常用函数的傅里叶变换汇总

• 快速链接:
  • 【总目录】信号与系统 学习笔记 Signals and Systems with Python
  • (1) 简介 Intro
  • (2) 傅里叶 Fourier
    • 常用函数的傅里叶变换汇总
  • (3) LTI 系统 与 滤波器

常用函数的傅里叶变换汇总

$\begin{aligned} \displaystyle f({\color{blue}t}) \longleftarrow& \longrightarrow F({\color{blue}j\omega}) \\ F(j t) \longleftarrow& \longrightarrow {\color{blue}2\pi }f(-\omega)\\ f({\color{blue}\alpha} t) \longleftarrow& \longrightarrow {\color{blue}\frac{1}{\lvert \alpha \rvert}}F(j\frac{\omega}{{\color{blue}\alpha}})\\ {\color{blue}a}\cdot f_1 + {\color{blue}b}\cdot f_2 \longleftarrow& \longrightarrow {\color{blue}a}\cdot F_1 + {\color{blue}b}\cdot F_2 \\ f(t {\color{blue}\pm t_0}) \longleftarrow& \longrightarrow {\color{blue}e^{\pm j \omega t_0}}F(j\omega)\\ f(t {\color{blue}\pm t_0}) \longleftarrow& \longrightarrow \lvert F(j\omega)\rvert {\color{blue}e^{j[\varphi(\omega)\pm \omega t_0]}}\\ {\color{blue}e^{\mp j\omega_0 t}}f(t)\longleftarrow& \longrightarrow F\big[j(\omega{\color{blue}\pm\omega_0})\big]\\ f_1(t) {\color{blue}\star} f_2(t) \longleftarrow& \longrightarrow F_1(j\omega){\color{blue}\cdot} F_2(j\omega)\\ f_1(t){\color{blue}\cdot} f_2(t) \longleftarrow& \longrightarrow {\color{blue}\frac{1}{2\pi}}F_1(j\omega){\color{blue}\star} F_2(j\omega)\\ f^{{\color{blue}(n)}} (t) \longleftarrow& \longrightarrow {\color{blue}(j\omega)^n} F(j\omega)\\ \int^{t}_{-\infty} f(x) dx \longleftarrow& \longrightarrow \pi F(0)\delta(\omega) + \frac{F(j\omega)}{j\omega}\\ {\color{blue}(-jt)^n} f (t) \longleftarrow& \longrightarrow F^{{\color{blue}(n)}}(j\omega)\\ \pi f(0)\delta(t) + \frac{f(t)}{{\color{red}-}jt} \longleftarrow& \longrightarrow \int^{\omega}_{-\infty}F(jx)dx\\ e^{-\alpha t} \varepsilon(t)\longleftarrow& \longrightarrow \frac{1}{\alpha + j\omega}\\ e^{-\alpha \lvert t\rvert} \longleftarrow& \longrightarrow \frac{2\alpha}{\alpha^2 + \omega^2} \\ g_{\color{blue}\tau}(t) \longleftarrow& \longrightarrow {\color{blue}\tau} \text{Sa} \Big\lgroup \displaystyle \frac{\omega{\color{blue}\tau}}{2} \Big\rgroup\\ {\color{red}1} \longleftarrow& \longrightarrow {\color{blue}2\pi}\delta{(\omega)}\\ {\color{red}\delta} \longleftarrow& \longrightarrow 1 \\ \delta^{\color{blue}\prime} \longleftarrow& \longrightarrow {\color{blue}j\omega} \\ \delta^{{\color{blue}(n)}} \longleftarrow& \longrightarrow (j\omega)^{\color{blue}n} \\ {\color{red}\varepsilon}(t)\longleftarrow& \longrightarrow \pi \delta(\omega) + \frac{1}{j\omega}\\ {\color{blue}\text{sgn}}(t)\longleftarrow& \longrightarrow \frac{2}{j\omega}\\ \downarrow R(\tau) \longleftarrow& \longrightarrow {\color{red}E}(\omega) \downarrow \\ {\int^{\infty}_{-\infty}f(t)f(t-\tau)dt} \longleftarrow& \longrightarrow \lvert F(j\omega) \rvert ^2\\ \downarrow R(\tau) \longleftarrow& \longrightarrow {\color{red}P}(\omega)\downarrow \\ \lim_{T\to\infty} \big[ \frac{1}{T} \int^{\frac{T}{2}}_{-\frac{T}{2}} f(t)f(t-\tau)dt \big] \longleftarrow& \longrightarrow \lim_{T\to\infty} \frac{\lvert F_T(j\omega)\rvert ^2}{T}\\ e^{j{\color{blue}\omega_0} t} \longleftarrow& \longrightarrow 2\pi \delta (\omega {\color{blue}- \omega_0}) \\ e^{-j\omega_0 t} \longleftarrow& \longrightarrow 2\pi \delta (\omega + \omega_0) \\ {\color{blue}\cos} ( \omega_0 t )\longleftarrow& \longrightarrow \pi \big[ \delta(\omega + \omega_0) {\color{blue}+} \delta(\omega-\omega_0)\big] \\ {\color{blue}\sin} (\omega_0 t) \longleftarrow& \longrightarrow {\color{blue}j}\pi \big[ \delta(\omega + \omega_0){\color{blue} -} \delta(\omega-\omega_0)\big] \\ f_{\color{blue}T}(t) \longleftarrow& \longrightarrow F_{\color{blue}T}(j\omega)\\ \delta_T(t) \star f_0(t) \longleftarrow& \longrightarrow \Omega \delta_\Omega(\omega) F_0(j\omega)\\ \delta_T(t) \star f_0(t) \longleftarrow& \longrightarrow \Omega \sum_{n=-\infty}^{\infty} F_0(jn\Omega) \delta (\omega- n\Omega)\\ \sum_{n=-\infty}^{\infty} F_n e^{jn\Omega t} \longleftarrow& \longrightarrow 2\pi \sum_{n=-\infty}^{\infty} F_n \delta (\omega- n\Omega) \\ \end{aligned}$