Energy Spectal Density of rectangular pulse

As we all know the Rectangular pulse is defined as x(t) = 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) = \int_{\frac{-T}{2}}^{\frac{T}{2}}x(t)e^{-j2\pi ft}dt

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

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

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

X(f) = \frac{T \sin \pi fT}{\pi fT}

X(f) = X(f) = sinc (\pi fT) 

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

s(f)= T^{2}sinc^{2}(\pi fT).

 

Author: Lakshmi Prasanna

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