Engineering Tables/Fourier Transform Table

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  Time Domain Frequency Domain
x(t) = \mathcal{F}^{-1}\left\{ X(\omega) \right\} X(\omega) = \mathcal{F} \left\{ x(t) \right\}
1 x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \omega)e^{j \omega t}d \omega X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t
2 1 \, 2\pi\delta(\omega) \,
3 -0.5 + u(t) \, \frac{1}{j \omega} \,
4 \delta (t) \, 1 \,
5 \delta (t-c) \, e^{-j \omega c} \,
6 u(t) \, \pi \delta(\omega)+\frac{1}{j \omega} \,
7 e^{-bt}u(t) \, \frac{1}{j \omega+b} \,
8 \cos \omega_0 t \, \pi \left[ \delta(\omega+\omega_0)+\delta(\omega-\omega_0) \right] \,
9 \cos (\omega_0 t + \theta) \, \pi \left[ e^{-j \theta}\delta(\omega+\omega_0)+e^{j \theta}\delta(\omega-\omega_0) \right] \,
10 \sin \omega_0 t \, j \pi \left[ \delta(\omega +\omega_0)-\delta(\omega-\omega_0) \right] \,
11 \sin (\omega_0 t + \theta) \, j \pi \left[ e^{-j \theta}\delta(\omega +\omega_0)-e^{j \theta}\delta(\omega-\omega_0) \right] \,
12 \mbox{rect} \left( \frac{t}{\tau} \right) \, \tau \mbox{sinc} \left( \frac{\tau \omega}{2 \pi} \right) \,
13 \tau \mbox{sinc} \left( \frac{\tau t}{2 \pi} \right) \, 2 \pi p_\tau(\omega)\,
14 \left( 1-\frac{2 |t|}{\tau} \right) p_\tau (t) \, \frac{\tau}{2} \mbox{sinc}^2 \left( \frac{\tau \omega}{4 \pi} \right) \,
15 \frac{\tau}{2} \mbox{sinc}^2 \left( \frac{\tau t}{4 \pi} \right) \, 2 \pi \left( 1-\frac{2|\omega|}{\tau} \right) p_\tau (\omega) \,
Notes:
  1. sinc(x) = sin(x) / x
  2. pτ(t) is the rectangular pulse function of width τ
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