Engineering Tables/Fourier Transform Properties

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Signal Fourier transform
unitary, angular frequency
Fourier transform
unitary, ordinary frequency
Remarks






1 Linearity
2 Shift in time domain
3 Shift in frequency domain, dual of 2
4 If is large, then is concentrated around 0 and spreads out and flattens
5 Duality property of the Fourier transform. Results from swapping "dummy" variables of and .
6 Generalized derivative property of the Fourier transform
7 This is the dual to 6
8 denotes the convolution of and — this rule is the convolution theorem
9 This is the dual of 8
10 For a purely real even function is a purely real even function is a purely real even function
11 For a purely real odd function is a purely imaginary odd function is a purely imaginary odd function