Fourier Transform

Building signals from sinusoids — synthesis and analysis animated

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f(t) = Σ aₙ sin(nωt + φₙ)  |  Fourier series with N harmonics

The rotating circles on top each represent one Fourier harmonic. Their combined tip traces the reconstructed waveform (right panel). The spectrum below shows amplitude vs. frequency. Increase harmonics to add fidelity — a square wave needs infinitely many to form perfect corners (Gibbs phenomenon).