1连续傅里叶变换∫∫∞∞−∞∞−−==ωωπωωωd e j F t f dtet f j F t j tj )(21)()()(2连续拉普拉斯变换连续拉普拉斯变换((单边单边))∫∫∞+∞−∞−==−j j st stds e s F jt f dte tf s F σσπ)(21)()()(03离散Z 变换变换((单边单边))∫∑≥==−∞=−Lk k kk dz z z F j k f z k f z F 0,)(21)()()(1π4离散傅里叶变换∫∑==∞−∞=−πθθθθθπ2)(21)()()(d e e F 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 et t f t j ±↔±时移)()(00s F e t t f st ±↔±时移)()(z F zm k f m±↔±（双边）时移)()(θθj m j e F e m k f ±↔±频移))(()(00ωωω∓j F t f etj ↔±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nn nd j F d d j dF jt f jt t tf ωωωω)()()()()(↔−S 域微分nn nds s F d s F t f t t tf )()()()()(′−↔−Z 域微分dzz dF zk kf )()(−↔频域微θθd e dF jk kf j )()(↔分分时域积分)()0()()(,)(ωδπωωFjjFfdxxft+↔=−∞∫∞−时域积分sfssFdxxft)0()()()1(−−∞−+↔∫部分求和1)()(*)(−↔=∑−∞=zzifkkfkiε时域累加∑∑∞−∞=∞−∞=−+−↔kjjjkkeFeeFkf)2()(1)()(0πθδπθθ频域积分)(,)()()()0(=−∞↔−+∫∞−FdjFjttftfωττπS域积分∫∞↔sdFttfηη)()(Z域积分ηηηdFzmkkfz mm∫∞+↔+1)()()(lim)0(zFfz→∞=,)]0()([lim)1(zfzzFfz−=→∞对称)(2)(ωπ−↔fjtF初值)(),(lim)0(sFssFfs→∞+=为真分式初值)(lim)(zFzMf Mz∞→=（右边信号）,)()([lim)1(1MzfzFzMf Mz−=++∞→帕斯瓦尔∫∫∞∞−∞∞−==ωωπdjFdttfE22|)(|21|)(|终值),(lim)(==∞→sssFfs在收敛域内终值)()1(lim)(1zFzfz−=∞→（右边信号）帕斯瓦尔∫∑∞−∞==πθθπ222|)(|21|)(|deFkf jk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、常用连续傅里叶变换、拉普拉斯变换、ZZ 变换对一览表连续傅里叶变换对∫∞∞−−=dtet f j F tj ωω)()(拉普拉斯变换对（单边）∫∞−−=0)()(dte tf s F stZ 变换对（单边）∑∞=−=0)()(k kz 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),(≥k k f 象函数1)(t δ)(21ωπδ)(t δ1)(k δ10),(≥−m m k δm 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 t 2)(11ωαωαj j ++)()(t te t e t t εεαα−−2)(11αα++s s )(k a k εa z z −)(1k ka k ε−2)(a z z −)sin()cos(00t t ωω)]()([)]()([0000ωωδωωδπωωδωωδπ−−+−++j )()cos(t t εβ22β+s s )(k e k εααe z z −)(k ka k ε2)(a z az−t 1)sgn(ωπj −)()sin(t t εβ22ββ+s )(k ekj εββj e z z −)(2k a k kε322)(a z z a az −+||t 22ω−)()cosh(t t εβ22β−s s )(2)(k aa a kk ε−−22a z z −)(2)(k aa a kk ε−+222a z z −t j e 0ω±)(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 e t εβα−22)(βαωαω+++j j )()cos(t t e t εβα−22)(βαα+++s s )(k b a 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 z 0),(||>−αεαt et 222ωαα+)()(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 zn t t )()(2)(2)(ωδπωδπn n j j ′)()(100t e b bb t εααα−−−)(01α++s s b s b )()cos(k k a k εβ22cos 2)cos (aaz 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 εβ22cosh 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+s 0),(>k k k a kε⎟⎠⎞⎜⎝⎛−a z z ln )(!k k a kεza e ⎪⎪⎩⎪⎪⎨⎧><=2||,02||),cos()(τττπt t t t f 22)2()2()2cos(2ωτπωτπτ−⋅)()sin(21t t t εββ222)(β+s s )(!)(ln k k a kεza 1)!2(1k z1cosh∑∞−∞=Ωn tjn n e F T n 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+11ln 21−+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 22ωτωωττSa teb t b b αα−+−])[(110201)(α++s b s b )(]))(())(())(([221022102210t e b b b eb b b e b b b ttt εγβγαγγβγβαββαγαβααγβα−−−−−+−+−−+−+−−+−))()((0122γβα+++++s s s b s b s b tWt Wt Sa W ππ)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 e b b b te b b b eb b b t ttεαβαβαβαβααβαββααβ−−−−−+−−⋅−+−+−+−)()(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 Aej ++−=))((220122βγ++++s s b s b s b⎪⎪⎩⎪⎪⎨⎧><+=2||,02||),2(1)(ττττt t t t f ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛−−212ωτωωτSa e j j ⎥⎦⎤⎢⎣⎡−×⎥⎦⎤⎢⎣⎡+−↔⎪⎪⎪⎩⎪⎪⎪⎨⎧><<−−<=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 Aej +−−+−−=)]))[((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∫∞∞−−=dte tf 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 δ1，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 ε0}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 n n)(t −−ε0}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 −−−−ε0}Re{,1<s s n)1()!1(!)!1(−−−−+−k n k n k ε1||,)1(<−z z z nn)(t e at ε−}Re{}Re{,1a s as −>+)(k a k ε||||,a z az z>−)(t te atε−}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 atn −−−−−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 εβ1cos 2cos 22+−−ββz z z z )()sin(t t εβ0}Re{,22>+s s ββ)()sin(k k εβ1cos 2sin 2+−ββz z z )()cos(t t etεβα−}Re{}Re{,)(22a s s s −>+++βαα)()cos(k k a kεβ1cos 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 za 0}Re{,||>−a et α}Re{}Re{}Re{,222a s a as a−>>−−1||,||<a a k |1|||||,)1)(()1(2az a az a z z a <<−−−0}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 a a k ε)(1k a k ε)(2k a kε21211211),(a a k a a a a k k ≠−−++ε)(k a kε)(k a k ε)()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。
