Fourier series

Fourier's scandalous 1807 claim: any repeating signal — square waves, sawtooth ramps, a violin's waveform — is a sum of plain sinusoids at multiples of one base frequency. The establishment scoffed. The establishment was wrong.

Build the intuition

The recipe: fundamental plus harmonics

A signal repeating f times per second can be built from sinusoids at f, 2f, 3f, … — the fundamental and its harmonics — each with its own amplitude and phase. The amplitude list is the signal's recipe. Play A440 on two instruments: same fundamental, different harmonic amplitudes — that difference is timbre.

x(t) = a_0 + \sum_{n=1}^{\infty} a_n \cos(n\omega t) + b_n \sin(n\omega t)

Finding coefficients = projecting

How much of harmonic n? Inner product with that harmonic (multiply and integrate) — exactly the projection move from last lesson, made continuous. Orthogonality guarantees each harmonic's coefficient comes out clean, untouched by the others. The intimidating integral formulas are dot products in robes.

b_n = \frac{2}{T}\int_0^T x(t)\sin(n\omega t)\,dt

Sharp corners need high frequencies

A square wave needs only odd harmonics, at amplitudes 1, ⅓, ⅕, … — but infinitely many, because corners are fast events and fast events live at high frequencies. Truncate the series and corners ring slightly (the Gibbs phenomenon) — the overshoot you can watch in the demo. Smoothness in time ↔ rapid decay in frequency: the duality that runs the whole subject.

See it move

InteractiveBuilding a square from circles

Harmonics2

2 harmonics: corners are forming. Each added harmonic is smaller and faster, sharpening the shoulders.

Fourier's claim, performed: stack odd harmonics at 1/n strength and watch roundness conspire into corners — Gibbs ringing included.

A worked example

Read a square wave's recipe

Square wave at 100 Hz. Its series:
$\frac{4}{\pi}\left(\sin\omega t + \tfrac{1}{3}\sin 3\omega t + \tfrac{1}{5}\sin 5\omega t + \cdots\right)$
Reading: energy at 100, 300, 500, 700 Hz… nothing at even multiples. An analyzer pointed at a square wave shows exactly these spikes.
Practical payoff: a “digital” square wave radiates interference at all those odd harmonics — why fast circuits need careful shielding, and why audio amps spec “harmonic distortion.”

Out in the world

Why instruments sound like themselves

A clarinet's closed pipe suppresses even harmonics; a violin string supplies a full ladder of them. Same note, different coefficient lists, instantly distinguishable to your ear — which is, in effect, a running Fourier analyzer (the cochlea sorts sound by frequency along its length).

Common confusion, cleared

“Adding smooth waves can't truly make sharp corners.”

With infinitely many it can — convergence does the impossible-sounding work. Finite sums get arbitrarily close, with the small Gibbs overshoot as the honest residue.

“The series is just for square waves and textbook shapes.”

Any periodic signal qualifies: engine vibrations, ECG beats, vowel sounds. The textbook shapes are demos; the theorem is universal.

Check yourself

PracticeQuick check

Two instruments play the same note but sound different because…

Recap

Periodic signals = fundamental + harmonics, each with amplitude and phase.
Coefficients are projections — inner products with each harmonic.
Sharp features demand high harmonics; smoothness means fast spectral decay.

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