(完整word版)信号与系统的公式汇总分类

1连续傅里叶变换 ⎰⎰∞∞-∞∞--==ωωπωωωd e j F t f dtet f j F t j tj )(21)()()(2连续拉普拉斯变换(单边) ⎰⎰∞+∞-∞-==-j j ststdse s F j tf dte tf s F σσπ)(21)()()(0 3离散Z 变换(单边) ⎰∑≥==-∞=-Lk k kk dz z z F j k f z k f z F 0,)(21)()()(10π4离散傅里叶变换 ⎰∑==∞-∞=-πθθθθθπ2)(21)()()(d e eF k f e k f e F k j j k kj j线性 )()()()(2121ωωj bF j aF t bf t af +↔+线性 )()()()(2121s bF s aF t bf t af +↔+线性 )()()()(2121z bF z aF k bf k af +↔+ 线性 )()()()(2121θθj j e bF e aF k bf k af +↔+时移)()(00ωωj F e t t f t j ±↔± 时移)()(00s F e t t f st ±↔± 时移)()(z F z m k f m ±↔±（双边）时移)()(θθj m j e F e m k f ±↔±频移 ))(()(00ωωω j F t f e t j ↔±频移 )()(00s s F t f e t s ↔±频移 )()(00z e F k f e j k j ωω ↔±（尺度变换）频移 )()()(00θθθ j jk e F k f e ↔±尺度 变换 )(||1)(aj F e a b at f a bj ωω↔+尺度 变换 )(||1)(asF e a b at f s a b↔+尺度 变换 )()(azF k f a k ↔尺度 变换 )(0)/()()(θjn n e F n k f k f ↔⎩⎨⎧=反转 )()(ωj F t f -↔- 反转 )()(s F t f -↔- 反转 )()(1-↔-z F k f （仅限双边） 反转 )()(θj e F k f -↔-时域 卷积 )()()(*)(2121ωωj F j F t f t f ↔时域 卷积)()()(*)(2121s F s F t f t f ↔时域 卷积)()()(*)(2121z F z F t f t f ↔时域 卷积 )()()(*)(2121θθj j e F e F k f k f ↔频域 卷积 )(*)(21)()(2121ωωπj F j F t f t f ↔时域 微分)0()0()()()0()()(2---'--↔''-↔'y sy s F s t f f s sF t f时域 差分)1()0()()2()0()()1()2()1()()2()1()()1(22121zf f z z F z k f zf z zF k f f f z z F z k f f z F z k f --↔+-↔+-+-+↔--+↔----频域卷积 ψπθψπψd e F eF k f k f j j )()(21)()()(22121-⎰↔时域 微分 )()()()()()(ωωωωj F j j F j t f t f n n ↔'时域 差分 )()1()1()(θθj j e F e k f k f -↔--频域 微分 nn n d j F d d j dF jt f jt t tf ωωωω)()()()()(↔-S 域 微分 nn n ds s F d s F t f t t tf )()()()()('-↔-Z 域 微分 dzz dF zk kf )()(-↔频域 微分θθd e dF jk kf j )()(↔时域 积分 )()0()(0)(,)(ωδπωωF j j F f dx x f t+↔=-∞⎰∞-时域 积分 sf s s F dx x f t)0()()()1(--∞-+↔⎰部分 求和 1)()(*)(-↔=∑-∞=z zi f k k f ki ε时域 累加∑∑∞-∞=∞-∞=-+-↔k j j j k k e F e e F k f )2()(1)()(0πθδπθθ频域 积分 0)(,)()()()0(=-∞↔-+⎰∞-F d j F jt t f t f ωττπS 域 积分 ⎰∞↔s d F tt f ηη)()(Z 域 积分 ηηηd F z mk k f z m m⎰∞+↔+1)()()(lim )0(z F f z →∞=,)]0()([lim )1(zf z zF f z -=→∞对称 )(2)(ωπ-↔f jt F初值)(),(lim )0(s F s sF f s →∞+=为真分式初值)(lim )(z F z M f M z ∞→=（右边信号）,)()([lim )1(1M zf z F z M f M z -=++∞→帕斯 瓦尔⎰⎰∞∞-∞∞-==ωωπd j F dt t f E 22|)(|21|)(|终值0),(lim )(0==∞→s s sF f s 在收敛域内终值)()1(lim )(1z F z f z -=∞→（右边信号）帕斯 瓦尔⎰∑∞-∞==πθθπ222|)(|21|)(|d e F k f 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⎰∞∞--=dt et f j F tj ωω)()(拉普拉斯变换对（单边）⎰∞--=0)()(dt e t f s F stZ 变换对（单边） ∑∞=-=0)()(k k z k f z F函数 )(t f傅里叶变换)(ωj F 函数)(t f象函数)(s F函数0),(≥k k f象函数函数0),(≥k k f象函数1)(t δ )(21ωπδ)(t δ1)(k δ10),(≥-m m k δ m z -)()()(t t n δδ'n j j )(ωω)(t δ' s11-z z 0),(≥-m m k εm z z z-⋅-1 )(t ε)(1ωπδω+j )(t εs1)(k ε1-z z )(2k k ε32)1(-+z z z )(t t ε21)(ωωδπ-'j )()(t t t t nεε12!1+n s n s )(k k ε2)1(-z z )()1(k a k kε+22)(a z z - 0,)()(>--αεεααt te t e t t2)(11ωαωαj j ++)()(t te t e t t εεαα--2)(11αα++s s)(k a k εaz z - )(1k ka k ε-2)(a z z - )sin()cos(00t t ωω)]()([)]()([0000ωωδωωδπωωδωωδπ--+-++j)()cos(t t εβ 22β+s s)(k e k εα αe z z - )(k ka k ε2)(a z az - t1)sgn(ωπj -)()sin(t t εβ 22ββ+s )(k e k j εββj e z z - )(2k a k k ε322)(a z z a az -+||t22ω-)()cosh(t t εβ 22β-s s )(2)(k aa a kk ε-- 22a z z - )(2)(k aa a kk ε-+222a z z - tj e0ω± )(20ωωπδ)()sinh(t t εβ22ββ-s )(2)1(k k k ε- 3)1(-z z)(2)1(k kk ε+ 32)1(-z z )()cos(t t etεβα-22)(βαωαω+++j j )()cos(t t etεβα-22)(βαα+++s s )(k ba b a kk ε-- ))((b z a z z--)(11k ba b a k k ε--++))((2b z a z z -- )()sin(t t e t εβα-22)(βαωβ++j)()sin(t t e t εβα-22)(βαβ++s)()cos(k k εβ1cos 2)cos (2+--ββz z z z)()sin(k k εβ1cos 2sin 2+-ββz z z0),(||>-αεαt et222ωαα+)()(10t b t b ε+ 210s s b b + )()cos(k k εθβ+1cos 2)cos(cos 22+---βθβθz z z z )()sin(k k εθβ+1cos 2)sin(sin 22+--+βθβθz z z znt t )()(2)(2)(ωδπωδπn n j j ')()(10t e b b b t εααα---)(01α++s s b s b )()cos(k k a k εβ22cos 2)cos (a az z a z z +--ββ )()sin(k k a k εβ22cos 2sin a az z az +-ββ )sgn(t ωj 2)()]sin([13t t t εβββ-)(1222β+s s)()cosh(k k a k εβ 22cosh 2)cosh (aaz z a z z +--ββ)()sinh(k k a k εβ22cosh 2sinh a az z az +-ββ)0(,0,0,>⎪⎩⎪⎨⎧><--αααt e t e tt 222ωαω+-j)()sin()]1[213t t t εβββ- 222)(1β+s0),(>k k ka kε⎪⎭⎫ ⎝⎛-a z z ln )(!k k a kε za e⎪⎪⎩⎪⎪⎨⎧><=2||,02||),cos()(τττπt t t t f22)2()2()2cos(2ωτπωτπτ-⋅)()sin(21t t t εββ222)(β+s s)(!)(ln k k a kε za 1)!2(1kz1cosh∑∞-∞=Ωn tjn n e FTn F n n πωδπ2,)(2=ΩΩ-∑∞-∞=)()]cos()[sin(21t t t t εββββ+ 2222)(β+s s)(11k k ε+ ⎪⎭⎫ ⎝⎛-1ln z z z )(121k k ε+ 11ln21-+z z z ∑∞-∞=-=n T nT t t )()(δδTn n πωδωδ2)()(=ΩΩ-Ω=∑∞-∞=Ω )()cos(t t t εβ22222)(ββ+-s s)(])([1010t e b b e b b tt εαββαβαβα----+--))((01βα+++s s b s b⎪⎪⎩⎪⎪⎨⎧><=2||,02||,1)(τττt t t g⎪⎭⎫⎝⎛=⎪⎭⎫⎝⎛2sin 22ωτωωττSateb t b b αα-+-])[(110201)(α++s b s b)(]))(())(())(([221022102210t e b b b eb b b e b b b t tt εγβγαγγβγβαββαγαβααγβα-----+-+--+-+--+-))()((0122γβα+++++s s s b s b s btW t W t Sa Wππ)sin()(=⎪⎪⎩⎪⎪⎨⎧><=2||,02||,1)(W W j F ωωω)()sin(t t Ae t εθβα+-,其中ββαθ)(10j b b Ae j --=2201)(βα+++s b s b)(])()2()([2210221022210t eb b b te b b b eb b b tttεαβαβαβαβααβαββααβ-----+--⋅-+-+-+-)()(20122βα++++s s b s b s b⎪⎪⎩⎪⎪⎨⎧><-=∆2||,02||,||21)(τττt t t t f⎪⎭⎫⎝⎛422ωττSa )(])(21)2([22210212t e t b b b teb b eb t ttεαααααα---+-+-+3122)(α+++s b s b s b)()]sin([222210t t A e b b b t εθββγγγγ++++--其中)()(1220βγβββθj jb b b Ae j ++-=))((220122βγ++++s s b s b s b⎪⎪⎩⎪⎪⎨⎧><+=2||,02||),2(1)(ττττt t t t f⎥⎥⎦⎤⎢⎢⎣⎡⎪⎭⎫⎝⎛--212ωτωωτSa ejj ⎥⎦⎤⎢⎣⎡-⨯⎥⎦⎤⎢⎣⎡+-↔⎪⎪⎪⎩⎪⎪⎪⎨⎧><<--<=4)(sin 4)(sin )(82||,02||2),||21(2||,1)(1112111ττωττωττωττττττττt t t t t f)()]sin()([222210t t Ae e b b b t t εθββγαγγαγ+++-+---其中)()()(2210βαγββαβαθj j b j b b Ae j +--+--=)]))[((220122βαγ+++++s s b s b s b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 ⎰∞∞--=dt e t f s F st)()(双边Z 变换对∑∞-∞=-=k kzk f z F )()(函数象函数)(s F 和收敛域 函数象函数)(z F 和收敛域 )(t δ 1，整个S 平面 )(k δ1，整个Z 平面)()(t n δns ，有限S 平面)(k nδ∆0||,)1(>-z z z nn)(t ε 0}Re{,1>s s)(k ε1||,1>-z z z)(t t ε0}Re{,12>s s )()1(k k ε+ 1||,)1(22>-z z z)()!1(1t n t n ε-- 0}Re{,1>s s n)()!1(!)!1(k n k n k ε--+1||,)1(>-z z z nn)(t --ε 0}Re{,1<s s)1(---k ε1||,1<-z z z)(t t --ε0}Re{,12<s s )1()1(--+-k k ε1||,)1(22<-z z z)()!1(1t n t n ----ε 0}Re{,1<s s n)1()!1(!)!1(----+-k n k n k ε 1||,)1(<-z z z nn)(t e at ε-}Re{}Re{,1a s as ->+ )(k a k ε ||||,a z az z>- )(t teatε-}Re{}Re{,)(12a s a s ->+)()1(k a n nε+||||,)(22a z a z z >-)()!1(1t e n t atn ε--- }Re{}Re{,)(1a s a s n->+)()!1(!)!1(k a n k n k nε--+||||,)(a z a z z nn>- )(t e at ---ε }Re{}Re{,1a s as -<+ )1(---k a k ε||||,a z az z<- )()!1(1t e n t at n -----ε }Re{}Re{,)(1a s a s n-<+)1()!1(!)!1(----+-k a n k n k n ε ||||,)(a z a z z nn<-)()cos(t t εβ0}Re{,22>+s s sβ)()cos(k k εβ 1cos 2cos 22+--ββz z z z)()sin(t t εβ 0}Re{,22>+s s ββ)()sin(k k εβ 1cos 2sin 2+-ββz z z)()cos(t t etεβα-}Re{}Re{,)(22a s s s ->+++βαα)()cos(k k a kεβ 1cos 2cos 22+--ββza z za z)()sin(t t e t εβα- }Re{}Re{,)(22a s s ->++βαβ)()sin(k k a k εβ1cos 2sin 2+-ββza z za0}Re{,||>-a et α}Re{}Re{}Re{,222a s a a s a->>-- 1||,||<a a k |1|||||,)1)(()1(2a z a az a z z a <<---}Re{),sgn(||>-a t e t α}Re{}Re{}Re{,222a s a a s s->>-1||sgn,||<a a k|1|||||,)1)(()(2az a az a z z z a <<---卷积积分一览表⎰∞∞--=τττd t f f t f t f )()()(*)(121)(1t f)(2t f)(*)(21t f t f)(1t f)(2t f)(*)(21t f t f)(t f )(t δ')(t f ')(t f)(t δ)(t f )(t f)(t ελλd f t⎰∞-)()(t ε )(t ε )(t t ε)(t e t εα-)(t ε)()1(1t e t εαα--)(t ε)(t t ε)(212t t ε )(1t e t εα-)(2t e t εα-2112),()(121ααεαααα≠----t e e t t)(t e t εα-)(t e t εα-)(t te t εα-)(1t te t εα- )(2t e t εα-1221221212)()(1)(1)(21ααεαααααααα≠⎥⎥⎦⎤⎢⎢⎣⎡-+-----t e e t t t )(t t ε )(t e t εα-)(1122t e t t εαααα⎪⎪⎭⎫ ⎝⎛+--)()cos(1t t e t εθβα+- )(2t e t εα-⎪⎪⎭⎫⎝⎛-=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡+---+--+--1222122212arctan )()()cos()()cos(21ααβϕεβααϕθβααϕθβααt e e t t t)(t te t εα-)(t e t εα-)(212t e t tεα-卷积和一览表∑∞-∞=-=i i k f i ft f t f )()()(*)(121)(1t f)(2t f)(*)(21t f t f)(1t f)(2t f)(*)(21t f t f)(k f)(k δ )(k f)(k f )(k ε ∑-∞=ki i f )()(k ε)(k ε )()1(k k ε+)(k k ε)(k ε)()1(21k k k ε+)(k a kε)(k ε0),(111≠--+a k aa k ε)(1k a k ε)(2k a kε21211211),(a a k a a a a k k ≠--++ε )(k a k ε)(k a kε)()1(k a k kε+)(k k ε)(k a kε)()1()1()(12k a a a k a k k εε--+- )(k k ε )(k k ε)()1()1(61k k k k ε-+ )()cos(1k k a k εθβ+)(k a k ε⎥⎦⎤⎢⎣⎡-=-+---++++2112122211211cos sin arctan )(cos )cos(])1(cos[a a a k a a a a a k a k k ββϕεβϕθϕθβ关于)(t δ、)(k δ函数公式一览表)()0()()(t f t t f δδ=)()()()(000t t t f t t t f -=-δδ)()()()(t t t t δδδδ'-=-'=-)()0()()0()()(t f t f t t f δδδ'-'=')0()()(f dt t t f =⎰∞∞-δ)()()(00t f dt t t t f =-⎰∞∞-δ)(|)(|1)([1i ni i t t t f t f -'=∑=δδ)0()1()()()()(n n n f dt t t f -=⎰∞∞-δ)(||1)(t a at δδ=⎰⎰∞-∞∞-==tt d dt t )()(1)(εττδδ)()(0)(t d dt t tδττδδ='='⎰⎰∞-∞∞-)()()()()()(00000t t t f t t t f t t t f -'--'=-'δδδ)(1||1)()()(t a a at n nn δδ⋅=)()()()(k k k ak δδδδ=-=∑∞-∞===k f k k f k f k k f )0()()()()0()()(δδδ)()()(00t f dt t t t f '-=-'⎰∞∞-δ。