# Công thức hoán chuyển Fourier

 ${\displaystyle x(t)={\mathcal {F}}^{-1}\left\{X(\omega )\right\}}$ ${\displaystyle X(\omega )={\mathcal {F}}\left\{x(t)\right\}}$ 1 ${\displaystyle X(j\omega )=\int _{-\infty }^{\infty }x(t)e^{-j\omega t}dt}$ ${\displaystyle x(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }X(\omega )e^{j\omega t}d\omega }$ 2 ${\displaystyle 1\,}$ ${\displaystyle 2\pi \delta (\omega )\,}$ 3 ${\displaystyle -0.5+u(t)\,}$ ${\displaystyle {\frac {1}{j\omega }}\,}$ 4 ${\displaystyle \delta (t)\,}$ ${\displaystyle 1\,}$ 5 ${\displaystyle \delta (t-c)\,}$ ${\displaystyle e^{-j\omega c}\,}$ 6 ${\displaystyle u(t)\,}$ ${\displaystyle \pi \delta (\omega )+{\frac {1}{j\omega }}\,}$ 7 ${\displaystyle e^{-bt}u(t)\,(b>0)}$ ${\displaystyle {\frac {1}{j\omega +b}}\,}$ 8 ${\displaystyle \cos \omega _{0}t\,}$ ${\displaystyle \pi \left[\delta (\omega +\omega _{0})+\delta (\omega -\omega _{0})\right]\,}$ 9 ${\displaystyle \cos(\omega _{0}t+\theta )\,}$ ${\displaystyle \pi \left[e^{-j\theta }\delta (\omega +\omega _{0})+e^{j\theta }\delta (\omega -\omega _{0})\right]\,}$ 10 ${\displaystyle \sin \omega _{0}t\,}$ ${\displaystyle j\pi \left[\delta (\omega +\omega _{0})-\delta (\omega -\omega _{0})\right]\,}$ 11 ${\displaystyle \sin(\omega _{0}t+\theta )\,}$ ${\displaystyle j\pi \left[e^{-j\theta }\delta (\omega +\omega _{0})-e^{j\theta }\delta (\omega -\omega _{0})\right]\,}$ 12 ${\displaystyle {\mbox{rect}}\left({\frac {t}{\tau }}\right)\,}$ ${\displaystyle \tau {\mbox{sinc}}\left({\frac {\tau \omega }{2\pi }}\right)\,}$ 13 ${\displaystyle \tau {\mbox{sinc}}\left({\frac {\tau t}{2\pi }}\right)\,}$ ${\displaystyle 2\pi {\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,}$ 14 ${\displaystyle \left(1-{\frac {2|t|}{\tau }}\right){\mbox{rect}}\left({\frac {t}{\tau }}\right)\,}$ ${\displaystyle {\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau \omega }{4\pi }}\right)\,}$ 15 ${\displaystyle {\frac {\tau }{2}}{\mbox{sinc}}^{2}\left({\frac {\tau t}{4\pi }}\right)\,}$ ${\displaystyle 2\pi \left(1-{\frac {2|\omega |}{\tau }}\right){\mbox{rect}}\left({\frac {\omega }{\tau }}\right)\,}$ 16 ${\displaystyle e^{-a|t|},\Re \{a\}>0\,}$ ${\displaystyle {\frac {2a}{a^{2}+\omega ^{2}}}\,}$ Notes: ${\displaystyle {\mbox{sinc}}(x)=\sin(x)/x}$ ${\displaystyle {\mbox{rect}}\left({\frac {t}{\tau }}\right)}$ is the rectangular pulse function of width ${\displaystyle \tau }$ ${\displaystyle u(t)}$ is the Heaviside step function ${\displaystyle \delta (t)}$ is the Dirac delta function
 ${\displaystyle f(x)}$ ${\displaystyle {\frac {d}{dt}}}$ ${\displaystyle \int dx}$ ${\displaystyle F(j\omega )=\int f(x)e^{j\omega x}dx}$ ${\displaystyle j\omega }$ ${\displaystyle {\frac {1}{j\omega }}}$