Fourier Series

3 min read#fourier

A periodic signal is an exact sum of harmonics — sinusoids at integer multiples of one fundamental frequency.

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Fourier Series

A Fourier series expresses any reasonable periodic signal as a sum of sinusoids whose frequencies are integer multiples — the Harmonics — of a single fundamental. If a signal f(t) repeats with period T, then

f(t) = a_0 + \sum_{n=1}^{\infty}\Big[a_n\cos(n\omega_0 t) + b_n\sin(n\omega_0 t)\Big],\qquad \omega_0 = \frac{2\pi}{T}.

Each coefficient measures how much of that harmonic the signal contains, recovered by the orthogonality of sines and cosines:

a_n = \frac{2}{T}\int_0^T f(t)\cos(n\omega_0 t)\,dt,\qquad b_n = \frac{2}{T}\int_0^T f(t)\sin(n\omega_0 t)\,dt.

The remarkable part is that the building blocks are nothing exotic: every term is the displacement of a Simple Harmonic Oscillator, and the series is their superposition.

Building a square wave

The clearest demonstration is the square wave, which turns out to be a sum over the odd harmonics only, each weighted by 1/n:

f(t) = \frac{4}{\pi}\left(\sin t + \frac{1}{3}\sin 3t + \frac{1}{5}\sin 5t + \cdots\right).

Add a few terms and the flat plateaus and steep edges of a square wave emerge out of pure curves:

Square wave from 1, 2, and 5 odd harmonics
More harmonics → flatter tops and sharper edges. The wiggles never fully vanish — see Gibbs Phenomenon.

Epicycles: synthesis made visible

There is a beautiful mechanical picture of the same sum. Attach a circle to the rim of a bigger circle, and another to that one, and so on — one circle per harmonic, each spinning at its own integer rate with radius \tfrac{4}{\pi}\cdot\tfrac1n. The tip of the final arm traces the signal. Below, the chained circles for the odd harmonics turn, and the height of their tip is drawn out to the right: a square wave, written by spinning wheels.

Chained epicycles for the odd harmonics 1, 1/3, 1/5, … Their tip traces a square wave to the right.

Watch what each wheel does: the big one sets the basic back-and-forth, and each smaller, faster wheel sharpens the corners. Stop adding wheels and the corners stay slightly rounded and rippling — the unavoidable Gibbs Phenomenon.

See also