** Fourier Analysis and Fourier Synthesis**

The French Mathematician Baptiste Joseph Fourier (1768-1830) invented a type of mathematical analysis by which it can be proved that any periodic wave can be represented as a sum of sine waves having the appropriate amplitude, frequency and phase. The function that sums all the harmonics into a sound wave is called a Fourier integral. Furthermore for harmonic spectra, the frequencies of the component waves are related in a simple way: they are whole number multiples of a single frequency, f0, 2f0,3f0, and so on.

A square (or pulsed wave with a mark to space ratio of 1:1) requires the sum of
an infinite number of sine components whose frequencies are odd whole number multiples
of the fundamental (f0, 3f0, 5f0, 7f, ...) and whose amplitudes decrease in proportion
to the inverse of the harmonic number (1, 1/3, 1/5, 1/7, ...) and the proper phases.

Most periodic sound waves consist of both odd and even frequency components although
closed organ pipes and some wind instruments (eg clarinets) do have predominantly
odd frequency components. A triangular wave also has only odd numbered frequency
components. Their amplitudes are different to those of a square wave however, reducing
to the inverse of the square of the harmonic number (1, 1/9, 1/25, 1/49 ...)

A Fourier representation of a complex wave of finite duration (as musical sounds
are) requires an infinite number of different harmonics. Trying to represent actual
sounds as sums of true sine waves, which persist from an infinite past into an infinite
future, is a mathematical artifice.

Consider the nearly periodic sounds produced by musical instruments. A sum of harmonically
related sine waves doesn't correctly represent such a sound, because the sound starts,
persists a while and dies away.

"Noisy" sounds such as the hiss of escaping air or the "sh" or "s" sounds of speech can be represented as the sum of sine waves (a Fourier integral) that have slightly different frequencies. When the sound is repeated the waveforms won't be exactly the same. The power of the sound in any narrow range of frequencies will be about the same, but the amplitudes and phases of the individual frequency components won't be identical. Nevertheless the two different "sh" sounds will sound the same; we will hear them as being identical. It may however, for certain purposes, be adequate to use fewer components.