Fourier series

Idea

A Fourier series splits a periodic function into a sum of sine and cosine waves of different frequencies. The low-frequency terms draw the rough outline, and the high-frequency terms fill in the detail. Many common functions with period $2\pi$ can be written this way:

The coefficients $a_n,b_n$ give the weight of each frequency in $f$. Adding only the first $N$ terms gives an approximation of $f$, and the larger $N$ is, the closer the fit.

Interaction

• Number of terms N: use the slider to change how many terms are added, and watch the stacked circles approach a square wave or a 2D periodic orbit
• 1D square wave / 2D orbit: switch between the waveform and the path in the plane, and compare how the same harmonics look in the two views

Suggested order: a few harmonics → increase N and watch the fit → compare the oscillation at the discontinuous edges.

What to notice

• A periodic path that is regular enough can be split into a sum of basic frequencies (circular motions).
• The smoother the function, the faster the coefficients decay, and the faster a finite-N approximation gets close to the target.
• At a discontinuity, adding terms still leaves the Gibbs overshoot, about 9% of the jump height.

Related

• Spirograph curve

Further reading

• Fourier series (Wikipedia)
• Gibbs phenomenon (Wikipedia)