The Discrete Fourier Transform (DFT) of a time series maps it to the the frequency space and the inverse (IDFT) transformation brings back the original signal. Each time series has a unique representation in the frequency space, from which the time series can be reconstructed.

The DFT and IDFT are respectively: \( X_k = \sum_{n=0}^{N-1} x_n \cdot e^{(-2 \pi i\, k\, n) / N} \) and \(x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{(2 \pi i\, k\, n) / N}\)

The animation below is designed to help understand how a sum of monofrequency signals, represented by complex \(e^{2 i \pi f n}\), can approximate an arbitrary given time series, based on a few typical examples. The complex exponentials are projected to the real axis (vertical) to obtain the time series. The numbers below each single frequency signal are respectively the amplitude and the phase (as a multiple of \(\pi\)).

Press Right Arrow to speed up transitions, Left Arrow to slow down, Down Arrow for a temporary slow motion, Up Arrow to go back to initial speed, L/S to lengthen/shorten the trail