信号与系统公式分类

信号与系统公式汇总分类信号与系统公式汇总分类PAGEPAGE6/10连续傅里叶变换

连续拉普拉斯变 离散Z变单边

离散傅里叶变换F(

f(t)e

jtdt

换(单边)

F(z)k0

f(k)zk

F(ej)

k

f(k)e

f(t)

1

F(j)ed

f(k)

1 F(z)zk1dz,k0

f(k)1

F(e)edjjjk F(s)ft)edt Lst0

2 2ft) 1jF(s)estds2jj线性

(t)

(t)aF(j)

(j)

线性 af(t)bf(t)aF(s)1 2

(s) 线性

af(k)bf(k)aF(z)bF1 2

(z) 线性

af(k)bf

(k)aF(ej)bF(ej)1 2 1 2

1 2 1 2

1 2 1 2时移 ftt)ejtF(j)0

时移 f(tt)estF(s)000

时移 f(km)zmF(z)（双边）

时移 f(km)ejmF(ej)频移

频移 es0tf(t)F(ss

频移 ejkf(k)F(e

z)（尺度变换） 频移

ejk

f(k)F(ej())0e f(t)F(j(00

0 0 0 0 0尺度 1 jb 尺度f(atb) eaF(j )

1 bs s 尺度f(atb) eaF()

akf(k)F(z)

尺度 f (k)f(k/n)F(ejn)(n)变换 |a| a 变换反转 f(t)F(反转时域 时域

|a| af(t)F(s)

变换 a 变换反转 f(k)F(z1)（仅限双边） 反时域 时

0f(k)F(ej)f(t)*f(t)F(j)F

(j)

f(t)*f(t)F(s)F(s) f(t)*

(t)F(z)F(z)

f(k)*f(k)F(ej)F

(ej)卷积 1 2 1 2

卷积 1 2 1 2 卷积

1 2 1 2

卷积 1 2 1 2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

(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

1 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(ed时域

f(x)dx,f()0F(j)(0)() 时域 t

F(s) f(1(0) 部分

f(k)*(k)

kfi) z

时域 f(k)F(e)F(ej0)

(2k)积分

j 积分

f(x)dx s s 求和

i

z1

累加 k

1ej

k频域频域(0)tf(t)F(j)d,F()0S积分(jt)积分f(t)t F()ds Zf(k)zmF)d,积分kmzm1f(0)limF(z) flim[zF(z)zf(0)]zz对称F(jt)初值f(0limsF(sF(s为真分式s初值f(M)limzMF(z)（右边信号）,f(M1)lim[zM1F(z)zf(M)z z帕斯E|ft)|2dt |F(j)|2d1帕斯终值f(limsF(ss0在收敛域内s0终值瓦尔2f(lim(z1)F(z（右边信号）z1|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 Z变换对（单边）F(j)

f(t)edt 边）

F(z)k0

f(k)zk函数f(t)

傅里叶变换F(j)

F(s)0函数f(t)

f(t)estdt

1F(s)1

函数1f(k),k01

函数f(k),k

象函数(t)1 1

(t)

(k)1

(km),m0 zm(t)(n)(t) j(j)n (t) s

zz1

(km),m0 z zmz1(t)

1()j

(t)

1 (k)s

zz1

k2(k)

z2z(z1)3t(t)

j()12

t(t)t

n(t)

1 zs2 sn1z

k(k)

z(z1)

(k

k(k

z2z(za)2zt

1 1t

1 1e(t)te(t)

ak(k)

kak1(k)e (t)

(t), 0

j j)2

s

(s)2

za

(za)2cos(t)0

[(0

)()]0

cos(t)(t)

s (k) z kak(k) azsin(t)0

j[(0

)()]0

s22

ze

(za)21 j

sin(t)(t)

z

az2a2zt s

2 ej

(k)

zej

k2ak

(k

(za)3|t| 22

cosh(t)(t)

ss22

ak(a)k2a

(k

zz2a2

ak(a)k2a

(k

z2z2a 2( )

sinh(t)(t)

k(k(k

z (k(k) z2jt

0j

s22s

2akbk

(zz

2ak1bk1

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

(j)22

etcos(t)(t)

(s)22

ab

(k)

(z

ab

(k

(za)(zb)

cos(k)(k)

z(zcos)

sin(k)(k) zsine t

t)(t)

(j)22

e t

t)(t)

(s)22

z22zcos1

z22zcos1ett),0

(btb)(t)

bbs0 1

cos(k)(k)

z2coszcos()

sin(k)(k)

z2sinzsin()22 0 b b

s2bsb

z22zcos0z(zacos)0

z22zcos1azsinttn

j2()2(j)n(n)()

0(

b)ett1

1s(s

akcos(k)(k)

z2

2azcosa

aksin(k)(k)

z2

2azcosa2sgn(t)

2 1[tsin(t)](t)

1akcosh(k)(k)

z(zacosh)

aksinh(k)(k)

azsinhj 3 s2(s22

z22azcosha2

z22azcosha2t,t0

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

1 ak

z ak a

,( 0)

j22

23

(s22)2

(k),k0

ln za

(k)eze ,t0

k k! 2cos(2f(t)

t),|t|2

cos( 2

1tsin(t)(t)

s (lna)

(k) 1

1 1cosh

2 2

(s22)2 k!

az (2k)! z|t|2

( )( )2 2Fejntn

F(n

n),T

1[sin(t)tcos(t)](t)2

s2(s22)2

1 (k)k1

zln z z1

1 2k1

(k

1 z1zln2 z12n2

n

(

( 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)g(t)t|[0 1bb b2 2et 0 1 2bbb2te2Sa sin 2[(bb)tb]etbsb1 0()())( ) bs2bsb20,|t| 2 20 11(s)2bbb(s)(s)(s)1 02)()0 1 22et](t)WF(j)W2Aetsin(tt)bb(j)bsb[bb 0 1b2 2t 0bb ()2e 1b22tetbsbsb2t0,||WAej1 0(s21 0)2 20 1bbb(2)0 1 2(s)2(s)2()2e t](t) 2|t|f(t) ,|t2[bb 0 1 2b2etAsin()]()tt0,|t|2Sa2[be (b)te t421 2tbs2bsb22bs2bsb122(bbb2)t2et](t)(s)31021 0(s)(s22)20 12其中Aej(bb2)jb0 2(j)1t11(t),|t| f(t)220,|t|j1ej2222|t2[bb 0 1 2b2e Ae sin(t)](t)tt f(t)),1t2()221bs2bsb21 0220,|t|其中Ae jbb(j)b(j)2(s)[(s)22)]0 1282()() ()sin 1 sin 1(j) 14 4双边拉普拉斯变换对双边拉普拉斯变换对ZF(s)f(t)estdtF(z)f(k)zk函数象函数和收F(s)象函数F(z)敛域函数(t)1S平面(k)和收敛域Z面(n)(t)snS(k)nzn(zz,|z|0n(t)tn1(n(t)1,Re{s}0s1,Re{s}0s21,Re{s}0sn1,Re{s}0s1,Re{s}0s21,Re{s}0sn(k)t(t)(k(k)z1z2,|z|1,|z|1(kn1)!(k)k!(n1)!(z2zn ,|z|(zn(t)(k(k1)(k1)zzz2,|z1t(t),|z1tn1(n(t)(kn1)!(k1)k!(n1)!(zzn(z,|z1eat(t)1s1,Re{s}Re{a}ak(k)teat(t),zzz2,|za|,|za|tn1eat(t)(sa)21(sa)n(n1)an(k),(kn1)!k!(n1)!an(k)(za)zn,|za|eetcos(tt)s(s)22,akcos(k)(k)etsin(tt)()(s),z2zacosz22zacoszasin22aksin(kk)()z 2zacos12e,Re{a}0|t|2as2a2ss2a,Re{s}Re{a}a ,|a1|k|(a21)z(za(z2z)(z,|az1|aesgn(t),Re{a}0,Re{s}Re{a}a|k|sgn,|a1,|az|1a卷积积分一览表1f(t)2(t)1f(t)21f(t)2(t)1f(t)2f(t)(t)tf()d(t)(t)t(t)f(tf(t)*f(t)f()f(t)d1 21f(t)f(t)f(t)*f(t)f(t)f(t)f(t)*f(t)et(t)(t)e(t)1(t)t(t)1t2e1t(t)et2(t)1(etet(t),121 2et(t)et(t)te(t)2 1()t1tet(t)1et2(t)2 1( ) et121e