An interactive online graphing calculator to visualize a pulse \( f(t) \) and its Fourier transform \( F(\omega) \) is presented. \( t \) is the time and \( \omega \) is the angular frequency.

Fourier Transform of a Pulse

The pulse is defined as follows and its graph is shown below.
\[
f(t) =
\begin{cases}
1 \quad \text{if } -T/2 \le t \le T/2 \\ \\
0 \quad \text{if } t \lt - T/2 \; \text{or} \; t \gt T/2
\end{cases}
\]
The Fourier of \( f(t) \) is defines by
\[ F(\omega) = \int_{-\infty}^{+\infty} f(t) e^\left( - j \omega t\right) dt \\
= \int_{-T/2}^{+T/2} 1 \cdot e^\left( - j \omega t\right) dt \\
= \left[ \dfrac{ e^\left( - j \omega t \right)}{-j \omega} \right]^{T/2}_{-T/2}
\]
Evaluate and simplify to obtain
\[ F(\omega) = \dfrac{\sin(\omega(T/2))}{\omega/2}
\]

Interactive Tutorial

Increase and decrease the width \( T \) of the pulse and notice the changes to both the pulse (blue) and its Fourier transform (green). Explain what is happening.