Fast Fourier Transforms

快速傅里叶变换

Fourier analysis of a periodic function refers to the extraction of the series of sines and cosines which when superimposed will reproduce the function. This analysis can be expressed as a Fourier series. The fast Fourier transform is a mathematical method for transforming a function of time into a function of frequency. Sometimes it is described as transforming from the time domain to the frequency domain. It is very useful for analysis of time-dependent phenomena.

对周期函数的傅里叶分析指的是提取一组正弦和余弦函数，当这些函数叠加时可以复现原函数。这种分析可以表示为傅里叶级数。快速傅里叶变换是一种将时间函数转换为频率函数的数学方法。有时它被描述为从时域转换到频域。它对于分析时间相关的现象非常有用。

One important application is for the analysis of sound. It is important to assess the frequency distribution of the power in a sound because the human ear exercises that capacity in the hearing process. The following illustrations describe the sound of a London police whistle both in the time domain and in the frequency domain (by means of the FFT).

一个重要应用是声学分析。评估声音中功率的频率分布很重要，因为人类的耳朵在听觉过程中行使这种能力。以下示例描述了伦敦警察哨声在时域和频域中的声音（通过FFT实现）。

Whistle A

哨子 A

Labeling the two pipes of the whistle A and B, the illustration at left shows the sound of Whistle A alone. The top graph is the ordinary display of signal voltage from the microphone vs time. The bottom graph is the fast Fourier transform (FFT) of that signal. It shows that most of the power is at one frequency, approximating a sine wave. The fact that the peak showing most of the power is at position four just reflects the fact that four periods were chosen for the FFT sample,

给哨子的两个管子标上A和B，左边的示意图显示了仅用Whistle A发出的声音。顶部的图是麦克风接收到的信号电压随时间的变化普通显示。底部的图是该信号的快速傅里叶变换（FFT）。它显示大部分的功率集中在某一频率，近似于正弦波。峰值显示大部分功率位于位置四，这仅仅反映了FFT样本中选择了四个周期。

Whistle B

哨声 B

This is the same kind of display with whistle B alone. Three periods were chosen for the FFT this time, resulting in a main peak at position 3.

这种显示与仅使用哨子B的情况相同。这次FFT选择了三个周期，结果在位置3处出现主峰。

Whistles A and B

哨音A和B

When whistles A and B are sounded simultaneously the time plot shows the characteristic beat frequency pattern. The FFT shows the two distinct frequencies of the individual pipes.

当A和B号哨子同时吹响时，时间图显示了典型的拍频模式。FFT显示了两个独立管乐器的 distinct 频率。

These illustrations show the essential nature of the FFT. For a sine wave with a single frequency, the FFT consists of a single peak. Combining two sound waves produces a complex pattern in the time domain, but the FFT clearly shows it as consisting almost entirely of two frequencies.

这些示意图展示了FFT的本质特性。对于单一频率的正弦波，FFT仅产生一个峰值。将两个声波叠加会产生时域中的复杂模式，但FFT清晰地显示出它几乎由两个频率组成。