积分变换公式

1/6积分变换傅立叶级数f(t)=a02+∑(ancosnπtl∞n=1+bnsinnπtl)a0=1l∫f(τ)dτl−lan=1l∫f(τ)cosnπτll−ldτbn=1l∫f(τ)sinnπτll−ldτn=1,2,…傅立叶积分公式f̂(ω)=∫f(τ)e−jωτ+∞−∞dτf(t)=12π∫f̂+∞−∞(ω)ejωtdω狄里克雷积分公式∫sinωω+∞0dω=π2ℱ[e−βt2]=√πβe−ω24β对称公式f(t)↔f̂(ω)f̂(t)↔2πf(−ω)欧拉公式cosnω0t=12(ejnω0t+e−jnω0t)sinnω0t=j2(e−jnω0t−ejnω0t)f̂(ω)为f(t)的频谱密度函数，模|f̂(ω)|称为振幅频谱，简称频谱，φ(ω)=argf̂(ω)为相位频谱。2/6δ函数(i)δ(t−t0)={+∞t=t00t≠t0(ii)∫δ(t−t0)+∞−∞dt=1δ函数的筛选性质∫δ(t−t0)baφ(t)dt=φ(t0),at0𝑏δ函数性质1.δ(t)是偶函数。2.α(t)在t0邻域内连续α(t)δ(t−t0)=α(t0)δ(t−t0)海维赛函数H(t)={1,t≥00,t0H′(t)=δ(t)∫δ(n)(t−t0)φ(t)dt+∞−∞=(−1)nφ(n)(t0)ℱ[δ(t−t0)]=∫δ(t−t0)+∞−∞e−jωtdt=e−jωt0ℱ−1[δ(ω−ω0)]=12π∫δ(ω−ω0)ejωt+∞−∞dω=12πejω0t{δ(t−t0)↔e−jωt0δ(t)↔1{ejω0t↔2πδ(ω−ω0)1↔2πδ(ω)|ℱ[δ(t−t0)]|=|e−jωt0|=1H(t)↔1jω+πδ(ω)ℱ[ejat]=2πδ(ω−a)ℱ[cosat]=π[δ(ω+a)+δ(ω−a)]ℱ[sinat]=πj[δ(ω+a)−δ(ω−a)]sgnt={1,t0−1,t03/6sgnt=2H(t)−1ℱ[sgnt]=2jωℱ[e−βtH(t)]=1β+jωδ(at)=1|a|δ(t),a≠0δ(t2−a2)=12|a|[δ(t+a)−δ(t−a)],a≠0傅立叶变换性质线性性质ℱ[αf(t)+βg(t)]=αf̂(ω)+βĝ(ω)ℱ−1[αf̂(ω)+βĝ(ω)]=αf(t)+βg(t)位移性质ℱ[f(t−t0)]=e−jωt0f̂(ω)ℱ−1[f̂(ω−a)]=ejatf(t)相似性质ℱ[f(at)]=1|a|f̂(ωa)微分性质ℱ[f’(t)]=jωf̂(ω)ℱ[f(n)(t)]=(jω)nf̂(ω)ℱ[−jtf(t)]=ddωf̂(ω)(−j)nℱ[tnf(t)]=dndωnf̂(ω)ℱ[tnf(t)]=jndndωnf̂(ω)积分性质ℱ[∫f(τ)dτt−∞]=1jωf̂(ω)+πf̂(0)δ(ω)卷积f(t)∗g(t)=∫f(τ)g(τ−t)dτ+∞−∞f∗g=g∗f(f∗g)∗h=f∗(g∗h)f∗(g+h)=f∗g+f∗h4/6卷积定理ℱ[f(t)∗g(t)]=f̂(ω)ĝ(ω)ℱ[f1(t)∗f2(t)∗⋯∗fn(t)]=f̂1(ω)f̂2(ω)⋯f̂n(ω)ℱ[f(t)g(t)]=12πf̂(ω)∗ĝ(ω)ℱ[f1(t)f2(t)⋯fn(t)]=1(2π)n−1f̂1(ω)∗f̂2(ω)∗⋯∗f̂n(ω)δ(t−a)∗f(t)=f(t−a)δ(t−a)∗δ(t−b)=δ(t−a−b)拉普拉斯变换F(s)=ℒ[f(t)]=∫f(t)e−stdt+∞0ℒ[f(t)]=ℱ[f(t)e−βtH(t)]逆变换反演积分公式f(t)=ℒ−1[F(s)]=12πj∫F(s)estdsβ+j∞β−j∞(t0)周期函数的拉普拉斯变换：f(t)在[0,+∞)内是以T为周期的函数F(s)=11−e−sT∫f(t)e−stdtT0拉普拉斯变换性质1.线性性质ℒ[αf(t)+βg(t)]=αF(s)+βG(s);ℒ−1[aF(s)+βG(s)]=αf(t)+βg(t)2.相似性质ℒ[f(at)]=1aF(sa)a03.微分性质导数的象函数ℒ[f′(t)]=sF(s)−f(0)ℒ[f(n)(t)]=snF(s)−sn−1f(0)−sn−2f′(0)−⋯−f(n−1)(0)象函数的导数L[−tf(t)]=F′(s)(−1)nℒ[tnf(t)]=F(n)(s)4.积分性质积分的象函数ℒ[∫f(t)dtt0]=1sF(s)ℒ[∫dtt0∫dtt0⋯∫f(t)dtt0]=1snF(s)象函数的积分5/6ℒ[f(t)t]=∫F(s)ds∞sℒ[f(t)tn]=∫ds∞s∫ds∞s⋯∫F(s)ds∞s位移性质ℒ[eatf(t)]=F(s−a)a是复常数延迟性质ℒ[f(t−τ)H(t−τ)]=e−sτF(s)卷积与卷积定理ℒ[f1(t)∗f2(t)∗⋯∗fn(t)]=F1(s)F2(s)⋯Fn(s)初值定理与终值定理初值定理f(0+)=limt→0+f(t)=lims→∞sF(s)终值定理f(+∞)=limt→+∞f(t)=lims→0sF(s)Re(s)−𝑐解析幂函数的拉式变换ℒ[tm]=Γ(m+1)sm+1Re(s)0若当定理s=β+Rej(θ+π2)0≤θ≤π在区域Re(s)≤β内，lims→∞F(s)=0，函数F(s)est沿半圆CR的积分存在limR→∞∫F(s)estdsCR展开定理F(s)在复平面s上有限个奇点在Re(s)𝛽内，设s→∞时，F(s)→0f(t)=12πj∫F(s)β+j∞β−j∞estds=∑Res[F(s)est,sk]nk=16/6常见拉氏变换：ℒ[H(t)]=1sRe(s)0ℒ−1[1s]=1t0eat↔1s−aRe(s)𝑎;e−at↔1s+aRe(s)−𝑎;ejωt↔1s−jωRe(s)0sinat↔as2+a2Re(s)0;𝑐𝑜𝑠𝑎𝑡↔ss2+a2Re(s)0δ(t−a)↔e−at(a≥0);δ(t)↔1