# Fourier Transforms Properties

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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

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