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