Example 4.12 - Rectangular pulse

Rectangular pulse – Fourier transform for a given rectangular signal course with duration $\Delta t$. The width of dominant frequencies is inversely proportional to the pulse duration.

After calculating the Fourier transform, i.e., expressing the given pulse $f(t)$ as

\[ f(t) = \int_0^{+\infty} B(\omega) \cos(\omega t) dt, \]

we get the form of function $B(\omega)$:

\[ B(\omega) = \frac{2 \sin(\frac{\Delta t}{2} \omega)}{\pi \Delta t \, \omega}. \]

The largest frequency amplitudes are concentrated near zero. The point where $B(\omega) = 0$ for the first time determines the width of the frequency spectrum:

\[ \sin(\frac{\Delta t}{2} \omega) = 0 \quad \rightarrow \quad \frac{\Delta t}{2} \Delta \omega = \pi \quad \rightarrow \quad \Delta t \, \Delta \omega = 2\pi. \]

An interactive graph showing the change in the shape of the frequency spectrum depending on the pulse duration $\Delta t$ can be obtained by downloading the following source code:

Source code: nb