Engineering Tables/Fourier Transform Properties

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

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

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

${\displaystyle \int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}dt\,}$
1 ${\displaystyle a\cdot g(t)+b\cdot h(t)\,}$ ${\displaystyle a\cdot G(\omega )+b\cdot H(\omega )\,}$ ${\displaystyle a\cdot G(f)+b\cdot H(f)\,}$ Linearity
2 ${\displaystyle g(t-a)\,}$ ${\displaystyle e^{-ia\omega }G(\omega )\,}$ ${\displaystyle e^{-i2\pi af}G(f)\,}$ Shift in time domain
3 ${\displaystyle e^{iat}g(t)\,}$ ${\displaystyle G(\omega -a)\,}$ ${\displaystyle G\left(f-{\frac {a}{2\pi }}\right)\,}$ Shift in frequency domain, dual of 2
4 ${\displaystyle g(at)\,}$ ${\displaystyle {\frac {1}{|a|}}G\left({\frac {\omega }{a}}\right)\,}$ ${\displaystyle {\frac {1}{|a|}}G\left({\frac {f}{a}}\right)\,}$ If ${\displaystyle |a|\,}$ is large, then ${\displaystyle g(at)\,}$ is concentrated around 0 and ${\displaystyle {\frac {1}{|a|}}G\left({\frac {\omega }{a}}\right)\,}$ spreads out and flattens
5 ${\displaystyle G(t)\,}$ ${\displaystyle g(-\omega )\,}$ ${\displaystyle g(-f)\,}$ Duality property of the Fourier transform. Results from swapping "dummy" variables of ${\displaystyle t\,}$ and ${\displaystyle \omega \,}$.
6 ${\displaystyle {\frac {d^{n}g(t)}{dt^{n}}}\,}$ ${\displaystyle (i\omega )^{n}G(\omega )\,}$ ${\displaystyle (i2\pi f)^{n}G(f)\,}$ Generalized derivative property of the Fourier transform
7 ${\displaystyle t^{n}g(t)\,}$ ${\displaystyle i^{n}{\frac {d^{n}G(\omega )}{d\omega ^{n}}}\,}$ ${\displaystyle \left({\frac {i}{2\pi }}\right)^{n}{\frac {d^{n}G(f)}{df^{n}}}\,}$ This is the dual to 6
8 ${\displaystyle (g*h)(t)\,}$ ${\displaystyle {\sqrt {2\pi }}G(\omega )H(\omega )\,}$ ${\displaystyle G(f)H(f)\,}$ ${\displaystyle g*h\,}$ denotes the convolution of ${\displaystyle g\,}$ and ${\displaystyle h\,}$ — this rule is the convolution theorem
9 ${\displaystyle g(t)h(t)\,}$ ${\displaystyle (G*H)(\omega ) \over {\sqrt {2\pi }}\,}$ ${\displaystyle (G*H)(f)\,}$ This is the dual of 8
10 For a purely real even function ${\displaystyle g(t)\,}$ ${\displaystyle G(\omega )\,}$ is a purely real even function ${\displaystyle G(f)\,}$ is a purely real even function
11 For a purely real odd function ${\displaystyle g(t)\,}$ ${\displaystyle G(\omega )\,}$ is a purely imaginary odd function ${\displaystyle G(f)\,}$ is a purely imaginary odd function