Fourier Series

Mathematical Principles & Theorems

Decomposes any periodic function \(f(x)\) of period \(2\pi\) into orthogonal harmonic sinusoids: \(f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty [a_n \cos(nx) + b_n \sin(nx)]\) where \(a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos(nx)dx\) and \(b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin(nx)dx\). Complex exponential form \(f(x) = \sum c_n e^{i n x}\) represents epicyclic rotating vectors. Gibbs phenomenon produces an asymptotic \(\approx 8.95\%\) overshoot near jump discontinuities.

Operating Instructions

• Select target periodic waveform: Square Wave, Sawtooth, Triangle Wave, or Pulse.
• Set the number of harmonic terms \(N\) using the precise numeric value box.
• Adjust epicycle animation speed to observe circular phasor vector addition in real time.
• Observe Gibbs phenomenon ringing near steep step transitions on the signal reconstruction canvas.