Engineering Tables/Fourier Transform Properties

	Signal	Fourier transform unitary, angular frequency	Fourier transform unitary, ordinary frequency	Remarks
	$g(t)\equiv$ $\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}G(\omega )e^{i\omega t}d\omega$	$G(\omega )\equiv$ $\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}g(t)e^{-i\omega t}dt$	$G(f)\equiv$ $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}g(t)e^{-i2\pi ft}dt$
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	$e^{iat}g(t)$	$G(\omega -a)$	$G(f-\frac{a}{2\pi })$	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	$\frac{d^{n}g(t)}{d{t}^n}$	$(i\omega {)}^{n}G(\omega )$	$(i2\pi f{)}^{n}G(f)$	Generalized derivative property of the Fourier transform
7	$t^{n}g(t)$	$i^n\frac{d^{n}G(\omega )}{d{\omega }^n}$	${\left(\frac{i}{2\pi }\right)}^n\frac{d^{n}G(f)}{d{f}^n}$	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	$g(t)h(t)$	$\frac{(G*H)(\omega )}{\sqrt{2\pi }}$	$(G*H)(f)$	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