What is a Fourier series?

A Fourier series represents a periodic function f(x) as an infinite sum of sinusoids: f(x) = a₀/2 + Σ(n=1..∞)[aₙcos(nx) + bₙsin(nx)]. The coefficients are aₙ = (1/π)∫f(x)cos(nx)dx and bₙ = (1/π)∫f(x)sin(nx)dx. Any piecewise-smooth periodic function can be represented this way, making Fourier series fundamental to signal processing, physics, and engineering.

What is the Gibbs phenomenon?

At a jump discontinuity (like the edges of a square wave), the partial Fourier sum always overshoots by approximately 9% of the jump height, no matter how many harmonics are included. This overshoot does not disappear as N→∞; it only becomes narrower. It was first noted by Henry Wilbraham in 1848 and later studied by J. Willard Gibbs in 1899. It is caused by the slow 1/n decay of Fourier coefficients near discontinuities.

What does each rotating circle represent?

Each rotating circle (phasor) represents one harmonic of the Fourier series. The radius equals the harmonic's amplitude |cₙ| = √(aₙ²+bₙ²), and the rotation speed equals the harmonic frequency n. The tip of the chain of phasors traces a point; its y-coordinate at each moment equals the Fourier partial sum at the current phase. This is exactly the same mechanism used in the Discrete Fourier Transform-based epicycle drawings.

How does the Custom preset work?

When Custom is selected, click and drag on the waveform panel (right side of the canvas) to draw one period of any waveform. The simulator computes the Discrete Fourier Transform (DFT) of your drawn samples to extract the Fourier coefficients aₙ and bₙ. The phasor chain and spectrum are updated in real time as you draw. This demonstrates that ANY periodic waveform can be decomposed into sinusoidal components.

Fourier Series Synthesiser

Animated phasor chains build classic waveforms harmonic by harmonic — adjust N and watch the Gibbs phenomenon

Preset: Square Wave Frame: 0 N harmonics active

f(x) = (4/π)·[sin x + sin 3x/3 + sin 5x/5 + ...]

Harmonics N 8

Speed 1.0×

Fourier Series: Any Periodic Function as Sines and Cosines

In 1822, Joseph Fourier proved that virtually any periodic function can be reconstructed as an infinite sum of sines and cosines. This is one of the most powerful and far-reaching results in all of mathematics, with applications in signal processing, quantum mechanics, heat conduction, image compression (JPEG), audio (MP3), and hundreds of other fields.

Fourier series definition: For a function f(x) with period 2π, f(x) = a₀/2 + Σ(n=1..∞) [aₙ cos(nx) + bₙ sin(nx)]
where the Fourier coefficients are:
aₙ = (1/π) ∫₋π^π f(x) cos(nx) dx bₙ = (1/π) ∫₋π^π f(x) sin(nx) dx

Analytic Formulae for the Six Presets

Waveform	Fourier Series	Coeff decay
Square	(4/π)·[sin x + sin 3x/3 + sin 5x/5 + …] (odd harmonics only)	O(1/n)
Sawtooth	(2/π)·[sin x − sin 2x/2 + sin 3x/3 − …] (all harmonics, alternating sign)	O(1/n)
Triangle	(8/π²)·[sin x − sin 3x/9 + sin 5x/25 − …] (odd n, alternating)	O(1/n²)
Half-Wave Rect	1/π + sin x/2 + (2/π)·[cos 2x/3 − cos 4x/15 + …]	O(1/n²)
Pulse (d=25%)	0.25 + Σ (2sin(nπd)/nπ)·cos(nx), d=¼	O(1/n)
Custom	Numeric DFT of drawn curve (512 samples)	Depends on smoothness

Notice the coefficient decay rates. For smooth functions (like the triangle wave, which is C⁰ but not C¹), coefficients decay at least as fast as O(1/n²). For discontinuous functions (square, sawtooth), they decay only as O(1/n). This slow decay is directly responsible for the Gibbs phenomenon — the partial sums need many terms to "resolve" the sharp jump.

The Gibbs Phenomenon

Set the preset to Square wave and increase N from 1 to 50. The partial sum approximation becomes sharper and sharper near the discontinuity — but the overshoot peak never disappears. No matter how large N becomes, the maximum overshoot remains approximately 8.9% of the jump height.

Gibbs constant: lim_{N→∞} max overshoot / jump height = (2/π) ∫₀^π sinc(x) dx − 1 = (Si(π)/π) − ½ ≈ 0.0894898... i.e. about 8.95% where Si(x) = ∫₀^x sin(t)/t dt (sine integral)

The overshoot does shrink in width — it concentrates into an ever-thinner spike near the discontinuity — but its height converges to the Gibbs constant, not zero. This matters for signal processing: Gibbs ringing in bandlimited reconstructions causes artefacts, motivating window functions (Hann, Hamming) and Fejér summation methods.

Phasor Representation

Every sinusoidal term aₙcos(nx) + bₙsin(nx) can be written as a single rotating vector (phasor) of radius Aₙ =√(aₙ²+bₙ²) rotating at angular speed n. Chaining these phasors tip-to-tail — as shown on the left panel — is exactly the animation in the simulator. The y-coordinate of the final phasor tip traces the Fourier partial sum over time.

Complex Fourier form: f(x) = Σ_{n=−∞}^{+∞} cₙ · e^{inx} where cₙ = (aₙ − i·bₙ)/2 for n>0, c₀ = a₀/2, c₋ₙ = (aₙ + i·bₙ)/2 |cₙ| = √(aₙ²+bₙ²)/2 = amplitude of nth phasor arg(cₙ) = initial phase of nth phasor

Convergence Theory

Dirichlet conditions (1829): if f is bounded, has finitely many extrema and discontinuities in each period, the Fourier series converges to f(x) at continuity points and to [f(x⁺)+f(x⁻)]/2 at jump discontinuities.
Parseval's identity: (1/π)∫|f(x)|² dx = a₀²/2 + Σ(aₙ²+bₙ²) — the total energy splits across harmonics.
L² completeness: the trigonometric system {1, cos nx, sin nx} forms a complete orthonormal basis for L²(−π,π). Every square-integrable function is its Fourier series in the L² sense.
Uniform convergence: if f is continuous and piecewise smooth, the series converges uniformly (no Gibbs phenomenon).
Fejér's theorem: Cesàro means (averages of partial sums) converge uniformly for any continuous periodic function, providing a summation method that avoids Gibbs overshoot.

Applications

Signal processing: Audio equalisation uses Fourier decomposition; MP3 compression discards inaudible harmonics; waveform synthesis in synthesisers adds harmonics to build timbres
Heat equation: Fourier introduced his series specifically to solve ∂u/∂t = α∇²u — separating variables yields Fourier modes that decay exponentially, and any initial condition u(x,0) = f(x) expands in the Fourier series
Quantum mechanics: Schrödinger eigenstates in a box are Fourier modes; any initial wavefunction ψ(x,0) is a Fourier series in those eigenstates
Image compression: JPEG uses the Discrete Cosine Transform (a relative of Fourier series); images are stored as 8×8-block DCT coefficients
Optics: A diffraction grating's far-field pattern is the Fourier transform of the aperture function
Number theory: Dirichlet characters and L-functions use Fourier analysis on finite groups (Z/nZ)

Frequently Asked Questions

Why do square and sawtooth waves have only O(1/n) decay?

Integration by parts gives the rule: if f has a jump discontinuity, its Fourier coefficients decay as 1/n (from the boundary terms in the integration). Each additional degree of smoothness adds one power of 1/n to the decay rate. The square wave has 0 continuous derivatives (it's discontinuous), so 1/n. The triangle wave is continuous but has a discontinuous derivative, so 1/n². The parabola is C¹ with discontinuous second derivative, so 1/n³. A C∞ function (e.g. sin x itself) has superpolynomial decay. A C∞ periodic analytic function has exponential decay |cₙ| ≤ C·r^n for some r<1.

What is the difference between Fourier series and the Fourier transform?

The Fourier series represents a periodic function as a discrete sum of harmonics (integer frequencies only). The Fourier transform f̂(ξ) = ∫f(x)e^{−2πiξx}dx decomposes an aperiodic function (defined on all of ℝ) into a continuous spectrum of frequencies. The Fourier series is the discrete special case: you can derive it by taking the Fourier transform of a periodised version of f, which produces a Dirac comb of delta functions at integer frequencies. Conversely, the Fourier series coefficients of a periodic function are samples of the Fourier transform spaced at intervals of 1/(period).

Why can the same phasor idea draw arbitrary shapes (like letters)?

A closed curve parametrised as z(t) = x(t) + iy(t) for t ∈ [0,2π] can also be expanded in a Fourier series: z(t) = Σ cₙ e^{int}. Each cₙ e^{int} is a circle of radius |cₙ| rotating at frequency n (positive frequencies rotate counterclockwise, negative frequencies clockwise). Chaining them produces the epicycle animations seen in 3Blue1Brown-style videos. The only difference from waveform synthesis is that both the x and y components are synthesised simultaneously via complex Fourier coefficients, so any 2D closed curve — including letters, portraits, and drawings — can be traced.

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