For Simon, my piano teacher, 4 October 2002
Add together two simple waves and you produce a third, more complicated wave. Add another simple wave and the waveform becomes even more complicated and so on. In fact, the process of adding waves together allows you to produce a waveform of almost any shape and in principle any sound can be created like this, even complex pieces of music.
But the reverse process of analysing complex pieces of music is much more difficult. When starting with a very complicated sound it is not immediately clear what simple waves have been added together to create it. But the mathematical process shown here does this for you. No matter how intricate the sound, this equation teases apart its waveform to reveal its basic components.
The equation is a called a Fourier Transform after the 19th century French mathematician and engineer Joseph Fourier. It takes a complicated wave represented by x(t) and breaks it into simple waves of frequency ω. The entire spectrum of these waves is represented by X(ω). The symbol e is a constant number, i is an imaginary number, t is time, ∞ is infinity and ∫ … dt represents a mathematical process called integration which is related to addition.
A derivation of the Fourier transform is at: