【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
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\(\begin{gather*} e^{-jat}\stackrel{\mathscr{F}}{\longrightarrow}2\pi\delta(\omega +a) \end{gather*}\)
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\(\begin{gather*} \delta(t +a)\stackrel{\mathscr{F}}{\longrightarrow}e^{-ja\omega} \end{gather*}\)
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\(\begin{gather*} e^{-\beta t}(\beta \gt 0,t\geq 0)\stackrel{\mathscr{F}}{\longrightarrow}2\pi\delta(\omega +a) \end{gather*}\)
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\(\begin{gather*} \sin\omega_0t\stackrel{\mathscr{F}}{\longrightarrow}j\pi[\delta(\omega +\omega_0)-\delta(\omega -\omega_0)] \end{gather*}\)
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\(\begin{gather*} \cos\omega_0t\stackrel{\mathscr{F}}{\longrightarrow} \pi[\delta(\omega +\omega_0)+\delta(\omega -\omega_0)] \end{gather*}\)
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\(\begin{gather*} E(|t|\leq \frac{\tau }{2})\stackrel{\mathscr{F}}{\longrightarrow}E\tau Sa(\frac{\omega \tau}{2}) \end{gather*}\)
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\(\begin{gather*} \delta'(t)\stackrel{\mathscr{F}}{\longrightarrow}j\omega \end{gather*}\)
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\(\begin{gather*} Sa(\omega_0 t)\stackrel{\mathscr{F}}{\longrightarrow}\frac{\pi}{\omega_0}[u(\omega+ \omega_0)-u(\omega- \omega_0)] \end{gather*}\)
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\(\begin{gather*} sgn(t)\stackrel{\mathscr{F}}{\longrightarrow}\frac{2}{j\omega} \end{gather*}\)
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\(\begin{gather*} u(t)=\frac{1}{2}+\frac{1}{2}sgn(t)\stackrel{\mathscr{F}}{\longrightarrow}\pi\delta(\omega)+\frac{1}{j\omega} \end{gather*}\)
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调制定理
\(\begin{gather*} f(t)\cos(\omega_0t)\stackrel{\mathscr{F}}{\longrightarrow}\frac{1}{2}[F(\omega- \omega_0 )+F(\omega+ \omega_0 )] \end{gather*}\)
\(\begin{gather*} f(t)\sin(\omega_0t)\stackrel{\mathscr{F}}{\longrightarrow}\frac{1}{2}[F(\omega- \omega_0 )-F(\omega+ \omega_0 )] \end{gather*}\)
