Fourier's idea: a sum of sinusoids

A mind-blowing idea

At the beginning of the 19th century, Joseph Fourier put forward an idea that seemed crazy: any periodic signal, even an angular shape like a square wave, can be written as a sum of well-chosen sinusoids. These sinusoids have frequencies that are multiples of a base frequency: the fundamental and its harmonics.

Rebuilding a square wave

By taking the fundamental, then adding harmonic 3, then harmonic 5, and so on, a wave closer and closer to a square wave appears. The more harmonics you add, the sharper the corners become.

Step-by-step construction of a square wave:

fundamental alone       :   .-''-.      _.-''-._      (a simple sinusoid)
+ harmonic 3            :   .-""-.__.-""-.            (the troughs flatten out)
+ harmonics 3,5,7...    :   +------+    +------+       (more and more "square")

(each added harmonic sharpens the corners; in the limit, you get the perfect square wave)

The vocabulary

• The fundamental sets the period of the signal, its base frequency f.
• The harmonics are the sinusoids of frequency 2f, 3f, 4f... Each has its own amplitude, which says "how much" of that frequency is present.
• The set of these amplitudes makes up the Fourier series of the signal.

Why it is so useful

This decomposition changes everything: instead of studying a complicated signal in time, you study it frequency by frequency. Adjusting the bass and treble of a piece of music, filtering out noise, compressing a sound or an image: all of this consists of modifying certain frequencies without touching the others.

Classic pitfall

Only periodic signals have a Fourier series made of well-separated harmonics (spectral lines). An arbitrary, non-periodic signal can also be decomposed, but over a continuum of frequencies: that is the subject of the Fourier transform, covered in the next lesson. Confusing the two leads to looking for harmonics where there is only a continuous spectrum.