Fourier Synthesis

In 1807, Joseph Fourier made a revolutionary claim: any periodic function — no matter how complicated — can be represented as a (possibly infinite) sum of simple sinusoids. This idea, now known as Fourier's theorem, underpins an enormous range of science and engineering, from audio compression (MP3) to quantum mechanics.

The Fourier Series

A periodic function $f(t)$ with period $T$ can be written as:

$f(t) = \frac{a_0}{2} + \sum_{n=1}^{\infty}\bigl[a_n \cos(n\omega t) + b_n \sin(n\omega t)\bigr]$

where $\omega = 2\pi / T$ is the fundamental angular frequency and the coefficients $a_n$ , $b_n$ determine how much of each harmonic is present.

How Each Wave Is Built

Target Wave	Non-zero Terms	Series
Square wave	Odd harmonics only ( $n = 1, 3, 5, \ldots$ )	$\frac{4}{\pi} \sum \frac{\sin(n t)}{n}$
Sawtooth wave	All harmonics	$\frac{2}{\pi} \sum \frac{(-1)^{n+1} \sin(n t)}{n}$
Triangle wave	Odd harmonics only	$\frac{8}{\pi^2} \sum \frac{(-1)^{(n-1)/2} \sin(n t)}{n^2}$

Notice that the triangle wave coefficients fall off as $1/n^2$ (much faster than the square wave's $1/n$ ), which is why it converges more smoothly.

The Gibbs Phenomenon

No matter how many harmonics you add, a sharp discontinuity (like the jump in a square wave) will always have an overshoot of about 9% near the edge. This is called the Gibbs phenomenon and is a fundamental limitation of Fourier series for discontinuous functions.

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Wave Type

Number of Harmonics5

Using 5 harmonics. More harmonics → better approximation.

Things to Try

Square wave with 1 harmonic — you see a single sine wave. Add more harmonics and watch the waveform sharpen into a square shape.
Square wave with 30 harmonics — zoom in on the discontinuity to see the Gibbs overshoot (~9% ripple that never goes away).
Triangle wave — notice how quickly it converges even with just 3-5 harmonics, because its coefficients decay as $1/n^2$ .
Sawtooth wave — uses all harmonics (both even and odd), so it converges differently from the square and triangle waves.
Compare all three at the same number of harmonics to appreciate how smoothness of the target function affects convergence speed.

Where Fourier Analysis Appears

Audio & music: Every musical instrument produces a unique mix of harmonics (timbre). The equalizer on your stereo adjusts individual Fourier components.
Image compression (JPEG): Images are decomposed into 2-D cosine components; discarding high-frequency ones compresses the file.
Signal processing: Filtering noise, designing antennas, and analysing seismic data all rely on Fourier transforms.
Quantum mechanics: The wave function in momentum space is the Fourier transform of the wave function in position space.
Heat conduction: Fourier originally developed his series to solve the heat equation — each harmonic decays at a rate proportional to $n^2$ .

From series to transform — The Fourier series works for periodic functions. For non-periodic signals, we generalize to the Fourier transform, which decomposes a function into a continuous spectrum of frequencies rather than discrete harmonics.