- Signals and Systems Tutorial
- Signals & Systems Home
- Signals & Systems Overview
- Signals Basic Types
- Signals Classification
- Signals Basic Operations
- Systems Classification
- Signals Analysis
- Fourier Series
- Fourier Series Properties
- Fourier Series Types
- Fourier Transforms
- Fourier Transforms Properties
- Distortion Less Transmission
- Hilbert Transform
- Convolution and Correlation
- Signals Sampling Theorem
- Signals Sampling Techniques
- Laplace Transforms
- Laplace Transforms Properties
- Region of Convergence
- Z-Transforms (ZT)
- Z-Transforms Properties
- Signals and Systems Resources
- Signals and Systems - Resources
- Signals and Systems - Discussion
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) $