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一维傅里叶变换
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什么是傅里叶变换
笔记
G(f)=∫−∞∞g(t)e−i2πftdt=Fg(t)
Inverse Fourier Transform
g(t)=∫−∞∞G(f)ei2πftdf=F−1G(f)
其中,i=−1,表示复数虚部
单位
Temporal Coordinates, e.g. t in seconds, f in cycles/second
G(f)g(t)=∫−∞∞g(t)e−i2πftdt=∫−∞∞G(f)ei2πftdf Fourier Transform Inverse Fourier Transform
Spatial Coordinates, e.g. x in cm,kx is spatial frequency in cycles/cm
G(kx)=∫−∞∞g(x)e−i2πkxxdxg(x)=∫−∞∞G(kx)ei2πkxxdkx Fourier Transform Inverse Fourier Transform
eiθ=cosθ+isinθz=x+iy=∣z∣eiθ
根据欧拉公式将傅里叶变换展开:
G(kx)=∫−∞∞g(x)e−i2πkxxdx=∫−∞∞g(x)(cos(−2πkxx)+isin(−2πkxx))dx=∫−∞∞g(x)cos(2πkxx)dx−i∫−∞∞g(x)sin(2πkxx)dx
Spatial Coordinates, e.g. x in cm,kx is spatial frequency in cycles/cm
一维傅里叶变换的Python实现
来源:python实现一维傅里叶变换和逆变换_bilibili
代码如下:
附原视频:
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