Main properties

Section: Fourier

Linearity
Energy and inner product preservation (Plancherel-Parseval theorem)
Translation in time becomes modulation in frequency
Modulation in time becomes translation in frequency
Time scale
Derivation
If $s$ is real, $\hat s$ is complex with hermitian symmetrie
Convolution : $$ \widehat{u\star v}(\nu) = \hat u(\nu) \hat v(\nu) $$ For digital signals of size $N$, the convolution is supposed to be N-periodic (using zero padding if necessary)