知识点公式总结

Flourier变换总结
- Flourier积分表达式
- 傅里叶变换和傅里叶逆变换
- 一些常用函数的Fourier变换
- - 变换中可能用到的公式
  - - 诱导公式
    - 指数三角转换
    - 和差化积积化和差
- δ函数的相关性质
- 傅里叶变换的相关性质
- 能量积分
- 卷积
Laplace变换总结
- 常用Laplace变换
- 性质
- 初值定理和终值定理
- 卷积定理

Flourier变换总结

Flourier积分表达式

f(t)=12π∫−∞+∞[∫−∞+∞f(τ)e−jωτdτ]ejωtdωf(t)=2π∫0+∞[∫0+∞f(τ)sin⁡ωτdτ]sin⁡ωtdωf(t)=2π∫0+∞[∫0+∞f(τ)sin⁡ωτdτ]sin⁡ωtdωf(t)=2π∫0+∞[∫0+∞f(τ)cos⁡ωτdτ]cos⁡ωtdω\begin{aligned} &f(t)=\frac{1}{2 \pi} \int_{-\infty}^{+\infty}\left[\int_{-\infty}^{+\infty} f(\tau) \mathrm{e}^{-j \omega \tau} \mathrm{d} \tau\right] \mathrm{e}^{\mathrm{j} \omega t} \mathrm{~d} \omega\\ &f(t)=\frac{2}{\pi} \int_{0}^{+\infty}\left[\int_{0}^{+\infty} f(\tau) \sin \omega \tau \mathrm{d} \tau\right] \sin \omega t \mathrm{~d} \omega\\ &f(t)=\frac{2}{\pi} \int_{0}^{+\infty}\left[\int_{0}^{+\infty} f(\tau) \sin \omega \tau \mathrm{d} \tau\right] \sin \omega t \mathrm{~d} \omega\\ &f(t)=\frac{2}{\pi} \int_{0}^{+\infty}\left[\int_{0}^{+\infty} f(\tau) \cos \omega \tau \mathrm{d} \tau\right] \cos \omega t \mathrm{~d} \omega\\ \end{aligned}f(t)=2π1∫−∞+∞[∫−∞+∞f(τ)e−jωτdτ]ejωt dωf(t)=π2∫0+∞[∫0+∞f(τ)sinωτdτ]sinωt dωf(t)=π2∫0+∞[∫0+∞f(τ)sinωτdτ]sinωt dωf(t)=π2∫0+∞[∫0+∞f(τ)cosωτdτ]cosωt dω

分别是Fourier积分表达式的复数、三角、正弦、余弦形式，表达式中有e的几次幂的，复数形式最简单，后面的形式，其实通常用欧拉公式化为e的幂函数反而更好算。cos⁡θ=ejθ+e−iθ2sin⁡θ=eiθ−e−iθ2i\cos \theta=\frac{e^{j \theta}+e^{-i \theta}}{2}\quad \sin \theta=\frac{e^{i \theta}-e^{-i \theta}}{2 i}cosθ=2ejθ+e−iθsinθ=2ieiθ−e−iθ
另外，利用后两个公式时，原本是奇函数就进行奇延拓，偶函数就偶延拓，傅里叶变换后，把区间去掉负的一半即可。

傅里叶变换和傅里叶逆变换

F(w)=∫−∞+∞f(t)e−jwtdtFs(w)=∫0+∞f(t)sin⁡wtdtFc(w)=∫0+∞f(t)cos⁡wtdtF(w)=−2jFs(w)F(w)=2F(w)F−1(w)=12π∫−∞+∞F(w)ejwtdtFs−1(w)=2π∫0t∞Fs(w)sin⁡wtdtFc−1(w)=2π∫0+∞Fc(w)cos⁡wtdt\begin{aligned} &F(w)=\int_{-\infty}^{+\infty} f(t) e^{-j w t} d t\\ &F_{s}(w)=\int_{0}^{+\infty} f(t) \sin w t d t\\ &F_{c}(w)=\int_{0}^{+\infty} f(t) \cos w t d t\\ &F(w)=-2 j F_{s}(w)\\ &F(w)=2 F(w)\\ &F^{-1}(w)=\frac{1}{2 \pi} \int_{-\infty}^{+\infty} F(w) e^{j w t} d t\\ &F_{s}^{-1}(w)=\frac{2}{\pi} \int_{0}^{t \infty} F_{s}(w) \sin w t d t\\ &F_{c}^{-1}(w)=\frac{2}{\pi} \int_{0}^{+\infty} F_{c}(w) \cos w t d t\\ \end{aligned}F(w)=∫−∞+∞f(t)e−jwtdtFs(w)=∫0+∞f(t)sinwtdtFc(w)=∫0+∞f(t)coswtdtF(w)=−2jFs(w)F(w)=2F(w)F−1(w)=2π1∫−∞+∞F(w)ejwtdtFs−1(w)=π2∫0t∞Fs(w)sinwtdtFc−1(w)=π2∫0+∞Fc(w)coswtdt

一些常用函数的Fourier变换

1→2πδ(w)u(t)⟶1jw+πδ(w)t→2πjδ′(w)tn→2πjnδ(n)(w)sin⁡w0t→jπ[δ(w+w0)−f(w−w0)]cos⁡w0t→π[δ(w+w0)+f(w−w0)]e−βt⟶1β+jw=β−jwβ2+w2eβt→1−β+jw=−(β+jw)β2+w2e−β∣t∣→2ββ2+w2e−βt2→πβe−w24βF[sgn⁡t]=2jω\begin{aligned} &1 \rightarrow 2 \pi \delta(w)\\ &u(t) \longrightarrow \frac{1}{j w}+\pi \delta(w)\\ &t \rightarrow 2 \pi j \delta^{\prime}(w)\\ &t^{n} \rightarrow 2 \pi j^{n} \delta^{(n)}(w)\\ &\sin w_{0} t \rightarrow j \pi\left[\delta\left(w+w_{0}\right)-f\left(w-w_{0}\right)\right]\\ &\cos w_{0} t \rightarrow \pi\left[\delta\left(w+w_{0}\right)+f\left(w-w_{0}\right)\right]\\ &e^{-\beta t} \longrightarrow \frac{1}{\beta+j w}=\frac{\beta-j w}{\beta^{2}+w^{2}}\\ &e^{\beta t} \rightarrow \frac{1}{-\beta+j w}=\frac{-(\beta+j w)}{\beta^{2}+w^{2}}\\ &e^{-\beta|t|} \rightarrow \frac{2 \beta}{\beta^{2}+w^{2}}\\ &e^{-\beta t^{2}} \rightarrow \sqrt{\frac{\pi}{\beta}} e^{-\frac{w^{2}}{4 \beta}}\\ &\mathscr{F}[\operatorname{sgn} t]=\frac{2}{j \omega} \end{aligned}1→2πδ(w)u(t)⟶jw1+πδ(w)t→2πjδ′(w)tn→2πjnδ(n)(w)sinw0t→jπ[δ(w+w0)−f(w−w0)]cosw0t→π[δ(w+w0)+f(w−w0)]e−βt⟶β+jw1=β2+w2β−jweβt→−β+jw1=β2+w2−(β+jw)e−β∣t∣→β2+w22βe−βt2→βπe−4βw2F[sgnt]=jω2

sgn⁡t=2u(t)−1\operatorname{sgn} t=2u(t)-1sgnt=2u(t)−1

∫0+∞sin⁡wtwdw={−π2t<00t=0π2t>0\int_{0}^{+\infty} \frac{\sin w t}{w} d w=\left\{\begin{array}{cl}-\frac{\pi}{2} & t<0 \\ 0 & t=0 \\ \frac{\pi}{2} & t>0\end{array}\right.∫0+∞wsinwtdw=⎩⎨⎧−2π02πt<0t=0t>0

u(t)=12+1π∫0+∞sin⁡wtwdwu(t)=\frac{1}{2}+\frac{1}{\pi} \int_{0}^{+\infty} \frac{\sin w t}{w} d wu(t)=21+π1∫0+∞wsinwtdw

∫−∞+∞sin⁡2ww2=π\int_{-\infty}^{+\infty} \frac{\sin ^{2} w}{w^{2}}=\pi∫−∞+∞w2sin2w=π

sin⁡3t=34sin⁡t−14sin⁡3t\sin ^{3} t=\frac{3}{4} \sin t-\frac{1}{4} \sin 3 tsin3t=43sint−41sin3t

∫0+∞cos⁡wtβ2+w2dw=π2βe−β∣t∣\int_{0}^{+\infty} \frac{\cos w t}{\beta^{2}+w^{2}} d w=\frac{\pi}{2 \beta} e^{-\beta|t|}∫0+∞β2+w2coswtdw=2βπe−β∣t∣

变换中可能用到的公式

诱导公式

sin⁡(α+π2)=cos⁡αsin⁡(a+π)=−sin⁡αsin⁡(π2−α)=cos⁡αsin⁡(π−α)=sin⁡α\begin{aligned} &\sin \left(\alpha+\frac{\pi}{2}\right)=\cos \alpha\\ &\sin (a+\pi)=-\sin \alpha\\ &\sin \left(\frac{\pi}{2}-\alpha\right)=\cos \alpha\\ &\sin (\pi-\alpha)=\sin \alpha\\ \end{aligned}sin(α+2π)=cosαsin(a+π)=−sinαsin(2π−α)=cosαsin(π−α)=sinα

cos⁡(a+π2)=−sin⁡αcos⁡(α+π)=−cos⁡αcos⁡(π2−a)=sin⁡αcos⁡(π−a)=−cos⁡α\begin{aligned} &\cos \left(a+\frac{\pi}{2}\right)=-\sin \alpha\\ &\cos (\alpha+\pi)=-\cos \alpha\\ &\cos \left(\frac{\pi}{2}-a\right)=\sin \alpha\\ &\cos (\pi-a)=-\cos \alpha\\ \end{aligned}cos(a+2π)=−sinαcos(α+π)=−cosαcos(2π−a)=sinαcos(π−a)=−cosα

指数三角转换

eiθ=cos⁡θ+isin⁡θcos⁡θ=eiθ+e−iθ2sin⁡θ=eiθ−e−iθ2isinh⁡x=ex−e−x2cosh⁡x=ex+e−x2tanh⁡x=ex−e−xex+e−x\begin{aligned} &e^{i \theta}=\cos \theta+i \sin \theta\\ &\cos \theta=\frac{e^{i \theta}+e^{-i \theta}}{2}\\ &\sin \theta=\frac{e^{i \theta}-e^{-i \theta}}{2 i}\\ &\sinh x=\frac{e^{x}-e^{-x}}{2}\\ &\cosh x=\frac{e^{x}+e^{-x}}{2}\\ &\tanh x=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\\ \end{aligned}eiθ=cosθ+isinθcosθ=2eiθ+e−iθsinθ=2ieiθ−e−iθsinhx=2ex−e−xcoshx=2ex+e−xtanhx=ex+e−xex−e−x

和差化积积化和差

sin⁡α+sin⁡β=2sin⁡α+β2cos⁡α−β2cos⁡α+cos⁡β=2cos⁡α+β2cos⁡α−β2sin⁡α+cos⁡β=2sin⁡α−β2cos⁡(α+β2−π4)sin⁡αcos⁡β=12[sin⁡(α+β)+sin⁡(α−β)]cos⁡αcos⁡β=12[cos⁡(α−β)+cos⁡(α+β)]sin⁡αsin⁡β=12[cos⁡(α−β)−cos⁡(α+β)]\begin{aligned} &\sin \alpha+\sin \beta=2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}\\ &\cos \alpha+\cos \beta=2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}\\ &\sin \alpha+\cos \beta=2 \sin \frac{\alpha-\beta}{2} \cos \left(\frac{\alpha+\beta}{2}-\frac{\pi}{4}\right)\\ &\sin \alpha \cos \beta=\frac{1}{2}[\sin (\alpha+\beta)+\sin (\alpha-\beta)]\\ &\cos \alpha \cos \beta=\frac{1}{2}[\cos (\alpha-\beta)+\cos (\alpha+\beta)]\\ &\sin \alpha \sin \beta=\frac{1}{2}[\cos (\alpha-\beta)-\cos (\alpha+\beta)]\\ \end{aligned}sinα+sinβ=2sin2α+βcos2α−βcosα+cosβ=2cos2α+βcos2α−βsinα+cosβ=2sin2α−βcos(2α+β−4π)sinαcosβ=21[sin(α+β)+sin(α−β)]cosαcosβ=21[cos(α−β)+cos(α+β)]sinαsinβ=21[cos(α−β)−cos(α+β)]

δ函数的相关性质

∫−∞+∞δ(t)f(t)dt=f(0)∫−∞+∞δ(t−t0)f(t)dt=f(t0)∫−∞+∞δ′(t)f(t)dt=−f′(0)∫−∞+∞δ(n)(t)f(t)dt=(−1)nf(n)(0)∫−∞+∞δ(n)(t−t0)f(t)dt=(−1)nf(n)(t0)\begin{aligned} &\int_{-\infty}^{+\infty} \delta(t) f(t) d t=f(0)\\ &\int_{-\infty}^{+\infty} \delta\left(t-t_{0}\right) f(t) d t=f\left(t_{0}\right)\\ &\int_{-\infty}^{+\infty} \delta^{\prime}(t) f(t) d t=-f^{\prime}(0)\\ &\int_{-\infty}^{+\infty} \delta^{(n)}(t) f(t) d t=(-1)^{n} f^{(n)}(0)\\ &\int_{-\infty}^{+\infty} \delta^{(n)}\left(t-t_{0}\right) f(t) d t=(-1)^{n} f^{(n)}\left(t_{0}\right)\\ \end{aligned}∫−∞+∞δ(t)f(t)dt=f(0)∫−∞+∞δ(t−t0)f(t)dt=f(t0)∫−∞+∞δ′(t)f(t)dt=−f′(0)∫−∞+∞δ(n)(t)f(t)dt=(−1)nf(n)(0)∫−∞+∞δ(n)(t−t0)f(t)dt=(−1)nf(n)(t0)

还可以推导出
tδ′(t)=−f(t)(1−et)δ′(t)=δ(t)g(t)δ(t)=g(0)δ(t)g(t)δ′(t)=−g′(0)δ(t)\begin{aligned} &t \delta^{\prime}(t)=-f(t)\\ &\left(1-e^{t}\right) \delta^{\prime}(t)=\delta(t)\\ &g(t) \delta(t)=g(0) \delta(t)\\ &g(t) \delta^{\prime}(t)=-g^{\prime}(0) \delta(t)\\ \end{aligned}tδ′(t)=−f(t)(1−et)δ′(t)=δ(t)g(t)δ(t)=g(0)δ(t)g(t)δ′(t)=−g′(0)δ(t)

傅里叶变换的相关性质

F[f(at)]=1∣a∣⁡F(wa)F[f(−t)]=F(−w)F[ejw0tf(t)]=F(w−w0)F[tf(t)]=j⋅df(w)dwF[tnf(t)]=jn⋅dnF(w)dwnF[f(t−t0)]=e−jw+oF[f(t)]F[f′(t)]=jwF[f(t)]F[f′′(t)]=(jw)nF[f(t)]F[∫−∞tf(t)dt]=1jwF(f(t)]\begin{aligned} &F[f(a t)]=\frac{1}{\operatorname{|a|} } F\left(\frac{w}{a}\right)\\ &F[f(-t)]=F(-w)\\ &F\left[e^{j w_{0} t} f(t)\right]=F\left(w-w_{0}\right)\\ &F[t f(t)]=j \cdot \frac{d f(w)}{d w}\\ &F\left[t^{n} f(t)\right]=j^{n} \cdot \frac{d^{n} F(w)}{d w^{n}}\\ &F\left[f\left(t-t_{0}\right)\right]=e^{-j w+o} F[f(t)]\\ &F\left[f^{\prime}(t)\right]=j w F[f(t)]\\ &F\left[f^{\prime \prime}(t)\right]=(j w)^{n} F[f(t)]\\ &F\left[\int_{-\infty}^{t} f(t) d t\right]=\frac{1}{j w} F(f(t)]\\ \end{aligned}F[f(at)]=∣a∣1F(aw)F[f(−t)]=F(−w)F[ejw0tf(t)]=F(w−w0)F[tf(t)]=j⋅dwdf(w)F[tnf(t)]=jn⋅dwndnF(w)F[f(t−t0)]=e−jw+oF[f(t)]F[f′(t)]=jwF[f(t)]F[f′′(t)]=(jw)nF[f(t)]F[∫−∞tf(t)dt]=jw1F(f(t)]

dnF(w)dwn=(−j)nF(tnf(t)]dF(w)dw=F[−jtf(t)]F[f(t)cos⁡w0t]=12[F(w−w0)+F(w+w0)]F[f(t)sin⁡w0t]=12j[f(w−w0)−F(w+w0)]\begin{aligned} &\frac{d^{n} F(w)}{d w^{n}}=(-j)^{n} F\left(t^{n} f(t)\right]\\ &\frac{d F(w)}{d w}=F[-j t f(t)]\\ &F\left[f(t) \cos w_{0} t\right]=\frac{1}{2}\left[F\left(w-w_{0}\right)+F\left(w+w_{0}\right)\right]\\ &F\left[f(t) \sin w_{0} t\right]=\frac{1}{2 j}\left[f\left(w-w_{0}\right)-F\left(w+w_{0}\right)\right]\\ \end{aligned}dwndnF(w)=(−j)nF(tnf(t)]dwdF(w)=F[−jtf(t)]F[f(t)cosw0t]=21[F(w−w0)+F(w+w0)]F[f(t)sinw0t]=2j1[f(w−w0)−F(w+w0)]

能量积分

振幅频率|F(w)|

相角频率
φ(w)=arctan⁡φ=arctan⁡∫−∞+∞f(t)sin⁡w+dt∫−∞+∞f(t)cos⁡wtdt\varphi(w)=\arctan \varphi=\arctan \frac{\int_{-\infty}^{+\infty} f(t) \sin w+d t}{\int_{-\infty}^{+\infty} f(t) \cos w t d t}φ(w)=arctanφ=arctan∫−∞+∞f(t)coswtdt∫−∞+∞f(t)sinw+dt

∫−∞+∞[f(t)]2dt=12π∫−∞+∞∣F(w)∣2dw\int_{-\infty}^{+\infty}[f(t)]^{2} d t=\frac{1}{2 \pi} \int_{-\infty}^{+\infty}|F(w)|^{2} d w∫−∞+∞[f(t)]2dt=2π1∫−∞+∞∣F(w)∣2dw

该式称为Parseval等式=能量积分

∫(w)=∣F(w)∣2\int(w)=|F(w)|^{2}∫(w)=∣F(w)∣2

该式称为能量密度函数/能量谱函数，是偶函数

卷积

卷积的一些性质
f1(t)∗f2(t)=∫−∞+∞f1(τ)f2(t−τ)dτf1(t)∗f2(t)=f2(t)∗f1(t);f1(t)∗[f2(t)∗f3(t)]=[f1(t)∗f2(t)]∗f3(t)a[f1(t)∗f2(t)]=[af1(t)]∗f2(t)=f1(t)∗[af2(t)](a为常数 );eat[f1(t)∗f2(t)]=[eatf1(t)]∗[eatf2(t)](a为常数 );[f1(t)+f2(t)]∗[g1(t)+g2(t)]=f1(t)∗g1(t)+f2(t)∗g1(t)+f1(t)∗g2(t)+f2(t)∗g2(t)ddt[f1(t)∗f2(t)]=ddtf1(t)∗f2(t)=f1(t)∗ddtf2(t)∫−∞t[f1(ξ)∗f2(ξ)]dξ=f1(t)∗∫−∞tf2(ξ)dξ=∫−∞tf1(ξ)dξ∗f2(t)f(t)∗δ(t−t0)=f(t−t0)f(t)∗δ′(t)=f′(t)f(t)∗u(t)=∫−∞tf(τ)dτ∣f1(t)∗f2(t)∣⩽∣t1∣t)∣∗∣t2(t)∣\begin{aligned} &f_{1}(t) * f_{2}(t)=\int_{-\infty}^{+\infty} f_{1}(\tau) f_{2}(t-\tau) d \tau\\ &\\ &f_{1}(t) * f_{2}(t)=f_{2}(t) * f_{1}(t) ;\\ &f_{1}(t) *\left[f_{2}(t) * f_{3}(t)\right]=\left[f_{1}(t) * f_{2}(t)\right] * f_{3}(t)\\ &a\left[f_{1}(t) * f_{2}(t)\right]=\left[a f_{1}(t)\right] * f_{2}(t)=f_{1}(t) *\left[a f_{2}(t)\right](a \text { 为常数 }) ;\\ &\mathrm{e}^{a t}\left[f_{1}(t) * f_{2}(t)\right]=\left[\mathrm{e}^{a t} f_{1}(t)\right] *\left[\mathrm{e}^{a t} f_{2}(t)\right](a \text { 为常数 }) ;\\ &\left[f_{1}(t)+f_{2}(t)\right] *\left[g_{1}(t)+g_{2}(t)\right]=f_{1}(t) * g_{1}(t)+f_{2}(t) * g_{1}(t)+f_{1}(t) * g_{2}(t)+f_{2}(t) * g_{2}(t)\\ &\frac{d}{d t}\left[f_{1}(t) * f_{2}(t)\right]=\frac{\mathrm{d}}{\mathrm{d} t} f_{1}(t) * f_{2}(t)=f_{1}(t) * \frac{\mathrm{d}}{\mathrm{d} t} f_{2}(t)\\ &\int_{-\infty}^{t}\left[f_{1}(\xi) * f_{2}(\xi)\right] \mathrm{d} \xi=f_{1}(t) * \int_{-\infty}^{t} f_{2}(\xi) \mathrm{d} \xi=\int_{-\infty}^{t} f_{1}(\xi) \mathrm{d} \xi * f_{2}(t) \\ &f(t) * \delta\left(t-t_{0}\right)=f\left(t-t_{0}\right) \\ &f(t) * \delta^{\prime}(t)=f^{\prime}(t) \\ &f(t) * u(t)=\int_{-\infty}^{t} f(\tau) \mathrm{d} \tau\\ &\left.\left|f_{1}(t) * f_{2}(t)\right| \leqslant\left|t_{1}\right| t\right)|*| t_{2}(t) \mid\\ \end{aligned} f1(t)∗f2(t)=∫−∞+∞f1(τ)f2(t−τ)dτf1(t)∗f2(t)=f2(t)∗f1(t);f1(t)∗[f2(t)∗f3(t)]=[f1(t)∗f2(t)]∗f3(t)a[f1(t)∗f2(t)]=[af1(t)]∗f2(t)=f1(t)∗[af2(t)](a 为常数 );eat[f1(t)∗f2(t)]=[eatf1(t)]∗[eatf2(t)](a 为常数 );[f1(t)+f2(t)]∗[g1(t)+g2(t)]=f1(t)∗g1(t)+f2(t)∗g1(t)+f1(t)∗g2(t)+f2(t)∗g2(t)dtd[f1(t)∗f2(t)]=dtdf1(t)∗f2(t)=f1(t)∗dtdf2(t)∫−∞t[f1(ξ)∗f2(ξ)]dξ=f1(t)∗∫−∞tf2(ξ)dξ=∫−∞tf1(ξ)dξ∗f2(t)f(t)∗δ(t−t0)=f(t−t0)f(t)∗δ′(t)=f′(t)f(t)∗u(t)=∫−∞tf(τ)dτ∣f1(t)∗f2(t)∣⩽∣t1∣t)∣∗∣t2(t)∣

卷积定理
F[f1(t)∗f2(t)]=F1(w)⋅F2(w)F−1[F1(w)⋅F2(w)]=f1(t)∗f2(t)\begin{aligned} &F\left[f_{1}(t) * f_{2}(t)\right]=F_{1}(w) \cdot F_{2}(w)\\ &F^{-1}\left[F_{1}(w) \cdot F_{2}(w)\right]=f_{1}(t) * f_{2}(t)\\ \end{aligned}F[f1(t)∗f2(t)]=F1(w)⋅F2(w)F−1[F1(w)⋅F2(w)]=f1(t)∗f2(t)

自相关函数和互相关函数
∫−∞+∞f1(t)f2(t+τ)dt=R12(τ)∫−∞+∞f(t)f(t+τ)dτ=R(τ)∫(w)=∫−∞+∞R(1)e−jwtdt\begin{aligned} &\int_{-\infty}^{+\infty} f_{1}(t) f_{2}(t+\tau) d t=R_{12}(\tau)\\ &\int_{-\infty}^{+\infty} f(t) f(t+\tau) d \tau=R(\tau)\\ &\int(w)=\int_{-\infty}^{+\infty} R(1) e^{-j w t} d t \end{aligned}∫−∞+∞f1(t)f2(t+τ)dt=R12(τ)∫−∞+∞f(t)f(t+τ)dτ=R(τ)∫(w)=∫−∞+∞R(1)e−jwtdt

Laplace变换总结

常用Laplace变换

δ′(t)→sδ(t)→11→15t→1s212t2→1s3tn→Γ(n+1)sn+1tn→n!sn+1sinat →as2+a2cos⁡at→ss2+a2eat→1s−a\begin{aligned} &\delta^{\prime}(t) \rightarrow s\\ &\delta(t) \rightarrow 1\\ &1 \rightarrow \frac{1}{5}\\ &t \rightarrow \frac{1}{s^{2}}\\ &\frac{1}{2} t^{2} \rightarrow \frac{1}{s^{3}}\\ &t^{n} \rightarrow \frac{\Gamma(n+1)}{s^{n+1}}\\ &t^{n} \rightarrow \frac{n !}{s^{n+1}}\\ &\text { sinat } \rightarrow \frac{a}{s^{2}+a^{2}}\\ &\cos a t \rightarrow \frac{s}{s^{2}+a^{2}}\\ &e^{a t} \rightarrow \frac{1}{s-a}\\ \end{aligned}δ′(t)→sδ(t)→11→51t→s2121t2→s31tn→sn+1Γ(n+1)tn→sn+1n! sinat →s2+a2acosat→s2+a2seat→s−a1
对于上面的Gamma函数有
Γ(x)=∫0+∞+x−1e−tdt(x)0)Γ(z)=∫0+∞tz−1e−tdt(Re(z)>0)∫0+∞e−t2dt=π2Γ(12)=2∫0+∞e−t2dt=π\begin{aligned} &\left.\Gamma(x)=\int_{0}^{+\infty}+x-1 e^{-t} d t(x) 0\right) \\ &\Gamma(z)=\int_{0}^{+\infty} t^{z-1} e^{-t} d t(R e(z)>0) \\ &\int_{0}^{+\infty} e^{-t^{2}} d t=\frac{\sqrt{\pi}}{2} \\ &\Gamma\left(\frac{1}{2}\right)=2 \int_{0}^{+\infty} e^{-t^{2}} d t=\sqrt{\pi} \end{aligned}Γ(x)=∫0+∞+x−1e−tdt(x)0)Γ(z)=∫0+∞tz−1e−tdt(Re(z)>0)∫0+∞e−t2dt=2πΓ(21)=2∫0+∞e−t2dt=π

性质

L[tf(t)]=−F′(s)L[tnf(t)]=(−1)nF(n)sL[f′(t)]=∫F(s)−f(a)L[f′′(t)]=∫2f(s)−sf(0)−f′(0)L[f(n)(t)]=∫1nf(s)−sn−1f(0)−sn−2f′(0)−⋯−f(n−1)(0)=∫nF(s)−∑i=0n−1sn+1−if(i)(0)(Re(s))c)L[∫0tf(t)dt]=1s⋅f(s)L[f(t)t]=∫s∞F(s)dsL[eatf(t)]=F(s−a)(Re(s−a)>c)L[f(t−a)]=e−asF(s)L[f(t−a)u(t−a)]=e−asf(s)（注意上下收敛域，一般这个准没错）L[f(a+)=1aF(sa)\begin{aligned} &\mathscr{L}[tf(t)]=-F^{\prime}(s)\\ &\mathscr{L}\left[t^{n} f(t)\right]=(-1)^{n} F^{(n)} s\\ &\mathscr{L}\left[f^{\prime}(t)\right]=\int F(s)-f(a)\\ &\mathscr{L}\left[f^{\prime \prime}(t)\right]=\int^{2} f(s)-s f(0)-f^{\prime}(0)\\ &\mathscr{L}\left[f^{(n)}(t)\right] \\ &=\int_{1}^{n} f(s)-s^{n-1} f(0)-s^{n-2} f^{\prime}(0)-\cdots-f^{(n-1)}(0) \\ &\left.=\int^{n} F(s)-\sum_{i=0}^{n-1} s^{n+1-i} f^{(i)}(0)(R e(s)) c\right) \\ &\mathscr{L}\left[\int_{0}^{t} f(t) d t\right]=\frac{1}{s} \cdot f(s)\\ &\mathscr{L}\left[\frac{f(t)}{t}\right]=\int_{s}^{\infty} F(s) d s \mathscr{L}\left[e^{a t} f(t)\right]=F(s-a)(R e(s-a)>c)\\ &\mathscr{L}[f(t-a)]=e^{-a s} F(s) \\ & \mathscr{L}[f(t-a) u(t-a)]=e^{-a s} f(s)（注意上下收敛域，一般这个准没错）\\ &\mathscr{L}\left[f(a+)=\frac{1}{a} F\left(\frac{s}{a}\right)\right.\\ \end{aligned}L[tf(t)]=−F′(s)L[tnf(t)]=(−1)nF(n)sL[f′(t)]=∫F(s)−f(a)L[f′′(t)]=∫2f(s)−sf(0)−f′(0)L[f(n)(t)]=∫1nf(s)−sn−1f(0)−sn−2f′(0)−⋯−f(n−1)(0)=∫nF(s)−i=0∑n−1sn+1−if(i)(0)(Re(s))c)L[∫0tf(t)dt]=s1⋅f(s)L[tf(t)]=∫s∞F(s)dsL[eatf(t)]=F(s−a)(Re(s−a)>c)L[f(t−a)]=e−asF(s)L[f(t−a)u(t−a)]=e−asf(s)（注意上下收敛域，一般这个准没错）L[f(a+)=a1F(as)

初值定理和终值定理

lim⁡t→0f(t)=lim⁡s→∞sin⁡(s)\lim \limits_{t \rightarrow 0} f(t)=\lim \limits_{s \rightarrow \infty} \sin (s)t→0limf(t)=s→∞limsin(s)
f(0)=lim⁡s→∞sF(s)f(0)=\lim \limits_{s \rightarrow \infty} s F(s)f(0)=s→∞limsF(s)
lim⁡t→∞f(t)=lim⁡s→0sF(s)\lim \limits_{t \rightarrow \infty} f(t)=\lim \limits _{s \rightarrow 0} s F(s)t→∞limf(t)=s→0limsF(s)
f(∞)=lim⁡s→0sF(s)f( \infty)=\lim \limits_{s \rightarrow 0} s F(s)f(∞)=s→0limsF(s)

卷积定理

L[t1(t)∗f2(t)]=F1(s):F2(s)\mathscr{L}\left[t_{1}(t) * f_{2}(t)\right]=F_{1}(s): F_{2}(s)L[t1(t)∗f2(t)]=F1(s):F2(s)
L−1[F1(s)⋅f2(s)]=f1(t)∗t2(t)\mathscr{L}^{-1}\left[F_{1}(s) \cdot f_{2}(s)\right]=f_{1}(t) * t_{2}(t)L−1[F1(s)⋅f2(s)]=f1(t)∗t2(t)

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