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(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t  x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \omega)e^{j \omega t}d \omega
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) \, (b > 0)  \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 \mbox{rect} \left( \frac{ \omega }{ \tau } \right) \,
14  \left( 1-\frac{2 |t|}{\tau} \right) \mbox{rect} \left( \frac{ t }{ \tau } \right) \,  \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) \mbox{rect} \left( \frac{ \omega }{ \tau } \right) \,
16  e^{-a|t|}, \Re\{a\}>0 \,  \frac{2a}{a^2 + \omega^2} \,
Notes:
  1.  \mbox{sinc}(x)=\sin(x)/x
  2.  \mbox{rect} \left( \frac{ t }{ \tau } \right) is the rectangular pulse function of width  \tau
  3.  u(t) is the Heavyside step function
  4.  \delta (t) is the Dirac delta function