# Fourier Analysis and Synthesis

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http://hyperphysics.phy-astr.gsu.edu/hbase/audio/Fourier.html#c1

**Fourier analysis**is the process of decomposing a sound or any other periodic function into its constituent sine or cosine waves. This analysis helps reveal the underlying frequency content of a signal, making it valuable for tasks such as filtering, compression, and understanding the behavior of systems in the frequency domain. Once you know the harmonic content of a sustained sound from Fourier analysis, you have the capability of synthesizing that sound from a series of pure tone generators by properly adjusting their amplitudes and phases and adding them together. This is called

**Fourier synthesis**.

## History

In 1822, the mathematician Fourier proved that any continuous function could be produced as an infinite sum of sine and cosine waves. His result has far-reaching implications for the reproduction and synthesis of sound.

## Fourier analysis in music

A pure sine wave can be converted into sound by a loudspeaker and will be perceived to be a steady, pure tone of a single pitch. The sounds from orchestral instruments usually consists of a fundamental and a complement of harmonics, which can be considered to be a superposition of sine waves of a fundamental frequency

*f*and integer multiples of that frequency.This set of numbers tells you the harmonic content of the sound and is sometimes referred to as the harmonic spectrum of the sound. The harmonic content is the most important determiner of the quality or timbre of a sustained musical note.

One of the important ideas for sound reproduction which arises from Fourier analysis is that it takes a high quality audio reproduction system to reproduce percussive sounds or sounds with fast transients. The sustained sound of a trombone can be reproduced with a limited range of frequencies because most of the sound energy is in the first few harmonics of the fundamental pitch. But if you are going to synthesize the sharp attack of a cymbal, you need a broad range of high frequencies to produce the rapid change. You can visualize the task of adding up a bunch of sine waves to produce a sharp pulse and perhaps you can see that you need large amplitudes of waves with very short rise times (high frequencies) to produce the sharp attack of the cymbal. This insight from Fourier analysis can be generalized to say that any sound with a sharp attack, or a sharp pulse, or rapid changes in the waveform like a square wave will have a lot of high frequency content.

One of the important ideas for sound reproduction which arises from Fourier analysis is that it takes a high quality audio reproduction system to reproduce percussive sounds or sounds with fast transients. The sustained sound of a trombone can be reproduced with a limited range of frequencies because most of the sound energy is in the first few harmonics of the fundamental pitch. But if you are going to synthesize the sharp attack of a cymbal, you need a broad range of high frequencies to produce the rapid change. You can visualize the task of adding up a bunch of sine waves to produce a sharp pulse and perhaps you can see that you need large amplitudes of waves with very short rise times (high frequencies) to produce the sharp attack of the cymbal. This insight from Fourier analysis can be generalized to say that any sound with a sharp attack, or a sharp pulse, or rapid changes in the waveform like a square wave will have a lot of high frequency content.

As an example of what you learn from a Fourier transform, the transform of a square wave shows that it has only odd harmonics and that the amplitude of those harmonics drops in a geometric fashion, with the nth harmonic having 1/n times the amplitude of the fundamental.

- [REF]
- [HP] HyperPhysics: http://hyperphysics.phy-astr.gsu.edu/hbase/audio/Fourier.html#c1
- Fourier, J.B. Joseph (1822),
*Théorie analytique de la chaleur*(in French), Paris: Firmin Didot, père et fils, OCLC 2688081