Fourier Series

An N-periodic signal can be represented as a series of harmonically related sinusoids.

Fourier has shown that any periodic signal can be decomposed into harmonically related sinusoids:

This analysis seems to fits the needs of music coding by using a Fourier series for every tempo unit. It has several drawbacks:

• there may be points where the Fourier series does not converge
• a periodic continuous signal with a period different from 2p is represented as a discontinuous signal over . This causes a bad convergence of its Fourier series.
• More generally, it cannot efficiently represent signals which are not synchronized with a tempo, such as speech recordings or images.

A solution consists in using a transform where the frequency is allowed to have continuous variations. The Fourier series becomes the Fourier transform. The concept of period disappears in the process, and the frequency contents is actually a sinusoid contents.

Fourier transform