信号与系统答案 西北工业大学 段哲民 第五章

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1、第五章习题5.1求下列各时间函数f()t的像函数F(s)。()(−at)()(1)ft=1−eUt(2)f(t)()=sinωt+φU(t)()1()−at()()−at()()ft=1−eUt(3)ft=e1−atUt(4)a()2()(5)ft=tUt(6)f(t)=(t+2)()Ut+3δ()t()()−at(7)ft=tcosωtUt(8)f(t)=(e+at−1)U()t答案11αF(s)=−=(1)ss+αs(s+α)ssinψ+ωcosψF(s)=2(2)s+ωsF(s)=2(3)(s+α)1α1F(s)=×=(4)αs(s+α)s(s+α)2F(s)

2、=2(5)s2123s+2s+1F(s)=++3=22(6)sss(7)⎡⎤11jtωω−−jtjtω1jtωFsLt()==+[]cosωtLte⎢⎥(e)=+=Lte⎡⎣⎤⎡⎤⎦⎣⎦Lte⎣⎦222221111s−ω×+×=222222(sj−++ωω)2(sj)(sω)(8)2−αt−αααFs()=+−=−Le[]ααt1L⎡⎤(1−+=e)Lt[]+=⎣⎦22ss()++αsss()α5.2求下列各像函数F()s的原函数f(t)。()()2s+1s+32s+16(1)F()s=(2)F()s=()()2ss+2s+4()s+5s+6()s+12232s+9

3、s+9s(3)F()s=(4)F()s=22s+3s+2()s+3s+2s答案(1)KKK123F(s)=++ss+2s+4(s+1)(s+3)3K=×s=1s(s+2)(s+4)8s=0(s+1)(s+3)1K=(s+2)=2s(s+2)(s+4)4s=−2(s+1)(s+3)3K=(s+4)=3s(s+2)(s+4)8s=−4313F(s)=8+4+8ss+2s+431*2t3−4tf(t)=(+e+e)U(t)848(2)1234152−KKK1235945F(s)=++=++s+2s+3s+12s+2s+3s+1212−2t34−3t152−12tf(t)=

4、(e−e+e)U(t)5945(3)3s+521F(s)=2+=2++(s+1)(s+2)s+1s+2−t−2tf(t)=2δ(t)+(2e+e)U(t)(4)2s3s+214F(s)==1−=1+−2s+3s+2(s+1)(s+2)s+1s+2−t−2tf(t)=δ(t)+(e−4e)U(t)5.3求下列各像函数F()s的原函数f(t)。32s+6s+6s1(1)F()s=(2)F()s=222s+6s+8s()s+1答案(1)2−4F(s)=s++s+2s+4−2t−4tf(t)=δ′(t)+(2e−4e)U(t)(2)KKKKK1231−31112132122

5、F(s)=++++=++++322322(s+1)(s+1)s+1ss(s+1)(s+2)s+1ss12−t−t−t12−tf(t)=(te+2te+3e+t−3)U(t)=(t+t+3)eU(t)+(t−3)U(t)225.4求下列各像函数F()s的原函数f(t)。−(s−1)2+e1(1)F()s=(2)F()s=2()−s()s−1+4s1−e2−s⎡1−e⎤(3)F()s=⎢⎥⎣s⎦答案(1)−(s−1)212eF(s)=+×2222因(s−1)+22(s−1)+22sin2tU(t)↔2又因有s+42−ssin2(t−1)U(t−1)↔e2故由时移性有s+

6、4又由复频移性有t2−(s−1)esin2(t−1)U(t−1)↔e2(s−1)+4t1tf(t)=esin2tU(t)+esin2(t−1)U(t−1)故2(2)11F(s)=×−ss1−e∞∞f(t)=U(t)∗∑∑δ(t−K)=U(t−K)故nk=00=,K∈N(3)−s−s1−e1−eF(s)=×2s1−sU(t)−U(t−1)↔(1−e)因有s故f(t)=[]U(t)−U(t−1)∗[]U(t)−U(t−1)=tU(t)−2(t−1)U(t−1)+(t−2)U(t−2)25.5用留数法求像函数()4s+17s+16f(t)。Fs=的原函数()2()s+2s

7、+3答案2s+s+=，得到一个单极点s=−3令F(s)的分母(2)(3)01和一个二重极点s=−22。下面求各极点上的留数。2[]st⎡4s+17s+16st⎤−3tRes=F(s)(s+3)e=e⎥=e1s=−3⎢2⎣(s+2)⎦s=−31⎧d[st]⎫Res2=⎨F(s)(s+2)e⎬=(2−1)!⎩dt⎭s=−22⎡d4s+17s+16st⎤⎢(e)⎥=⎣dss+3⎦s=−222⎡4s+24s+35st4s+17s+16st⎤⎢2e+te⎥=⎣s+6s+9s+3⎦s=−2−2t−2t−2t3e+(−2)te=(3−2t)e[−3t−2t]故f(t)=e+(