常用傅里叶变换/拉普拉斯变换¶
傅里叶变换¶
变换对的三种形式¶
\[\begin{equation} \tag{FT.1} \label{FT} \left\{ \begin{aligned} F(\omega) &= \frac{1}{2\pi} \int_{-\infty}^{+\infty} f(t) e^{-\mathrm{i}\omega t} \mathrm{d}t \\ f(t) &= \int_{-\infty}^{+\infty} F(\omega) e^{\mathrm{i}\omega t} \mathrm{d}\omega \\ \end{aligned} \right. \end{equation}\]
\[\begin{equation} \tag{FT.2} \left\{ \begin{aligned} F(\omega) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(t) e^{-\mathrm{i}\omega t} \mathrm{d}t \\ f(t) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} F(\omega) e^{\mathrm{i}\omega t} \mathrm{d}\omega \\ \end{aligned} \right. \end{equation}\]
\[\begin{equation} \tag{FT.3} \left\{ \begin{aligned} F(\omega) &= \int_{-\infty}^{+\infty} f(t) e^{\mathrm{i}\omega t} \mathrm{d}t \\ f(t) &= \frac{1}{2\pi} \int_{-\infty}^{+\infty} F(\omega) e^{-\mathrm{i}\omega t} \mathrm{d}\omega \\ \end{aligned} \right. \end{equation}\]
注意指数项负号的位置。
常用变换表¶
以下变换均采用 \eqref{FT} 形式。
| 原函数 \(f(t)\) | 傅里叶变换 \(F(\omega)\) |
|---|---|
| \(1\) | \(\delta(\omega)\) |
| \(\text{rect}(x) = \begin{cases} 1, & \|x\| < \frac{1}{2} \\ 0, & \|x\| > \frac{1}{2} \end{cases}\) | \(\text{sinc}(\omega)\) |
| \(\delta(x)\) | \(\frac{1}{2\pi}\) |
| \(H(x) = \begin{cases} 1, & x > 0 \\ 0, & x < 0 \end{cases}\) | \(\frac{1}{2\pi} \left[\frac{1}{\mathrm{i} \omega} + \pi \delta(\omega) \right]\) |