Example 4.11 - Wave packet

Wave packet – Fourier transform for a given rectangular course of harmonic wave amplitudes. The width of the wave packet is inversely proportional to the width of the spectrum.

The time course of the signal is as follows:

\[ f(t) = \frac{2\sin(\frac{\Delta \omega}{2} t)}{t \, \Delta \omega} \cos (\omega_0 t) \]

The parameter $\omega_0$ determines the frequency of the carrier wave ("filling" wave) and the parameter $\Delta \omega$ determines the width of the frequency spectrum used, which inversely determines the width of the wave packet (through changing the frequency of the "modulating" sine).

Wave packet with shown amplitude envelopes $\frac{2}{t \, \Delta \omega}$ (dotted) and $\frac{2\sin(\frac{\Delta \omega}{2} t)}{t \, \Delta \omega}$ (dashed):

Wave packet alone without amplitude envelopes:

For an interactive graph where you can change the parameters $\omega_0$ and $\Delta \omega$, download the following source code (Wolfram Mathematica notebook):

Source code: nb