【信号与系统】常用的傅里叶变换对
推荐背下来，这样用起来会很快[*]\(\begin{gather*}e^{-jat}\stackrel{\mathscr{F}}{\longrightarrow}2\pi\delta(\omega +a)\end{gather*}\)
[*]\(\begin{gather*}\delta(t +a)\stackrel{\mathscr{F}}{\longrightarrow}e^{-ja\omega}\end{gather*}\)
[*]\(\begin{gather*}e^{-\beta t}(\beta \gt 0,t\geq 0)\stackrel{\mathscr{F}}{\longrightarrow}2\pi\delta(\omega +a)\end{gather*}\)
[*]\(\begin{gather*}\sin\omega_0t\stackrel{\mathscr{F}}{\longrightarrow}j\pi[\delta(\omega +\omega_0)-\delta(\omega -\omega_0)]\end{gather*}\)
[*]\(\begin{gather*}\cos\omega_0t\stackrel{\mathscr{F}}{\longrightarrow} \pi[\delta(\omega +\omega_0)+\delta(\omega -\omega_0)]\end{gather*}\)
[*]\(\begin{gather*}E(|t|\leq \frac{\tau }{2})\stackrel{\mathscr{F}}{\longrightarrow}E\tau Sa(\frac{\omega \tau}{2})\end{gather*}\)
[*]\(\begin{gather*}\delta'(t)\stackrel{\mathscr{F}}{\longrightarrow}j\omega\end{gather*}\)
[*]\(\begin{gather*}Sa(\omega_0 t)\stackrel{\mathscr{F}}{\longrightarrow}\frac{\pi}{\omega_0}\end{gather*}\)
[*]\(\begin{gather*}sgn(t)\stackrel{\mathscr{F}}{\longrightarrow}\frac{2}{j\omega}\end{gather*}\)
[*]\(\begin{gather*}u(t)=\frac{1}{2}+\frac{1}{2}sgn(t)\stackrel{\mathscr{F}}{\longrightarrow}\pi\delta(\omega)+\frac{1}{j\omega}\end{gather*}\)
[*]调制定理
\(\begin{gather*}f(t)\cos(\omega_0)\stackrel{\mathscr{F}}{\longrightarrow}\frac{1}{2}\end{gather*}\)
\(\begin{gather*}f(t)\sin(\omega_0)\stackrel{\mathscr{F}}{\longrightarrow}\frac{1}{2}\end{gather*}\)
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