信号与系统常用变换对及性质梳理

常用的连续傅里叶变换对及其对偶关系j1()()ed2πtftFωωω+∞−∞=∫j()()edtFfttωω+∞−−∞=∫1(0)()d2πfFωω+∞−∞=∫(0)()dFftt+∞−∞=∫连续傅里叶变换对对偶的连续傅里叶变换对重要连续时间函数)(tf傅里叶变换)(ωF连续时间函数)(tf傅里叶变换)(ωF重要√冲激)(tδ1直流12π()δω√√冲激偶'()tδjωt2πj'()δω()()ntδ(j)nωnt()2πj()nnδω√阶跃)(tu1π()jδωω+11()22πjttδ−)(ωu单位斜变)(ttu21jπ'()δωω−√符号1,0sgn()0,01,0tttt⎧⎪==⎨⎪−⎩2jω1,0πtt≠j,0()0,0j,0Fωωωω−⎧⎪==⎨⎪⎩√冲激延时)(0tt−δ0jetω−复指数信号0jetω02π()δωω−√√余弦0cos()tω00π[()()]δωωδωω++−)()(00tttt−++δδ02cos()tω√正弦0sin()tω00jπ[()()]δωωδωω+−−)()(00tttt−−+δδ0j2sin()tω√门脉冲1,/2()0,/2tGttτττ⎧⎪=⎨≥⎪⎩抽样函数Sa()2ωττ抽样脉冲ccSa()πtωω低通cc2c1,()0,Gωωωωωω⎧⎪=⎨≥⎪⎩√√三角1,()0,ttfttτττ⎧−⎪=⎨≥⎪⎩2Sa()2ωττ2ccSa()2π2tωωccc1,()0,Fωωωωωωω⎧−⎪=⎨≥⎪⎩√单边指数e(),0atuta−1jaω+1jtτ−2πe(),0uτωωτ−√双边指数e,0ata−222aaω+22ττ+tπe,0τωτ−√0ecos()(),0attutaω−220j(j)aaωωω+++√0esin()(),0attutaω−0220(j)aωωω++指数脉冲e(),0attuta−21(j)aω+21,0(j)tττ−2πe()uτωωω−1e(),0(1)!kattutak−−−1(j)kaω+√时域周期冲激序列11111()()()()TnnttnTnωδδωδωωδω+∞+∞=−∞=−∞=−↔−=∑∑频域周期冲激序列√√钟形脉冲2()etτ−钟形脉冲2()2πeωττ−√矩形调幅0()()22cosututtττω⎡⎤+−−⎢⎥⎣⎦00()()SaSa222ωωτωωττ+−⎡⎤+⎢⎥⎣⎦√1j1()()entnftFnωω+∞=−∞=∑11()2π()()nFFnnωωδωω+∞=−∞=−∑,11011()()nFnFTωωωω==,1100()()()()()22TTftftututFω⎡⎤=+−−↔⎢⎥⎣⎦连续傅里叶变换性质及其对偶关系j1()()ed2πtftFωωω+∞−∞=∫j()()edtFfttωω+∞−−∞=∫1(0)()d2πfFωω+∞−∞=∫(0)()dFftt+∞−∞=∫连续傅里叶变换对对偶的连续傅里叶变换对重要名称连续时间函数)(tf傅里叶变换)(ωF名称连续时间函数)(tf傅里叶变换)(ωF重要√线性)()(21tftfβα+)()(21ωβωαFF+√尺度变换0),(≠aatf)(1aFaω尺度+时移(),0fatba−≠)(1aFaωjebaω−√√对偶性()()ftFω↔互易性()2π()Ftfω↔−√√时移)(0ttf−0j()etFωω−频移0j()etftω)(0ωω−F√√时域微分'()ftj()Fωω频域微分(j)()tft−'()Fω√√时域积分(1)()()dtftfττ−−∞=∫()π(0)()jFFωδωω+频域积分()π(0)()jftfttδ+−()dFωσσ−∞∫√时域卷积)(*)(thtf)()(ωωHF频域卷积)()(tptf1()*()2πFPωω√)(tf为实函数()()j()FRXωωω=+实部()Rω为偶函数虚部()Xω为奇函数√反褶共轭对称性)(tf−时域反褶)(*tf共轭)(*tf−共轭取反)(ω−F频域反褶)(*ω−F共轭取反)(*ωF共轭奇偶虚实性{}e()even()ftft=实偶{}o()odd()ftft=实奇()()FRωω=实偶()j()FXωω=虚奇√希尔伯特变换)()()(tutftf=()()j()FRIωωω=+1()()*πRIωωω=√时域抽样s()()nfttnTδ+∞=−∞−∑ss1()nFnTωω+∞=−∞−∑频域抽样ss1()nftnTω+∞=−∞−∑s()()nFnωδωω+∞=−∞−∑√帕塞瓦尔定理221()d()d2πfttFωω∞∞−∞−∞=∫∫双边拉普拉斯变换对与双边z变换对的类比关系()()edstFsftt+∞−−∞=∫()()nnXzxnz+∞−=−∞=∑双边拉普拉斯变换对双边z变换对重要连续时间函数)(tf象函数)(sF和收敛域离散时间序列()xn象函数()Xz和收敛域重要√)(tδ1,整个s平面()nδ1,整个z平面√n阶导数()()ntδns,有限s平面k阶后向差分()knδ∇(1)kkzz−,0z√)(tu1s,0σ()un1zz−,1z√√)(ttu21s,0σ()nun2(1)zz−,1z√√(),ntutn+∈]1!nns+,0σ!()()!!nunnkk−1(1)kzz+−,1z√()ut−−1s,0σ(1)un−−−1zz−,1z()tut−−21s,0σ(1)nun−−−2(1)zz−,1z(),ntutn+−−∈]1!nns+,0σ!(1)()!!nunnkk−−−−1(1)kzz+−,1z√e()atut−1sa+,aσ−()naunzza−,za√1()nnaun−2()zza−,za√√e()attut−21()sa+,aσ−(1)()nnaun+22()zza−,za√√e(),nattutn−+∈]1!()nnsa++,aσ−(1)!()(1)!!nnaunnkk++−11()kkzza++−,za√e()atut−−−1sa+,aσ−(1)naun−−−zza−,zae()attut−−−21()sa+,aσ−(1)(1)nnaun−+−−22()zza−,zae(),nattutn−+−−∈]1!()nnsa++,aσ−(1)!(1)(1)!!nnaunnkk+−−−+−11()kkzza++−,za√0cos()()tutω220ssω+,0σ0cos()()nunω020(cos)2cos1zzzzωω−−+,1z√√0sin()()tutω0220sωω+,0σ0sin()()nunω020sin2cos1zzzωω−+,1z√√0ecos()()attutω−220()sasaω+++,aσ−0cos()()nanunω0220(cos)2coszzazazaωω−−+,za√√0esin()()attutω−0220()saωω++,aσ−0sin()()nanunω0220sin2cosazzazaωω−+,za√eat−,0a222asa−−,aaσ−na,1a11()()()aazzaza−−−−−,1aza−esgn()att−,0a222ssa−,aaσ−sgn()nan,1a211()()zzaza−−−−,1aza−