# Visualization of the Fourier Transform of a Pulse

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.

T =