Fourier Transforms Properties

Here are the properties of Fourier Transform:

Linearity Property

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

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

Then pnearity property states that

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

Time Shifting Property

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

Then Time shifting property states that

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

Frequency Shifting Property

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

Then frequency shifting property states that

$e^{jomega_0 t} . x (t) stackrel{mathrm{F.T}}{longleftrightarrow} X(omega - omega_0)$

Time Reversal Property

$ ext{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 Scapng Property

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

Then Time scapng 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} jomega . 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 (jomega)^n} X(omega) $

Multippcation and Convolution Properties

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

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

Then multippcation 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) $