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

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

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

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

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

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

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

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

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

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

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

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