The Fourier Transform of a periodic signal f(t) can be computed in the following way. In this animation, the signal is depicted in blue.
For each frequency \tau, wrap the signal around the origin such that [0,2\pi/\tau] is mapped onto one turn. Repeat the signal k times (in this animation k=3).
The center of gravity of the wrapped signal describes a curve in the complex plane (depicted in orange). This curves stays mostly near the origin, except for a few wild swings. The amplitude and phase of those excursions are the Fourier coefficients of the signal f(t).
Note that we do not really need the original signal to be periodic ; the construction holds for any properly windowed signal. For each frequency \tau, the center of gravity of the wrapped signal, read as a complex number, is the value of the Fourier transform \hat{f}(\tau).
This animation is part of a joint article by Hajer Bahouri & François Vigneron about signal analysis (images.math.cnr....
28 июл 2018
