TOP MENUWhat is Fourier transform and Fast Fourier transform (FFT)? |
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■Fourier series
Any periodic signal can be expressed using trigonometric functions (sin, cos), and these functions are called Fourier series. The formula is as follows.
■Complex fourier series
For Fourier series, it is sometimes more convenient to use complex exponential functions rather than trigonometric functions. From Euler's formula; ■Fourier Transform:FT
While the Fourier series expresses a periodic function, the Fourier transform expresses the frequency components of an aperiodic function.
The idea is that even an aperiodic function can be treated as a periodic function if the period is treated as infinite. The formula is as follows. ■Discrete Fourier Transform:DFT
When real signal data is imported into a computer, etc., the data is discretized, so the Fourier transform described above cannot be used.
Therefore, the method used to perform Fourier transform on a discretized signal is called discrete Fourier transform.
Among discrete Fourier transforms, a method that performs calculations at high speed is called fast Fourier transform (FFT), and I think that when we say DFT, we often refer to FFT.
A constraint on fast Fourier transform is that the number of data must be 2 to the n power (for example, 128,256,512).
The formula for fast Fourier transform is as follows.
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