The FFT Algorithm

其他 2018-09-28 02:53:11 阅读次数: 0

Fourier Transform(DFT)

There is an entire theory of expanding functions in terms of sin(kwx) called Fourier Analysis. We are interested in extracting frequencies from DATA.

$x_{0}, x_{1}, x_{2}, x_{3}, x_{4} ... f_{0}, f_{1}, f_{2}, f_{3}, f_{4} ...$

$\left \{ f_{0}, f_{1}, ..., f_{N-1} \right \}\overset{DFT}{\rightarrow} \left \{ \hat{f_{0}}, \hat{f_{1}}, ..., \hat{f_{N-1}}\right \}$

$\hat{f_{k}} = \sum_{n=0}^{N-1} f_{n}\cdot e^{-2\pi ikn/N}$

$\omega = 2\pi kn/N$

$\left \{ \hat{f_{0}}, \hat{f_{1}}, ..., \hat{f_{N-1}}\right \}\overset{IDFT}{\rightarrow} \left \{ f_{0}, f_{1}, ..., f_{N-1} \right \}$

$f_{n} =\frac{1}{N} \sum_{k=0}^{N-1}\hat{f_{k}}\cdot e^{2\pi ikn/N}$

Fast Fourier Transform

A fast Fourier transform (FFT) is an algorithm that samples a signal over a period of time (or space) and divides it into its frequency components.

DFT may be implemented much more efficiently if N the # of samples is the power of 2:

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转载自blog.csdn.net/Da_tianye/article/details/82793447

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