# 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 |