常用傅里叶变换对

表6.3常用的连续傅里叶变换对及其对偶关系∫+∞∞−=ωωπωdeFtftj)(21)(dtetfFtj∫+∞∞−−=ωω)()(连续傅里叶变换对相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√)(tδ11)(2ωπδ√√)(tdtdδωjt)(2ωδωπddj)(tdtdkkδkj)(ωkt)(2ωδωπkkkddj√)(tu)(1ωπδω+jtjtπδ21)(21−)(ωu)(ttu21)(ωωδωπ−ddj⎩⎨⎧−=0,10,1)sgn(tttωj20,1≠tπ⎩⎨⎧−=0,0,)(ωωωjjF√)(0tt−δ0tjeω−tje0ω)(20ωωπδ−√t0cosω)]()([00ωωδωωδπ−++)()(00tttt−++δδ0cos2tωt0sinω)]()([00ωωδωωδπ−−+j)()(00tttt−−+δδ0sin2tjω√⎪⎩⎪⎨⎧=ττtttf,0,1)()2(ωττSa)(WtSaWπ⎪⎩⎪⎨⎧=WWFωωω,0,1)(√√⎪⎩⎪⎨⎧−=τττttttf,0,1)()2(2ωττSa)2(22WtSaWπ⎪⎩⎪⎨⎧−=ωωωω,0,1)(√0}Re{),(−atueatωja+1jt−τ10),(2−τωπτωue0}Re{,−aeta222aa+ω22ττ+t0,−τπωτe√0}Re{),(cos0−attueatω202)(ωωω+++jaja√0}Re{),(sin0−attueatω2020)(ωωω++ja0}Re{),(−atuteat2)(1ωja+0,)(12−ττjt)(2ωπωτωue−0}Re{),()!1(1−−−atuketatkkja)(1ω+√∑+∞−∞=−=lTlTtt)()(δδ∑+∞−∞=−kTkT)2(2πωδπ√2)(τte−2)2(ωττπ−e√ttutu0cos)]2()2([ωττ−−+]2)0(2)0([2τωωτωωτ−++SaSa∑+∞−∞=ktjkkeF0ω∑+∞−∞=−kkkF)(20ωωδπ2连续傅里叶变换性质及其对偶关系∫+∞∞−=ωωπωdeFtftj)(21)(dtetfFtj∫+∞∞−−=ωω)()(1(0)()2fFdωωπ+∞−∞=∫(0)()Fftdt+∞−∞=∫连续傅里叶变换对相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+√尺度比例变换0),(≠aatf)(1aFaω对偶性)(tf)(ωg)(tg)(2ωπ−f√√时移)(0ttf−0)(tjeFωω−频移tjetf0)(ω)(0ωω−F√时域微分性质)(tfdtd)(ωωFj频域微分性质)(tjtf−)(ωωFdd√时域积分性质∫∞−tdfττ)()()0()(ωδπωωFjF+频域积分性质)()0()(tfjttfδπ+−σσωdF∫∞−)(√时域卷积性质)(*)(thtf)()(ωωHF频域卷积性质)()(tptf)(*)(21ωωπPF√√对称性)(tf−)(*tf)(*tf−)(ω−F)(*ω−F)(*ωF奇偶虚实性质)(tf是实函数{})()(tfOdtfo={})()(tfEvtfe={})(ImωFj{})(ReωF希尔伯特变换)()()(tutftf=)()()(ωωωjIRF+=πωωω1*)()(IR=√时域抽样∑+∞−∞=−nnTttf)()(δ∑+∞−∞=−kTkFT)2(1πω频域抽样∑+∞−∞=−nntf)2(100ωπω∑+∞−∞=−kkF)()(0ωωδω√帕什瓦尔公式ωωπdFdttf22)(21)(∫∫∞∞−∞∞−=取反----------取反共轭----共轭取反共轭取反----共轭3基本的离散傅里叶级数对∫+∞∞−=ωωπωdeFtftj)(21)(dtetfFtj∫+∞∞−−=ωω)()(1(0)()2fFdωωπ+∞−∞=∫(0)()Fftdt+∞−∞=∫离散傅里叶级数对相对偶的离散傅里叶级数对重要周期N的序列~][nf傅里叶级数系数~kF周期N的序列~][nf傅里叶级数系数~kF重要√√√√√√√√√√√4双边拉氏变换对与双边Z变换对的类比关系()()stFsftedt+∞−−∞=∫()[]nnFzfnz+∞−=−∞=∑双边拉氏变换对双边Z变换对重要连续时间函数)(tf像函数)(sF和收敛域离散时间序列][nf像函数)(zF和收敛域重要√)(tδ1,整个s平面[]nδ1，整个Z平面√)(tk）（δks，有限s平面[]knδΔ1(1)kz−−，0z√)(tus1，0}Re{s[]un11(1)z−−，1z√√)(ttu21s，0}Re{s(1)[]nun+211(1)z−−，1z√1()(1)!ktutk−−1ks,0}Re{s(1)![]!(1)!nkunnk+−−11(1)kz−−，1z()ut−−s1，Re{}0s[1]un−−−11(1)z−−，1z()tut−−21s，Re{}0s(1)[1]nun−+−−211(1)z−−，1z1()(1)!ktutk−−−−1ks,Re{}0s(1)![1]!(1)!nkunnk+−−−−−11(1)kz−−，1z√()ateut−1sa+,Re{}Re()sa−[]naun11(1)az−−，za√√()atteut−21()sa+,Re{}Re()sa−(1)[]nnaun+211(1)az−−，za1()(1)!katteutk−−−1()ksa+,Re{}Re()sa−(1)![]!(1)!nnkaunnk+−−11(1)kaz−−，za()ateut−−−1sa+,Re{}Re()sa−[1]naun−−−11(1)az−−,za1()(1)!katteutk−−−−−1()ksa+,Re{}Re()sa−(1)![1]!(1)!nnkaunnk+−−−−−11(1)kaz−−,za√0cos()tutω220ssω+，0}Re{s0cos[]nunΩ101201(cos)1(2cos)zzz−−−−Ω−Ω+√√0sin()tutω0220sωω+，0}Re{s0sin[]nunΩ10120(sin)1(2cos)zzz−−−Ω−Ω+√√0cos()atetutω−220()ssaω++，Re{}sa−0cos[]nanunΩ101201(cos)1(2cos)azazz−−−−Ω−Ω+√0sin()atetutω−0220()saωω++，Re{}sa−0sin[]nanunΩ10120(sin)1(2cos)azazz−−−Ω−Ω+ate−，Re{}0a222asa−−，Re{}Re{}Re{}asa−na，1a11111()(1)(1)aazazaz−−−−−−−−，1azasgn()atet−，Re{}0a222ssa−，Re{}Re{}Re{}asa−sgn[]nan，1a21111(1)(1)zazaz−−−−−−−，1aza