# properties of Fourier Transforms

1.Linearity Property:-

$x_{1}(t)\leftrightarrow&space;X_{1}(j\omega&space;)$

$x_{2}(t)\leftrightarrow&space;X_{2}(j\omega)$

$a\&space;x_{1}(t)+b\&space;x_{2}(t)\leftrightarrow&space;\&space;\&space;?$.

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

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

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

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

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

2.Time-shifting Property:-

$x(t)\leftrightarrow&space;X(j\omega&space;)$

$x(t-t_{o})\leftrightarrow&space;\&space;?$.

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

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

$t&space;\&space;limits&space;\&space;:&space;\&space;-\infty&space;\&space;to&space;\&space;\infty,&space;\&space;\&space;\&space;\lambda&space;\&space;\&space;limits&space;\&space;:&space;\&space;\infty&space;\&space;to&space;\&space;-\infty$

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

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

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

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

3.Frequency-shifting Property:-

$x(t)\leftrightarrow&space;X(\omega&space;)$

$?\&space;\leftrightarrow&space;\&space;X(\omega&space;-\omega&space;_{o})$.

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

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

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

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

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

4. Differentiation in time-domain:-

$x(t)\leftrightarrow&space;X(\omega&space;)$

$\frac{dx(t)}{dt}\leftrightarrow&space;\&space;?$.

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

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

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

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

$\frac{dx(t)}{dt}\leftrightarrow&space;\&space;j&space;\omega&space;\&space;X(\omega&space;)$.

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

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

5.Differentiation in w-domain:-

$x(t)\leftrightarrow&space;X(\omega&space;)$

$?\leftrightarrow&space;\frac{dX(\omega&space;)}{d\omega&space;}$.

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

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

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

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

$\frac{dX(\omega&space;)}{d\omega&space;}\leftrightarrow&space;\&space;-jt\&space;x(t)$.

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

6. Conjugation property:-

$x(t)\leftrightarrow&space;X(\omega&space;)$

$x^{*}(t)\leftrightarrow&space;\&space;\&space;?$.

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

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

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

$x^{*}(t)\leftrightarrow&space;\&space;X^{*}(-\omega&space;)$.

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

7. Time-Scaling property:-

$x(t)\leftrightarrow&space;X(\omega&space;)$

$x(at)\leftrightarrow&space;\&space;\&space;?$.

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

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

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

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

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

8. Convolution in Time-domain:-

$x_{1}(t)\leftrightarrow&space;X_{1}(\omega&space;)$

$x_{2}(t)\leftrightarrow&space;X_{2}(\omega&space;)$

$x_{1}(t)&space;*&space;x_{2}(t)\leftrightarrow&space;\&space;\&space;?$.

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

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

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

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

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

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

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

$x_{1}(t)&space;*&space;x_{2}(t)\leftrightarrow&space;\&space;X_{1}(\omega&space;)&space;\&space;X_{2}(\omega&space;)$.

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## Author: Lakshmi Prasanna Ponnala

Completed M.Tech in Digital Electronics and Communication Systems.

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