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PAGEPAGE5/9连续傅里叶变换

连续拉普拉斯变换(单边)

离散Z变换(单边)

离散傅里叶变换()

F(s)

f(t)estdt

F(z)

f(k)zk

F(e)

f(k)e Fj f(t)e

jtdt0

k0

jk

jkf(t)12

F(j)ejt

f(t)

1

jF(s)estdsj

f(k)

1

F(z)zk1dz,k01 2 1 1 2 1

f(k)12

F(e2

j)e

jkd1线性 af1

(t)bf

(t)aF(j)bF

j)

af1(t)bf2(t)(s)bF2(s) 线性

af1(kbf2(k)zbF2z) 线性

af(k)bf(k)aF(ej)bF(ej)102时移 ftt)ejtF(j)1020频移 ef(t) F(j(00

时移 f(tt0)est0F(s)频移 es0tf(t)F(ss0

时移 f(km)zmF(z)〔双边〕 时移频移 ej0kf(k)F(ej0z)〔尺度变换〕 频移

f(km)ejmF(ej)ejk0f(k)F(ej(0))尺度 1 jb 尺度

1 bs s 尺度

k() z

尺度 f (k)f(k/n)F(ejn)f(atb) eaF(j )

f(atb) eaF()

afk F()

(n) 变换 |a| a反转 f(t)F(时域1 2 1 f(t)*f(t)F(j)F(j1 2 1 卷积

变换 |a| a 变换反转 f(t)F(s) 反转时域 时域f1(t)*f2(t)F1(s)F2(s)卷积 卷积

a 变换f(k)F(z1)〔仅限双边〕 反转时域f1(t)*f2(t)F1(z)F2(z)卷积

0f(k)F(ej)1 2 1 f(k)*f(k)F(ej)F(ej1 2 1 频域f(t)f

(t)1F(j)*F

(j)

f(k1)z1F(z)f(1)

频域f(k)f

(k)1

F(ej)F

(ej())d卷积 1 2 1 2

时域 f(t)sF(s)f(0)

1 2时域 f(k2)z2F(z)z1ff(2) 卷积

21 2时域f(t)f(n)(t)jF(j)(j)nF(j)微分

微分 f(t)s2F(s)sy(0)y(0

差分 f(kzF(z)zf(0) 时域f(k2)z2F(zz2f(0zf差分

f(k)f(k1)(1ej)F(ej)频域tf(t)(jt)nf(t)j微分

dF(d

dnF(dn

S域tf(t)(t)nf(t)F(s)微分

dnF(s) Zdsn 微分

kf(k)z

dF(z) 频域dz 微分

kf(k)

dF(ej)d时域

f(x)dx,f()0

F(j)F(0) 时域 t

F(s) f(1)(0) 局部

()*

)

() z

时域

()

F(ej)(

j0)

(2)积分

j 积分

f(x)dx s s

fk k求和

fii

z1

fk累加 k

1e

Fe kk频域 (0)tf(t)F(j)d,F()0 S域

f(t) Z

f(k)zmF)

f(0)limF(z),f(1)lim[zF(z)zf(0)]积分 (jt)

积分 t

F()ds

积分 km

zm1

z

z对称 F(jt)

初值 f(0)limsF(s),F(s)为真分式 初值

f(M)limzMF(z)〔右边信号〕,f(M1)lim[zM1F(z)zf(M)s

z

z帕斯帕斯E|f(t)|2dt |F(j)|2d1瓦尔2终值f()limsF(s),s0在收敛域内s0终值f()lim(z1)F(z)〔右边信号〕z1帕斯|f(k)|2122|F(ej)|2d瓦尔k常用连续傅里叶变换、拉普拉斯变换、Z变换对一览表连续傅里叶变换对 拉普拉斯变换对〔单边〕 Z变换对〔单边〕F(j)函数f(t)

f(t)ejtdt傅里叶变换F(j)

F(s)0函数f(t)

f(t)estdt

象函数F(s)

函数f(k),k

F(z)k0象函数

f(k)zk函数f(k),k

象函数(t)1

12()

(t)

1 (k) 1

(km),m

zm(t)(n)(t) j(j)n

(t)

s 1 zz1

(km),m0

z zmz1(t)

1()j

(t)

1 (k)s

zz1

k2

(k)

z2z(z1)3t(t)

j()12

t(t)tn(t)

1 s2 sn1

k(k) z(z1)2

(k1)ak(k

z2(za)2t()

1 1t(),0

1 1et(t)tet(t)

ak(k)

z kak1(k) ze t te t

j j)2

s

(s)2

za

(za)20cos(t)0

[(

)()]

cos(t)(t)

s (k)

z kak(k) az0sin(t)0

j[(

)()]

s22

ze

(za)200001 j0000t

sin(t)(t)

s2

ejk

zk kak kajze

az2a2z(za)3|t| 22

cosh(t)(t)

ss2

ak(a)k(k)2a

zz2a2

ak(a)k2a

(k

z2z2aj

t

j

sinh(t)(t)

s22s

k(k(k2akbk

z(zz

(k(k2ak1bk1

z2(z1)3z2e tcos(t)(t)

(j)22 e

tcos(t)(t)

(s)22

ab

(k)

(z

ab

(k

(za)(zb)etsin(

t)

(t)

(j)22

t

sin(

t)

(t)

(s)2

cos(k)(k)

z(zcos)z22zcos

sin(k)(k)

zsinz22zcos1(

(btb)(t) b0

)(k) z2coszcos()

sin(k)(k) z2sinzsin()e t), 0

22

0 1 s2

z22zcos1

z22zcos1n ()

)n(n)

b0(b0

b)et(t) 1s

kcos(

)

z(zacos))

ksin(

)

azsin)tt j j

1

s(s) a k

z22azcosa2 a k

z22azcosa22j

1[tsin(t)](t) 13 s2(s22

akcosh(k)(k)

z(zacosh)z22azcosha

aksinh(k)(k)

azsinhz22azcosha2t,t

0)

1[1t)]sin(t)(t)

1 ak

ln z ak at ,(

j22

23

(s22)2

(k),k0

za

(k)eze ,t0

k k! t cos( ),| t f(t) 2

cos( )222

1tsin(t)(t)

s (lna)

(k) 1

1 cosh 1|t|

()2

()2 2

(s22)2 k!

az (2k)! z 2 2 2

1[sin(t)tcos(t)](t) s2

1 ()

z 1

1 z1Fenn

2 F nn

n),T 2

(s22)2

k1 k

zlnz1

2k1

(k

zln2

z1

)

n)

s22

bb

bb

bsb(t) (tnT)

n

t

t)(t)

[0 1 e(0 1)et(t) 1 0Tn

2T

(s22)2

(s)(s) [bb2etb22et1,|t|

2

bsb

1(

2) )()

bs2bsbg(t) 2

Sa

sin

[(b

bb

1 0

2 1 0

2 2

0 1 1

(s)2

bbb2

(s)(s)(s)0,|t|

0 1 2 et(t) 2

Aetsin(t(t)

bs

()()1[b0b1b22etb0bb22tet12W sin(Wt)Sa(Wt)

F(j) 2

1 0 )2

b2s

W Aejb0j)

(s)22

bbb(2)

(s)2(s)0,|| 2

0 1

21()1

et](t) 2|t|

t

[0bb22etAsin(t)t) ,|t

(b12b2)te

bs2bs

22

bs2bsbf(t) 2

Sa2

2 1 0

2 1 0

2 4

1(bbb2)t2et](t)

(s)3

(s)(s22)0,|t|

2 0 1 2

其中

(bb2)jb0 0 2

Ae

(

1j)t12f(t)1(t ),|t22 10,|t|j ej2Sa f(t)2|t|),1t22[bb 0 1 2b2e Ae sin(t)](tt)22 t)bs2bsb 22121 00,|t|2其中Aejbb(j)b(j)2(s)[(s)22)]0 122) 8 sin(1)sin(1)(j) 14 4双边拉普拉斯变换与双边Z变换对一览表双边拉普拉斯变换对

双边Z变换对F(s)

f(t)estdt

F(z)

k

f(k)zk函数(t)(n)(t)(t)t(t)tn1 (t)(n1)!(t)t(t)

F(s1，整个S平面sn，有限S1,Re{s}0s1,Re{s}0s21,Re{s}0sn1,Re{s}0s1,Re{s}0s2

函数(k)n(k)(k)(k1)(k)(kn1)!(k)k!(n1)!(k1)(k1)(k1)

F(z和收敛域1，整个Z平面zn ,|z|(znz ,|z1z1z2 ,|z|(z2zn ,|z|(znz ,|z1z1z2 ,|z(z2tn1(n

(t)

1,Re{s}0sn1

(kn1)!(k1)k!(n1)!

zn ,|z(znzeat(t)

sa

,

ak(k)

za

,|z||a|teat(t)

1 ,(sa)2

(n1)a

n(k)

z2 ,|za(za)2tn1(n

eat(t)

1(sa)n1

,

(knk!(n1)!

an(k)

zn(za)z

,|z||a|eat(t) eat(t)

s1

,,

ak(k1)(kn1)! an(k1)

zzn

,|z||a|,|z||a|(n

(sa)n

k!(n1)!

(za)ncos(t)(t)

s ,0s22

cos(k)(k) z2zcosz22zcos1卷积积分一览表ff(t)*f(t)f()f(t)d1 21f(t)1f(t)2f(t)*f(t)1 2f(t)1f(t)f(t)f(t)(t)(t)f(t)2(t)(t)f(t)*f(t)1 2f(t)f(t)tf()d(t)t(t)et(t)(t)1ett)1(t)t(t)12t(t)2e11t(t)et2(t)(etet(t),121 2et/