Fourier Analysis and Synthesis

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. 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.

The process of decomposing a musical instrument sound or any other periodic function into its constituent sine or cosine waves is called Fourier analysis. You can characterize the sound wave in terms of the amplitudes of the constituent sine waves which make it up. 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.

Once you know the harmonic content of a sustained musical 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.

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 is 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. 作为一个傅里叶变换的例子，正方形波的变换显示它只包含奇次谐波，且这些谐波的幅度以几何方式递减，第n次谐波的幅度是基频幅度的1/n倍。