Fourier Transforms Properties

Here are the properties of Fourier Transform:

Linearity Property

$\text{If}\,\,x (t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} X(\omega) $

$ \text{&} \,\, y(t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} Y(\omega) $

Then linearity property states that

$a x (t) + b y (t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} a X(\omega) + b Y(\omega) $

Time Shifting Property

$\text{If}\, x(t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} X (\omega)$

Then Time shifting property states that

$x (t-t_0) \stackrel{\mathrm{F.T}}{\longleftrightarrow} e^{-j\omega t_0 } X(\omega)$

Frequency Shifting Property

$\text{If}\,\, x(t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} X(\omega)$

Then frequency shifting property states that

$e^{j\omega_0 t} . x (t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} X(\omega - \omega_0)$

Time Reversal Property

$ \text{If}\,\, x(t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} X(\omega)$

Then Time reversal property states that

$ x (-t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} X(-\omega)$

Time Scaling Property

$ \text{If}\,\, x (t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} X(\omega) $

Then Time scaling property states that

$ x (at) {1 \over |\,a\,|} X { \omega \over a}$

Differentiation and Integration Properties

$ If \,\, x (t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} X(\omega)$

Then Differentiation property states that

$ {dx (t) \over dt} \stackrel{\mathrm{F.T}}{\longleftrightarrow} j\omega . X(\omega)$

$ {d^n x (t) \over dt^n } \stackrel{\mathrm{F.T}}{\longleftrightarrow} (j \omega)^n . X(\omega) $

and integration property states that

$ \int x(t) \, dt \stackrel{\mathrm{F.T}}{\longleftrightarrow} {1 \over j \omega} X(\omega) $

$ \iiint ... \int x(t)\, dt \stackrel{\mathrm{F.T}}{\longleftrightarrow} { 1 \over (j\omega)^n} X(\omega) $

Multiplication and Convolution Properties

$ \text{If} \,\, x(t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} X(\omega) $

$ \text{&} \,\,y(t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} Y(\omega) $

Then multiplication property states that

$ x(t). y(t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} X(\omega)*Y(\omega) $

and convolution property states that

$ x(t) * y(t) \stackrel{\mathrm{F.T}}{\longleftrightarrow} {1 \over 2 \pi} X(\omega).Y(\omega) $