常用傅氏变换.pdf

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1、P1:RPU/XXXP2:RPU/XXXQC:RPU/XXXT1:RPUCUUK852-Mandal&AsifMay25,200720:5204PartIIContinuous-timesignalsandsystemsTable5.2.CTFTpairsforelementaryCTsignalsTimedomainFrequencydomain∞∞1CTsignalsx(t)=X(ω)ejωtdtX(ω)=x(t)e−jωtdtComments2π−∞−∞(1)Constant12πδ(ω)(2)Impulsefunctionδ(t)11(3)Units

2、tepfunctionu(t)πδ(ω)+jω1(4)Causaldecayinge−atu(t)a>0a+jωexponentialfunction2a(5)Two-sideddecayinge−a

3、t

4、a>0a2+ω2exponentialfunction1(6)First-ordertime-risingte−atu(t)a>0(a+jω)2causaldecayingexponentialfunctionn!(7)Nth-ordertime-risingtne−atu(t)a>0(a+jω)n+1causaldecayingexponentialfunc

5、tion1t>02(8)Signfunctionsgn(t)=−1t<0jω(9)Complexexponentialejω0t2πδ(ω−ω)0(10)Periodiccosinefunctioncos(ω0t)π[δ(ω−ω0)+δ(ω+ω0)]π(11)Periodicsinefunctionsin(ω0t)[δ(ω−ω0)−δ(ω+ω0)]jπjω(12)Causalcosinefunctioncos(ω0t)u(t)2[δ(ω−ω0)+δ(ω+ω0)]+ω2−ω20πω0(13)Causalsinefunctionsin(ω0t)u(t)2j[δ(ω

6、−ω0)−δ(ω+ω0)]+ω2−ω20a+jω(14)Causaldecayinge−atcos(ωt)u(t)a>00(a+jω)2+ω2exponentialcosine0function−atω0(15)Causaldecayingesin(ω0t)u(t)a>0(a+jω)2+ω2exponentialsinefunction0t1

7、t

8、≤τ/2ωτ(16)Rectangularfunctionrect=τsincτ=0τ0

9、t

10、>τ/22πWWtω1

11、ω

12、≤W(17)Sincfunctionsincrect=ππ2W0

13、ω

14、>W

15、

16、t

17、t1−

18、t

19、≤τ2ωτ(18)Triangularfunction△=ττsincτ>0τ0otherwise2π∞∞(19)Impulsetrainδ(t−kT0)ω0δ(ω−mω0)angulark=−∞m=−∞frequency√ω0=2π/T0−t2/2σ2−σ2ω2/2(20)Gaussianfunctioneσ2πeP1:RPU/XXXP2:RPU/XXXQC:RPU/XXXT1:RPUCUUK852-Mandal&AsifMay25,200720:5)(cont.w)

20、wω/2w/2)=0pp)=0(w(w−−w

21、X(w

22、w

23、(w)(wpd)

24、1jw(w2

25、Xwpd)(waa2+(w

26、X=)2a)=2)

27、)=1X(w=)(w(w(wXX

28、XX00(wX00p102a2/))(w(w

29、(w/2/2

30、Xppttttt)(tuat−))=e

31、)(t(tx

32、ta(tu−d0)=1)=(t)=e(tx)=(tx(txx0000111)(tδ)=(t)u(t=x)(tx1=))(t(tu

33、

34、txata−−eeMagnitudeandphasespectraforselectedelementaryCTfunctions==))(t(txx(1)Const

35、ant(2)Unitimpulsefunction(3)Unitstepfunction(4)Decayingexponential(5)Two-sideddecayingexponentialTable5.3.MagnitudespectraareshownaslinesandphasespectraareshownasdashedlinesFunctionTime-domainwaveformMagnitudeandphasespectra205P1:RPU/XXXP2:RPU/XXXQC:RPU/XXXT1:RPUCUUK852-Mandal&AsifMa

36、y25,200720:5