# Energy Spectal Density of rectangular pulse

As we all know the Rectangular pulse is defined as $x(t)&space;=&space;rect(\frac{t}{T})$, exists for a duration of T sec symmetrical with respect to y-axis as shown

Fourier Transform is $X(f)$

$X(f)&space;=&space;\int_{\frac{-T}{2}}^{\frac{T}{2}}x(t)e^{-j2\pi&space;ft}dt$

$X(f)&space;=&space;\int_{\frac{-T}{2}}^{\frac{T}{2}}&space;1.e^{-j2\pi&space;ft}dt$

$X(f)&space;=&space;\left&space;[&space;\frac{e^{-j2\pi&space;ft}}{j2\pi&space;f}&space;\right&space;]&space;^{\frac{T}{2}}_{\frac{-T}{2}}$

$X(f)&space;=&space;\frac{1}{\pi&space;f}\left&space;[&space;\frac{e^{j\pi&space;ft}-e^{-j\pi&space;ft}}{2j}&space;\right&space;]$

$X(f)&space;=&space;\frac{T&space;\sin&space;\pi&space;fT}{\pi&space;fT}$

$X(f)&space;=&space;X(f)&space;=&space;sinc&space;(\pi&space;fT)$

$\therefore$ The Energy Spectral Density of the given signal $x(t)$ will be $S(f)&space;=&space;\left&space;|&space;X(f)&space;\right&space;|^{2}$

$s(f)=&space;T^{2}sinc^{2}(\pi&space;fT)$.

## Author: Lakshmi Prasanna

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