信號與系統(tǒng)第二版課后習(xí)題解答(3-4)奧本海姆

SignalandSystemSignalandSystemPAGEPAGE10Chap3Acontinuous-tieriodisignax(tisreavalueandhasafundamentalperiodT=8.ThenonzeroFourierseriescoefficientsx(t)area a a* 41 1 3 3Expressx(t)intheformSolution:

Akk0

cos(t )k kFundamentalperioT8. 28 40x(t)

aej0kt aej0t aej0t aej30t ak 1 1 3 3k

ej30t2ej0t 2ej0t 40t 4je0t34cos(8sin(4 4AisrealvaluedandhasafundamentaperioN=5.Thenonzerox[n]area 1ej/4ej/4a* 2ej0 2 2 4 4Expressx[n]intheformSolution:

x[n]A0

Asin(n )k k kk1for,a 1,a ej/4 ,a ej/4, a 2ej,0 2 2 4a 2ej4x[n]

ae/Nkk Na ae0 2

ea2

e

ej(8/5)n41ej/4eej/4e2ej/3e2ej/3e4 812cos(n )4cos(n 5 4 5 34 3 8 512sin(n )4sin(n 5 4 5 6Forthecontinuous-timeperiodicsignalx(t)2cos( t)4sin( t)3 3Determinethefundamentalfrequency andtheFourierseries0coefficientsa suchthatkxt)

aejk0t.kSolution:for theperiodofcos(

k5t)is T 3,theperiodofsin(

t)is3 3T6sotheperiodofx(t)is6,i.e. w /6/30x(t)2cos( t)4sin( t)3 32

1cos(22 1

t)4t)02 (ej2tej2t)2j(ej5tej5t)2 0 0 0 01then, a0

2, a a2

, a2

2j, a5

2j3.5Let x1

beacontinuous-timeperiodicsignalwithfundamentalfrequency andFouriercoefficientsa.Giventhat1 kx(t)x2 1

(1t)x1

(t1)Howisthefundamentalfrequency2

of x2

(t) relatedto?Also,findarelationshipbetweentheFourierseriescoefficients b ofkx(t) andthecoefficients a Youmayusethepropertieslistedin2 kTable3.1.Solution:Because x2

(t)x1

(1t)x1

(t,then x2

(t) hasthesameperiodas x1

(t),thatis T2

TT, w w1 2 1b

1

x(t)ejkwtdt (x

(1t)

(t1))ejkwtdt21k T T 2 T 1 1211

x(1t)ejkwtdt

1

(t1)ejkwtdt11T T 1 T T 111e1a jkwae1k

jkwe1e

(ak

a)ejkw1k1Supposegiventhefollowinginformationaboutasignalx(t):x(t)isrealandodd.x(t)isperiodicwithperiodT=2andhasFouriercoefficientsa.k3. a 0for|k1.k4 12|xt)|2dt1.20Specifytwodifferentsignalsthatsatisfytheseconditions.Solution:x(t)

akk

jkte0ewhile: x(t) isrealandodd,then a ispurelyimaginaryandkodd，a 0, a a ,.0 k kT2,then0

2/2and

a 0for k1kso x(t)

ekta

ejtaejt0kk

0 1 0 1 0a(ejt1

ejt)2a1

sin(t)for

12

x(t)2dta

2

2a

2

21202 a21

/2

0 1 12 x(t)2

sin(t)3.13 Consider a continuous-time LTIsystem whose responseisH(j)

h(t)edt

sin(4)Iftheinputtothissystemisaperiodicsignal1,0t4x(t)1,4t8WithperiodT=8,determinethecorrespondingsystemoutputy(t).Solution:FundamentalperiodT8.2/8/40x(t)

ak0k0

jkte0e y(t)

k

aH(jkk

)ejktH(jk)0

sin(4kk 0

)4,k00,0,k0 y(t)

akk

H(

)ejkwt4a00 00Because

1xt)dt1

dt1

(1)dt00 T T 80 84另：x(t)為實(shí)奇信號，則ak為純虛奇函數(shù)，也可以得到a0為0So y(t)0.3.15 Consider a continuous-time ideal lowpass filter S frequencyresponseis0,H()1,0, Whentheinputtothisfilterisasignalx(t)withfundamentalperiodT/6andFourierseriescoefficientsa,itisfoundthatkxt)Syt)xt).Forwhatvaluesofkisitguaranteedthata 0?kSolution:for

x(t)

akk

jkte0e y(t)

k

aH(jkk

)ejkt0即對于所有的k，H(jk0)10for

H()1,1000,0,也就是說k0

100,T/60

12即12k<100，k<=8，故當(dāng)k>8時(shí)，ak＝0。3.35.Consideracontinuous-timeLTIsystemSwhosefrequencyresponseis0,otherwiseH(j)1,||0,otherwiseWhentheinputtothissystemisasignalx(t)withfundamentalperiodT/7andFourierseriescoefficients a,itisfoundthatthekoutputy(t)isidenticaltox(t).Forwhatvaluesofkisitguaranteedthata 0?kSolution:T=/7,2/T14.0x(t)

akk

jwkte0e y(t)

k

aH(jkwk

)ejkwt0 b a0k k

H(jkw)0for

w250H(jw) ,0,. otherwise1,k17H(jkw)0 0,. otherwisethatisk0

250,k250, 14

k is integer, sok18..or..k17.Let y(t)x(t) , b ak k

, it needs ak

0 ,fork18..or..k17.Chap4UsetheFouriertransformanalysisequation(4.9)tocalculateFouriertransformsof;(a)e2(tu(t(b)e2|tSketchandlabelthemagnitudeofeachFouriertransform.Solution:(a). X()

x(t)ejtdt

e2(t1)u(t1)ejtdte2(t1)etdte(2)te2dt1 1e(2j)t2j e(e(2j)t2je2

e21

2j

2(b). X()

x(t)ejtdte2t1ejtdte2t2ejtdt1e22tejtdt 1 1 1e2

e(2j)t|e2 e(2j)t2j 1 2j ej ej 4ej2j 2j 42UsetheFouriertransformanalysisequation(4.9)tocalculateFouriertransformsof:(a)(t(t(b)d{u(2t)u(t2)}dtSketchandlabelthemagnitudeofeachFouriertransform.Solution:(a). X()

x(t)ejtdt(t1)(t-1)]-tdt-(t1)e-tdt

(t-1)e-jtdt-ej

ej2cos(b). X()

x(t)ejtdtd{u(2t)u(t2)}edtdt

(2t)(tdt

(2t)etdt

(t2)ejtdtej2ej22jsinUsetheFouriertransformsynthesisequation(4.8)todeterminetheinverseFouriertransformof X(j)X(e(,where|X(j)|2{u(3)u(3)}X(j)32Useyouranswertodeterminethevaluesoftforwhichx(t)=0.Solution:xt) 1

X(jw)ejwtdw 13 j(13 j(

X(jw)ejX(jw)ejwtdw 1

u(w2 ejwtdw13

j(3w) ej 2e ejwtdw2

ej(t)wdw32 3 332212

3ej(t2)w3j(t3)23ej(t2)w3j(t3)23

ej3(t3) j(t3)233 32jsin ) 2jsin ) 1

2 2j(t

3) (t3)2 2 t3)k,k3( 2Ifx(t)=0then(t

3)02Thatistk

3,k03 2Giventhatx(t)hastheFouriertransform X(j),expressFouriertransformsof the signalslistedbelowinthe termsofX(j).YoumayfindusefultheFouriertransformpropertieslistedinTable4.1.(a)x1

(t)t)x(1t)(b)x(t)x(3t6)2(c) x3

(t)

d2dt2

x(tSolution:AccorrdingtothepropertiesoftheFouriertransform,wllget:(a).

xt) FT

X(j) x1t) FT X()ex(1t) FT X()ethen

xt)x1t)x(1t) FT1X(j)X(j)ejX(j)ej2X(j)cos1(b).

xt) FT

X(j)x(atb) FT X(j )eaa1 a1 bj xt)xt6) F X2

(j)1X(j)ej23 3(c).

xt) FT

X(

xt) FT

X(j)ejd2dt2

xt) FT ()2X()x(t)3

d2dt2

xt)FT

X(j)2X(j)ej34.11Giventherelationshipsy(t)x(t)h(t)，And g(t)x(3t)h(3t)Andgiventhatx(t)hasFouriertransform X(j)andh(t)hasFouriertransform H(j),useFouriertransformpropertiestoshowthatg(t)hastheformg(t)Ay(Bt)DeterminethevaluesofAandB.Solution:for yt)xt)ht) FTY()X()H()and

g(t)x(3t)h(3t)1 x(t) FT

X(j 3 31 h(t) FT H(j )3 3thenG(j)1X(j)1H(j)1Y(j)13 3 3 3 9 31FT gt) yt)31 A ,B314.14Considerasignalx