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

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