复变函数与积分变换系列（五） - Fourier变换
Fourier变换
Author : Benjamin142857
Date : 2018/10/2
[TOC]
Fourier 变换式
正变换 ： F [ f ( t ) ] \mathscr{F}[f(t)] F[f(t)]
F ( ω ) = ∫ − ∞ ∞ f ( t ) e − j ω t d t F(\omega) = \int_{-\infty}^{\infty}f(t)e^{-j\omega t}dt F(ω)=∫−∞∞f(t)e−jωtdt
逆变换 ： F − 1 [ F ( ω ) ] \mathscr{F}^{-1}[F(\omega)] F−1[F(ω)]
f ( t ) = 1 2 π ∫ − ∞ ∞ F ( ω ) e j ω t d ω f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{j\omega t}d\omega f(t)=2π1∫−∞∞F(ω)ejωtdω
Fourier 变换四大性质
1. 线性性质
正变换
F [ α f 1 ( t ) + β f 2 ( t ) ] = α F [ f 1 ( t ) ] + β F [ f 2 ( t ) ] \mathscr{F}[\alpha f_1(t)+\beta f_2(t)] = \alpha \mathscr{F}[f_1(t)] + \beta \mathscr{F}[f_2(t)] F[αf1(t)+βf2(t)]=αF[f1(t)]+βF[f2(t)]
逆变换
F − 1 [ α F 1 ( ω ) + β F 2 ( ω ) ] = α F − 1 [ F 1 ( ω ) ] + β F − 1 [ F 2 ( ω ) ] \mathscr{F}^{-1}[\alpha F_1(\omega) + \beta F_2(\omega)] = \alpha \mathscr{F}^{-1}[F_1(\omega)] + \beta \mathscr{F}^{-1}[F_2(\omega)] F−1[αF1(ω)+βF2(ω)]=αF−1[F1(ω)]+βF−1[F2(ω)]
2. 位移性质
正变换
F [ f ( t + t 0 ) ] = e j ω t 0 F [ f ( t ) ] \mathscr{F}[f(t+t_0)] = e^{j\omega t_0}\mathscr{F}[f(t)] F[f(t+t0)]=ejωt0F[f(t)]
逆变换
F − 1 [ F ( ω + ω 0 ) ] = e − j ω 0 t F − 1 [ F ( ω ) ] \mathscr{F}^{-1}[F(\omega+\omega_0)] = e^{-j\omega_0 t}\mathscr{F}^{-1}[F(\omega)] F−1[F(ω+ω0)]=e−jω0tF−1[F(ω)]
3. 微分性质
前提条件 f ( t ) f(t) f(t) 可去间断点有限， ∣ t ∣ → ∞ |t|\rightarrow \infty ∣t∣→∞ 时， f ( t ) → 0 f(t) \rightarrow 0 f(t)→0
先微后正变
F [ f ′ ( t ) ] = j ω F [ f ( t ) ] \mathscr{F}[f\prime(t)] = j\omega\mathscr{F}[f(t)] F[f′(t)]=jωF[f(t)]
F [ f ( n ) ( t ) ] = ( j ω ) ( n ) F [ f ( t ) ] \mathscr{F}[f^{(n)}(t)] = (j\omega)^{(n)}\mathscr{F}[f(t)] F[f(n)(t)]=(jω)(n)F[f(t)]
先正变后微
d d ω F ( ω ) = ( − j ) F [ t f ( t ) ] \frac{d}{d\omega}F(\omega) = (-j)\mathscr{F}[tf(t)] dωdF(ω)=(−j)F[tf(t)]
d n d ω n F ( ω ) = ( − j ) n F [ t n f ( t ) ] \frac{d^n}{d\omega^n}F(\omega) = (-j)^n\mathscr{F}[t^nf(t)] dωndnF(ω)=(−j)nF[tnf(t)]
先正变后微 - 意义推广
F [ t n f ( t ) ] = 1 ( − j ) n d n d ω n F ( ω ) \mathscr{F}[t^nf(t)] = \frac{1}{(-j)^n}\frac{d^n}{d\omega^n}F(\omega) F[tnf(t)]=(−j)n1dωndnF(ω)
4. 积分性质
前提条件， t → + ∞ t\rightarrow +\infty t→+∞ 时， ∫ − ∞ t f ( t ) d t → 0 \int_{-\infty}^tf(t)dt \rightarrow 0 ∫−∞tf(t)dt→0
F [ ∫ − ∞ t f ( t ) d t ] = 1 j w F [ f ( t ) ] \mathscr{F}[\int_{-\infty}^tf(t)dt] = \frac{1}{jw}\mathscr{F}[f(t)] F[∫−∞tf(t)dt]=jw1F[f(t)]
一些重要的Fourier变换（变换简表）
1. 钟形脉冲 f ( t ) = A e − β t 2 f(t) = Ae^{-\beta t^2} f(t)=Ae−βt2
( β >0 ) f ( t ) = A e − β t 2 f(t) = Ae^{-\beta t^2}\tag{$\beta$>0} f(t)=Ae−βt2(β>0)
F ( ω ) = π β A e − ω 2 4 β F(\omega) = \sqrt{\frac{\pi}{\beta}}Ae^{-\frac{\omega^2}{4\beta}} F(ω)=βπAe−4βω2
2. 指数衰减 f ( t ) = e − β t f(t) = e^{-\beta t} f(t)=e−βt
f ( t ) = { 0 , ( t < 0 ) e − β t , ( t ≥ 0 ) f(t) = \begin{cases}0,\ \ \ \ \ \ \ (t<0) \\ e^{-\beta t},\ \ \ \ \ (t\geq 0)\end{cases} f(t)={0, (t<0)e−βt, (t≥0)
F ( ω ) = 1 β + j ω F(\omega) = \frac{1}{\beta + j\omega} F(ω)=β+jω1
3. 余弦 f ( t ) = cos ω 0 t f(t) = \cos \omega_0t f(t)=cosω0t
f ( t ) = cos ω 0 t f(t) = \cos{\omega_0t} f(t)=cosω0t
F ( ω ) = π [ δ ( ω + ω 0 ) + δ ( ω − ω 0 ) ] F(\omega) = \pi[\delta(\omega+\omega_0) + \delta(\omega - \omega_0)] F(ω)=π[δ(ω+ω0)+δ(ω−ω0)]
4. 正弦 f ( t ) = sin ω 0 t f(t) = \sin \omega_0t f(t)=sinω0t
f ( t ) = sin ω 0 t f(t) = \sin{\omega_0t} f(t)=sinω0t
F ( ω ) = j π [ δ ( ω + ω 0 ) − δ ( ω − ω 0 ) ] F(\omega) = j\pi[\delta(\omega+\omega_0) - \delta(\omega - \omega_0)] F(ω)=jπ[δ(ω+ω0)−δ(ω−ω0)]