Properties of Fourier Transforms

1.Linearity Property:-

x_{1}(t)\leftrightarrow X_{1}(j\omega )

x_{2}(t)\leftrightarrow X_{2}(j\omega)

a\ x_{1}(t)+b\ x_{2}(t)\leftrightarrow \ \ ?.

we know that  Fourier Transform of a signal  x(t)  is X(j\omega) = \int_{-\infty }^{\infty }\ x(t) \ e^{-j\omega t} \ dt .

F\left \{a\ x_{1}(t)+b\ x_{2}(t)\right \} = \int_{-\infty }^{\infty }\ \left \{ a\ x_{1}(t)+b\ x_{2}(t) \right \} \ e^{-j\omega t}\ dt.

F\left \{a\ x_{1}(t)+b\ x_{2}(t)\right \} = \int_{-\infty }^{\infty }\ \left \{ a\ x_{1}(t)\ e^{-j\omega t}\ dt+b\ x_{2}(t) \ e^{-j\omega t}\ dt\right \}

F\left \{a\ x_{1}(t)+b\ x_{2}(t)\right \} = \ \ a \int_{-\infty }^{\infty }\ x_{1}(t)\ e^{-j\omega t}\ dt+b \int_{-\infty }^{\infty }\ x_{2}(t) \ e^{-j\omega t}\ dt.

F\left \{a\ x_{1}(t)+b\ x_{2}(t)\right \} = \ a \ X_{1}(j\omega) +b \ X_{2}(j\omega) .

2.Time-shifting Property:-

x(t)\leftrightarrow X(j\omega )

x(t-t_{o})\leftrightarrow \ ?.

we know that  X(j\omega) = \int_{-\infty }^{\infty }\ x(t) \ e^{-j\omega t} \ dt .

L\left \{ x(t-t_{o}) \right \} = \int_{-\infty }^{\infty }\ x(t-t_{o}) \ e^{- j\omega t}\ dt.    Lett-t_{o}=\lambda \Rightarrow dt= d\lambda

t \ limits \ : \ -\infty \ to \ \infty, \ \ \ \lambda \ \ limits \ : \ \infty \ to \ -\infty

L\left \{ x(t-t_{o}) \right \} = \int_{\lambda =-\infty }^{\infty }\ x(\lambda ) \ e^{- j\omega (\lambda +t_{o})}\ d\lambda .

L\left \{ x(t-t_{o}) \right \} =e^{-j\omega t_{o}} \int_{\lambda =-\infty }^{\infty }\ x(\lambda ) \ e^{-j\omega \lambda }\ d\lambda.

x(t-t_{o})\leftrightarrow \ e^{- j\omega t_{o}}\ X(j\omega).

from the above equation x(t-t_{o})  forms  Fourier Transform pair with e^{- j\omega t_{o}} \ X(j\omega).

3.Frequency-shifting Property:-

x(t)\leftrightarrow X(\omega )

?\ \leftrightarrow \ X(\omega -\omega _{o}).

we know that  X(\omega ) \ or X(j\omega ) = \int_{-\infty }^{\infty }\ x(t) \ e^{-j\omega t} \ dt .

L\left \{ e^{j\omega _{o}t}\ x(t) \right \} = \int_{t=-\infty }^{\infty }\ e^{j \omega _{o}t}\ x(t) \ e^{-j\omega t}\ dt.

L\left \{ e^{j\omega _{o}t}\ x(t) \right \} = \int_{t=-\infty }^{\infty }\ \ x(t) \ e^{-(\omega -\omega _{o})t}\ dt .

e^{j\omega _{o}t}x(t)\leftrightarrow \ X(\omega -\omega _{o}).

from the above equation e^{j\omega _{o}t}\ x(t)  forms  Fourier Transform pair with X(\omega -\omega _{o}).

4. Differentiation in time-domain:-

x(t)\leftrightarrow X(\omega )

\frac{dx(t)}{dt}\leftrightarrow \ ?.

we know that  inverse Fourier  Transform  x(t) =\frac{1}{2\pi } \int_{\omega =\infty }^{\infty }\ X( \omega ) \ e^{j\omega t} \ d\omega .

\frac{dx(t)}{dt} =\frac{1}{2\pi } \int_{\omega =\infty }^{\infty }\ X(\omega ) \ \frac{d(e^{j\omega t})}{dt} \ d\omega.

\frac{dx(t)}{dt} =\frac{1}{2\pi } \int_{\omega =\infty }^{\infty }\ X(\omega ) \ j \omega \ e^{j\omega t} \ d\omega .

\frac{dx(t)}{dt} =\frac{1}{2\pi } \int_{\omega =\infty }^{\infty }\ (\ j \omega \ X(\omega )) \ e^{j\omega t} \ d\omega.

\frac{dx(t)}{dt}\leftrightarrow \ j \omega \ X(\omega ).

from the above equation \frac{dx(t)}{dt}  forms Fourier Transform pair with \ j \omega \ X(\omega )

Similarly  \frac{d^{n}x(t)}{dt^{n}}\leftrightarrow \ \ (j \omega) ^{n}\ X(\omega ).

5.Differentiation in w-domain:-

x(t)\leftrightarrow X(\omega )

?\leftrightarrow \frac{dX(\omega )}{d\omega }.

we know that  X(\omega ) = \int_{t =-\infty }^{\infty }\ x(t) \ e^{-j\omega t} \ dt .

\frac{dX(\omega )}{d\omega } = \int_{t=-\infty }^{\infty }\ x(t) \ \frac{d(e^{-j\omega t}) }{d\omega } \ dt.

\frac{dX(\omega )}{d\omega } = \int_{t=-\infty }^{\infty }\ x(t) \ e^{-j\omega t} \ (-j t)\ dt .

\frac{dX(\omega )}{d\omega } = \int_{t=-\infty }^{\infty }\ (-jt \ x(t)) \ e^{-j\omega t} \ dt.

\frac{dX(\omega )}{d\omega }\leftrightarrow \ -jt\ x(t).

from the above equation \frac{dX(\omega )}{d\omega }  forms Fourier Transform pair with -jt\ x(t).

6. Conjugation property:-

x(t)\leftrightarrow X(\omega )

x^{*}(t)\leftrightarrow \ \ ?.

we know that  X(\omega ) = \int_{t=-\infty }^{\infty }\ x(t) \ e^{-j\omega t} \ dt .

F\left \{ x^{*}(t) \right \} = \int_{-\infty }^{\infty }\ x^{*}(t) \ e^{-j\omega t} \ dt.

F\left \{ x^{*}(t) \right \} = \int_{t =-\infty }^{\infty }( x(t ) \ e^{j \omega t} \ dt )^{*} .

x^{*}(t)\leftrightarrow \ X^{*}(-\omega ).

from the above equation x^{*}(t)  forms Fourier Transform pair with X^{*}(-\omega ).

7. Time-Scaling property:-

x(t)\leftrightarrow X(\omega )

x(at)\leftrightarrow \ \ ?.

we know that  X(\omega ) = \int_{-\infty }^{\infty }\ x(t) \ e^{-j\omega t} \ dt .

F\left \{ x(at) \right \} = \int_{-\infty }^{\infty }\ x(at) \ e^{-j\omega t} \ dt.          Let  at=\ \lambda ,      dt=\ \frac{d\lambda}{a}t \ limits \ : \ -\infty \ to \ \infty, \ \ \ \lambda \ \ limits \ : \ -\infty \ to \ \infty.

F\left \{ x(at) \right \} = \frac{1}{a}\int_{\lambda =-\infty }^{\infty }\ x(\lambda ) \ e^{-j(\frac{\omega }{a})\lambda } \ d\lambda .

x(at)\leftrightarrow \frac{1}{a} \ X(\frac{\omega }{a}), \ \ \ if \ a>0.

x(-at)\leftrightarrow \frac{1}{a} \ X(\frac{-\omega }{a}), \ \ \ if \ a<0 \ and \ (a\neq -1).

8. Convolution in Time-domain:-

x_{1}(t)\leftrightarrow X_{1}(\omega )

x_{2}(t)\leftrightarrow X_{2}(\omega )

x_{1}(t) * x_{2}(t)\leftrightarrow \ \ ?.

we know that  X(\omega ) = \int_{t=-\infty }^{\infty }\ x(t) \ e^{-j\omega t} \ dt .

F\left \{ x_{1}(t) * x_{2}(t) \right \} = \int_{t=-\infty }^{\infty }\ \left \{ x_{1}(t) * x_{2}(t) \right \}\ e^{-j\omega t} \ dt.

F\left \{ x_{1}(t) * x_{2}(t) \right \} = \int_{t=-\infty }^{\infty }\ \int_{\tau =-\infty }^{\infty }\ x_{1}(\tau )\left \{ x_{2}(t-\tau ) \right \}\ e^{-j\omega t} \ dt \ d\tau.

F\left \{ x_{1}(t) * x_{2}(t) \right \} = \int_{\tau =-\infty }^{\infty }\ x_{1}(\tau )\left \{ \int_{t=-\infty }^{\infty } x_{2}(t-\tau )\ e^{-j\omega t} \ dt \right \} \ d\tau.

F\left \{ x_{1}(t) * x_{2}(t) \right \} = \int_{\tau =-\infty }^{\infty }\ x_{1}(\tau )\left \{ e^{-j\omega \tau } X_{2}(\omega ) \right \} \ d\tau.

F\left \{ x_{1}(t) * x_{2}(t) \right \} = \left \{ \int_{\tau =-\infty }^{\infty }\ x_{1}(\tau ) e^{-j\omega \tau } \ d\tau \right \} \ X_{2}(\omega ).

F\left \{ x_{1}(t) * x_{2}(t) \right \} = \ X_{1}(\omega ) \ X_{2}(\omega )

x_{1}(t) * x_{2}(t)\leftrightarrow \ X_{1}(\omega ) \ X_{2}(\omega ).

 

1 Star2 Stars3 Stars4 Stars5 Stars (No Ratings Yet)

Loading...

Author: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.