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1、1 连续傅里叶变换 dejFtf dtetfjF tj tj )( 2 1 )( )()( 2 连续拉普拉斯变换(单边 ) j j st st dsesF j tf dtetfsF )( 2 1 )( )()( 0 3 离散 Z 变换 (单边 ) L k k k kdzzzF j kf zkfzF 0,)( 2 1 )( )()( 1 0 4 离散傅里叶变换 2 )( 2 1 )( )()( deeFkf ekfeF kjj k kjj 线性 )()()()( 2121 jbFjaFtbftaf 线性)()()()( 2121 sbFsaFtbftaf线性)()()()( 2121 zbFza
2、Fkbfkaf线性)()()()( 2121 jj ebFeaFkbfkaf 时移 )()( 0 0 jFettf tj时移)()( 0 0 sFettf st 时移)()(zFzmkf m (双边)时移)()( jmj eFemkf 频移 )()( 0 0 jFtfe tj频移)()( 0 0 ssFtfe ts 频移)()( 00 zeFkfe jkj (尺度变换) 频移)()( )( 00 jjk eFkfe 尺度 变换 )( | 1 )( a jFe a batf a b j尺度 变换 )( | 1 )( a s Fe a batf s a b 尺度 变换 )()( a z Fkfak
3、 尺度 变换 )( 0 )/( )( )( jn n eF nkf kf 反转 )()(jFtf 反转 )()(sFtf 反转)()( 1 zFkf(仅限双边)反转)()( j eFkf 时域 卷积 )()()(*)( 2121 jFjFtftf 时域 卷积 )()()(*)( 2121 sFsFtftf 时域 卷积 )()()(*)( 2121 zFzFtftf 时域 卷积 )()()(*)( 2121 jj eFeFkfkf 频域 卷积 )(*)( 2 1 )()( 2121 jFjFtftf 时域 微分 )0()0()()( )0()()( 2 ysysFstf fssFtf 时域 差分
4、 ) 1()0()()2( )0()() 1( )2() 1()()2( )1()() 1( 22 12 1 zffzzFzkf zfzzFkf ffzzFzkf fzFzkf 频域 卷积 deFeFkfkf jj )()( 2 1 )()( )( 2 2 121 时域 微分 )()()()()( )( jFjjFjtftf nn 时域 差分 )()1() 1()( jj eFekfkf 频域 微分 n n n d jFd d jdF jtfjtttf )()( )()()( S域 微分 n n n ds sFd sFtftttf )( )()()()( Z 域 微分 dz zdF zkkf
5、)( )( 频域 微分 d edF jkkf j )( )( 时域 积分 )()0( )( 0)(,)(F j jF fdxxf t时域 积分 s f s sF dxxf t )0()( )( )1(部分 求和 1 )()(*)( z z ifkkf k i 时域 累加 k j j j k keF e eF kf)2()( 1 )( )( 0 频域 积分 0)(,)( )( )( )0(FdjF jt tf tf S域 积分 s dF t tf )( )( Z 域 积分 d F z mk kf z m m 1 )()( )(lim)0(zFf z , )0()(lim)1(zfzzFf z 对
6、称)(2)(fjtF初值 )(),(lim)0(sFssFf s 为真分式 初值 )(lim)(zFzMf M z (右边信号) ,)()(lim) 1( 1 MzfzFzMf M z 帕斯 瓦尔 djFdttfE 22 | )(| 2 1 | )(|终值 0),(lim)( 0 sssFf s 在收敛域内 终值 )()1(lim)( 1 zFzf z (右边信号) 帕斯 瓦尔 2 22 |)(| 2 1 |)(|deFkf j k 信号与系统公式性质一览表 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 dtetfjF tj )()( 拉普拉斯变换对(单边) 0 )()
7、(dtetfsF st Z 变换对(单边) 0 )()( k k zkfzF 函数 )(tf 傅里叶变换 )( jF 函数 )(tf 象函数 )(sF 函数 0),(kkf 象函数 函数 0),(kkf 象函数 1)(t)(21)(t 1 )(k 1 0),(mmk m z )()( )( tt nn jj)( )(ts 1 1z z 0),(mmk m z z z 1 )(t)( 1 j )(t s 1 )(k 1z z )( 2 kk 3 2 )1(z zz )(tt 2 1 )(j )()(tttt n 12 !1 n s n s )(kk 2 ) 1( z z )()1(kak k 2
8、 2 )(az z 0, )()(ttete tt 2 )( 11 j j )()(ttete tt 2 )( 11 s s )(kak az z )( 1 kkak 2 )(az z )sin( )cos( 0 0 t t )()( )()( 00 00 j )()cos(tt 22 s s )(ke k ez z )(kkak 2 )(az az t 1 )sgn(j)()sin(tt 22 s )(ke kj j ez z )( 2 kak k 3 22 )(az zaaz |t 2 2 )()cosh(tt 22 s s )( 2 )( k a aa kk 22 az z )( 2 )
9、( k a aa kk 22 2 az z tj e 0 )(2 0 )()sinh(tt 22 s )( 2 ) 1( k kk 3 )1( z z )( 2 )1( k kk 3 2 )1(z z )()cos(tte t 22 )( j j )()cos(tte t 22 )(s s )(k ba ba kk )(bzaz z )( 11 k ba ba kk )( 2 bzaz z )()sin(tte t 22 )( j )()sin(tte t 22 )(s )()cos(kk 1cos2 )cos( 2 zz zz )()sin(kk 1cos2 sin 2 zz z 0),(
10、| te t 22 2 )()( 10 tbtb 2 10 s sbb )()cos(kk 1cos2 )cos(cos 2 2 zz zz )()sin(kk 1cos2 )sin(sin 2 2 zz zz n tt)()(2)(2 )(nn jj)()( 1 00 teb bb t )( 01 ss bsb )()cos(kka k 22 cos2 )cos( aazz azz )()sin(kka k 22 cos2 sin aazz az )sgn(t j 2 )()sin( 1 3 ttt )( 1 222 ss )()cosh(kkak 22 cosh2 )cosh( aazz
11、azz )()sinh(kka k 22 cosh2 sinh aazz az )0( , 0, 0, te te t t 22 2 j )()sin()1 2 1 3 ttt 222 )( 1 s 0),(kk k a k az z ln )( ! k k a k z a e 2 | ,0 2 |),cos( )( t tt tf 22 ) 2 () 2 ( ) 2 cos( 2 )()sin( 2 1 ttt 222 )(s s )( ! )(ln k k a k z a 1 )!2( 1 k z 1 cosh n tjn ne F T nF n n 2 , )(2)()cos()sin(
12、 2 1 tttt 222 2 )(s s )( 1 1 k k1 ln z z z)( 12 1 k k 1 1 ln 2 1 z z z n T nTtt)()( T n n 2 )()( )()cos(ttt 222 22 )(s s )()( 1010 te bb e bb tt )( 01 ss bsb 2 | ,0 2 | , 1 )( t t tg 2 sin 2 2 Sa t ebtbb)( 110 2 01 )(s bsb )( )( )()( 2 210 2 210 2 210 te bbb e bbb e bbb t tt )()( 01 2 2 sss bsbsb t
13、W t W tSa W)sin( )( 2 | , 0 2 | , 1 )( W W jF )()sin(ttAe t ,其中 )( 10 jbb Ae j 22 01 )(s bsb )( )( )2( )( 2 210 2 210 2 2 210 te bbb te bbb e bbb t tt )()( 2 01 2 2 ss bsbsb 2 | , 0 2 | , |2 1 )( t t t tf 42 2 Sa )()( 2 1 )2( 22 210 212 tetbbb tebbeb t tt 3 01 2 2 )(s bsbsb )()sin( 22 2 210 ttAe bbb
14、 t 其中 )( )( 1 2 20 j jbbb Ae j )( 22 01 2 2 ss bsbsb 2 | ,0 2 |), 2 ( 1 )( t tt tf 2 1 2 Saej j 4 )( sin 4 )( sin )( 8 2 | ,0 2 | 2 ), |2 1( 2 | , 1 )( 11 1 2 1 1 1 t t t t tf )()sin( )( 22 2 210 ttAee bbb tt 其中 )( )()( 2 210 j jbjbb Ae j )( 22 01 2 2 ss bsbsb 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 dtetfsF s
15、t )()( 双边 Z 变换对 k k zkfzF)()( 函数象函数)(sF和收敛域函数象函数)(zF和收敛域 )(t 1,整个 S 平面 )(k 1,整个 Z 平面 )( )( t n n s ,有限 S平面 )(k n 0| , ) 1( z z z n n )(t 0Re, 1 s s )(k 1| , 1 z z z )(tt 0Re, 1 2 s s )()1(kk1| , ) 1( 2 2 z z z )( )!1( 1 t n t n 0Re, 1 s s n )( )!1( ! )!1( k nk nk 1| , ) 1( z z z n n )( t0Re, 1 s s )
16、 1( k1| , 1 z z z )( tt0Re, 1 2 s s ) 1() 1(kk1| , ) 1( 2 2 z z z )( )!1( 1 t n t n 0Re, 1 s s n ) 1( )!1( ! )!1( k nk nk 1| , ) 1( z z z n n )(te at ReRe, 1 as as )(ka k | ,az az z )(tte at ReRe, )( 1 2 as as )() 1(kan n | , )( 2 2 az az z )( )!1( 1 te n t at n ReRe, )( 1 as as n )( )!1( ! )!1( ka
17、nk nk n | , )( az az z n n )( te at ReRe, 1 as as ) 1( ka k | ,az az z )( )!1( 1 te n t at n ReRe, )( 1 as as n ) 1( )!1( ! )!1( ka nk nk n | , )( az az z n n )()cos(tt0Re, 22 s s s )()cos(kk 1cos2 cos 2 2 zz zz )()sin(tt0Re, 22 s s )()sin(kk 1cos2 sin 2 zz z )()cos(tte t ReRe, )( 22 as s s )()cos(k
18、ka k 1cos2 cos 2 2 zaz zaz )()sin(tte t ReRe, )( 22 as s )()sin(kka k 1cos2 sin 2 zaz za 0Re, | | ae t ReReRe, 2 22 asa as a 1| , | aa k | 1 | , ) 1)( ) 1( 2 a za azaz za 0Re ),sgn( | | a te t ReReRe, 2 22 asa as s 1|sgn, | aa k | 1 | , ) 1)( )( 2 a za azaz zza 卷积积分一览表 dtfftftf)()()(*)( 121 )( 1 tf)
19、( 2 tf)(*)( 21 tftf)( 1 tf)( 2 tf)(*)( 21 tftf )(tf)(t )(tf)(tf )(t )(tf )(tf)(tdf t )()(t)(t)(tt )(te t )(t )()1( 1 te t )(t)(tt )( 2 1 2 tt )( 1 te t )( 2 te t 21 12 ),()( 1 21 tee tt )(te t )(te t )(tte t )( 1 tte t )( 2 te t 12 2 12 2 12 12 )( )( 1 )( 1)( 21 tee t tt )(tt)(te t )( 11 22 te t t )
20、()cos( 1 tte t )( 2 te t 12 22 12 22 12 arctan )( )( )cos( )( )cos( 21 tee t tt )(tte t )(te t )( 2 1 2 tet t 卷积和一览表 i ikfiftftf)()()(*)( 121 )( 1t f)( 2 tf)(*)( 21 tftf)( 1 tf)( 2 tf)(*)( 21 tftf )(kf)(k)(kf)(kf)(k k i if)( )(k)(k)()1(kk)(kk)(k)() 1( 2 1 kkk )(kak)(k0),( 1 1 1 ak a a k )( 1 kak)( 2
21、 kak 21 21 1 2 1 1 ),(aak aa aa kk )(kak)(kak)() 1(kak k )(kk)(kak)( )1( ) 1( )( 1 2 k a aa k a k k )(kk)(kk)() 1()1( 6 1 kkkk )()cos( 1 kkak )(kak 21 1 21 2 2 2 1 1 2 1 1 cos sin arctan )( cos )cos() 1(cos aa a k aaaa aka kk 关于)(t、)(k函数公式一览表 )()0()()(tfttf)()()()( 000 tttftttf)()()()(tttt)()0()()0()()(tftfttf )0()()(fdtttf)()()( 00 tfdttttf )( |)(| 1 )( 1 i n i i tt tf tf )0()1()()( )()(nnn fdtttf )( | 1 )(t a at t tddtt)()(1)()()(0)(tddtt t )()()()()()( 00000 tttftttftttf )( 1 | 1 )( )()( t a a at n n n )()()()(kkkak k fkkf kfkkf )0()()( )()0()()( )()()( 00 tfdttttf
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