Fourier & Sound Lab

Every signal is a sum of sine waves. Draw with circles, build sounds from harmonics, and see what your ears do with frequencies.

Sound

off

Master volume 20%

Starts quiet. All output passes a soft limiter and every gain change is ramped, so nothing clicks or blasts. Sound only starts after you press a button.

Shape

Letter

Or draw your own closed shape directly on the canvas. The path is resampled to 512 points and transformed with a discrete Fourier transform.

Epicycles

Number of circles 60

Speed 1.0× Trail 70%

Circles Original shape Approximation

Frequency table

#	Freq k	Amplitude	Phase

Biggest circles first. Freq k is how many turns each circle makes per loop; amplitude is its radius; phase is where it starts.

Fourier series

Any repeating waveform of period T is a sum of sines at multiples of the fundamental f₀ = 1/T: x(t) = Σ aₙ sin(2π n f₀ t + φₙ). A square wave needs only odd harmonics with amplitudes 1/n; a saw uses all of them; a triangle uses odd ones falling as 1/n². Sharp jumps need infinitely many terms, and truncating gives ringing that never shrinks below ≈9% of the jump (Gibbs).

Discrete Fourier transform

Given N samples zₙ, the DFT gives cₖ = (1/N) Σ zₙ e^−2πikn/N. Reading the sum backwards, zₙ = Σ cₖ e^+2πikn/N: each cₖ is a circle of radius |cₖ| spinning k turns per loop starting at angle arg cₖ. Sorting by radius and keeping the biggest M circles is the best M-term approximation, and the squared error equals the discarded energy Σ|cₖ|² (Parseval). The spectrum analyser uses the same maths via a fast algorithm (FFT).

Sampling and Nyquist

A signal sampled at rate fₛ can only represent frequencies below the Nyquist limit fₛ/2. Anything higher folds back (aliases) to a false lower frequency. The 512-point loop above holds at most 256 turns per loop; with fewer circles you drop the fine detail first. Your audio device runs at 44.1–48 kHz, so the spectrum ends near 22–24 kHz.