In 1807 Joseph Fourier announced a claim that many thought was crazy: any periodic waveform — the sound of a violin, a heartbeat, even the outline of a drawing — can be built from sine waves alone. See for yourself: build a wave, draw a shape with circles, wind a signal around a wheel and look into its spectrum.
Sum of harmonics
two periods · hover to see values
Spectrum and harmonics
drag the bars or the sliders
The slider sets the amplitude An, the knob sets the phase φn (drag in a circle or use the arrow keys; double-click resets).
Draw your own closed shape with the mouse or pick a preset.
Draw one closed shape in a single stroke
Signal over time
4 seconds · gold line = point weights
Wound around a circle
pink dot = center of mass
Center-of-mass distance
as a function of winding frequency
Space — sweep · drag along the plot to pick a frequency
Signal components
g(t) = Σ A cos(2π f t)
The signal is hidden: it contains 2 components. Sweep the winding frequency, and when you hit a peak, click Mark peak.
Time
draw one period of a wave
DFT →DFT ↓← synthesis↑ synthesis
Frequency
drag a bar — the wave will change
Did you know…?
Gauss knew the fast Fourier transform around 1805 — he used it to compute the orbits of the asteroids Pallas and Juno. His notes were published only after his death and then forgotten; the algorithm was rediscovered by Cooley and Tukey in 1965. Today an FFT runs in every phone.
A machine spotted Gibbs's overshoot. In 1898 Albert Michelson built a mechanical harmonic analyzer that summed 80 sine waves — and at jumps it drew strange overshoots. J. Willard Gibbs explained them (1899), although Henry Wilbraham had described the effect as early as 1848.
Your ear does Fourier analysis. In the cochlea of the inner ear, the basilar membrane vibrates at different places for different frequencies: near the base — high tones (up to about 20 kHz), near the apex — low ones (about 20 Hz). Thousands of hair cells act like the bars of a spectrum.
Challenges 0 / 3
A square wave from 5 harmonics
Set harmonics 1, 3, 5, 7 and 9 in the proportions 1 : ⅓ : ⅕ : ⅐ : ⅑ (the rest at zero, phases 0°). to do
Find the hidden frequencies
Click “Random signal”, sweep with the winding machine and mark every peak. to do
Thrifty artist
Draw your own shape and reproduce it with at most 20 circles at ≥ 95% accuracy (a plain circle doesn't count!). to do