# Engineering Tables/Fourier Transform Properties

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks
$g(t)\!\equiv\!$

$\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!G(\omega) e^{i \omega t} d \omega \,$
$G(\omega)\!\equiv\!$

$\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty}\!\!g(t) e^{-i \omega t} dt \,$
$G(f)\!\equiv$

$\int_{-\infty}^{\infty}\!\!g(t) e^{-i 2\pi f t} dt \,$
1 $a\cdot g(t) + b\cdot h(t)\,$ $a\cdot G(\omega) + b\cdot H(\omega)\,$ $a\cdot G(f) + b\cdot H(f)\,$ Linearity
2 $g(t - a)\,$ $e^{- i a \omega} G(\omega)\,$ $e^{- i 2\pi a f} G(f)\,$ Shift in time domain
3 $e^{ iat} g(t)\,$ $G(\omega - a)\,$ $G \left(f - \frac{a}{2\pi}\right)\,$ Shift in frequency domain, dual of 2
4 $g(a t)\,$ $\frac{1}{|a|} G \left( \frac{\omega}{a} \right)\,$ $\frac{1}{|a|} G \left( \frac{f}{a} \right)\,$ If $|a|\,$ is large, then $g(a t)\,$ is concentrated around 0 and $\frac{1}{|a|}G \left( \frac{\omega}{a} \right)\,$ spreads out and flattens
5 $G(t)\,$ $g(-\omega)\,$ $g(-f)\,$ Duality property of the Fourier transform. Results from swapping "dummy" variables of $t \,$ and $\omega \,$.
6 $\frac{d^n g(t)}{dt^n}\,$ $(i\omega)^n G(\omega)\,$ $(i 2\pi f)^n G(f)\,$ Generalized derivative property of the Fourier transform
7 $t^n g(t)\,$ $i^n \frac{d^n G(\omega)}{d\omega^n}\,$ $\left (\frac{i}{2\pi}\right)^n \frac{d^n G(f)}{df^n}\,$ This is the dual to 6
8 $(g * h)(t)\,$ $\sqrt{2\pi} G(\omega) H(\omega)\,$ $G(f) H(f)\,$ $g * h\,$ denotes the convolution of $g\,$ and $h\,$ — this rule is the convolution theorem
9 $g(t) h(t)\,$ $(G * H)(\omega) \over \sqrt{2\pi}\,$ $(G * H)(f)\,$ This is the dual of 8
10 For a purely real even function $g(t)\,$ $G(\omega)\,$ is a purely real even function $G(f)\,$ is a purely real even function
11 For a purely real odd function $g(t)\,$ $G(\omega)\,$ is a purely imaginary odd function $G(f)\,$ is a purely imaginary odd function