Table of Fourier Transforms

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Definition of Fourier Transforms

If \( f(t) \) is a function of the real variable \( t \), then the Fourier transform \( F(\omega) \) of \( f \) is given by the integral \[ F(\omega) = \int_{-\infty}^{+\infty} e^{-j \omega t} f(t) \, dt \] where \( j = \sqrt{-1} \), the imaginary unit.
In what follows, \( u(t) \) is the unit step function defined by
\( u(t) = 1 \) for \( t \geq 0 \) and \( u(t) = 0 \) for \( t < 0 \). (see figure below).

unit step function

Table of Fourier Transforms

\( f(t) \) \( F(\omega) \)
\( u(t) e^{-a t} \), \( a > 0 \) \( \dfrac{1}{a + j \omega} \)
\( f(t) = 1 \) for \( -a \leq t \leq a \)
and \( 0 \) otherwise		 \( \dfrac{2 \sin (\omega a)}{\omega} \)
\( f(t) = A \) (constant) \( 2 \pi A \delta (\omega) \)
\( \delta (t) \) 1
\( \delta (t - a) \) \( e^{-j \omega a} \)
\( \cos (a t) \) \( \pi [\delta (\omega + a) + \delta (\omega - a)] \)
\( \sin (a t) \) \( -j \pi [\delta (\omega - a) - \delta (\omega + a)] \)
\( e^{j a t} \) \( 2 \pi [\delta (\omega - a)] \)
\( f'(t) \) \( j \omega F(\omega) \)
\( f''(t) \) \( (j \omega)^2 F(\omega) \)
\( t f(t) \) \( j \dfrac{d F(\omega)}{d \omega} \)
\( t^2 f(t) \) \( j^2 \dfrac{d^2 F(\omega)}{d \omega^2} \)

More References and links

integrals and their applications in calculus.