# Engineering Tables/Fourier Transform Properties

Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks
$g(t)\!\equiv \!$ ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!G(\omega )e^{i\omega t}d\omega \,$ $G(\omega )\!\equiv \!$ ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\!\!g(t)e^{-i\omega t}dt\,$ $G(f)\!\equiv$ $\int _{-\infty }^{\infty }\!\!g(t)e^{-i2\pi ft}dt\,$ 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 $g(t-a)\,$ $e^{-ia\omega }G(\omega )\,$ $e^{-i2\pi af}G(f)\,$ 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 $g(at)\,$ ${\frac {1}{|a|}}G\left({\frac {\omega }{a}}\right)\,$ ${\frac {1}{|a|}}G\left({\frac {f}{a}}\right)\,$ If $|a|\,$ is large, then $g(at)\,$ 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 ${\frac {d^{n}g(t)}{dt^{n}}}\,$ $(i\omega )^{n}G(\omega )\,$ $(i2\pi f)^{n}G(f)\,$ 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