransform(5.0-5.2)6.5.PropertiesofDiscrete-TimeFourierTransform(5.3-5.7)6.6.FrequencyResponse(3.2,3.8,5.4)6.7.LinearConstant-CoefficientDifferenceEquations(5.8)6.1.PropertiesofSequenceAexp(jwn)6.1.1.PeriodicityofSequenceAexp(jwn)Let’sconsidersequenceAexp(jωn),whereAisacomplexnumbergenerally.Thesequenceisperiodicifandonlyifωcanbeexpressedas,Nk2π=ω(6.1)wherekandNaretwointegers.ItcanbeshownthatNisaperiodofthesequence.IfN0andkandNhavenofactorsincommon,Nwillbethefundamentalperiodofthesequence.Example.Determinetheperiodicityofthefollowingsignals:(1)x(t)=cos(ωt).(2)x(t)=exp(jωt).(3)x(n)=cos(ωn).Example.Findthefundamentalperiodofthefollowingsequence:6.1.2.FrequencyofSequenceAexp(jwn)ωisreferredtoasthefrequencyofsequenceAexp(jωn).Itisthevariationofthephaseωninasamplinginterval,anddescribeshowfastthephaseωnchanges.ItcanbeshownthatAexp(jω1n)=Aexp(jω2n)ifω1-ω2=2πk,wherekisaninteger.Example.Severalpairsofsignalsaregivenasfollows.Determinewhetherthetwosignalsineachpairareidentical:(1)x1(t)=cos(ω1t)andx2(t)=cos(ω2t),whereω1≠ω2..n43jexpn32jexp)n(x⎟⎠⎞⎜⎝⎛π+⎟⎠⎞⎜⎝⎛π=(6.2)(2)x1(t)=exp(jω1t)andx2(t)=exp(jω2t),whereω1≠ω2.(3)x1(n)=cos(jω1n)andx2(n)=cos(jω2n),whereω1≠ω2.Twoconceptsneedtobeclarified:(1)nisthenormalizedtime.IftandTarethephysicaltimeandthesamplinginterval,respectively,thenn=t/T.(6.3)(2)ωisactuallythenormalizedfrequency.AssumethatΩandTarethephysicalfrequencyandthesamplinginterval,respectively.Then,ω=ΩT.(6.4)6.2.DefinitionofDiscrete-TimeFourierSeries6.2.1.DefinitionAnysequencex(n)withperiodNcanberepresentedbyadiscrete-timeFourierseries,i.e.,,knN2jexp)k(X)n(xNk∑〉〈=⎟⎠⎞⎜⎝⎛π=(6.5)whereX(k)isgivenbyX(k)iscalledthespectrumofx(n).NotethatX(k)hasperiodN.From(6.5)and(6.6),wecanseethataperiodicsequencecanbedecomposedintoasetofelementarysequences.Anyoftheelementarysequences,X(k)exp(j2πkn/N),hasthefrequency2πk/NandthecoefficientX(k).6.2.2.DerivationTherightsideof(6.5)is.knN2jexp)n(xN1)k(XNn∑〉〈=⎟⎠⎞⎜⎝⎛π-=(6.6).)nn(kN2jexpN1)n(x)nn(kN2jexpN1)n(xknN2jexpnkN2jexp)n(xN1knN2jexp)k(X1Nnnn1N0kNnNkNkNnNk∑∑∑∑∑∑∑-+=′-=〉〈=′〉〈=〉〈=〉〈=′〉〈=⎥⎦⎤⎢⎣⎡′-π′=⎥⎦⎤⎢⎣⎡′-π′=⎟⎠⎞⎜⎝⎛π⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛′π-′=⎟⎠⎞⎜⎝⎛π(6.7)Since,1Nnnn,0nn,1)nn(kN2jexpN11N0k⎩⎨⎧-+≤′=′=⎥⎦⎤⎢⎣⎡′-π∑-=(6.8)(6.7)isequaltox(n).Example.DeterminetheFourierseriescoefficientsforeachofthefollowingsignals:(1)x(n)=sin(2πMn/N),whereMandNhavenocommonfactorsandMN.(2)x(n)=1+sin(2πn/N)+3cos(2πn/N)+cos(4πn/N+π/2).(3)x(n)isshowninfigure6.1.0……Figure6.1.APeriodicSequence.n16.3.PropertiesofDiscrete-TimeFourierSeries6.3.1.LinearityLet’sassumethatx1(n)andx2(n)havethesameperiod,anda1anda2aretwoarbitraryconstants.Ifx1(n)↔X1(k)andx2(n)↔X2(k),then-MMN-Na1x1(n)+a2x2(n)↔a1X1(k)+a2X2(k).(6.9)6.3.2.ShiftingIfx(n)↔X(k),then,knN2jexp)k(X)nn(x00⎟⎠⎞⎜⎝⎛π-↔-(6.10)wheren0isanarbitraryinteger.Ifx(n)↔X(k),then),kk(XnkN2jexp)n(x00-↔⎟⎠⎞⎜⎝⎛π(6.11)wherek0isanarbitraryinteger.6.3.3.ReversalIfx(n)↔X(k),thenx(-n)↔X(-k).(6.12)From(6.12),thefollowingconclusionscanbedrawn:(1)x(n)even⇔X(k)even.(2)x(n)odd⇔X(k)odd.6.3.4.ConjugationIfx(n)↔X(k),thenx*(n)↔X*(-k).(6.13)From(6.13),thefollowingconclusionscanbedrawn:(1)Im[x(n)]=0⇔X(k)=X*(-k).(2)Re[x(n)]=0⇔X(k)=-X*(-k).(3)Im[X(k)]=0⇔x(n)=x*(-n).(4)Re[X(k)]=0⇔x(n)=-x*(-n).6.3.5.SymmetryIfx(n)↔X(k),then).k(xN1)n(X-↔(6.14)6.3.6.ConvolutionLetx1(n)andx2(n)havethesameperiodN.Ifx1(n)↔X1(k)andx2(n)↔X2(k),then).k(X)k(NX)mn(x)m(x21Nm21↔-∑〉〈=(6.15)Thissumiscalledtheperiodicconvolutionsumofx1(n)andx2(n).Letx1(n)andx2(n)havethesameperiodN.Ifx1(n)↔X1(k)andx2(n)↔X2(k),thenThissumistheperiodicconvolutionsumofX1(k)andX2(k).6.3.7.Parseval’sEquationIfx(n)↔X(k),then.)mk(X)m(X)n(x)n(xNm2121∑〉〈=-↔(6.16).)k(X)n(xN1Nk2Nn2∑∑〉〈=〉〈==(6.17)6.4.DefinitionofDiscrete-TimeFourierTransformAsequencex(n)canberepresentedbyadiscrete-timeFourierintegral,i.e.,(),dnjexp)(X21)n(x2∫πωωωπ=(6.18)whereX(ω)isgivenby(6.19)().njexp)n(x)(Xn∑∞-∞=ω-=ω(6.19)iscalledthediscrete-timeFouriertransform.(6.18)iscalledtheinversediscrete-timeFouriertransform.X(ω)isreferredtoasthespectrumofx(n).NotethatX(ω)hasperiod2π.(6.18)and(6.19)showthatasequencecanbedecomposedintoasetofelementarysequences.AnyelementarysequencehastheformX(ω)exp(jωn)dω/(2π),whichhasthefrequencyωandthecoefficientX(ω)dω/(2π).6.4.1.DerivationofDiscrete-TimeFourierTransformTherightsideof(6.18)equals()()∫∑π∞-∞=′ωω⎥⎦⎤⎢⎣⎡′ω-′π2ndnjexpnjexp)n(x21[][].d)nn(jexp21)n(xd)nn(jexp21)n(xnn2∑∫∑∫∞-∞=′ππ-∞-∞=′πω′-ωπ′=ω′-ωπ′=(6.20)[]),nn(d)nn(jexp21-′δ=ω′-ωπ∫ππ-(6.21)(6.20)becomes).n(x)nn()n(xn=-′δ′∑∞-∞=′(6.22)6.4.2.ConvergenceofDiscrete-TimeFourierTransformTheseriesin(6.19)convergeswhenx(n)isabsolutelysummable.Thatis,thereexistsafiniteconstantBsuchthatSinceItneedbenotedthatthisconditionissufficientfortheconvergenceoftheseriesin(6.19)butnotnecessary.Example.FindtheFouriertransformsofthefollowingsequences:(1)x(n)=δ(n)..N|n|0,N|n|,1x(n)(2)⎩⎨⎧≤=(3)x(n)=anu(n),0|a|1.(4)x(n)=-anu(-n-1),|a