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−jωtdt 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)=12π∫−∞∞F(ω)ejωtdω 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[αf1(t)+βf2(t)]=αF[f1(t)]+βF[f2(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[αF1(ω)+βF2(ω)]=αF−1[F1(ω)]+βF−1[F2(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)+βF