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P1:RPU/XXXP2:RPU/XXXQC:RPU/XXXT1:RPUCUUK852-Mandal&AsifMay25,200720:5204PartIIContinuous-timesignalsandsystemsTable5.2.CTFTpairsforelementaryCTsignalsTimedomainFrequencydomainCTsignalsx(t)=12π∞−∞X(ω)ejωtdtX(ω)=∞−∞x(t)e−jωtdtComments(1)Constant12πδ(ω)(2)Impulsefunctionδ(t)1(3)Unitstepfunctionu(t)πδ(ω)+1jω(4)Causaldecayingexponentialfunctione−atu(t)1a+jωa0(5)Two-sideddecayingexponentialfunctione−a|t|2aa2+ω2a0(6)First-ordertime-risingcausaldecayingexponentialfunctionte−atu(t)1(a+jω)2a0(7)Nth-ordertime-risingcausaldecayingexponentialfunctiontne−atu(t)n!(a+jω)n+1a0(8)Signfunctionsgn(t)=1t0−1t02jω(9)Complexexponentialejω0t2πδ(ω−ω0)(10)Periodiccosinefunctioncos(ω0t)π[δ(ω−ω0)+δ(ω+ω0)](11)Periodicsinefunctionsin(ω0t)πj[δ(ω−ω0)−δ(ω+ω0)](12)Causalcosinefunctioncos(ω0t)u(t)π2[δ(ω−ω0)+δ(ω+ω0)]+jωω20−ω2(13)Causalsinefunctionsin(ω0t)u(t)π2j[δ(ω−ω0)−δ(ω+ω0)]+ω0ω20−ω2(14)Causaldecayingexponentialcosinefunctione−atcos(ω0t)u(t)a+jω(a+jω)2+ω20a0(15)Causaldecayingexponentialsinefunctione−atsin(ω0t)u(t)ω0(a+jω)2+ω20a0(16)Rectangularfunctionrecttτ=1|t|≤τ/20|t|τ/2τsincωτ2πτ =0(17)SincfunctionWπsincWtπrectω2W=1|ω|≤W0|ω|W(18)Triangularfunction△tτ=1−|t|τ|t|≤τ0otherwiseτsinc2ωτ2πτ0(19)Impulsetrain∞k=−∞δ(t−kT0)ω0∞m=−∞δ(ω−mω0)angularfrequencyω0=2π/T0(20)Gaussianfunctione−t2/2σ2σ√2πe−σ2ω2/2P1:RPU/XXXP2:RPU/XXXQC:RPU/XXXT1:RPUCUUK852-Mandal&AsifMay25,200720:5Table5.3.MagnitudeandphasespectraforselectedelementaryCTfunctionsMagnitudespectraareshownaslinesandphasespectraareshownasdashedlinesFunctionTime-domainwaveformMagnitudeandphasespectra(1)Constantx(t)=1t0x(t)=11w0X(w)=2pd(w)2pX(w)=0|X(w)|(2)Unitimpulsefunctionx(t)=δ(t)t0x(t)=d(t)1X(w)=0|X(w)|w0X(w)=11(3)Unitstepfunctionx(t)=u(t)t0x(t)=u(t)10ωjwX(w)=pd(w)+1|X(w)|X(w)p/2−p/2(4)Decayingexponentialx(t)=e−atu(t)t0x(t)=e−atu(t)w0X(w)p/2−p/2a+jwX(w)=1|X(w)|(5)Two-sideddecayingexponentialx(t)=e−a|t|t0x(t)=e−a|t|w0X(w)=02/aa2+w2X(w)=2a|X(w)|(cont.)205P1:RPU/XXXP2:RPU/XXXQC:RPU/XXXT1:RPUCUUK852-Mandal&AsifMay25,200720:5Table5.3.(cont.)FunctionTime-domainwaveformMagnitudeandphasespectra(6)First-ordertime-risingdecayingexponentialfunctionx(t)=te−atu(t)t0x(t)=te−atu(t)w0X(w)p−p(a+jw)2X(w)=1a21|X(w)|(7)Nth-ordertime-risingdecayingexponentialfunctionx(t)=tne−atu(t)t0x(t)=tne−atu(t)w0X(w)(a+jw)n+1n!X(w)=an+1n!−2(n+1)p2(n+1)p|X(w)|(8)Signfunctionsgn(t)=1t0−1t0t0x(t)=sgn(t)1−1wX(w)jwX(w)=22p2p−|X(w)|(9)Complexexponentialfunctionx(t)=ejω0tt0x(t)=ejw0t1|x(t)|=1x(t)=w0w0w0X(w)=2pd(w−w0)X(w)=0|X(w)|w02p(10)Cosinefunctionx(t)=cos(ω0t)t0x(t)=cos(w0t)1w0X(w)=p[d(w−w0)+d(w+w0)]X(w)=0w0−w0pp206P1:RPU/XXXP2:RPU/XXXQC:RPU/XXXT1:RPUCUUK852-Mandal&AsifMay25,200720:5(11)Sinefunctionx(t)=sin(ω0t)t0x(t)=sin(w0t)1w0X(w)=jp[d(w+w0)−d(w−w0)]w0−w0ppp/2−p/2|X(w)|X(w)(12)Causalcosinefunctionx(t)=cos(ω0t)u(t)t0x(t)=cos(w0t)u(t)1w0X(w)=[d(w−w0)+d(w−w0)]+w0−w0|X(w)|p2jw/w20−w2(13)Causalsinefunctionx(t)=sin(ω0t)u(t)t0x(t)=sin(w0t)u(t)1w0X(w)=[d(w−w0)+d(w−w0)]+w0−w0|X(w)|p2jw0/w20−w2(14)Causaldecayingexponentialcosinefunctionx(t)=e−atcos(ω0t)u(t)t0x(t)=e−atcos(w0t)u(t)w0(a+jw)2+w02(a+jw)X(w)=X(w)p/2−p/2|X(w)|(15)Causaldecayingexponentialsinefunctionx(t)=e−atsin(ω0t)u(t)t0w0(a+jw)2+w02w0X(w)=X(w)p−p|X(w)|x(t)=e−atsin(w0t)u(t)(cont.)207P1:RPU/XXXP2:RPU/XXXQC:RPU/XXXT1:RPUCUUK852-Mandal&AsifMay25,200720:5Table5.3.(cont.)FunctionTime-domainwaveformMagnitudeandphasespectra(16)Gatefunctionx(t)=recttτt01ttx(t)=rect()2t−2t0t2pt2p−wX(w)wt2pX(w)=tsinc()|X(w)|p(17)Sincfunctionx(t)=WπsincWtπ0WpWp−tWpWtpx(t)=()sinc()tt012WwX(w)=rect()−WWX(w)=0|X(w)|(18)Triangularfunctiontτ=1−|t|τ|t|≤τ0otherwiset01ttx(t)=Δ()−tt0t2pt2p−wX(w)=0wt2pX(w)=tsinc2()|X(w)|(19)Impulsetrainx(t)=∞k=−∞δ(t−kT)t0∞k=−∞x(t)=∑d(t−kT)−TT|X(w)|w0∑∞d(w−)k=−∞TTX2kp2p(w)=T2pT−2pT2pT4pT−4pX(w)=0(20)Gaussianfunctionx(t)=e−t2/2σ2t0x(t)=e−t2/2s21w0X(w)=s√2pe−s2w2/2X(w)=0|X(w)|s√2p208