🌊 Fourier Series
Any periodic signal can be decomposed into a sum of pure sine waves — this is the Fourier series. The left panel shows rotating phasors: each circle spins at a harmonic frequency, and the tip of the last phasor traces the waveform. The right panel shows the signal built so far (green) vs the ideal target (dim). The bottom panel is the frequency spectrum — amplitude of each harmonic. Notice the Gibbs phenomenon: the ~9% overshoot at discontinuities that never fully disappears even with infinitely many harmonics. 🇺🇦 Українська
Waveform
Frequency Spectrum
How It Works
The Fourier coefficients for a square wave (±1, period 2π) are: a_n = 4/(nπ) for odd n, zero for even n. Each rotating phasor has length equal to its amplitude. As you increase N, the partial sum converges to the target waveform — but never quite reaches the corners. This 9% overshoot at discontinuities is the Gibbs phenomenon, named after J. W. Gibbs (1899). It affects audio engineering (ringing near transients), image compression (JPEG ringing at edges), and numerical PDE solvers.