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{array}{rl}f(t)⟵& ⟶F(j\omega )\\ F(jt)⟵& ⟶2\pi f(-\omega )\\ f(\alpha t)⟵& ⟶\frac{1}{|\alpha |}F(j\frac{\omega }{\alpha })\\ a\cdot f_1+b\cdot f_2⟵& ⟶a\cdot F_1+b\cdot F_2\\ f(t\pm t_0)⟵& ⟶e^{\pm j\omega t_0}F(j\omega )\\ f(t\pm t_0)⟵& ⟶|F(j\omega )|e^{j[\phi (\omega )\pm \omega t_0]}\\ e^{\mp j{\omega }_{0}t}f(t)⟵& ⟶F[j(\omega \pm {\omega }_0)]\\ f_1(t)\star f_2(t)⟵& ⟶F_1(j\omega )\cdot F_2(j\omega )\\ f_1(t)\cdot f_2(t)⟵& ⟶\frac{1}{2\pi }{F}_1(j\omega )\star F_2(j\omega )\\ f^{(n)}(t)⟵& ⟶(j\omega {)}^{n}F(j\omega )\\ {\int }_{-\infty }^{t}f(x)dx⟵& ⟶\pi F(0)\delta (\omega )+\frac{F(j\omega )}{j\omega }\\ (-jt{)}^{n}f(t)⟵& ⟶F^{(n)}(j\omega )\\ \pi f(0)\delta (t)+\frac{f(t)}{-jt}⟵& ⟶{\int }_{-\infty }^{\omega }F(jx)dx\\ e^{-\alpha t}\epsilon (t)⟵& ⟶\frac{1}{\alpha +j\omega }\\ e^{-\alpha |t|}⟵& ⟶\frac{2\alpha }{{\alpha }^2+{\omega }^2}\\ g_{\tau }(t)⟵& ⟶\tau \text{Sa}⟮\frac{\omega \tau }{2}⟯\\ 1⟵& ⟶2\pi \delta (\omega )\\ \delta ⟵& ⟶1\\ {\delta }^{\prime }⟵& ⟶j\omega \\ {\delta }^{(n)}⟵& ⟶(j\omega {)}^n\\ \epsilon (t)⟵& ⟶\pi \delta (\omega )+\frac{1}{j\omega }\\ \text{sgn}(t)⟵& ⟶\frac{2}{j\omega }\\ \downarrow R(\tau )⟵& ⟶E(\omega )\downarrow \\ {\int }_{-\infty }^{\infty }f(t)f(t-\tau )dt⟵& ⟶|F(j\omega ){|}^2\\ \downarrow R(\tau )⟵& ⟶P(\omega )\downarrow \\ \underset{T\to \infty }{\lim }[\frac{1}{T}{\int }_{-\frac{T}{2}}^{\frac{T}{2}}f(t)f(t-\tau )dt]⟵& ⟶\underset{T\to \infty }{\lim }\frac{|F_T(j\omega ){|}^2}{T}\\ e^{j{\omega }_{0}t}⟵& ⟶2\pi \delta (\omega -{\omega }_0)\\ e^{-j{\omega }_{0}t}⟵& ⟶2\pi \delta (\omega +{\omega }_0)\\ \cos ({\omega }_{0}t)⟵& ⟶\pi [\delta (\omega +{\omega }_0)+\delta (\omega -{\omega }_0)]\\ \sin ({\omega }_{0}t)⟵& ⟶j\pi [\delta (\omega +{\omega }_0)-\delta (\omega -{\omega }_0)]\\ f_T(t)⟵& ⟶F_T(j\omega )\\ {\delta }_T(t)\star f_0(t)⟵& ⟶\Omega {\delta }_{\Omega }(\omega )F_0(j\omega )\\ {\delta }_T(t)\star f_0(t)⟵& ⟶\Omega \sum _{n=-\infty }^{\infty }{F}_0(jn\Omega )\delta (\omega -n\Omega )\\ \sum _{n=-\infty }^{\infty }{F}_n{e}^{jn\Omega t}⟵& ⟶2\pi \sum _{n=-\infty }^{\infty }{F}_n\delta (\omega -n\Omega )\end{array}$$