2022年信號(hào)與系統(tǒng)的公式匯總分類

1、線性1 連續(xù)傅里葉變換)2 連續(xù)拉普拉斯變換(單邊 ) 線性F3 離散 Z 變換 (單邊 ) ff4 離散傅里葉變換j)k)F(j)f(t)ejtdtF(s )0f( t)estdt(z)f(k)zkF(ej)f(k)ejkk0kf( t)1F(j) ejtdf(t)1jjjF(sst ) edsf( k)1jLF(z )zk1dz ,k0f(k)12F( ej)ekd2222af 1( t)bf2( t)aF 1(j)bF 2(j)線性af 1( t)bf2( t)aF 1 (s )bF 2(s )af 1( k)bf2(k )aF 1 (z )bF 2(z )線性af 1 (k)bf2 (

2、k )aF 1 (ej)bF 2 (e時(shí)移時(shí)移f(tt0)est0F(s )時(shí)移f(km )zmF(z )（雙邊）時(shí)移f(km )ejm F( ej)f( tt0)ejt 0F(j)頻移頻移es 0tf(t)F(ss 0)頻移ej0kf( k)F( ej0z )（尺度變換）頻移ejk0f(k)F(ej(0)ej0 tf( t)F(j(0)尺度f(atb )|1|ejbF(ja)尺度f(atb )|1|ebsF(s)尺度akf( k)F(z)尺度f (n)(k)f 0(k/n )F(ejn)aa變換變換變換a變換) daaaf(k)F( ej)反轉(zhuǎn)f(t)F(j)反轉(zhuǎn)f(t)F(s )反轉(zhuǎn)f(k

3、)F(z1 )（僅限雙邊）反轉(zhuǎn)時(shí)域時(shí)域f1 ( t)*f2(t)F 1( s )F 2(s )時(shí)域f 1 (t)*f2( t)F 1(z )F 2(z )時(shí)域f 1 ( t)*f2( t)F 1(j)F2(j)f 1 (k)*f2( k)F 1( ej)F2( ej)卷積卷積卷積卷積f1( t)f2(t)1F 1(j)*F2(j)1 ( k)f2( k)12F 1( ej)F 2(e j(頻域f(k)1z1F(z)f(1 )頻域卷積2時(shí)域f( t)sF( s )f(0)時(shí)域2)f(k2)z2F(z )z1f()1f(2)卷積微分f( t)s2F(s )sy ( 0)y(0)f(k)1zF(z

4、)zf(0)時(shí)域時(shí)域f(t)f(n )(t)jF(j)(j)nF(j差分f(k)f(k)1( 1ej j )F(e jjf(k2)z2F(z)z2f(0)zf( )1差分微分S 域tf(t)(t)nf(t)F(s)dnF(s)Z 域tf(t)(jt)nf(t)jdF(j)dnF(j)kf(k)jdF(ej)頻域kf( k)zdF(z )頻域微分ddn微分dsn微分dz微分kd)2(時(shí)域tf(x)dx,f()0F(j)F(0)時(shí)域tf(x) dxF(s)f(1)(0)部分f(k)*( k)ikf( i)zz1時(shí)域f( k)F(ejj)F( ej0)(積分j積分ss求和累加1ek頻域f(0 ) t

5、(f(t)F(j)d,F()0S 域f(t)F s() dZ 域f(k)zmzF()d(0 )lim zF(z ),f( 1 )lim zzF(z )zf(0積分jt)積分積分kmm1t對(duì)稱F(jt)2f()初值f(0)lim ssF(s ),F(s )為真分式初值f(M)lim zzMF(z )（右邊信號(hào)） ,f( M)1lim zzM1F(z )zf( M)帕斯E|f(t|) 2dt1 2|F(j|) 2d終值f()lim s 0sF(s),s0在收斂域內(nèi)終值f()lim (z 1z1 )F(z )（右邊信號(hào)）帕斯k|f(k )| 212|F( ej)| 2d瓦爾瓦爾2信號(hào)與系統(tǒng)公式性質(zhì)一

6、覽表et(t連續(xù)傅里葉變換對(duì)2)2)常用連續(xù)傅里葉變換、拉普拉斯變換、Z 變換對(duì)一覽表z(象函數(shù))1拉普拉斯變換對(duì)（單邊）Z 變換對(duì)（單邊）F(j)f(t)ejtdtF(s )0f( t)estdtF(z)k0f(k)zk函數(shù)傅里葉變換函數(shù)象函數(shù)f(函數(shù)0象函數(shù)f函數(shù)0f(t)F( j)f(t)F(s )k ),k( k),k(t)112()(t)1(k )1 (km ),m0zm( t)(n )(t)j(j)n(t)s1 zz1( km ),m0zz1zm(t)1()(t)1(k )zz1k 2(k)z 2zjs(z1 )3t(t)j()1t( t)tn(t )1sn !1k(k)( zz2

7、( k1 ) ak(k)z2)12s2n(za)2)tet(t,)01j(1et( t)tet( t)s1(s1)2ak(k)zzakak1( k)zj(za2)ecos(0t)(0)(0cos(t)(t)s2s2ek(k)zzkak(k)azsin(0t)j(0)(0)e(za )21jsgn()sin(t)( t)s22ejk(k)zzjk2ak(k)az2a2zte( za)3|t|2cosh(t)( t)s2s2ak(a )k(k)2 zza2ak(a )k(k)z22z2a22 a2 aej0 t2(0)sinh(t)(t)s22k (k)1(k)( zz( k1 ) k(k)z22

8、1 )32(z1 )3tcos(t)( t)( jj)2etcos(t)(t)(ss)22akb k(k)(zzzb )ak1bk1(k )z 2aba )(abza)(zbetsin(t)(t )( j)22etsin(t)(t)(s)22cos(k)(k)z2z (zzcos)1sin(k )(k)2zsin2cos2zcosfe| t|(t),00 )22j2(2)n(n )()1(b0 tb 1)(t)(t)(t)b0b 1scos(k)( k)(b 0z2coszcos()sin(k)( k)b 1s)(z2sinzsin()22s2zz22zcos1z2z22zcos1b0(b 0

9、b 1) ettntb 1sb 0akcos(k)(k )z(zacos)aksin(k)(k )azsin)2(ja2s (s)22azcosa22azcossgn(t)|tn222)221tsin(t)(t)1akcosh(k )( k)aksinh(k )( k)z(zacosh)azsinhzz2a2j3s2 s22)22 azcosha22 azcosheett,t0(,213 1t)sin(t(tak(k),kakj2 210lnzza(k)a(s22)2ez,t0kk!|cos(2)cos(t),1tsin(t)(t)s(lnak(k)11)cosh1 z1)!(t)t)(2)2

10、2(s22)2az(2kk !(20 |,t|2ts2k11(k)zlnzz111zlnzFsin(t)tcos(kFnejnn(n,)T2(s 22)22k12z1nT( t)( tnT)n(n)tcos(t)( t)s22b0b 1etb 1) et(t)(sb0)(s 22)2s2nTfg( t)|,1t|22Saj2Sa2sin2(b 0b 1)tb 1 et)b 1 sb 020b0b 1)(b 22ettb 0b 1)b 22ett(b2s2sb 1sb 0)()(s)2tb 0b 1)(b 22es)()(s0|,t|2(t)WSa (W t )F()|,1|WAetsin(t)

11、( t),其中b 1sb 0()b22teb22 s)b 1sb 0b 0b 1b 0(b 1b22esin(W t )2)2Aejb 0b 1(j)et(tt|,0|W(s)2b 0b 1b 2(2(s2(s)2()212|t|,|t|224 b 2et( b 12 b2) tetb 2s 2b 1 sbb 0b 1b22etA sin(tjb1)(t)b2s2)(b 1 sb 022(t )b 01( b 0b 1b 22)t2 et( t(b22)t|2(s)3(ss 22)|,02其中Aej(j)f( t)1(t|2),|t|2j1ej2Sa2f( t)2(|,1t|112|t|),1

12、|t|2(41)b0b 1b 22et(Aejtsin(t)( t)(sb 2s2sb 1 s2b 02)2( 12()220 |,t20 |,t|)2(41)sin其中Aejb 0b 1()b 2(j)2)()j8sin1雙邊拉普拉斯變換與雙邊Z 變換對(duì)一覽表(雙邊拉普拉斯變換對(duì)a 雙邊 Z 變換對(duì)|F(s)f(t)estdtF(z)f(k)zkk函數(shù)象函數(shù)F(s )和收斂域函數(shù)象函數(shù)F(z )和收斂域(t)1，整個(gè) S 平面(k)1，整個(gè) Z 平面(n)(t)n s ，有限 S 平面n(k)(zznn|,z|0)1(t)1,Res 0(k)zz1|,z|1st(t)1,Res 0(k1 )

13、(k)(zz22|,z|12 s)1tn1( t)1,Res 0(kn1 )!(k)(zznn|,z|11 )!n sk (!n1 )!)1( n( t)1,Res 0( k)1zz1|,z|1st( t)1,Res 0( k)1(k)1(zz22|,z|12 s)1(tn1(t)1,Res 0(k(!n1 )!(k)1(zznn|,z|11 )!n skn1 )!)1neat(t)s1a,Res Reaak(k)zza|,z|a|teat(t)(s12,Res Rea (n)1an(k)(z2 z)2|,z|a|a )atn1eat(t)(s1n,Res Rea (kn1 )!an(k)(z

14、n z)n|,z|a|1 )!a )k(!n1 )!naeat( t)s1a,Res Reaak( k)1zza|,z|a|tn1eat(t)(s1n,Res Rea ( kn1 )!an(k)1(zn z)n|,z|a|(1 )!a )k (!n1 )!anecos(t)(t)s2s2,Res 0cos(k)(k)z2z22zcos1zcossin(k)(k)z2zsin1sin(t)(t)s22,Res 02zcostcos(t)(t)( ss)22,Res Reakcos(k)(k )z2z22zacos1zacosetsin(t)(t)( s)22,Res Rea aksin(k)(k

15、)z2zasin12zacose|,Rea 0s22 a2,Rea Res Rea a| k|,a|1(z(a2)()1z)1|,a|z|1aaazae|sgn( t),s22s2,Rea Res Rea a| k|sgn,|a|1(a(z2z )1|,a|z|1|Rea0aza)(aza卷積積分一覽表e1tf1t)(t)ef2t)(22f1( t)*f2(t)f2(tf1()f2( t(t)d)f1t)f2t)tf1(t)*f2(t)(t)f1(t)*)f(t)(t)f(t)f(t)(t)f(t)f(t)(t)tf(t)(t)t(t t ) dt)2t)1( 1eet(t)(t)t)(t)(

16、t)1t2(tt2e1t(t)2t(t1(e1 te2t)(t),1et(t)et( t)tet(t2111)t21e1t(1)2et211ete1t(t)e2t(t)t(t)et( t)1)22cos(t2t11cos()2e2t(ttet(t)et(t)1t2 et( t)cos()2)2e1t(e2t(t)(21)222arctan 2 1卷積和一覽表f1t)f2t)f1(t)*f2(t)f1(t)*f2(t)if1(i)f(ki)f1(t)*f2(t)f1t)f2t)kf(k )(k)f(k )f(k )(k )f(i)i( k )( tf( 0 )ak(k)f1(k1 )(k)k(k)(k )a 1 k1cos1(k)1k(k)(k)2ak(k)(k)1ak1(k),a0ak 1( k)ak 2(k)k a 11k a 21(k),a1a2aa 1a 2a(aka)1(k)ak(k)(k )( k)k(k)ak(k)1ka(k)(k)1ak( 12k(k )k(k)1( k1 )k (k)1(k)ak 1cos(k)(k )ak(k)(k)1ak1cos(2a 1 2a2 2a 1a 2cosarctan6a 1sina 2a 1cosf( t)(t)( t)( t)(tt0)關(guān)于(t 、(k 函數(shù)公式一覽表f(t)(f(t)( t)f(0 )( t