Engineering Tables/Fourier Transform Properties

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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